Research ArticleLooking for Accurate Forecasting of Copper TCRCBenchmark Levels

Francisco J D-az-Borrego Mar-a del Mar Miras-Rodr-guez and Bernabeacute Escobar-Peacuterez

Universidad de Sevilla Seville Spain

Correspondence should be addressed to Marıa del Mar Miras-Rodrıguez mmirasuses

Received 22 November 2018 Revised 24 February 2019 Accepted 5 March 2019 Published 27 March 2019

Guest Editor Marisol B Correia

Copyright copy 2019 Francisco J Dıaz-Borrego et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Forecasting copper prices has been the objective of numerous investigations However there is a lack of research about the priceat which mines sell copper concentrate to smelters The market reality is more complex since smelters obtain the copper that theysell from the concentrate that mines produce by processing the ore which they have extracted It therefore becomes necessary tothoroughly analyse the price at which smelters buy the concentrates from the mines besides the price at which they sell the copperIn practice this cost is set by applying discounts to the price of cathodic copper the most relevant being those corresponding tothe smeltersrsquo benefit margin (Treatment Charges-TC and Refining Charges-RC) These discounts are agreed upon annually in themarkets and their correct forecasting will enable makingmore adequate models to estimate the price of copper concentrates whichwould help smelters to duly forecast their benefit margin Hence the aim of this paper is to provide an effective tool to forecastcopper TCRC annual benchmark levels With the annual benchmark data from 2004 to 2017 agreed upon during the LMECopperWeek a three-model comparison is made by contrasting different measures of error The results obtained indicate that the LES(Linear Exponential Smoothing) model is the one that has the best predictive capacity to explain the evolution of TCRC in boththe long and the short term This suggests a certain dependency on the previous levels of TCRC as well as the potential existenceof cyclical patterns in them This model thus allows us to make a more precise estimation of copper TCRC levels which makes ituseful for smelters and mining companies

1 Introduction

11 Background The relevance of copper trading is undeni-able In 2016 exports of copper ores concentrates coppermatte and cement copper increased by 15 reaching 473 b$USD while imports attained 439 b $USD [1] In additionthe global mining capacity is expected to rise by 10 from the235 million tonnes recorded in 2016 to 259 million tonnesin 2020 with smelter production having reached the recordfigure of 190 million tonnes in 2016 [2]

The worldrsquos copper production is essentially achievedthrough alternative processes which depend on the chemicaland physical characteristics of the copper ores extractedAccording to the USGSrsquo 2017 Mineral Commodity Summaryon Copper [3] global identified copper resources contained21 billion tonnes of copper as of 2014 of which about 80

are mainly copper sulphides whose copper content has to beextracted through pyrometallurgical processes [4] In 2010the average grades of ores being mined ranged from 05 to2 Cu which makes direct smelting unfeasible for economicand technical reasons So copper sulphide ores undergoa process known as froth-flotation to obtain concentratescontaining asymp 30 Cu which makes concentrates the mainproducts offered by copper mines [2 5] Concentrates arelater smelted and in most cases electrochemically refinedto produce high-purity copper cathodes (Figure 1) Coppercathodes are generally regarded as pure copper with aminimum copper content of 999935 Cu [6] Cathodesare normally produced by integrated smelters that purchaseconcentrates at a discounted price of the copper marketprice and sell cathodic copper at the market price adding apremium when possible

HindawiComplexityVolume 2019 Article ID 8523748 16 pageshttpsdoiorg10115520198523748

2 Complexity

Sulfide ores (05 - 20 Cu)

Comminution

Concentrates (20 - 30 Cu)

Other smeltingprocesseslowast

Flotation

Drying

Anodes (995 Cu)

Anode refiningand casting

Drying

Flash smelting Submerged tuyere

smelting

Matte (50-70Cu)

Converting

Blister Cu (99 Cu)

Electrorefining

Cathodes (9999 Cu)

Direct-to-copper smelting

Vertical lance smelting

Figure 1 Industrial processing of copper sulphides ore to obtain copper cathodes (Source Extractive Metallurgic of Copper pp2 [4])

12 TCRC and Their Role in Copper Concentrates TradeThe valuation of these copper concentrates is a recurrenttask undertaken by miners or traders following processes inwhich market prices for copper and other valuable metalssuch as gold and silver are involved as well as relevantdiscounts or coefficients that usually represent a major partof the revenue obtained for concentrate trading smeltingor refining The main deductions are applied to the marketvalue of the metal contained by concentrates such as CopperTreatment Charges (TC) Copper Refining Charges (RC) thePrice Participation Clause (PP) and Penalties for PunishableElements [7] These are fixed by the different parties involvedin a copper concentrates long-term or spot supply contractwhere TCRC are fixed when the concentrates are sold to acopper smelterrefinery The sum of TCRC is often viewedas the main source of revenue for copper smelters along withcopper premiums linked to the selling of copper cathodesFurthermore TCRC deductions pose a concern for coppermines as well as a potential arbitrage opportunity for traderswhose strong financial capacity and indepth knowledge of themarket make them a major player [8]

Due to their nature TCRC are discounts normallyagreed upon taking a reference which is traditionally set onan annual basis at the negotiations conducted by the major

market participants during LME Week every October andmore recently during the Asia Copper Week each Novemberan event that is focused more on Chinese smelters TheTCRC levels set at these events are usually taken as bench-marks for the negotiations of copper concentrate supplycontracts throughout the following year Thus as the yeargoes on TCRC average levels move up and down dependingon supply and demand as well as on concentrate availabilityand smeltersrsquo available capacity Consultants such as PlattsWoodMackenzie andMetal Bulletin regularly carry out theirown market surveys to estimate the current TCRC levelsFurthermoreMetal Bulletinhas created the first TCRC indexfor copper concentrates [9]

13 The Need for Accurate Predictions of TCRC The cur-rent information available for market participants may beregarded as sufficient to draw an accurate assumption ofmar-ket sentiment about current TCRC levels but not enoughto foresee potential market trends regarding these crucialdiscounts far less as a reliable tool which may be ultimatelyapplicable by market participants to their decision-makingframework or their risk-management strategies Hence froman organisational standpoint providing accurate forecastsof copper TCRC benchmark levels as well as an accurate

Complexity 3

mathematical model to render these forecasts is a researchtopic that is yet to be explored in depth This is a questionwith undeniable economic implications for traders minersand smelters alike due to the role of TCRC in the coppertrading revenue stream

Our study seeks to determine an appropriate forecastingtechnique for TCRC benchmark levels for copper concen-trates that meets the need of reliability and accuracy Toperform this research three different and frequently-appliedtechniques have been preselected from among the optionsavailable in the literature Then their forecasting accuracyat different time horizons will be tested and comparedThese techniques (Geometric Brownian Motion -GBM-the Mean Reversion -MR- Linear Exponential Smoothing-LES-) have been chosen primarily because they are commonin modelling commodities prices and their future expectedbehaviour aswell as in stock indicesrsquo predictiveworks amongother practical applications [10ndash13] The selection of thesemodels is further justified by the similarities shared byTCRCwith indices interest rates or some economic variables thatthese models have already been applied to Also in our viewthe predictive ability of these models in commodity pricessuch as copper is a major asset to take them into consider-ation The models have been simulated using historical dataof TCRC annual benchmark levels from 2004 to 2017 agreedupon during the LME Copper Week The dataset employedhas been split into two parts with two-thirds as the in-sampledataset and one-third as the out-of-sample one

The main contribution of this paper is to provide auseful and applicable tool to all parties involved in thecopper trading business to forecast potential levels of criticaldiscounts to be applied to the copper concentrates valuationprocess To carry out our research we have based ourselves onthe following premises (1)GBMwould deliver good forecastsif copper TCRC benchmark levels vary randomly over theyears (2) a mean-reverting model such as the OUP woulddeliver the best forecasts if TCRC levels were affected bymarket factors and consequently they move around a long-term trend and (3) a moving average model would givea better forecast than the other two models if there werea predominant factor related to precedent values affectingthe futures ones of benchmark TCRC In addition we havealso studied the possibility that a combination of the modelscould deliver the most accurate forecast as the time horizonconsidered is increased since there might thus be a limitedeffect of past values or of sudden shocks on future levels ofbenchmark TCRC So after some time TCRC levels couldbe ldquonormalizedrdquo towards a long-term average

The remainder of this article is structured as followsSection 2 revises the related work on commodity discountsforecasting and commodity prices forecasting techniques aswell as different forecasting methods Section 3 presents thereasoning behind the choice of each of the models employedas well as the historic datasets used to conduct the studyand the methodology followed Section 4 shows the resultsof simulations of different methods Section 5 indicates errorcomparisons to evaluate the best forecasting alternative forTCRC among all those presented Section 6 contains thestudyrsquos conclusions and proposes further lines of research

2 Related Work

The absence of any specific method in the specialised litera-ture in relation to copper TCRC leads us to revisit previousliterature in order to determine the most appropriate modelto employ for our purpose considering those that havealready been used in commodity price forecasting as thelogical starting point due to the application similarities thatthey share with ours

Commodity prices and their forecasting have been atopic intensively analysed in much research Hence there aremultiple examples in literature with an application to oneor several commodities such as Xiong et al [14] where theaccuracy of differentmodelswas tested to forecast the intervalof agricultural commodity future prices Shao and Dai [15]whose work employs the Autoregressive Integrated MovingAverage (ARIMA) methodology to forecast food crop pricessuch as wheat rice and corn and Heaney [16] who tests thecapacity of commodities future prices to forecast their cashprice were the cost of carry to be included in considerationsusing the LME Lead contract as an example study As wasdone in other research [17ndash19] Table 1 summarizes similarresearch in the field of commodity price behaviours

From the summary shown in Table 1 it is seen that mov-ing average methods have become increasingly popular incommodity price forecastingThemost broadly implementedmoving average techniques are ARIMA and ExponentialSmoothing They consider the entire time series data anddo not assign the same weight to past values than thosecloser to the present as they are seen as affecting greater tofuture valuesThe Exponential Smoothingmodelsrsquo predictiveaccuracy has been tested [20 21] concluding that there aresmall differences between them (Exponential Smoothing)and ARIMA models

Also GBM andMRmodels have been intensively appliedto commodity price forecasting Nonetheless MR modelspresent a significant advantage over GBM models whichallows them to consider the underlying price trend Thisadvantage is of particular interest for commodities thataccording to Dixit and Pindyck [22]mdashpp 74 regarding theprice of oil ldquoin the short run it might fluctuate randomly upand down (in responses to wars or revolutions in oil producingcountries or in response to the strengthening or weakeningof the OPEC cartel) in the longer run it ought to be drawnback towards the marginal cost of producing oil Thus onemight argue that the price of oil should be modelled as a mean-reverting processrdquo

3 Methodology

This section presents both the justification of the modelschosen in this research and the core reference dataset used asinputs for each model compared in the methodology as wellas the steps followed during the latter to carry out an effectivecomparison in terms of the TCRC forecasting ability of eachof these models

31 Models in Methodology GBM (see Appendix A for mod-els definition) has been used in much earlier research as a

4 Complexity

Table 1 Summary of previous literature research

AUTHOR RESEARCH

Shafiee amp Topal [17] Validates a modified version of the long-term trend reverting jump and dip diffusion model for forecastingcommodity prices and estimates the gold price for the next 10 years using historical monthly data

Li et al [18] Proposes an ARIMA-Markov Chain method to accurately forecast mineral commodity prices testing themethod using mineral molybdenum prices

Issler et al [19]Investigates several commoditiesrsquo co-movements such as Aluminium Copper Lead Nickel Tin and Zinc atdifferent time frequencies and uses a bias-corrected average forecast method proposed by Issler and Lima [43]to give combined forecasts of these metal commodities employing RMSE as a measure of forecasting accuracy

Hamid amp Shabri [44]Models palm oil prices using the Autoregressive Distributed Lag (ARDL) model and compares its forecastingaccuracy with the benchmark model ARIMA It uses an ARDL bound-testing approach to co-integration in

order to analyse the relationship between the price of palm oil and its determinant factorsDuan et al [45] Predicts Chinarsquos crude oil consumption for 2015-2020 using the fractional-order FSIGMmodel

Brennan amp Schwartz [46] Employs the Geometric Brownian Motion (GBM) to analyse a mining projectrsquos expected returns assuming itproduces a single commodity

McDonald amp Siegel [47] Uses GBM to model the random evolution of the present value of an undefined asset in an investment decisionmodel

Zhang et al [48] Models gold prices using the Ornstein-Uhlenbeck Process (OUP) to account for a potentially existent long-termtrend in a Real Option Valuation of a mining project

Sharma [49] Forecasts gold prices in India with the Box Jenkins ARIMA method

way of modelling prices that are believed not to follow anyspecific rule or pattern and hence seen as random Blackand Scholes [23] first used GBM to model stock prices andsince then others have used it to model asset prices as well ascommodities these being perhaps themost common of all inwhich prices are expected to increase over time as does theirvariance [11] Hence following our first premise concerningwhether TCRC might vary randomly there should not exista main driving factor that would determine TCRC futurebenchmark levels and therefore GBM could to a certainextent be a feasible model for them

However although GBM or ldquorandom walkrdquo may be wellsuited to modelling immediate or short-term price paths forcommodities or for TCRC in our case it lacks the abilityto include the underlying long-term price trend should weassume that there is oneThus in accordance with our secondpremise on benchmark TCRC behaviour levels would movein line with copper concentrate supply and demand as wellas the smeltersrsquo and refineriesrsquo available capacity to trans-form concentrates into metal copper Hence a relationshipbetween TCRC levels and copper supply and demand isknown to exist and therefore is linked to its market priceso to some extent they move together coherently Thereforein that case as related works on commodity price behavioursuch as Foo Bloch and Salim [24] do we have opted for theMR model particularly the OUP model

Both GBM and MR are Markov processes which meansthat future values depend exclusively on the current valuewhile the remaining previous time series data are not consid-eredOn the other handmoving averagemethods employ theaverage of a pre-established number of past values in differentways evolving over time so future values do not rely exclu-sively on the present hence behaving as though they had onlya limitedmemory of the pastThis trait of themoving averagemodel is particularly interestingwhen past prices are believedto have a certain though limited effect on present values

which is another of the premises for this research Existingalternatives ofmoving average techniques pursue consideringthis ldquomemoryrdquo with different approaches As explained byKalekar [25] Exponential Smoothing is suitable only for thebehaviours of a specific time series thus Single ExponentialSmoothing (SES) is reasonable for short-term forecastingwith no specific trend in the observed data whereas DoubleExponential Smoothing or Linear Exponential Smoothing(LES) is appropriate when data shows a cyclical pattern ora trend In addition seasonality in observed data can becomputed and forecasted through the usage of ExponentialSmoothing by the Holt-Winters method which adds an extraparameter to the model to handle this characteristic

32 TCRC Benchmark Levels and Sample Dataset TCRClevels for copper concentrates continuously vary through-out the year relying on private and individual agreementsbetween miners traders and smelters worldwide Nonethe-less the TCRC benchmark fixed during the LME week eachOctober is used by market participants as the main referenceto set actual TCRC levels for each supply agreed upon for thefollowing year Hence the yearrsquos benchmark TCRC is takenhere as a good indicator of a yearrsquos TCRC average levelsAnalysed time series of benchmark TCRC span from 2004through to 2017 as shown in Table 2 as well as the sourceeach valuewas obtained fromWe have not intended to reflectthe continuous variation of TCRC for the course of anygiven year though we have however considered benchmarkprices alone as we intuitively assume that annual variations ofactual TCRC in contracts will eventually be reflected in thebenchmark level that is set at the end of the year for the yearto come

33 TCRC Relation TC and RC normally maintain a 101relation with different units with TC being expressed in US

Complexity 5

GBM RC First 20 pathsGBM TC First 20 paths

0

50

100

150

200

250

300

TC (U

SDM

T)

2013 2014 2015 2016 20172012

GBM TC First 20 paths - Year Averaged

05

101520253035

RC (U

Sclb

)

0

5

10

15

20

25

30

RC (U

Slb

)

0

50

100

150

200

250

TC (U

SDM

T)

2013 2014 2015 2016 20172012

Observed TC Benchmark Levels

2013 2014 2015 2016 20172012

Observed RC Benchmark Levels

2013 2014 2015 2016 20172012

GBM RC First 20 paths - Year Averaged

Figure 2 The upper illustrations show the first 20 Monte Carlo paths for either TC or RC levels using monthly steps The illustrations belowshow the annual averages of monthly step forecasts for TC and RC levels

MR TC First 20 paths

2012 2013 2014 2015 2016 2017

MR RC First 20 paths

MR RC First 20 paths - Year Averaged

2013 2014 2015 2016 20172012

Observed

5

6

7

8

9

10

11

RC (U

Slb

)

2468

10121416

RC (U

Sclb

)

0

50

100

150

200

TC (U

SDM

T)

2013 2014 2015 2016 20172012

MR TC First 20 paths - Year Averaged

2013 2014 2015 2016 20172012

Observed

50

60

70

80

90

100

110

TC (U

SDM

T)

Figure 3 The first 20 Monte Carlo paths following the OUP model using monthly steps are shown on the upper illustrations for either TCor RC The annual averaged simulated paths every 12 values are shown below

dollars per metric tonne and RC in US cents per pound ofpayable copper content in concentrates In fact benchmarkdata show that of the last 14 years it was only in 2010that values of TC and RC did not conform to this relation

though it did remain close to it (46547) Nonetheless thisrelation has not been observed in the methodology hereinthus treating TC and RC independently developing separateunrelated forecasts for TC and RC to understand whether

6 Complexity

Table 2 TCRC year benchmark levels (Data available free ofcharge Source Johansson [50] Svante [51] Teck [52] Shaw [53]Willbrandt and Faust [54 55] Aurubis AG [56] Drouven and Faust[57] Aurubis AG [58] EY [59] Schachler [60] Nakazato [61])

YEAR TC (USDMT) RC (USclb)2004 45 452005 85 852006 95 952007 60 602008 45 452009 75 752010 465 472011 56 562012 635 6352013 70 702014 92 922015 107 1072016 9735 97352017 925 925

these individual forecasts for TC and RC would render betterresults than a single joint forecast for both TCRC levels thatwould take into account the 101 relation existent in the data

34 Models Comparison Method The works referred to inthe literature usually resort to different measures of errorsto conduct the testing of a modelrsquos forecasting capacityregardless of the specific nature of the forecasted value be itcotton prices or macroeconomic parameters Thus the mostwidely errors used includeMean Squared Error (MSE) MeanAbsolute Deviation (MAD) Mean Absolute Percentage Error(MAPE) and Root Mean Square Error (RMSE) [14 16 17 2125ndash29] In this work Geometric Brownian Motion (GBM)Ornstein-Uhlenbeck Process (OUP) and Linear ExponentialSmoothing (LES) models have been used to forecast annualTCRC benchmark levels using all the four main measuresof error mentioned above to test the predictive accuracy ofthese three models The GBM and OUP models have beensimulated and tested with different step sizes while the LESmodel has been analysed solely using annual time steps

The GBM and OUPmodels were treated separately to theLESmodel thus GBM andOUPwere simulated usingMonteCarlo (MC) simulations whereas LES forecasts were simplecalculations In a preliminary stage the models were firstcalibrated to forecast values from 2013 to 2017 using availabledata from 2004 to 2012 for TC and RC separately hencethe disregarding of the well-known inherent 101 relation (seeAppendix B for Models Calibration) The GBM and OUPmodels were calibrated for two different step sizes monthlysteps and annual steps in order to compare forecastingaccuracy with each step size in each model Following thecalibration of the models Monte Carlo simulations (MC)were carried out usingMatlab software to render the pursuedforecasts of GBM andOUPmodels obtaining 1000 simulatedpaths for each step size MC simulations using monthly stepsfor the 2013ndash2017 timespan were averaged every 12 steps to

deliver year forecasts of TCRC benchmark levels On theother hand forecasts obtained by MC simulations takingannual steps for the same period were considered as yearforecasts for TCRC annual benchmark levels without theneed for extra transformation Besides LES model forecastswere calculated at different time horizons to be able tocompare the short-term and long-term predictive accuracyLES forecasts were obtained for one-year-ahead hence usingknown values of TCRC from 2004 to 2016 for two yearsahead stretching the calibrating dataset from 2004 to 2015for five-year-ahead thus using the same input dataset as forthe GBM and OUP models from 2004 to 2012

Finally for every forecast path obtained we have calcu-lated the average of the squares of the errors with respect tothe observed valuesMSE the average distance of a forecast tothe observed meanMAD the average deviation of a forecastfrom observed values MAPE and the square root of MSERMSE (see Appendix C for error calculations) The results ofMSE MAD MAPE and RMSE calculated for each forecastpathwere averaged by the total number of simulations carriedout for each case Averaged values of error measures of allsimulated paths MSE MAD MAPE and RMSE for bothannual-step forecasts and monthly step forecasts have beenused for cross-comparison between models to test predictiveability at every step size possible

Also to test each modelrsquos short-term forecasting capacityagainst its long-term forecasting capacity one-year-aheadforecast errors of the LES model were compared with theerrors from the last year of the GBM and OUP simulatedpaths Two-year-ahead forecast errors of LES models werecomparedwith the average errors of the last two years ofGBMand OUP and five-year-ahead forecast errors of LES modelswere compared with the overall average of errors of the GBMand OUP forecasts

4 Analysis of Results

Monte Carlo simulations of GBM and OUP models render1000 different possible paths for TC and RC respectivelyat each time step size considered Accuracy errors for bothannual time steps and averaged monthly time steps for bothGBM and OUP forecasts were first compared to determinethe most accurate time step size for each model In additionthe LES model outcome for both TC and RC at differenttimeframes was also calculated and measures of errors for allthe three alternative models proposed at an optimum timestep were finally compared

41 GBM Forecast The first 20 of 1000 Monte Carlo pathsfor the GBM model with a monthly step size using Matlabsoftware may be seen in Figure 2 for both the TC and the RClevels compared to their averaged paths for every 12 valuesobtainedThe tendency for GBM forecasts to steeply increaseover time is easily observable in the nonaveraged monthlystep paths shown

The average values of error of all 1000MC paths obtainedthrough simulation for averaged monthly step and annual-step forecasts are shown in Table 3 for both TC and RC

Complexity 7

Table 3 Average of main measures of error for GBM after 1000 MC simulations from 2013 to 2017

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC Averaged Monthly Steps 560x103 4698 050 5638TC Annual Steps 520x103 4902 052 5857RC Averaged Monthly Steps 5874 455 048 546RC Annual Steps 5085 491 052 584

Table 4 Number of paths for which measures of error are higher for monthly steps than for annual steps in GBM

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC 4571000 4551000 4531000 4531000RC 4391000 4301000 4291000 4291000

discounts over the period from 2013 to 2017 which maylead to preliminary conclusions in terms of an accuracycomparison between averaged monthly steps and annualsteps

However a more exhaustive analysis is shown in Table 4where the number of times the values of each error measureare higher for monthly steps is expressed over the totalnumber of MC simulations carried out The results indicatethat better values of error are reached the majority of timesfor averagedmonthly step simulations rather than for straightannual ones

In contrast short-term accuracy was also evaluated byanalysing the error measures of one-year-ahead forecasts(2013) in Table 5 and of two-year-ahead forecasts (2013ndash2014)in Table 6 The results indicate as one may expect thataccuracy decreases as the forecasted horizon is widened withthe accuracy of averaged monthly step forecasts remaininghigher than annual ones as found above for the five-yearforecasting horizon

42 OUP Forecast Long-term levels for TC and RC 120583 thespeed of reversion 120582 and the volatility of the process 120590 arethe parameters determined at themodel calibration stage (seeAppendix B for the model calibration explanation) whichdefine the behaviour of the OUP shown in Table 7 Thecalibration was done prior to the Monte Carlo simulationfor both TC and RC with each step size using availablehistorical data from 2004 to 2012 The OUP model was fittedwith the corresponding parameters for each case upon MCsimulation

Figure 3 shows theMeanReversionMCestimations of theTCRC benchmark values from 2013 to 2017 using monthlystepsThemonthly forecastswere rendered from January 2012through to December 2016 and averaged every twelve valuesto deliver a benchmark forecast for each year The averagedresults can be comparable to actual data as well as to theannual Monte Carlo simulations following Mean ReversionThe lower-side figures show these yearly averaged monthlystep simulation outcomes which clearly move around a dash-dotted red line indicating the long-term run levels for TCRCto which they tend to revert

The accuracy of monthly steps against annual steps forthe TCRC benchmark levels forecast was also tested by

determining the number of simulations for which averageerror measures became higher Table 8 shows the numberof times monthly simulations have been less accurate thanannual simulations for five-year-ahead OUP forecasting bycomparing the four measures of errors proposed The resultsindicate that only 25-32 of the simulations drew a higheraverage error which clearly results in a better predictiveaccuracy for monthly step forecasting of TCRC annualbenchmark levels

The averaged measures of errors obtained after the MCsimulations of the OUP model for both averaged monthlysteps and annual steps giving TCRC benchmark forecastsfrom 2013 to 2017 are shown in Table 9

The error levels of the MC simulations shown in Table 9point towards a higher prediction accuracy of averagedmonthly step forecasts of the OUP Model yielding anaveraged MAPE value that is 129 lower for TC and RC 5-step-ahead forecasts In regard to MAPE values for monthlysteps only 266 of the simulations rise above annual MCsimulations for TC and 25 for RC 5-step-ahead forecastswhich further underpins the greater accuracy of thisOUP set-up for TCRC level forecasts A significant lower probabilityof higher error levels for TCRC forecasts with monthly MCOUP simulations is reached for the other measures providedIn addition short-term and long-term prediction accuracywere tested by comparing errors of forecasts for five-year-ahead error measures in Table 10 one-year-ahead in Table 11and two-year-ahead in Table 12

With a closer forecasting horizon errormeasures show animprovement of forecasting accuracy when average monthlysteps are used rather than annual ones For instance theMAPE values for 2013 forecast for TC are 68 lower foraveraged monthly steps than for annual steps and alsoMAPE for 2013ndash2014 were 30 lower for both TC and RCforecasts Similarly better values of error are achieved for theother measures for averaged monthly short-term forecaststhan in other scenarios In addition as expected accuracyis increased for closer forecasting horizons where the levelof errors shown above becomes lower as the deviation offorecasts is trimmed with short-term predictions

43 LES Forecast In contrast to GBM and OUP the LESmodel lacks any stochastic component so nondeterministic

8 Complexity

Table 5 Average for main measures of error for GBM after 1000 MC simulations for 2013

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC Averaged Monthly Steps 33658 1455 021 1455TC Annual Steps 69381 2081 030 2081RC Averaged Monthly Steps 297 138 020 138RC Annual Steps 657 205 029 205

Table 6 Average for main measures of error for GBM after 1000 MC simulations for 2013-2014

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC Averaged Monthly Steps 99492 2187 031 2435TC Annual Steps 133x103 2833 040 3104RC Averaged Monthly Steps 1035 240 034 271RC Annual Steps 1313 284 034 311

Table 7 OUP parameters obtained in the calibration process

120583 120582 120590TCMonthly 6345 4792x105 2534TC Annual 6345 2974 1308RC Monthly 635 4763x105 2519RC Annual 635 2972 1305

methods such as theMonte Carlo are not required to obtain aforecast Nonetheless the LES model relies on two smoothingconstants which must be properly set in order to deliveraccurate predictions hence the values of the smoothingconstants were first optimised (see Appendix B for LESModel fitting and smoothing constants optimisation) Theoptimisation was carried out for one-year-ahead forecaststwo-year-ahead forecasts and five-year-ahead forecasts byminimising the values of MSE for both TC and RC Thedifferent values used for smoothing constants as well asthe initial values for level and trend obtained by the linearregression of the available dataset from 2004 through to 2012are shown in Table 12

Compared one-year-ahead two-year-ahead and five-year-ahead LES forecasts for TC and RC are shown inFigure 4 clearly indicating a stronger accuracy for shorter-term forecasts as the observed and forecasted plotted linesoverlap

Minimumvalues for errormeasures achieved through theLESmodel parameter optimisation are shown in Table 13Thevalues obtained confirm the strong accuracy for shorter-termforecasts of TCRC seen in Figure 4

5 Discussion and Model Comparison

The forecasted values of TCRC benchmark levels couldeventually be applied to broader valuation models for copperconcentrates and their trading activities as well as to thecopper smeltersrsquo revenue stream thus the importance ofdelivering as accurate a prediction as possible in relation tothese discounts to make any future application possible Eachof the models presented may be a feasible method on its own

with eventual later adaptations to forecasting future valuesof benchmark TCRC Nonetheless the accuracy of thesemodels as they have been used in this work requires firstlya comparison to determine whether any of them could bea good standalone technique and secondly to test whethera combination of two or more of them would deliver moreprecise results

When comparing the different error measures obtainedfor all the three models it is clearly established that resultsfor a randomly chosen simulation of GBM or OUP wouldbe more likely to be more precise had a monthly step beenused to deliver annual forecasts instead of an annual-step sizeIn contrast average error measures for the entire populationof simulations with each step size employed showing thatmonthly step simulations for GBM and OUP models arealways more accurate than straight annual-step forecastswhen a shorter time horizon one or two-year-ahead is takeninto consideration However GBM presents a higher levelof forecasting accuracy when the average for error measuresof all simulations is analysed employing annual steps forlong-term horizons whereas OUP averaged monthly stepforecasts remain more accurate when predicting long-termhorizons Table 14 shows the error improvement for the aver-aged monthly step forecasts of each model Negative valuesindicate that better levels of error averages have been found instraight annual forecasts than for monthly step simulations

Considering the best results for each model and com-paring their corresponding error measures we can opt forthe best technique to employ among the three proposed inthis paper Hence GBM delivers the most accurate one-yearforecast when averaging the next twelve-month predictionsfor TCRC values as does the MR-OUP model Table 15shows best error measures for one-year-ahead forecasts forGBM OUP and LES models

Unarguably the LES model generates minimal errormeasures for one-year-ahead forecasts significantly less thanthe other models employed A similar situation is found fortwo-year-ahead forecasts whereminimumerrormeasures arealso delivered by the LES model shown in Table 16

Finally accuracy measures for five-year-ahead forecastsof the GBM model might result in somewhat contradictory

Complexity 9

Table 8 Number of paths for whichmeasures of error are higher for monthly steps than for annual steps inMR-OUP 5-StepsMC simulation

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC 3111000 2831000 2661000 2661000RC 3161000 2811000 2501000 2501000

Table 9 Average for main measures of error for MR-OUP after 1000 MC simulations 2013-2017

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC Averaged Monthly Steps 84254 2614 027 2895TC Annual Steps 114x103 2931 031 3325RC Averaged Monthly Steps 840 261 027 289RC Annual Steps 1148 294 031 333

Table 10 Average for main measures of error for MR-OUP after 1000 MC simulations 2013

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC Averaged Monthly Steps 3914 522 007 522TC Annual Steps 36643 1555 022 1555RC Averaged Monthly Steps 039 051 007 051RC Annual Steps 363 154 022 154

Table 11 Average for main measures of error for MR-OUP after 1000 MC simulations 2013-2014

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC Averaged Monthly Steps 37165 1567 018 1900TC Annual Steps 68244 2185 026 2324RC Averaged Monthly Steps 376 157 018 191RC Annual Steps 689 219 026 243

Table 12 LES Model Optimised Parameters

L0 T0 120572 120573TC 1 year 713611 -15833 -02372 01598RC 1 year 71333 -01567 -02368 01591TC 2 years 713611 -15833 -1477x10minus4 7774226RC 2 years 71333 -01567 -1448x10minus4 7898336TC 5 years 713611 -15833 -02813 01880RC 5 years 71333 -01567 -02808 01880

Table 13 LES error measures for different steps-ahead forecasts

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC 1 year 10746x10minus5 00033 46831x10minus5 00033RC 1 year 51462x10minus8 22685x10minus4 32408x10minus5 22685x10minus4

TC 2 years 189247 97275 01057 137567RC 2 years 18977 09741 01059 13776TC 5 years 1775531 78881 00951 108422RC 5 years 11759 07889 00952 10844

terms for MSE reaching better values for annual steps thanfor averaged monthly steps while the other figures do betteron averagedmonthly steps Addressing the definition ofMSEthis includes the variance of the estimator as well as its biasbeing equal to its variance in the case of unbiased estimators

Therefore MSE measures the quality of the estimator butalso magnifies estimator deviations from actual values sinceboth positive and negative values are squared and averagedIn contrast RMSE is calculated as the square root of MSEand following the previous analogy stands for the standard

10 Complexity

Table 14 Error Average Improvement for averaged monthly steps before annual steps

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864GBM TCRC 1 Year 53265653 32723587 33333548 32723587GBM TCRC 2 Years 26952635 25751597 2619286 24341369GBM TCRC 5 Years -1173-1104 873709 926769 747565OUP TCRC 1 Year 88088708 63716250 62916319 63686224OUP TCRC 2 Years 46284421 27712617 29993075 22162075OUP TCRC 5 Years 26022553 1093997 13121218 13061269

Table 15 TCRC best error measures for 2013 forecasts after 1000 MC simulations

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864GBM TCRC 1 Yearlowast 33111336 1425143 020020 1425143OUP TCRC 1 Yearlowast 4105042 536054 0076600777 536054LES TCRC 1 Year 107x10minus5514x10minus8 0003226x10minus4 468x10minus5324x10minus5 0003226x10minus4

lowastAveraged monthly steps

Table 16 TCRC Best error measures for 2013ndash2014 forecasts after 1000 MC simulations

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864GBM TCRC 2 Stepslowast 103x1031006 2200242 031034 2447271OUP TCRC 2 Stepslowast 37331376 1573158 0180201813 1900191LES TCRC 2 Steps 18924189 972097 0105701059 1375137lowastAveraged monthly steps

deviation of the estimator if MSE were considered to be thevarianceThough RMSE overreacts when high values of MSEare reached it is less prone to this than MSE since it iscalculated as its squared root thus not accounting for largeerrors as disproportionately as MSE does Furthermore aswe have compared an average of 1000 measures of errorscorresponding to each MC simulation performed the valuesobtained for average RMSE stay below the square root ofaverage MSE which indicates that some of these dispropor-tionate error measures are to some extent distorting thelatter Hence RMSE average values point towards a higheraccuracy for GBM five-year forecasts with averaged monthlysteps which is further endorsed by the average values ofMAD and MAPE thus being the one used for comparisonwith the other two models as shown in Table 17

The final comparison clearly shows how the LES modeloutperforms the other two at all average measures providedfollowed by the OUP model in accuracy although the lattermore than doubles the average MAPE value for LES

The results of simulations indicate that measures of errorstend to either differ slightly or not at all for either forecastsof any timeframe A coherent value with the 101 relation canthen be given with close to the same level of accuracy bymultiplying RC forecasts or dividing TC ones by 10

6 Conclusions

Copper TCRC are a keystone for pricing copper concen-trates which are the actual feedstock for copper smeltersThe potential evolution of TCRC is a question of botheconomic and technical significance forminers as their value

decreases the potential final selling price of concentratesAdditionally copper minersrsquo revenues are more narrowlyrelated to the market price of copper as well as to othertechnical factors such as ore dilution or the grade of theconcentrates produced Smelters on the contrary are hugelyaffected by the discount which they succeed in getting whenpurchasing the concentrates since that makes up the largestpart of their gross revenue besides other secondary sourcesIn addition eventual differences between TCRC may givecommodity traders ludicrous arbitrage opportunities Alsodifferences between short- and long-termTCRC agreementsoffer arbitrage opportunities for traders hence comprisinga part of their revenue in the copper concentrate tradingbusiness as well copper price fluctuations and the capacityto make economically optimum copper blends

As far as we are aware no rigorous research has beencarried out on the behaviour of these discounts Based onhistorical data on TCRC agreed upon in the LME CopperWeek from 2004 to 2017 three potentially suitable forecastingmodels for TCRC annual benchmark values have beencompared through four measures of forecasting accuracyat different horizons These models were chosen firstlydue to their broad implementation and proven capacityin commodity prices forecasting that they all share andsecondly because of the core differences in terms of pricebehaviour with each other

Focusing on the MAPE values achieved those obtainedby the LES model when TC and RC are treated indepen-dently have been significantly lower than for the rest ofthe models Indeed one-year-ahead MAPE measures for TCvalues for the GBM model (20) almost triple those of

Complexity 11

2004 2006 2008 2010 2012 2014405060708090

100

TC (U

SDM

T)

LES TC 1 Step Ahead 2013

2004 2006 2008 2010 2012 2014

LES TC 2 Steps Ahead 2013-2014

2004 2006 2008 2010 2012 2014 2016 2018

LES TC 5 Steps Ahead 2013-2017

2004 2006 2008 2010 2012 2014

LES RC 1 Step Ahead 2013

2004 2006 2008 2010 2012 2014456789

10LES RC 2 Steps Ahead 2013-2014

Observed Forecast

2004 2006 2008 2010 2012 2014 2016 2018

LES RC 5 Steps Ahead 2013-2017

405060708090

100110

405060708090

100

456789

10

RC (U

Sclb

)

456789

1011

Figure 4 One-step-ahead forecasts (2013) two-step-ahead forecasts (2013-2014) and five-step-ahead forecasts (2013-2017) for TC and RCusing the Linear Exponential Smoothing model (LES)

Table 17 TCRC Best error measures for 2013ndash2017 forecasts after 1000 MC simulations

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864GBM TCRC 5 Stepslowast 600x1036165 4651459 049048 5571551OUP TCRC 5 Stepslowast 84334843 2616262 0271502719 2896289LES TCRC 5 Steps 17755117 788078 0095100952 1084108lowastAveraged monthly steps

the OUP model (766) in contrast with the significantlylower values from the LES model (00046) This gaptends to be narrowed when TCRC values are forecastedat longer horizons when most measures of error becomemore even The GBM and OUP models have proven todeliver better accuracy performance when the TCRC valuesare projected monthly and then averaged to obtain annualbenchmark forecasts Even so the LES model remains themost accurate of all with MAPE values of 10 at two-year-ahead forecasts with 18 and 31 for TC for OUP and GBMrespectively

Finally despite TC and RC being two independentdiscounts applied to copper concentrates they are bothset jointly with an often 101 relation as our data revealsThis relation also transcends to simulation results and errormeasures hence showing a negligible discrepancy betweenthe independent forecasting of TC and RC or the jointforecasting of both values keeping the 101 relationThis is forinstance the case of the five-year-ahead OUP MAPE values(0271502719) which were obtained without observing the101 relation in the data A similar level of discrepancy wasobtained at any horizon with any model which indicates thatboth values could be forecasted with the same accuracy usingthe selected model with any of them and then applying the101 relation

Our findings thus suggest that both at short and atlong-term horizons TCRC annual benchmark levels tendto exhibit a pattern which is best fit by an LES model Thisindicates that these critical discounts for the copper tradingbusiness do maintain a certain dependency on past valuesThis would also suggest the existence of cyclical patterns incopper TCRC possibly driven by many of the same marketfactors that move the price of copper

This work contributes by delivering a formal tool forsmelters or miners to make accurate forecasts of TCRCbenchmark levels The level of errors attained indicatesthe LES model may be a valid model to forecast thesecrucial discounts for the copper market In addition ourwork further contributes to helping market participants toproject the price of concentrates with an acceptable degree ofuncertainty as now they may include a fundamental elementfor their estimation This would enable them to optimise theway they produce or process these copper concentrates Alsothe precise knowledge of these discountsrsquo expected behaviourcontributes to letting miners traders and smelters aliketake the maximum advantage from the copper concentratetrading agreements that they are part of As an additionalcontribution this work may well be applied to gold or silverRC which are relevant deduction concentrates when thesehave a significant amount of gold or silver

12 Complexity

Once TCRC annual benchmark levels are able to beforecasted with a certain level of accuracy future researchshould go further into this research through the explorationof the potential impact that other market factors may have onthese discounts

As a limitation of this research we should point out thetimespan of the data considered compared to those of otherforecasting works on commodity prices for example whichuse broader timespans For our case we have consideredthe maximum available sufficiently reliable data on TCRCbenchmark levels starting back in 2004 as there is noreliable data beyond this year Also as another limitationwe have used four measures of error which are among themost frequently used to compare the accuracy of differentmodels However other measures could have been used at anindividual level to test each modelrsquos accuracy

Appendix

A Models

A1 Geometric Brownian Motion (GBM) GBM can bewritten as a generalisation of a Wiener (continuous time-stochasticMarkov process with independent increments andwhose changes over any infinite interval of time are normallydistributed with a variance that increases linearly with thetime interval [22]) process

119889119909 = 120572119909119889119905 + 120590119909119889119911 (A1)

where according to Marathe and Ryan [30] the first termis known as the Expected Value whereas the second is theStochastic component with 120572 being the drift parameter and120590 the volatility of the process Also dz is the Wiener processwhich induces the abovementioned stochastic behaviour inthe model

119889119911 = isin119905radic119889119905 997888rarr isin119905 sim 119873 (0 1) (A2)

TheGBMmodel can be expressed in discrete terms accordingto

119909 = 119909119905 minus 119909119905minus1 = 120572119909119905minus1119905 + 120590119909119905minus1120598radic119905 (A3)

In GBM percentage changes in x 119909119909 are normally dis-tributed thus absolute changes in x 119909 are lognormallydistributed Also the expected value and variance for 119909(119905) are

E [119909 (119905)] = 1199090119890120572119905 (A4)

var [119909 (119905)] = 119909201198902120572119905 (1198901205902119905 minus 1) (A5)

A2 Orstein-Uhlenbeck Process (OUP) TheOUPprocess wasfirst defined by Uhlenbeck and Orstein [31] as an alternativeto the regular Brownian Motion to model the velocity ofthe diffusion movement of a particle that accounts for itslosses due to friction with other particles The OUP processcan be regarded as a modification of Brownian Motion incontinuous time where its properties have been changed

(Stationary Gaussian Markov and stochastic process) [32]These modifications cause the process to move towards acentral position with a stronger attraction the further it isfrom this position As mentioned above the OUP is usuallyemployed to model commodity prices and is the simplestversion of amean-reverting process [22]

119889119878 = 120582 (120583 minus 119878) 119889119905 + 120590119889119882119905 (A6)

where S is the level of prices 120583 the long-term average towhich prices tend to revert and 120582 the speed of reversionAdditionally in a similar fashion to that of the GBM 120590 isthe volatility of the process and 119889119882119905 is a Wiener processwith an identical definition However in contrast to GBMtime intervals in OUP are not independent since differencesbetween current levels of prices S and long-term averageprices 120583 make the expected change in prices dS more likelyeither positive or negative

The discrete version of the model can be expressed asfollows

119878119905 = 120583 (1 minus 119890minus120582Δ119905) + 119890minus120582Δ119905119878119905minus1 + 120590radic1 minus 119890minus2120582Δ1199052120582 119889119882119905 (A7)

where the expected value for 119909(119905) and the variance for (119909(119905) minus120583) areE [119909 (119905)] = 120583 + (1199090 minus 120583) 119890minus120582119905 (A8)

var [119909 (119905) minus 120583] = 12059022120582 (1 minus 119890minus2120582119905) (A9)

It can be derived from previous equations that as timeincreases prices will tend to long-term average levels 120583 Inaddition with large time spans if the speed of reversion 120582becomes high variance tends to 0 On the other hand if thespeed of reversion is 0 then var[119909(119905)] 997888rarr 1205902119905 making theprocess a simple Brownian Motion

A3Holtrsquos Linear Exponential Smoothing (LES) Linear Expo-nential Smoothing models are capable of considering bothlevels and trends at every instant assigning higher weights inthe overall calculation to values closer to the present than toolder ones LES models carry that out by constantly updatinglocal estimations of levels and trends with the intervention ofone or two smoothing constants which enable the models todampen older value effects Although it is possible to employa single smoothing constant for both the level and the trendknown as Brownrsquos LES to use two one for each known asHoltrsquos LES is usually preferred since Brownrsquos LES tends torender estimations of the trend ldquounstablerdquo as suggested byauthors such as Nau [33] Holtrsquos LES model comes definedby the level trend and forecast updating equations each ofthese expressed as follows respectively

119871 119905 = 120572119884119905 + (1 minus 120572) (119871 119905minus1 + 119879119905minus1) (A10)

119879119905 = 120573 (119871 119905 minus 119871 119905minus1) + (1 minus 120573) 119879119905minus1 (A11)

119905+119896 = 119871 119905 + 119896119879119905 (A12)

Complexity 13

With 120572 being the first smoothing constant for the levelsand 120573 the second smoothing constant for the trend Highervalues for the smoothing constants imply that either levels ortrends are changing rapidly over time whereas lower valuesimply the contrary Hence the higher the constant the moreuncertain the future is believed to be

B Model Calibration

Each model has been calibrated individually using two setsof data containing TC and RC levels respectively from 2004to 2012 The available dataset is comprised of data from 2004to 2017 which make the calibration data approximately 23 ofthe total

B1 GBM Increments in the logarithm of variable x aredistributed as follows

(ln119909) sim 119873((120572 minus 12059022 ) 119905 120590119905) (B1)

Hence if m is defined as the sample mean of the differenceof the natural logarithm of the time series for TCRC levelsconsidered for the calibration and n as the number ofincrements of the series considered with n=9

119898 = 1119899119899sum119905=1

(ln119909119905 minus ln119909119905minus1) (B2)

119904 = radic 1119899 minus 1119899sum119894=1

(ln119909119905 minus ln119909119905minus1 minus 119898)2 (B3)

119898 = 120572 minus 12059022 (B4)

119904 = 120590 (B5)

B2 OUP The OUP process is an AR1 process [22] whoseresolution is well presented by Woolridge [34] using OLStechniques fitting the following

119910119905 = 119886 + 119887119910119905minus1 + 120576119905 (B6)

Hence the estimators for the parameters of the OUP modelare obtained by OLS for both TC and RC levels indepen-dently

= minus ln 119887Δ119905 (B7)

120583 = 1198861 minus 119887 (B8)

= 120576119905radic 2 ln (1 + 119887)(1 + 119887)2 minus 1 (B9)

B3 LES A linear regression is conducted on the inputdataset available to find the starting parameters for the LES

model the initial Level 1198710 and the initial value of thetrend 1198790 irrespective of TC values and RC values Hereas recommended by Gardner [35] the use of OLS is highlyadvisable due to the erratic behaviour shown by trends inthe historic data so the obtaining of negative values of 1198780 isprevented Linear regression fulfils the following

119884119905 = 119886119905 + 119887 (B10)

1198710 = 119887 (B11)

1198790 = 119886 (B12)

By fixing the two smoothing constants the values for theforecasts 119905+119896 can be calculated at each step using the modelequations There are multiple references in the literature onwhat the optimum range for each smoothing constant isGardner [35] speaks of setting moderate values for bothparameter less than 03 to obtain the best results Examplespointing out the same may be found in Brown [36] Coutie[37] Harrison [38] and Montgomery and Johnson [39]Also for many applications Makridakis and Hibon [20] andChatfield [40] found that parameter values should fall withinthe range of 03-1 On the other hand McClain and Thomas[41] provided a condition of stability for the nonseasonal LESmodel given by

0 lt 120572 lt 20 lt 120573 lt 4 minus 2120572120572

(B13)

Also the largest possible value of 120572 that allows the avoidanceof areas of oscillation is proposed by McClain and Thomas[41] and McClain [42]

120572 lt 4120573(1 + 120573)2 (B14)

However according to Gardner there is no tangible proofthat this value improves accuracy in any form Nonethelesswe have opted to follow what Nau [33] refers to as ldquotheusual wayrdquo namely minimising the Mean Squared Error(MSE) of the one-step-ahead forecast of TCRC for eachinput data series previously mentioned To do so Matlabrsquosfminsearch function has been used with function and variabletolerance levels of 1x10minus4 as well as a set maximum numberof function iterations and function evaluations of 1x106 tolimit computing resources In Table 18 the actual number ofnecessary iterations to obtain optimum values for smoothingconstants is shown As can be seen the criteria are wellbeyond the final results which ensured that an optimumsolution was reached with assumable computing usage (thesimulation required less than one minute) and with a highdegree of certainty

14 Complexity

Table 18 Iterations on LES smoothing constants optimisation

Iterations Function EvaluationsTC 1-Step 36 40TC 2-Steps 249945 454211TC 5-Steps 40 77RC 1-Step 32 62RC 2-Steps 254388 462226RC 5-Steps 34 66

C Error Calculations

C1 Mean Square Error (MSE)

119872119878119864 = 1119899119899sum119894=1

(119894 minus 119884119894)2 (C1)

where 119894 are the forecasted values and 119884119894 those observedC2 Mean Absolute Deviation (MAD)

119872119860119863 = 1119899119899sum119894=1

10038161003816100381610038161003816119894 minus 11988410038161003816100381610038161003816 (C2)

where 119894 are the forecasted values and 119884 the average value ofall the observations

C3 Mean Absolute Percentage Error (MAPE)

119872119860119875119864 = 1119899119899sum119894=1

10038161003816100381610038161003816119894 minus 11988411989410038161003816100381610038161003816119884119894 (C3)

The above formula is expressed in parts-per-one and is theone used in the calculations conducted here Hence multi-plying the result by 100 would deliver percentage outcomes

C4 Root Mean Square Error (RMSE)

119877119872119878119864 = radic119872119878119864 = radic 1119899119899sum119894=1

(119894 minus 119884119894)2 (C4)

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Supplementary Materials

The Matlab codes developed to carry out the simulationsfor each of the models proposed as well as the datasetsused and the Matlab workspaces with the simulationsoutcomes as they have been reflected in this paper areprovided in separate folders for each model GBM foldercontains the code to make the simulations for the Geo-metric Brownian Motion model ldquoGBMmrdquo as well as aseparate script which allows determining the total numberof monthly step simulations for which errors levels havebeen lower than annual-step simulations ldquoErrCoutermrdquoThe datasets used are included in two excel files ldquoTCxlsxrdquoand ldquoRCxlsxrdquo ldquoGBM OK MonthlyStepsmatrdquo is the Matlabworkspace containing the outcome of monthly step simu-lations whereas ldquoGBM OK AnnualStepsmatrdquo contains theoutcome for annual-step simulations Also ldquoGBM OKmatrdquois the workspace to be loaded prior to performing GBMsimulations MR folder contains the code file to carryout the simulations for the Orstein-Uhlenbeck ProcessldquoMRmrdquo as well as the separate script to compare errorsof monthly step simulations with annual-step simula-tions ldquoErrCountermrdquo ldquoTCxlsxrdquo and ldquoRCxlsxrdquo containthe TCRC benchmark levels from 2004 to 2017 Finallymonthly step simulationsrsquo outcome has been saved inthe workspace ldquoMR OK MonthlyStepsmatrdquo while annual-step simulationsrsquo outcome has been saved in the Matlabworkspace ldquoMR OK AnnualStepsmatrdquo Also ldquoMR OKmatrdquois the workspace to be loaded prior to performing MRsimulations LES folder contains the code file to carry outthe calculations necessary to obtain the LES model forecastsldquoLESmrdquo Three separate Matlab functions are includedldquomapeLESmrdquo ldquomapeLES1mrdquo and ldquomapeLES2mrdquo whichdefine the calculation of the MAPE for the LESrsquo five-year-ahead one-year-ahead and two-year-ahead forecastsAlso ldquoLES OKmatrdquo is the workspace to be loaded prior toperforming LES simulations Finally graphs of each modelhave been included in separate files as well in a subfoldernamed ldquoGraphsrdquo within each of the modelsrsquo folder Figuresare included in Matlabrsquos format (fig) and TIFF format TIFFfigures have been generated in both colours and black andwhite for easier visualization (Supplementary Materials)

References

[1] United Nations International Trade Statistics Yearbook vol 22016

[2] International Copper StudyGroupTheWorld Copper Factbook2017

[3] K M Johnson J M Hammarstrom M L Zientek and CL Dicken ldquoEstimate of undiscovered copper resources of theworld 2013rdquo US Geological Survey 2017

[4] M E Schlesinger and W G Davenport Extractive Metallurgyof Copper Elsevier 5th edition 2011

[5] S Gloser M Soulier and L A T Espinoza ldquoDynamic analysisof global copper flows Global stocks postconsumer materialflows recycling indicators and uncertainty evaluationrdquo Envi-ronmental Science amp Technology vol 47 no 12 pp 6564ndash65722013

Complexity 3

mathematical model to render these forecasts is a researchtopic that is yet to be explored in depth This is a questionwith undeniable economic implications for traders minersand smelters alike due to the role of TCRC in the coppertrading revenue stream

Our study seeks to determine an appropriate forecastingtechnique for TCRC benchmark levels for copper concen-trates that meets the need of reliability and accuracy Toperform this research three different and frequently-appliedtechniques have been preselected from among the optionsavailable in the literature Then their forecasting accuracyat different time horizons will be tested and comparedThese techniques (Geometric Brownian Motion -GBM-the Mean Reversion -MR- Linear Exponential Smoothing-LES-) have been chosen primarily because they are commonin modelling commodities prices and their future expectedbehaviour aswell as in stock indicesrsquo predictiveworks amongother practical applications [10ndash13] The selection of thesemodels is further justified by the similarities shared byTCRCwith indices interest rates or some economic variables thatthese models have already been applied to Also in our viewthe predictive ability of these models in commodity pricessuch as copper is a major asset to take them into consider-ation The models have been simulated using historical dataof TCRC annual benchmark levels from 2004 to 2017 agreedupon during the LME Copper Week The dataset employedhas been split into two parts with two-thirds as the in-sampledataset and one-third as the out-of-sample one

The main contribution of this paper is to provide auseful and applicable tool to all parties involved in thecopper trading business to forecast potential levels of criticaldiscounts to be applied to the copper concentrates valuationprocess To carry out our research we have based ourselves onthe following premises (1)GBMwould deliver good forecastsif copper TCRC benchmark levels vary randomly over theyears (2) a mean-reverting model such as the OUP woulddeliver the best forecasts if TCRC levels were affected bymarket factors and consequently they move around a long-term trend and (3) a moving average model would givea better forecast than the other two models if there werea predominant factor related to precedent values affectingthe futures ones of benchmark TCRC In addition we havealso studied the possibility that a combination of the modelscould deliver the most accurate forecast as the time horizonconsidered is increased since there might thus be a limitedeffect of past values or of sudden shocks on future levels ofbenchmark TCRC So after some time TCRC levels couldbe ldquonormalizedrdquo towards a long-term average

The remainder of this article is structured as followsSection 2 revises the related work on commodity discountsforecasting and commodity prices forecasting techniques aswell as different forecasting methods Section 3 presents thereasoning behind the choice of each of the models employedas well as the historic datasets used to conduct the studyand the methodology followed Section 4 shows the resultsof simulations of different methods Section 5 indicates errorcomparisons to evaluate the best forecasting alternative forTCRC among all those presented Section 6 contains thestudyrsquos conclusions and proposes further lines of research

2 Related Work

The absence of any specific method in the specialised litera-ture in relation to copper TCRC leads us to revisit previousliterature in order to determine the most appropriate modelto employ for our purpose considering those that havealready been used in commodity price forecasting as thelogical starting point due to the application similarities thatthey share with ours

Commodity prices and their forecasting have been atopic intensively analysed in much research Hence there aremultiple examples in literature with an application to oneor several commodities such as Xiong et al [14] where theaccuracy of differentmodelswas tested to forecast the intervalof agricultural commodity future prices Shao and Dai [15]whose work employs the Autoregressive Integrated MovingAverage (ARIMA) methodology to forecast food crop pricessuch as wheat rice and corn and Heaney [16] who tests thecapacity of commodities future prices to forecast their cashprice were the cost of carry to be included in considerationsusing the LME Lead contract as an example study As wasdone in other research [17ndash19] Table 1 summarizes similarresearch in the field of commodity price behaviours

From the summary shown in Table 1 it is seen that mov-ing average methods have become increasingly popular incommodity price forecastingThemost broadly implementedmoving average techniques are ARIMA and ExponentialSmoothing They consider the entire time series data anddo not assign the same weight to past values than thosecloser to the present as they are seen as affecting greater tofuture valuesThe Exponential Smoothingmodelsrsquo predictiveaccuracy has been tested [20 21] concluding that there aresmall differences between them (Exponential Smoothing)and ARIMA models

Also GBM andMRmodels have been intensively appliedto commodity price forecasting Nonetheless MR modelspresent a significant advantage over GBM models whichallows them to consider the underlying price trend Thisadvantage is of particular interest for commodities thataccording to Dixit and Pindyck [22]mdashpp 74 regarding theprice of oil ldquoin the short run it might fluctuate randomly upand down (in responses to wars or revolutions in oil producingcountries or in response to the strengthening or weakeningof the OPEC cartel) in the longer run it ought to be drawnback towards the marginal cost of producing oil Thus onemight argue that the price of oil should be modelled as a mean-reverting processrdquo

3 Methodology

This section presents both the justification of the modelschosen in this research and the core reference dataset used asinputs for each model compared in the methodology as wellas the steps followed during the latter to carry out an effectivecomparison in terms of the TCRC forecasting ability of eachof these models

31 Models in Methodology GBM (see Appendix A for mod-els definition) has been used in much earlier research as a

4 Complexity

Table 1 Summary of previous literature research

AUTHOR RESEARCH

Shafiee amp Topal [17] Validates a modified version of the long-term trend reverting jump and dip diffusion model for forecastingcommodity prices and estimates the gold price for the next 10 years using historical monthly data

Li et al [18] Proposes an ARIMA-Markov Chain method to accurately forecast mineral commodity prices testing themethod using mineral molybdenum prices

Issler et al [19]Investigates several commoditiesrsquo co-movements such as Aluminium Copper Lead Nickel Tin and Zinc atdifferent time frequencies and uses a bias-corrected average forecast method proposed by Issler and Lima [43]to give combined forecasts of these metal commodities employing RMSE as a measure of forecasting accuracy

Hamid amp Shabri [44]Models palm oil prices using the Autoregressive Distributed Lag (ARDL) model and compares its forecastingaccuracy with the benchmark model ARIMA It uses an ARDL bound-testing approach to co-integration in

order to analyse the relationship between the price of palm oil and its determinant factorsDuan et al [45] Predicts Chinarsquos crude oil consumption for 2015-2020 using the fractional-order FSIGMmodel

Brennan amp Schwartz [46] Employs the Geometric Brownian Motion (GBM) to analyse a mining projectrsquos expected returns assuming itproduces a single commodity

McDonald amp Siegel [47] Uses GBM to model the random evolution of the present value of an undefined asset in an investment decisionmodel

Zhang et al [48] Models gold prices using the Ornstein-Uhlenbeck Process (OUP) to account for a potentially existent long-termtrend in a Real Option Valuation of a mining project

Sharma [49] Forecasts gold prices in India with the Box Jenkins ARIMA method

way of modelling prices that are believed not to follow anyspecific rule or pattern and hence seen as random Blackand Scholes [23] first used GBM to model stock prices andsince then others have used it to model asset prices as well ascommodities these being perhaps themost common of all inwhich prices are expected to increase over time as does theirvariance [11] Hence following our first premise concerningwhether TCRC might vary randomly there should not exista main driving factor that would determine TCRC futurebenchmark levels and therefore GBM could to a certainextent be a feasible model for them

However although GBM or ldquorandom walkrdquo may be wellsuited to modelling immediate or short-term price paths forcommodities or for TCRC in our case it lacks the abilityto include the underlying long-term price trend should weassume that there is oneThus in accordance with our secondpremise on benchmark TCRC behaviour levels would movein line with copper concentrate supply and demand as wellas the smeltersrsquo and refineriesrsquo available capacity to trans-form concentrates into metal copper Hence a relationshipbetween TCRC levels and copper supply and demand isknown to exist and therefore is linked to its market priceso to some extent they move together coherently Thereforein that case as related works on commodity price behavioursuch as Foo Bloch and Salim [24] do we have opted for theMR model particularly the OUP model

Both GBM and MR are Markov processes which meansthat future values depend exclusively on the current valuewhile the remaining previous time series data are not consid-eredOn the other handmoving averagemethods employ theaverage of a pre-established number of past values in differentways evolving over time so future values do not rely exclu-sively on the present hence behaving as though they had onlya limitedmemory of the pastThis trait of themoving averagemodel is particularly interestingwhen past prices are believedto have a certain though limited effect on present values

which is another of the premises for this research Existingalternatives ofmoving average techniques pursue consideringthis ldquomemoryrdquo with different approaches As explained byKalekar [25] Exponential Smoothing is suitable only for thebehaviours of a specific time series thus Single ExponentialSmoothing (SES) is reasonable for short-term forecastingwith no specific trend in the observed data whereas DoubleExponential Smoothing or Linear Exponential Smoothing(LES) is appropriate when data shows a cyclical pattern ora trend In addition seasonality in observed data can becomputed and forecasted through the usage of ExponentialSmoothing by the Holt-Winters method which adds an extraparameter to the model to handle this characteristic

32 TCRC Benchmark Levels and Sample Dataset TCRClevels for copper concentrates continuously vary through-out the year relying on private and individual agreementsbetween miners traders and smelters worldwide Nonethe-less the TCRC benchmark fixed during the LME week eachOctober is used by market participants as the main referenceto set actual TCRC levels for each supply agreed upon for thefollowing year Hence the yearrsquos benchmark TCRC is takenhere as a good indicator of a yearrsquos TCRC average levelsAnalysed time series of benchmark TCRC span from 2004through to 2017 as shown in Table 2 as well as the sourceeach valuewas obtained fromWe have not intended to reflectthe continuous variation of TCRC for the course of anygiven year though we have however considered benchmarkprices alone as we intuitively assume that annual variations ofactual TCRC in contracts will eventually be reflected in thebenchmark level that is set at the end of the year for the yearto come

33 TCRC Relation TC and RC normally maintain a 101relation with different units with TC being expressed in US

Complexity 5

GBM RC First 20 pathsGBM TC First 20 paths

0

50

100

150

200

250

300

TC (U

SDM

T)

2013 2014 2015 2016 20172012

GBM TC First 20 paths - Year Averaged

05

101520253035

RC (U

Sclb

)

0

5

10

15

20

25

30

RC (U

Slb

)

0

50

100

150

200

250

TC (U

SDM

T)

2013 2014 2015 2016 20172012

Observed TC Benchmark Levels

2013 2014 2015 2016 20172012

Observed RC Benchmark Levels

2013 2014 2015 2016 20172012

GBM RC First 20 paths - Year Averaged

Figure 2 The upper illustrations show the first 20 Monte Carlo paths for either TC or RC levels using monthly steps The illustrations belowshow the annual averages of monthly step forecasts for TC and RC levels

MR TC First 20 paths

2012 2013 2014 2015 2016 2017

MR RC First 20 paths

MR RC First 20 paths - Year Averaged

2013 2014 2015 2016 20172012

Observed

5

6

7

8

9

10

11

RC (U

Slb

)

2468

10121416

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Sclb

)

0

50

100

150

200

TC (U

SDM

T)

2013 2014 2015 2016 20172012

MR TC First 20 paths - Year Averaged

2013 2014 2015 2016 20172012

Observed

50

60

70

80

90

100

110

TC (U

SDM

T)

Figure 3 The first 20 Monte Carlo paths following the OUP model using monthly steps are shown on the upper illustrations for either TCor RC The annual averaged simulated paths every 12 values are shown below

dollars per metric tonne and RC in US cents per pound ofpayable copper content in concentrates In fact benchmarkdata show that of the last 14 years it was only in 2010that values of TC and RC did not conform to this relation

though it did remain close to it (46547) Nonetheless thisrelation has not been observed in the methodology hereinthus treating TC and RC independently developing separateunrelated forecasts for TC and RC to understand whether

6 Complexity

Table 2 TCRC year benchmark levels (Data available free ofcharge Source Johansson [50] Svante [51] Teck [52] Shaw [53]Willbrandt and Faust [54 55] Aurubis AG [56] Drouven and Faust[57] Aurubis AG [58] EY [59] Schachler [60] Nakazato [61])

YEAR TC (USDMT) RC (USclb)2004 45 452005 85 852006 95 952007 60 602008 45 452009 75 752010 465 472011 56 562012 635 6352013 70 702014 92 922015 107 1072016 9735 97352017 925 925

these individual forecasts for TC and RC would render betterresults than a single joint forecast for both TCRC levels thatwould take into account the 101 relation existent in the data

34 Models Comparison Method The works referred to inthe literature usually resort to different measures of errorsto conduct the testing of a modelrsquos forecasting capacityregardless of the specific nature of the forecasted value be itcotton prices or macroeconomic parameters Thus the mostwidely errors used includeMean Squared Error (MSE) MeanAbsolute Deviation (MAD) Mean Absolute Percentage Error(MAPE) and Root Mean Square Error (RMSE) [14 16 17 2125ndash29] In this work Geometric Brownian Motion (GBM)Ornstein-Uhlenbeck Process (OUP) and Linear ExponentialSmoothing (LES) models have been used to forecast annualTCRC benchmark levels using all the four main measuresof error mentioned above to test the predictive accuracy ofthese three models The GBM and OUP models have beensimulated and tested with different step sizes while the LESmodel has been analysed solely using annual time steps

The GBM and OUPmodels were treated separately to theLESmodel thus GBM andOUPwere simulated usingMonteCarlo (MC) simulations whereas LES forecasts were simplecalculations In a preliminary stage the models were firstcalibrated to forecast values from 2013 to 2017 using availabledata from 2004 to 2012 for TC and RC separately hencethe disregarding of the well-known inherent 101 relation (seeAppendix B for Models Calibration) The GBM and OUPmodels were calibrated for two different step sizes monthlysteps and annual steps in order to compare forecastingaccuracy with each step size in each model Following thecalibration of the models Monte Carlo simulations (MC)were carried out usingMatlab software to render the pursuedforecasts of GBM andOUPmodels obtaining 1000 simulatedpaths for each step size MC simulations using monthly stepsfor the 2013ndash2017 timespan were averaged every 12 steps to

deliver year forecasts of TCRC benchmark levels On theother hand forecasts obtained by MC simulations takingannual steps for the same period were considered as yearforecasts for TCRC annual benchmark levels without theneed for extra transformation Besides LES model forecastswere calculated at different time horizons to be able tocompare the short-term and long-term predictive accuracyLES forecasts were obtained for one-year-ahead hence usingknown values of TCRC from 2004 to 2016 for two yearsahead stretching the calibrating dataset from 2004 to 2015for five-year-ahead thus using the same input dataset as forthe GBM and OUP models from 2004 to 2012

Finally for every forecast path obtained we have calcu-lated the average of the squares of the errors with respect tothe observed valuesMSE the average distance of a forecast tothe observed meanMAD the average deviation of a forecastfrom observed values MAPE and the square root of MSERMSE (see Appendix C for error calculations) The results ofMSE MAD MAPE and RMSE calculated for each forecastpathwere averaged by the total number of simulations carriedout for each case Averaged values of error measures of allsimulated paths MSE MAD MAPE and RMSE for bothannual-step forecasts and monthly step forecasts have beenused for cross-comparison between models to test predictiveability at every step size possible

Also to test each modelrsquos short-term forecasting capacityagainst its long-term forecasting capacity one-year-aheadforecast errors of the LES model were compared with theerrors from the last year of the GBM and OUP simulatedpaths Two-year-ahead forecast errors of LES models werecomparedwith the average errors of the last two years ofGBMand OUP and five-year-ahead forecast errors of LES modelswere compared with the overall average of errors of the GBMand OUP forecasts

4 Analysis of Results

Monte Carlo simulations of GBM and OUP models render1000 different possible paths for TC and RC respectivelyat each time step size considered Accuracy errors for bothannual time steps and averaged monthly time steps for bothGBM and OUP forecasts were first compared to determinethe most accurate time step size for each model In additionthe LES model outcome for both TC and RC at differenttimeframes was also calculated and measures of errors for allthe three alternative models proposed at an optimum timestep were finally compared

41 GBM Forecast The first 20 of 1000 Monte Carlo pathsfor the GBM model with a monthly step size using Matlabsoftware may be seen in Figure 2 for both the TC and the RClevels compared to their averaged paths for every 12 valuesobtainedThe tendency for GBM forecasts to steeply increaseover time is easily observable in the nonaveraged monthlystep paths shown

The average values of error of all 1000MC paths obtainedthrough simulation for averaged monthly step and annual-step forecasts are shown in Table 3 for both TC and RC

Complexity 7

Table 3 Average of main measures of error for GBM after 1000 MC simulations from 2013 to 2017

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC Averaged Monthly Steps 560x103 4698 050 5638TC Annual Steps 520x103 4902 052 5857RC Averaged Monthly Steps 5874 455 048 546RC Annual Steps 5085 491 052 584

Table 4 Number of paths for which measures of error are higher for monthly steps than for annual steps in GBM

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC 4571000 4551000 4531000 4531000RC 4391000 4301000 4291000 4291000

discounts over the period from 2013 to 2017 which maylead to preliminary conclusions in terms of an accuracycomparison between averaged monthly steps and annualsteps

However a more exhaustive analysis is shown in Table 4where the number of times the values of each error measureare higher for monthly steps is expressed over the totalnumber of MC simulations carried out The results indicatethat better values of error are reached the majority of timesfor averagedmonthly step simulations rather than for straightannual ones

In contrast short-term accuracy was also evaluated byanalysing the error measures of one-year-ahead forecasts(2013) in Table 5 and of two-year-ahead forecasts (2013ndash2014)in Table 6 The results indicate as one may expect thataccuracy decreases as the forecasted horizon is widened withthe accuracy of averaged monthly step forecasts remaininghigher than annual ones as found above for the five-yearforecasting horizon

42 OUP Forecast Long-term levels for TC and RC 120583 thespeed of reversion 120582 and the volatility of the process 120590 arethe parameters determined at themodel calibration stage (seeAppendix B for the model calibration explanation) whichdefine the behaviour of the OUP shown in Table 7 Thecalibration was done prior to the Monte Carlo simulationfor both TC and RC with each step size using availablehistorical data from 2004 to 2012 The OUP model was fittedwith the corresponding parameters for each case upon MCsimulation

Figure 3 shows theMeanReversionMCestimations of theTCRC benchmark values from 2013 to 2017 using monthlystepsThemonthly forecastswere rendered from January 2012through to December 2016 and averaged every twelve valuesto deliver a benchmark forecast for each year The averagedresults can be comparable to actual data as well as to theannual Monte Carlo simulations following Mean ReversionThe lower-side figures show these yearly averaged monthlystep simulation outcomes which clearly move around a dash-dotted red line indicating the long-term run levels for TCRCto which they tend to revert

The accuracy of monthly steps against annual steps forthe TCRC benchmark levels forecast was also tested by

determining the number of simulations for which averageerror measures became higher Table 8 shows the numberof times monthly simulations have been less accurate thanannual simulations for five-year-ahead OUP forecasting bycomparing the four measures of errors proposed The resultsindicate that only 25-32 of the simulations drew a higheraverage error which clearly results in a better predictiveaccuracy for monthly step forecasting of TCRC annualbenchmark levels

The averaged measures of errors obtained after the MCsimulations of the OUP model for both averaged monthlysteps and annual steps giving TCRC benchmark forecastsfrom 2013 to 2017 are shown in Table 9

The error levels of the MC simulations shown in Table 9point towards a higher prediction accuracy of averagedmonthly step forecasts of the OUP Model yielding anaveraged MAPE value that is 129 lower for TC and RC 5-step-ahead forecasts In regard to MAPE values for monthlysteps only 266 of the simulations rise above annual MCsimulations for TC and 25 for RC 5-step-ahead forecastswhich further underpins the greater accuracy of thisOUP set-up for TCRC level forecasts A significant lower probabilityof higher error levels for TCRC forecasts with monthly MCOUP simulations is reached for the other measures providedIn addition short-term and long-term prediction accuracywere tested by comparing errors of forecasts for five-year-ahead error measures in Table 10 one-year-ahead in Table 11and two-year-ahead in Table 12

With a closer forecasting horizon errormeasures show animprovement of forecasting accuracy when average monthlysteps are used rather than annual ones For instance theMAPE values for 2013 forecast for TC are 68 lower foraveraged monthly steps than for annual steps and alsoMAPE for 2013ndash2014 were 30 lower for both TC and RCforecasts Similarly better values of error are achieved for theother measures for averaged monthly short-term forecaststhan in other scenarios In addition as expected accuracyis increased for closer forecasting horizons where the levelof errors shown above becomes lower as the deviation offorecasts is trimmed with short-term predictions

43 LES Forecast In contrast to GBM and OUP the LESmodel lacks any stochastic component so nondeterministic

8 Complexity

Table 5 Average for main measures of error for GBM after 1000 MC simulations for 2013

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC Averaged Monthly Steps 33658 1455 021 1455TC Annual Steps 69381 2081 030 2081RC Averaged Monthly Steps 297 138 020 138RC Annual Steps 657 205 029 205

Table 6 Average for main measures of error for GBM after 1000 MC simulations for 2013-2014

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC Averaged Monthly Steps 99492 2187 031 2435TC Annual Steps 133x103 2833 040 3104RC Averaged Monthly Steps 1035 240 034 271RC Annual Steps 1313 284 034 311

Table 7 OUP parameters obtained in the calibration process

120583 120582 120590TCMonthly 6345 4792x105 2534TC Annual 6345 2974 1308RC Monthly 635 4763x105 2519RC Annual 635 2972 1305

methods such as theMonte Carlo are not required to obtain aforecast Nonetheless the LES model relies on two smoothingconstants which must be properly set in order to deliveraccurate predictions hence the values of the smoothingconstants were first optimised (see Appendix B for LESModel fitting and smoothing constants optimisation) Theoptimisation was carried out for one-year-ahead forecaststwo-year-ahead forecasts and five-year-ahead forecasts byminimising the values of MSE for both TC and RC Thedifferent values used for smoothing constants as well asthe initial values for level and trend obtained by the linearregression of the available dataset from 2004 through to 2012are shown in Table 12

Compared one-year-ahead two-year-ahead and five-year-ahead LES forecasts for TC and RC are shown inFigure 4 clearly indicating a stronger accuracy for shorter-term forecasts as the observed and forecasted plotted linesoverlap

Minimumvalues for errormeasures achieved through theLESmodel parameter optimisation are shown in Table 13Thevalues obtained confirm the strong accuracy for shorter-termforecasts of TCRC seen in Figure 4

5 Discussion and Model Comparison

The forecasted values of TCRC benchmark levels couldeventually be applied to broader valuation models for copperconcentrates and their trading activities as well as to thecopper smeltersrsquo revenue stream thus the importance ofdelivering as accurate a prediction as possible in relation tothese discounts to make any future application possible Eachof the models presented may be a feasible method on its own

with eventual later adaptations to forecasting future valuesof benchmark TCRC Nonetheless the accuracy of thesemodels as they have been used in this work requires firstlya comparison to determine whether any of them could bea good standalone technique and secondly to test whethera combination of two or more of them would deliver moreprecise results

When comparing the different error measures obtainedfor all the three models it is clearly established that resultsfor a randomly chosen simulation of GBM or OUP wouldbe more likely to be more precise had a monthly step beenused to deliver annual forecasts instead of an annual-step sizeIn contrast average error measures for the entire populationof simulations with each step size employed showing thatmonthly step simulations for GBM and OUP models arealways more accurate than straight annual-step forecastswhen a shorter time horizon one or two-year-ahead is takeninto consideration However GBM presents a higher levelof forecasting accuracy when the average for error measuresof all simulations is analysed employing annual steps forlong-term horizons whereas OUP averaged monthly stepforecasts remain more accurate when predicting long-termhorizons Table 14 shows the error improvement for the aver-aged monthly step forecasts of each model Negative valuesindicate that better levels of error averages have been found instraight annual forecasts than for monthly step simulations

Considering the best results for each model and com-paring their corresponding error measures we can opt forthe best technique to employ among the three proposed inthis paper Hence GBM delivers the most accurate one-yearforecast when averaging the next twelve-month predictionsfor TCRC values as does the MR-OUP model Table 15shows best error measures for one-year-ahead forecasts forGBM OUP and LES models

Unarguably the LES model generates minimal errormeasures for one-year-ahead forecasts significantly less thanthe other models employed A similar situation is found fortwo-year-ahead forecasts whereminimumerrormeasures arealso delivered by the LES model shown in Table 16

Finally accuracy measures for five-year-ahead forecastsof the GBM model might result in somewhat contradictory

Complexity 9

Table 8 Number of paths for whichmeasures of error are higher for monthly steps than for annual steps inMR-OUP 5-StepsMC simulation

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC 3111000 2831000 2661000 2661000RC 3161000 2811000 2501000 2501000

Table 9 Average for main measures of error for MR-OUP after 1000 MC simulations 2013-2017

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC Averaged Monthly Steps 84254 2614 027 2895TC Annual Steps 114x103 2931 031 3325RC Averaged Monthly Steps 840 261 027 289RC Annual Steps 1148 294 031 333

Table 10 Average for main measures of error for MR-OUP after 1000 MC simulations 2013

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC Averaged Monthly Steps 3914 522 007 522TC Annual Steps 36643 1555 022 1555RC Averaged Monthly Steps 039 051 007 051RC Annual Steps 363 154 022 154

Table 11 Average for main measures of error for MR-OUP after 1000 MC simulations 2013-2014

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC Averaged Monthly Steps 37165 1567 018 1900TC Annual Steps 68244 2185 026 2324RC Averaged Monthly Steps 376 157 018 191RC Annual Steps 689 219 026 243

Table 12 LES Model Optimised Parameters

L0 T0 120572 120573TC 1 year 713611 -15833 -02372 01598RC 1 year 71333 -01567 -02368 01591TC 2 years 713611 -15833 -1477x10minus4 7774226RC 2 years 71333 -01567 -1448x10minus4 7898336TC 5 years 713611 -15833 -02813 01880RC 5 years 71333 -01567 -02808 01880

Table 13 LES error measures for different steps-ahead forecasts

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC 1 year 10746x10minus5 00033 46831x10minus5 00033RC 1 year 51462x10minus8 22685x10minus4 32408x10minus5 22685x10minus4

TC 2 years 189247 97275 01057 137567RC 2 years 18977 09741 01059 13776TC 5 years 1775531 78881 00951 108422RC 5 years 11759 07889 00952 10844

terms for MSE reaching better values for annual steps thanfor averaged monthly steps while the other figures do betteron averagedmonthly steps Addressing the definition ofMSEthis includes the variance of the estimator as well as its biasbeing equal to its variance in the case of unbiased estimators

Therefore MSE measures the quality of the estimator butalso magnifies estimator deviations from actual values sinceboth positive and negative values are squared and averagedIn contrast RMSE is calculated as the square root of MSEand following the previous analogy stands for the standard

10 Complexity

Table 14 Error Average Improvement for averaged monthly steps before annual steps

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864GBM TCRC 1 Year 53265653 32723587 33333548 32723587GBM TCRC 2 Years 26952635 25751597 2619286 24341369GBM TCRC 5 Years -1173-1104 873709 926769 747565OUP TCRC 1 Year 88088708 63716250 62916319 63686224OUP TCRC 2 Years 46284421 27712617 29993075 22162075OUP TCRC 5 Years 26022553 1093997 13121218 13061269

Table 15 TCRC best error measures for 2013 forecasts after 1000 MC simulations

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864GBM TCRC 1 Yearlowast 33111336 1425143 020020 1425143OUP TCRC 1 Yearlowast 4105042 536054 0076600777 536054LES TCRC 1 Year 107x10minus5514x10minus8 0003226x10minus4 468x10minus5324x10minus5 0003226x10minus4

lowastAveraged monthly steps

Table 16 TCRC Best error measures for 2013ndash2014 forecasts after 1000 MC simulations

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864GBM TCRC 2 Stepslowast 103x1031006 2200242 031034 2447271OUP TCRC 2 Stepslowast 37331376 1573158 0180201813 1900191LES TCRC 2 Steps 18924189 972097 0105701059 1375137lowastAveraged monthly steps

deviation of the estimator if MSE were considered to be thevarianceThough RMSE overreacts when high values of MSEare reached it is less prone to this than MSE since it iscalculated as its squared root thus not accounting for largeerrors as disproportionately as MSE does Furthermore aswe have compared an average of 1000 measures of errorscorresponding to each MC simulation performed the valuesobtained for average RMSE stay below the square root ofaverage MSE which indicates that some of these dispropor-tionate error measures are to some extent distorting thelatter Hence RMSE average values point towards a higheraccuracy for GBM five-year forecasts with averaged monthlysteps which is further endorsed by the average values ofMAD and MAPE thus being the one used for comparisonwith the other two models as shown in Table 17

The final comparison clearly shows how the LES modeloutperforms the other two at all average measures providedfollowed by the OUP model in accuracy although the lattermore than doubles the average MAPE value for LES

The results of simulations indicate that measures of errorstend to either differ slightly or not at all for either forecastsof any timeframe A coherent value with the 101 relation canthen be given with close to the same level of accuracy bymultiplying RC forecasts or dividing TC ones by 10

6 Conclusions

Copper TCRC are a keystone for pricing copper concen-trates which are the actual feedstock for copper smeltersThe potential evolution of TCRC is a question of botheconomic and technical significance forminers as their value

decreases the potential final selling price of concentratesAdditionally copper minersrsquo revenues are more narrowlyrelated to the market price of copper as well as to othertechnical factors such as ore dilution or the grade of theconcentrates produced Smelters on the contrary are hugelyaffected by the discount which they succeed in getting whenpurchasing the concentrates since that makes up the largestpart of their gross revenue besides other secondary sourcesIn addition eventual differences between TCRC may givecommodity traders ludicrous arbitrage opportunities Alsodifferences between short- and long-termTCRC agreementsoffer arbitrage opportunities for traders hence comprisinga part of their revenue in the copper concentrate tradingbusiness as well copper price fluctuations and the capacityto make economically optimum copper blends

As far as we are aware no rigorous research has beencarried out on the behaviour of these discounts Based onhistorical data on TCRC agreed upon in the LME CopperWeek from 2004 to 2017 three potentially suitable forecastingmodels for TCRC annual benchmark values have beencompared through four measures of forecasting accuracyat different horizons These models were chosen firstlydue to their broad implementation and proven capacityin commodity prices forecasting that they all share andsecondly because of the core differences in terms of pricebehaviour with each other

Focusing on the MAPE values achieved those obtainedby the LES model when TC and RC are treated indepen-dently have been significantly lower than for the rest ofthe models Indeed one-year-ahead MAPE measures for TCvalues for the GBM model (20) almost triple those of

Complexity 11

2004 2006 2008 2010 2012 2014405060708090

100

TC (U

SDM

T)

LES TC 1 Step Ahead 2013

2004 2006 2008 2010 2012 2014

LES TC 2 Steps Ahead 2013-2014

2004 2006 2008 2010 2012 2014 2016 2018

LES TC 5 Steps Ahead 2013-2017

2004 2006 2008 2010 2012 2014

LES RC 1 Step Ahead 2013

2004 2006 2008 2010 2012 2014456789

10LES RC 2 Steps Ahead 2013-2014

Observed Forecast

2004 2006 2008 2010 2012 2014 2016 2018

LES RC 5 Steps Ahead 2013-2017

405060708090

100110

405060708090

100

456789

10

RC (U

Sclb

)

456789

1011

Figure 4 One-step-ahead forecasts (2013) two-step-ahead forecasts (2013-2014) and five-step-ahead forecasts (2013-2017) for TC and RCusing the Linear Exponential Smoothing model (LES)

Table 17 TCRC Best error measures for 2013ndash2017 forecasts after 1000 MC simulations

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864GBM TCRC 5 Stepslowast 600x1036165 4651459 049048 5571551OUP TCRC 5 Stepslowast 84334843 2616262 0271502719 2896289LES TCRC 5 Steps 17755117 788078 0095100952 1084108lowastAveraged monthly steps

the OUP model (766) in contrast with the significantlylower values from the LES model (00046) This gaptends to be narrowed when TCRC values are forecastedat longer horizons when most measures of error becomemore even The GBM and OUP models have proven todeliver better accuracy performance when the TCRC valuesare projected monthly and then averaged to obtain annualbenchmark forecasts Even so the LES model remains themost accurate of all with MAPE values of 10 at two-year-ahead forecasts with 18 and 31 for TC for OUP and GBMrespectively

Finally despite TC and RC being two independentdiscounts applied to copper concentrates they are bothset jointly with an often 101 relation as our data revealsThis relation also transcends to simulation results and errormeasures hence showing a negligible discrepancy betweenthe independent forecasting of TC and RC or the jointforecasting of both values keeping the 101 relationThis is forinstance the case of the five-year-ahead OUP MAPE values(0271502719) which were obtained without observing the101 relation in the data A similar level of discrepancy wasobtained at any horizon with any model which indicates thatboth values could be forecasted with the same accuracy usingthe selected model with any of them and then applying the101 relation

Our findings thus suggest that both at short and atlong-term horizons TCRC annual benchmark levels tendto exhibit a pattern which is best fit by an LES model Thisindicates that these critical discounts for the copper tradingbusiness do maintain a certain dependency on past valuesThis would also suggest the existence of cyclical patterns incopper TCRC possibly driven by many of the same marketfactors that move the price of copper

This work contributes by delivering a formal tool forsmelters or miners to make accurate forecasts of TCRCbenchmark levels The level of errors attained indicatesthe LES model may be a valid model to forecast thesecrucial discounts for the copper market In addition ourwork further contributes to helping market participants toproject the price of concentrates with an acceptable degree ofuncertainty as now they may include a fundamental elementfor their estimation This would enable them to optimise theway they produce or process these copper concentrates Alsothe precise knowledge of these discountsrsquo expected behaviourcontributes to letting miners traders and smelters aliketake the maximum advantage from the copper concentratetrading agreements that they are part of As an additionalcontribution this work may well be applied to gold or silverRC which are relevant deduction concentrates when thesehave a significant amount of gold or silver

12 Complexity

Once TCRC annual benchmark levels are able to beforecasted with a certain level of accuracy future researchshould go further into this research through the explorationof the potential impact that other market factors may have onthese discounts

As a limitation of this research we should point out thetimespan of the data considered compared to those of otherforecasting works on commodity prices for example whichuse broader timespans For our case we have consideredthe maximum available sufficiently reliable data on TCRCbenchmark levels starting back in 2004 as there is noreliable data beyond this year Also as another limitationwe have used four measures of error which are among themost frequently used to compare the accuracy of differentmodels However other measures could have been used at anindividual level to test each modelrsquos accuracy

Appendix

A Models

A1 Geometric Brownian Motion (GBM) GBM can bewritten as a generalisation of a Wiener (continuous time-stochasticMarkov process with independent increments andwhose changes over any infinite interval of time are normallydistributed with a variance that increases linearly with thetime interval [22]) process

119889119909 = 120572119909119889119905 + 120590119909119889119911 (A1)

where according to Marathe and Ryan [30] the first termis known as the Expected Value whereas the second is theStochastic component with 120572 being the drift parameter and120590 the volatility of the process Also dz is the Wiener processwhich induces the abovementioned stochastic behaviour inthe model

119889119911 = isin119905radic119889119905 997888rarr isin119905 sim 119873 (0 1) (A2)

TheGBMmodel can be expressed in discrete terms accordingto

119909 = 119909119905 minus 119909119905minus1 = 120572119909119905minus1119905 + 120590119909119905minus1120598radic119905 (A3)

In GBM percentage changes in x 119909119909 are normally dis-tributed thus absolute changes in x 119909 are lognormallydistributed Also the expected value and variance for 119909(119905) are

E [119909 (119905)] = 1199090119890120572119905 (A4)

var [119909 (119905)] = 119909201198902120572119905 (1198901205902119905 minus 1) (A5)

A2 Orstein-Uhlenbeck Process (OUP) TheOUPprocess wasfirst defined by Uhlenbeck and Orstein [31] as an alternativeto the regular Brownian Motion to model the velocity ofthe diffusion movement of a particle that accounts for itslosses due to friction with other particles The OUP processcan be regarded as a modification of Brownian Motion incontinuous time where its properties have been changed

(Stationary Gaussian Markov and stochastic process) [32]These modifications cause the process to move towards acentral position with a stronger attraction the further it isfrom this position As mentioned above the OUP is usuallyemployed to model commodity prices and is the simplestversion of amean-reverting process [22]

119889119878 = 120582 (120583 minus 119878) 119889119905 + 120590119889119882119905 (A6)

where S is the level of prices 120583 the long-term average towhich prices tend to revert and 120582 the speed of reversionAdditionally in a similar fashion to that of the GBM 120590 isthe volatility of the process and 119889119882119905 is a Wiener processwith an identical definition However in contrast to GBMtime intervals in OUP are not independent since differencesbetween current levels of prices S and long-term averageprices 120583 make the expected change in prices dS more likelyeither positive or negative

The discrete version of the model can be expressed asfollows

119878119905 = 120583 (1 minus 119890minus120582Δ119905) + 119890minus120582Δ119905119878119905minus1 + 120590radic1 minus 119890minus2120582Δ1199052120582 119889119882119905 (A7)

where the expected value for 119909(119905) and the variance for (119909(119905) minus120583) areE [119909 (119905)] = 120583 + (1199090 minus 120583) 119890minus120582119905 (A8)

var [119909 (119905) minus 120583] = 12059022120582 (1 minus 119890minus2120582119905) (A9)

It can be derived from previous equations that as timeincreases prices will tend to long-term average levels 120583 Inaddition with large time spans if the speed of reversion 120582becomes high variance tends to 0 On the other hand if thespeed of reversion is 0 then var[119909(119905)] 997888rarr 1205902119905 making theprocess a simple Brownian Motion

A3Holtrsquos Linear Exponential Smoothing (LES) Linear Expo-nential Smoothing models are capable of considering bothlevels and trends at every instant assigning higher weights inthe overall calculation to values closer to the present than toolder ones LES models carry that out by constantly updatinglocal estimations of levels and trends with the intervention ofone or two smoothing constants which enable the models todampen older value effects Although it is possible to employa single smoothing constant for both the level and the trendknown as Brownrsquos LES to use two one for each known asHoltrsquos LES is usually preferred since Brownrsquos LES tends torender estimations of the trend ldquounstablerdquo as suggested byauthors such as Nau [33] Holtrsquos LES model comes definedby the level trend and forecast updating equations each ofthese expressed as follows respectively

119871 119905 = 120572119884119905 + (1 minus 120572) (119871 119905minus1 + 119879119905minus1) (A10)

119879119905 = 120573 (119871 119905 minus 119871 119905minus1) + (1 minus 120573) 119879119905minus1 (A11)

119905+119896 = 119871 119905 + 119896119879119905 (A12)

Complexity 13

With 120572 being the first smoothing constant for the levelsand 120573 the second smoothing constant for the trend Highervalues for the smoothing constants imply that either levels ortrends are changing rapidly over time whereas lower valuesimply the contrary Hence the higher the constant the moreuncertain the future is believed to be

B Model Calibration

Each model has been calibrated individually using two setsof data containing TC and RC levels respectively from 2004to 2012 The available dataset is comprised of data from 2004to 2017 which make the calibration data approximately 23 ofthe total

B1 GBM Increments in the logarithm of variable x aredistributed as follows

(ln119909) sim 119873((120572 minus 12059022 ) 119905 120590119905) (B1)

Hence if m is defined as the sample mean of the differenceof the natural logarithm of the time series for TCRC levelsconsidered for the calibration and n as the number ofincrements of the series considered with n=9

119898 = 1119899119899sum119905=1

(ln119909119905 minus ln119909119905minus1) (B2)

119904 = radic 1119899 minus 1119899sum119894=1

(ln119909119905 minus ln119909119905minus1 minus 119898)2 (B3)

119898 = 120572 minus 12059022 (B4)

119904 = 120590 (B5)

B2 OUP The OUP process is an AR1 process [22] whoseresolution is well presented by Woolridge [34] using OLStechniques fitting the following

119910119905 = 119886 + 119887119910119905minus1 + 120576119905 (B6)

Hence the estimators for the parameters of the OUP modelare obtained by OLS for both TC and RC levels indepen-dently

= minus ln 119887Δ119905 (B7)

120583 = 1198861 minus 119887 (B8)

= 120576119905radic 2 ln (1 + 119887)(1 + 119887)2 minus 1 (B9)

B3 LES A linear regression is conducted on the inputdataset available to find the starting parameters for the LES

model the initial Level 1198710 and the initial value of thetrend 1198790 irrespective of TC values and RC values Hereas recommended by Gardner [35] the use of OLS is highlyadvisable due to the erratic behaviour shown by trends inthe historic data so the obtaining of negative values of 1198780 isprevented Linear regression fulfils the following

119884119905 = 119886119905 + 119887 (B10)

1198710 = 119887 (B11)

1198790 = 119886 (B12)

By fixing the two smoothing constants the values for theforecasts 119905+119896 can be calculated at each step using the modelequations There are multiple references in the literature onwhat the optimum range for each smoothing constant isGardner [35] speaks of setting moderate values for bothparameter less than 03 to obtain the best results Examplespointing out the same may be found in Brown [36] Coutie[37] Harrison [38] and Montgomery and Johnson [39]Also for many applications Makridakis and Hibon [20] andChatfield [40] found that parameter values should fall withinthe range of 03-1 On the other hand McClain and Thomas[41] provided a condition of stability for the nonseasonal LESmodel given by

0 lt 120572 lt 20 lt 120573 lt 4 minus 2120572120572

(B13)

Also the largest possible value of 120572 that allows the avoidanceof areas of oscillation is proposed by McClain and Thomas[41] and McClain [42]

120572 lt 4120573(1 + 120573)2 (B14)

However according to Gardner there is no tangible proofthat this value improves accuracy in any form Nonethelesswe have opted to follow what Nau [33] refers to as ldquotheusual wayrdquo namely minimising the Mean Squared Error(MSE) of the one-step-ahead forecast of TCRC for eachinput data series previously mentioned To do so Matlabrsquosfminsearch function has been used with function and variabletolerance levels of 1x10minus4 as well as a set maximum numberof function iterations and function evaluations of 1x106 tolimit computing resources In Table 18 the actual number ofnecessary iterations to obtain optimum values for smoothingconstants is shown As can be seen the criteria are wellbeyond the final results which ensured that an optimumsolution was reached with assumable computing usage (thesimulation required less than one minute) and with a highdegree of certainty

14 Complexity

Table 18 Iterations on LES smoothing constants optimisation

Iterations Function EvaluationsTC 1-Step 36 40TC 2-Steps 249945 454211TC 5-Steps 40 77RC 1-Step 32 62RC 2-Steps 254388 462226RC 5-Steps 34 66

C Error Calculations

C1 Mean Square Error (MSE)

119872119878119864 = 1119899119899sum119894=1

(119894 minus 119884119894)2 (C1)

where 119894 are the forecasted values and 119884119894 those observedC2 Mean Absolute Deviation (MAD)

119872119860119863 = 1119899119899sum119894=1

10038161003816100381610038161003816119894 minus 11988410038161003816100381610038161003816 (C2)

where 119894 are the forecasted values and 119884 the average value ofall the observations

C3 Mean Absolute Percentage Error (MAPE)

119872119860119875119864 = 1119899119899sum119894=1

10038161003816100381610038161003816119894 minus 11988411989410038161003816100381610038161003816119884119894 (C3)

The above formula is expressed in parts-per-one and is theone used in the calculations conducted here Hence multi-plying the result by 100 would deliver percentage outcomes

C4 Root Mean Square Error (RMSE)

119877119872119878119864 = radic119872119878119864 = radic 1119899119899sum119894=1

(119894 minus 119884119894)2 (C4)

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Supplementary Materials

The Matlab codes developed to carry out the simulationsfor each of the models proposed as well as the datasetsused and the Matlab workspaces with the simulationsoutcomes as they have been reflected in this paper areprovided in separate folders for each model GBM foldercontains the code to make the simulations for the Geo-metric Brownian Motion model ldquoGBMmrdquo as well as aseparate script which allows determining the total numberof monthly step simulations for which errors levels havebeen lower than annual-step simulations ldquoErrCoutermrdquoThe datasets used are included in two excel files ldquoTCxlsxrdquoand ldquoRCxlsxrdquo ldquoGBM OK MonthlyStepsmatrdquo is the Matlabworkspace containing the outcome of monthly step simu-lations whereas ldquoGBM OK AnnualStepsmatrdquo contains theoutcome for annual-step simulations Also ldquoGBM OKmatrdquois the workspace to be loaded prior to performing GBMsimulations MR folder contains the code file to carryout the simulations for the Orstein-Uhlenbeck ProcessldquoMRmrdquo as well as the separate script to compare errorsof monthly step simulations with annual-step simula-tions ldquoErrCountermrdquo ldquoTCxlsxrdquo and ldquoRCxlsxrdquo containthe TCRC benchmark levels from 2004 to 2017 Finallymonthly step simulationsrsquo outcome has been saved inthe workspace ldquoMR OK MonthlyStepsmatrdquo while annual-step simulationsrsquo outcome has been saved in the Matlabworkspace ldquoMR OK AnnualStepsmatrdquo Also ldquoMR OKmatrdquois the workspace to be loaded prior to performing MRsimulations LES folder contains the code file to carry outthe calculations necessary to obtain the LES model forecastsldquoLESmrdquo Three separate Matlab functions are includedldquomapeLESmrdquo ldquomapeLES1mrdquo and ldquomapeLES2mrdquo whichdefine the calculation of the MAPE for the LESrsquo five-year-ahead one-year-ahead and two-year-ahead forecastsAlso ldquoLES OKmatrdquo is the workspace to be loaded prior toperforming LES simulations Finally graphs of each modelhave been included in separate files as well in a subfoldernamed ldquoGraphsrdquo within each of the modelsrsquo folder Figuresare included in Matlabrsquos format (fig) and TIFF format TIFFfigures have been generated in both colours and black andwhite for easier visualization (Supplementary Materials)

References

[1] United Nations International Trade Statistics Yearbook vol 22016

[2] International Copper StudyGroupTheWorld Copper Factbook2017

[3] K M Johnson J M Hammarstrom M L Zientek and CL Dicken ldquoEstimate of undiscovered copper resources of theworld 2013rdquo US Geological Survey 2017

[4] M E Schlesinger and W G Davenport Extractive Metallurgyof Copper Elsevier 5th edition 2011

[5] S Gloser M Soulier and L A T Espinoza ldquoDynamic analysisof global copper flows Global stocks postconsumer materialflows recycling indicators and uncertainty evaluationrdquo Envi-ronmental Science amp Technology vol 47 no 12 pp 6564ndash65722013

4 Complexity

Table 1 Summary of previous literature research

AUTHOR RESEARCH

Shafiee amp Topal [17] Validates a modified version of the long-term trend reverting jump and dip diffusion model for forecastingcommodity prices and estimates the gold price for the next 10 years using historical monthly data

Li et al [18] Proposes an ARIMA-Markov Chain method to accurately forecast mineral commodity prices testing themethod using mineral molybdenum prices

Issler et al [19]Investigates several commoditiesrsquo co-movements such as Aluminium Copper Lead Nickel Tin and Zinc atdifferent time frequencies and uses a bias-corrected average forecast method proposed by Issler and Lima [43]to give combined forecasts of these metal commodities employing RMSE as a measure of forecasting accuracy

Hamid amp Shabri [44]Models palm oil prices using the Autoregressive Distributed Lag (ARDL) model and compares its forecastingaccuracy with the benchmark model ARIMA It uses an ARDL bound-testing approach to co-integration in

order to analyse the relationship between the price of palm oil and its determinant factorsDuan et al [45] Predicts Chinarsquos crude oil consumption for 2015-2020 using the fractional-order FSIGMmodel

Brennan amp Schwartz [46] Employs the Geometric Brownian Motion (GBM) to analyse a mining projectrsquos expected returns assuming itproduces a single commodity

McDonald amp Siegel [47] Uses GBM to model the random evolution of the present value of an undefined asset in an investment decisionmodel

Zhang et al [48] Models gold prices using the Ornstein-Uhlenbeck Process (OUP) to account for a potentially existent long-termtrend in a Real Option Valuation of a mining project

Sharma [49] Forecasts gold prices in India with the Box Jenkins ARIMA method

way of modelling prices that are believed not to follow anyspecific rule or pattern and hence seen as random Blackand Scholes [23] first used GBM to model stock prices andsince then others have used it to model asset prices as well ascommodities these being perhaps themost common of all inwhich prices are expected to increase over time as does theirvariance [11] Hence following our first premise concerningwhether TCRC might vary randomly there should not exista main driving factor that would determine TCRC futurebenchmark levels and therefore GBM could to a certainextent be a feasible model for them

However although GBM or ldquorandom walkrdquo may be wellsuited to modelling immediate or short-term price paths forcommodities or for TCRC in our case it lacks the abilityto include the underlying long-term price trend should weassume that there is oneThus in accordance with our secondpremise on benchmark TCRC behaviour levels would movein line with copper concentrate supply and demand as wellas the smeltersrsquo and refineriesrsquo available capacity to trans-form concentrates into metal copper Hence a relationshipbetween TCRC levels and copper supply and demand isknown to exist and therefore is linked to its market priceso to some extent they move together coherently Thereforein that case as related works on commodity price behavioursuch as Foo Bloch and Salim [24] do we have opted for theMR model particularly the OUP model

Both GBM and MR are Markov processes which meansthat future values depend exclusively on the current valuewhile the remaining previous time series data are not consid-eredOn the other handmoving averagemethods employ theaverage of a pre-established number of past values in differentways evolving over time so future values do not rely exclu-sively on the present hence behaving as though they had onlya limitedmemory of the pastThis trait of themoving averagemodel is particularly interestingwhen past prices are believedto have a certain though limited effect on present values

which is another of the premises for this research Existingalternatives ofmoving average techniques pursue consideringthis ldquomemoryrdquo with different approaches As explained byKalekar [25] Exponential Smoothing is suitable only for thebehaviours of a specific time series thus Single ExponentialSmoothing (SES) is reasonable for short-term forecastingwith no specific trend in the observed data whereas DoubleExponential Smoothing or Linear Exponential Smoothing(LES) is appropriate when data shows a cyclical pattern ora trend In addition seasonality in observed data can becomputed and forecasted through the usage of ExponentialSmoothing by the Holt-Winters method which adds an extraparameter to the model to handle this characteristic

32 TCRC Benchmark Levels and Sample Dataset TCRClevels for copper concentrates continuously vary through-out the year relying on private and individual agreementsbetween miners traders and smelters worldwide Nonethe-less the TCRC benchmark fixed during the LME week eachOctober is used by market participants as the main referenceto set actual TCRC levels for each supply agreed upon for thefollowing year Hence the yearrsquos benchmark TCRC is takenhere as a good indicator of a yearrsquos TCRC average levelsAnalysed time series of benchmark TCRC span from 2004through to 2017 as shown in Table 2 as well as the sourceeach valuewas obtained fromWe have not intended to reflectthe continuous variation of TCRC for the course of anygiven year though we have however considered benchmarkprices alone as we intuitively assume that annual variations ofactual TCRC in contracts will eventually be reflected in thebenchmark level that is set at the end of the year for the yearto come

33 TCRC Relation TC and RC normally maintain a 101relation with different units with TC being expressed in US

Complexity 5

GBM RC First 20 pathsGBM TC First 20 paths

0

50

100

150

200

250

300

TC (U

SDM

T)

2013 2014 2015 2016 20172012

GBM TC First 20 paths - Year Averaged

05

101520253035

RC (U

Sclb

)

0

5

10

15

20

25

30

RC (U

Slb

)

0

50

100

150

200

250

TC (U

SDM

T)

2013 2014 2015 2016 20172012

Observed TC Benchmark Levels

2013 2014 2015 2016 20172012

Observed RC Benchmark Levels

2013 2014 2015 2016 20172012

GBM RC First 20 paths - Year Averaged

Figure 2 The upper illustrations show the first 20 Monte Carlo paths for either TC or RC levels using monthly steps The illustrations belowshow the annual averages of monthly step forecasts for TC and RC levels

MR TC First 20 paths

2012 2013 2014 2015 2016 2017

MR RC First 20 paths

MR RC First 20 paths - Year Averaged

2013 2014 2015 2016 20172012

Observed

5

6

7

8

9

10

11

RC (U

Slb

)

2468

10121416

RC (U

Sclb

)

0

50

100

150

200

TC (U

SDM

T)

2013 2014 2015 2016 20172012

MR TC First 20 paths - Year Averaged

2013 2014 2015 2016 20172012

Observed

50

60

70

80

90

100

110

TC (U

SDM

T)

Figure 3 The first 20 Monte Carlo paths following the OUP model using monthly steps are shown on the upper illustrations for either TCor RC The annual averaged simulated paths every 12 values are shown below

dollars per metric tonne and RC in US cents per pound ofpayable copper content in concentrates In fact benchmarkdata show that of the last 14 years it was only in 2010that values of TC and RC did not conform to this relation

though it did remain close to it (46547) Nonetheless thisrelation has not been observed in the methodology hereinthus treating TC and RC independently developing separateunrelated forecasts for TC and RC to understand whether

6 Complexity

Table 2 TCRC year benchmark levels (Data available free ofcharge Source Johansson [50] Svante [51] Teck [52] Shaw [53]Willbrandt and Faust [54 55] Aurubis AG [56] Drouven and Faust[57] Aurubis AG [58] EY [59] Schachler [60] Nakazato [61])

YEAR TC (USDMT) RC (USclb)2004 45 452005 85 852006 95 952007 60 602008 45 452009 75 752010 465 472011 56 562012 635 6352013 70 702014 92 922015 107 1072016 9735 97352017 925 925

these individual forecasts for TC and RC would render betterresults than a single joint forecast for both TCRC levels thatwould take into account the 101 relation existent in the data

34 Models Comparison Method The works referred to inthe literature usually resort to different measures of errorsto conduct the testing of a modelrsquos forecasting capacityregardless of the specific nature of the forecasted value be itcotton prices or macroeconomic parameters Thus the mostwidely errors used includeMean Squared Error (MSE) MeanAbsolute Deviation (MAD) Mean Absolute Percentage Error(MAPE) and Root Mean Square Error (RMSE) [14 16 17 2125ndash29] In this work Geometric Brownian Motion (GBM)Ornstein-Uhlenbeck Process (OUP) and Linear ExponentialSmoothing (LES) models have been used to forecast annualTCRC benchmark levels using all the four main measuresof error mentioned above to test the predictive accuracy ofthese three models The GBM and OUP models have beensimulated and tested with different step sizes while the LESmodel has been analysed solely using annual time steps

The GBM and OUPmodels were treated separately to theLESmodel thus GBM andOUPwere simulated usingMonteCarlo (MC) simulations whereas LES forecasts were simplecalculations In a preliminary stage the models were firstcalibrated to forecast values from 2013 to 2017 using availabledata from 2004 to 2012 for TC and RC separately hencethe disregarding of the well-known inherent 101 relation (seeAppendix B for Models Calibration) The GBM and OUPmodels were calibrated for two different step sizes monthlysteps and annual steps in order to compare forecastingaccuracy with each step size in each model Following thecalibration of the models Monte Carlo simulations (MC)were carried out usingMatlab software to render the pursuedforecasts of GBM andOUPmodels obtaining 1000 simulatedpaths for each step size MC simulations using monthly stepsfor the 2013ndash2017 timespan were averaged every 12 steps to

deliver year forecasts of TCRC benchmark levels On theother hand forecasts obtained by MC simulations takingannual steps for the same period were considered as yearforecasts for TCRC annual benchmark levels without theneed for extra transformation Besides LES model forecastswere calculated at different time horizons to be able tocompare the short-term and long-term predictive accuracyLES forecasts were obtained for one-year-ahead hence usingknown values of TCRC from 2004 to 2016 for two yearsahead stretching the calibrating dataset from 2004 to 2015for five-year-ahead thus using the same input dataset as forthe GBM and OUP models from 2004 to 2012

Finally for every forecast path obtained we have calcu-lated the average of the squares of the errors with respect tothe observed valuesMSE the average distance of a forecast tothe observed meanMAD the average deviation of a forecastfrom observed values MAPE and the square root of MSERMSE (see Appendix C for error calculations) The results ofMSE MAD MAPE and RMSE calculated for each forecastpathwere averaged by the total number of simulations carriedout for each case Averaged values of error measures of allsimulated paths MSE MAD MAPE and RMSE for bothannual-step forecasts and monthly step forecasts have beenused for cross-comparison between models to test predictiveability at every step size possible

Also to test each modelrsquos short-term forecasting capacityagainst its long-term forecasting capacity one-year-aheadforecast errors of the LES model were compared with theerrors from the last year of the GBM and OUP simulatedpaths Two-year-ahead forecast errors of LES models werecomparedwith the average errors of the last two years ofGBMand OUP and five-year-ahead forecast errors of LES modelswere compared with the overall average of errors of the GBMand OUP forecasts

4 Analysis of Results

Monte Carlo simulations of GBM and OUP models render1000 different possible paths for TC and RC respectivelyat each time step size considered Accuracy errors for bothannual time steps and averaged monthly time steps for bothGBM and OUP forecasts were first compared to determinethe most accurate time step size for each model In additionthe LES model outcome for both TC and RC at differenttimeframes was also calculated and measures of errors for allthe three alternative models proposed at an optimum timestep were finally compared

41 GBM Forecast The first 20 of 1000 Monte Carlo pathsfor the GBM model with a monthly step size using Matlabsoftware may be seen in Figure 2 for both the TC and the RClevels compared to their averaged paths for every 12 valuesobtainedThe tendency for GBM forecasts to steeply increaseover time is easily observable in the nonaveraged monthlystep paths shown

The average values of error of all 1000MC paths obtainedthrough simulation for averaged monthly step and annual-step forecasts are shown in Table 3 for both TC and RC

Complexity 7

Table 3 Average of main measures of error for GBM after 1000 MC simulations from 2013 to 2017

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC Averaged Monthly Steps 560x103 4698 050 5638TC Annual Steps 520x103 4902 052 5857RC Averaged Monthly Steps 5874 455 048 546RC Annual Steps 5085 491 052 584

Table 4 Number of paths for which measures of error are higher for monthly steps than for annual steps in GBM

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC 4571000 4551000 4531000 4531000RC 4391000 4301000 4291000 4291000

discounts over the period from 2013 to 2017 which maylead to preliminary conclusions in terms of an accuracycomparison between averaged monthly steps and annualsteps

However a more exhaustive analysis is shown in Table 4where the number of times the values of each error measureare higher for monthly steps is expressed over the totalnumber of MC simulations carried out The results indicatethat better values of error are reached the majority of timesfor averagedmonthly step simulations rather than for straightannual ones

In contrast short-term accuracy was also evaluated byanalysing the error measures of one-year-ahead forecasts(2013) in Table 5 and of two-year-ahead forecasts (2013ndash2014)in Table 6 The results indicate as one may expect thataccuracy decreases as the forecasted horizon is widened withthe accuracy of averaged monthly step forecasts remaininghigher than annual ones as found above for the five-yearforecasting horizon

42 OUP Forecast Long-term levels for TC and RC 120583 thespeed of reversion 120582 and the volatility of the process 120590 arethe parameters determined at themodel calibration stage (seeAppendix B for the model calibration explanation) whichdefine the behaviour of the OUP shown in Table 7 Thecalibration was done prior to the Monte Carlo simulationfor both TC and RC with each step size using availablehistorical data from 2004 to 2012 The OUP model was fittedwith the corresponding parameters for each case upon MCsimulation

Figure 3 shows theMeanReversionMCestimations of theTCRC benchmark values from 2013 to 2017 using monthlystepsThemonthly forecastswere rendered from January 2012through to December 2016 and averaged every twelve valuesto deliver a benchmark forecast for each year The averagedresults can be comparable to actual data as well as to theannual Monte Carlo simulations following Mean ReversionThe lower-side figures show these yearly averaged monthlystep simulation outcomes which clearly move around a dash-dotted red line indicating the long-term run levels for TCRCto which they tend to revert

The accuracy of monthly steps against annual steps forthe TCRC benchmark levels forecast was also tested by

determining the number of simulations for which averageerror measures became higher Table 8 shows the numberof times monthly simulations have been less accurate thanannual simulations for five-year-ahead OUP forecasting bycomparing the four measures of errors proposed The resultsindicate that only 25-32 of the simulations drew a higheraverage error which clearly results in a better predictiveaccuracy for monthly step forecasting of TCRC annualbenchmark levels

The averaged measures of errors obtained after the MCsimulations of the OUP model for both averaged monthlysteps and annual steps giving TCRC benchmark forecastsfrom 2013 to 2017 are shown in Table 9

The error levels of the MC simulations shown in Table 9point towards a higher prediction accuracy of averagedmonthly step forecasts of the OUP Model yielding anaveraged MAPE value that is 129 lower for TC and RC 5-step-ahead forecasts In regard to MAPE values for monthlysteps only 266 of the simulations rise above annual MCsimulations for TC and 25 for RC 5-step-ahead forecastswhich further underpins the greater accuracy of thisOUP set-up for TCRC level forecasts A significant lower probabilityof higher error levels for TCRC forecasts with monthly MCOUP simulations is reached for the other measures providedIn addition short-term and long-term prediction accuracywere tested by comparing errors of forecasts for five-year-ahead error measures in Table 10 one-year-ahead in Table 11and two-year-ahead in Table 12

With a closer forecasting horizon errormeasures show animprovement of forecasting accuracy when average monthlysteps are used rather than annual ones For instance theMAPE values for 2013 forecast for TC are 68 lower foraveraged monthly steps than for annual steps and alsoMAPE for 2013ndash2014 were 30 lower for both TC and RCforecasts Similarly better values of error are achieved for theother measures for averaged monthly short-term forecaststhan in other scenarios In addition as expected accuracyis increased for closer forecasting horizons where the levelof errors shown above becomes lower as the deviation offorecasts is trimmed with short-term predictions

43 LES Forecast In contrast to GBM and OUP the LESmodel lacks any stochastic component so nondeterministic

8 Complexity

Table 5 Average for main measures of error for GBM after 1000 MC simulations for 2013

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC Averaged Monthly Steps 33658 1455 021 1455TC Annual Steps 69381 2081 030 2081RC Averaged Monthly Steps 297 138 020 138RC Annual Steps 657 205 029 205

Table 6 Average for main measures of error for GBM after 1000 MC simulations for 2013-2014

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC Averaged Monthly Steps 99492 2187 031 2435TC Annual Steps 133x103 2833 040 3104RC Averaged Monthly Steps 1035 240 034 271RC Annual Steps 1313 284 034 311

Table 7 OUP parameters obtained in the calibration process

120583 120582 120590TCMonthly 6345 4792x105 2534TC Annual 6345 2974 1308RC Monthly 635 4763x105 2519RC Annual 635 2972 1305

methods such as theMonte Carlo are not required to obtain aforecast Nonetheless the LES model relies on two smoothingconstants which must be properly set in order to deliveraccurate predictions hence the values of the smoothingconstants were first optimised (see Appendix B for LESModel fitting and smoothing constants optimisation) Theoptimisation was carried out for one-year-ahead forecaststwo-year-ahead forecasts and five-year-ahead forecasts byminimising the values of MSE for both TC and RC Thedifferent values used for smoothing constants as well asthe initial values for level and trend obtained by the linearregression of the available dataset from 2004 through to 2012are shown in Table 12

Compared one-year-ahead two-year-ahead and five-year-ahead LES forecasts for TC and RC are shown inFigure 4 clearly indicating a stronger accuracy for shorter-term forecasts as the observed and forecasted plotted linesoverlap

Minimumvalues for errormeasures achieved through theLESmodel parameter optimisation are shown in Table 13Thevalues obtained confirm the strong accuracy for shorter-termforecasts of TCRC seen in Figure 4

5 Discussion and Model Comparison

The forecasted values of TCRC benchmark levels couldeventually be applied to broader valuation models for copperconcentrates and their trading activities as well as to thecopper smeltersrsquo revenue stream thus the importance ofdelivering as accurate a prediction as possible in relation tothese discounts to make any future application possible Eachof the models presented may be a feasible method on its own

with eventual later adaptations to forecasting future valuesof benchmark TCRC Nonetheless the accuracy of thesemodels as they have been used in this work requires firstlya comparison to determine whether any of them could bea good standalone technique and secondly to test whethera combination of two or more of them would deliver moreprecise results

When comparing the different error measures obtainedfor all the three models it is clearly established that resultsfor a randomly chosen simulation of GBM or OUP wouldbe more likely to be more precise had a monthly step beenused to deliver annual forecasts instead of an annual-step sizeIn contrast average error measures for the entire populationof simulations with each step size employed showing thatmonthly step simulations for GBM and OUP models arealways more accurate than straight annual-step forecastswhen a shorter time horizon one or two-year-ahead is takeninto consideration However GBM presents a higher levelof forecasting accuracy when the average for error measuresof all simulations is analysed employing annual steps forlong-term horizons whereas OUP averaged monthly stepforecasts remain more accurate when predicting long-termhorizons Table 14 shows the error improvement for the aver-aged monthly step forecasts of each model Negative valuesindicate that better levels of error averages have been found instraight annual forecasts than for monthly step simulations

Considering the best results for each model and com-paring their corresponding error measures we can opt forthe best technique to employ among the three proposed inthis paper Hence GBM delivers the most accurate one-yearforecast when averaging the next twelve-month predictionsfor TCRC values as does the MR-OUP model Table 15shows best error measures for one-year-ahead forecasts forGBM OUP and LES models

Unarguably the LES model generates minimal errormeasures for one-year-ahead forecasts significantly less thanthe other models employed A similar situation is found fortwo-year-ahead forecasts whereminimumerrormeasures arealso delivered by the LES model shown in Table 16

Finally accuracy measures for five-year-ahead forecastsof the GBM model might result in somewhat contradictory

Complexity 9

Table 8 Number of paths for whichmeasures of error are higher for monthly steps than for annual steps inMR-OUP 5-StepsMC simulation

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC 3111000 2831000 2661000 2661000RC 3161000 2811000 2501000 2501000

Table 9 Average for main measures of error for MR-OUP after 1000 MC simulations 2013-2017

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC Averaged Monthly Steps 84254 2614 027 2895TC Annual Steps 114x103 2931 031 3325RC Averaged Monthly Steps 840 261 027 289RC Annual Steps 1148 294 031 333

Table 10 Average for main measures of error for MR-OUP after 1000 MC simulations 2013

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC Averaged Monthly Steps 3914 522 007 522TC Annual Steps 36643 1555 022 1555RC Averaged Monthly Steps 039 051 007 051RC Annual Steps 363 154 022 154

Table 11 Average for main measures of error for MR-OUP after 1000 MC simulations 2013-2014

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC Averaged Monthly Steps 37165 1567 018 1900TC Annual Steps 68244 2185 026 2324RC Averaged Monthly Steps 376 157 018 191RC Annual Steps 689 219 026 243

Table 12 LES Model Optimised Parameters

L0 T0 120572 120573TC 1 year 713611 -15833 -02372 01598RC 1 year 71333 -01567 -02368 01591TC 2 years 713611 -15833 -1477x10minus4 7774226RC 2 years 71333 -01567 -1448x10minus4 7898336TC 5 years 713611 -15833 -02813 01880RC 5 years 71333 -01567 -02808 01880

Table 13 LES error measures for different steps-ahead forecasts

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC 1 year 10746x10minus5 00033 46831x10minus5 00033RC 1 year 51462x10minus8 22685x10minus4 32408x10minus5 22685x10minus4

TC 2 years 189247 97275 01057 137567RC 2 years 18977 09741 01059 13776TC 5 years 1775531 78881 00951 108422RC 5 years 11759 07889 00952 10844

terms for MSE reaching better values for annual steps thanfor averaged monthly steps while the other figures do betteron averagedmonthly steps Addressing the definition ofMSEthis includes the variance of the estimator as well as its biasbeing equal to its variance in the case of unbiased estimators

Therefore MSE measures the quality of the estimator butalso magnifies estimator deviations from actual values sinceboth positive and negative values are squared and averagedIn contrast RMSE is calculated as the square root of MSEand following the previous analogy stands for the standard

10 Complexity

Table 14 Error Average Improvement for averaged monthly steps before annual steps

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864GBM TCRC 1 Year 53265653 32723587 33333548 32723587GBM TCRC 2 Years 26952635 25751597 2619286 24341369GBM TCRC 5 Years -1173-1104 873709 926769 747565OUP TCRC 1 Year 88088708 63716250 62916319 63686224OUP TCRC 2 Years 46284421 27712617 29993075 22162075OUP TCRC 5 Years 26022553 1093997 13121218 13061269

Table 15 TCRC best error measures for 2013 forecasts after 1000 MC simulations

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864GBM TCRC 1 Yearlowast 33111336 1425143 020020 1425143OUP TCRC 1 Yearlowast 4105042 536054 0076600777 536054LES TCRC 1 Year 107x10minus5514x10minus8 0003226x10minus4 468x10minus5324x10minus5 0003226x10minus4

lowastAveraged monthly steps

Table 16 TCRC Best error measures for 2013ndash2014 forecasts after 1000 MC simulations

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864GBM TCRC 2 Stepslowast 103x1031006 2200242 031034 2447271OUP TCRC 2 Stepslowast 37331376 1573158 0180201813 1900191LES TCRC 2 Steps 18924189 972097 0105701059 1375137lowastAveraged monthly steps

deviation of the estimator if MSE were considered to be thevarianceThough RMSE overreacts when high values of MSEare reached it is less prone to this than MSE since it iscalculated as its squared root thus not accounting for largeerrors as disproportionately as MSE does Furthermore aswe have compared an average of 1000 measures of errorscorresponding to each MC simulation performed the valuesobtained for average RMSE stay below the square root ofaverage MSE which indicates that some of these dispropor-tionate error measures are to some extent distorting thelatter Hence RMSE average values point towards a higheraccuracy for GBM five-year forecasts with averaged monthlysteps which is further endorsed by the average values ofMAD and MAPE thus being the one used for comparisonwith the other two models as shown in Table 17

The final comparison clearly shows how the LES modeloutperforms the other two at all average measures providedfollowed by the OUP model in accuracy although the lattermore than doubles the average MAPE value for LES

The results of simulations indicate that measures of errorstend to either differ slightly or not at all for either forecastsof any timeframe A coherent value with the 101 relation canthen be given with close to the same level of accuracy bymultiplying RC forecasts or dividing TC ones by 10

6 Conclusions

Copper TCRC are a keystone for pricing copper concen-trates which are the actual feedstock for copper smeltersThe potential evolution of TCRC is a question of botheconomic and technical significance forminers as their value

decreases the potential final selling price of concentratesAdditionally copper minersrsquo revenues are more narrowlyrelated to the market price of copper as well as to othertechnical factors such as ore dilution or the grade of theconcentrates produced Smelters on the contrary are hugelyaffected by the discount which they succeed in getting whenpurchasing the concentrates since that makes up the largestpart of their gross revenue besides other secondary sourcesIn addition eventual differences between TCRC may givecommodity traders ludicrous arbitrage opportunities Alsodifferences between short- and long-termTCRC agreementsoffer arbitrage opportunities for traders hence comprisinga part of their revenue in the copper concentrate tradingbusiness as well copper price fluctuations and the capacityto make economically optimum copper blends

As far as we are aware no rigorous research has beencarried out on the behaviour of these discounts Based onhistorical data on TCRC agreed upon in the LME CopperWeek from 2004 to 2017 three potentially suitable forecastingmodels for TCRC annual benchmark values have beencompared through four measures of forecasting accuracyat different horizons These models were chosen firstlydue to their broad implementation and proven capacityin commodity prices forecasting that they all share andsecondly because of the core differences in terms of pricebehaviour with each other

Focusing on the MAPE values achieved those obtainedby the LES model when TC and RC are treated indepen-dently have been significantly lower than for the rest ofthe models Indeed one-year-ahead MAPE measures for TCvalues for the GBM model (20) almost triple those of

Complexity 11

2004 2006 2008 2010 2012 2014405060708090

100

TC (U

SDM

T)

LES TC 1 Step Ahead 2013

2004 2006 2008 2010 2012 2014

LES TC 2 Steps Ahead 2013-2014

2004 2006 2008 2010 2012 2014 2016 2018

LES TC 5 Steps Ahead 2013-2017

2004 2006 2008 2010 2012 2014

LES RC 1 Step Ahead 2013

2004 2006 2008 2010 2012 2014456789

10LES RC 2 Steps Ahead 2013-2014

Observed Forecast

2004 2006 2008 2010 2012 2014 2016 2018

LES RC 5 Steps Ahead 2013-2017

405060708090

100110

405060708090

100

456789

10

RC (U

Sclb

)

456789

1011

Figure 4 One-step-ahead forecasts (2013) two-step-ahead forecasts (2013-2014) and five-step-ahead forecasts (2013-2017) for TC and RCusing the Linear Exponential Smoothing model (LES)

Table 17 TCRC Best error measures for 2013ndash2017 forecasts after 1000 MC simulations

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864GBM TCRC 5 Stepslowast 600x1036165 4651459 049048 5571551OUP TCRC 5 Stepslowast 84334843 2616262 0271502719 2896289LES TCRC 5 Steps 17755117 788078 0095100952 1084108lowastAveraged monthly steps

the OUP model (766) in contrast with the significantlylower values from the LES model (00046) This gaptends to be narrowed when TCRC values are forecastedat longer horizons when most measures of error becomemore even The GBM and OUP models have proven todeliver better accuracy performance when the TCRC valuesare projected monthly and then averaged to obtain annualbenchmark forecasts Even so the LES model remains themost accurate of all with MAPE values of 10 at two-year-ahead forecasts with 18 and 31 for TC for OUP and GBMrespectively

Finally despite TC and RC being two independentdiscounts applied to copper concentrates they are bothset jointly with an often 101 relation as our data revealsThis relation also transcends to simulation results and errormeasures hence showing a negligible discrepancy betweenthe independent forecasting of TC and RC or the jointforecasting of both values keeping the 101 relationThis is forinstance the case of the five-year-ahead OUP MAPE values(0271502719) which were obtained without observing the101 relation in the data A similar level of discrepancy wasobtained at any horizon with any model which indicates thatboth values could be forecasted with the same accuracy usingthe selected model with any of them and then applying the101 relation

Our findings thus suggest that both at short and atlong-term horizons TCRC annual benchmark levels tendto exhibit a pattern which is best fit by an LES model Thisindicates that these critical discounts for the copper tradingbusiness do maintain a certain dependency on past valuesThis would also suggest the existence of cyclical patterns incopper TCRC possibly driven by many of the same marketfactors that move the price of copper

This work contributes by delivering a formal tool forsmelters or miners to make accurate forecasts of TCRCbenchmark levels The level of errors attained indicatesthe LES model may be a valid model to forecast thesecrucial discounts for the copper market In addition ourwork further contributes to helping market participants toproject the price of concentrates with an acceptable degree ofuncertainty as now they may include a fundamental elementfor their estimation This would enable them to optimise theway they produce or process these copper concentrates Alsothe precise knowledge of these discountsrsquo expected behaviourcontributes to letting miners traders and smelters aliketake the maximum advantage from the copper concentratetrading agreements that they are part of As an additionalcontribution this work may well be applied to gold or silverRC which are relevant deduction concentrates when thesehave a significant amount of gold or silver

12 Complexity

Once TCRC annual benchmark levels are able to beforecasted with a certain level of accuracy future researchshould go further into this research through the explorationof the potential impact that other market factors may have onthese discounts

As a limitation of this research we should point out thetimespan of the data considered compared to those of otherforecasting works on commodity prices for example whichuse broader timespans For our case we have consideredthe maximum available sufficiently reliable data on TCRCbenchmark levels starting back in 2004 as there is noreliable data beyond this year Also as another limitationwe have used four measures of error which are among themost frequently used to compare the accuracy of differentmodels However other measures could have been used at anindividual level to test each modelrsquos accuracy

Appendix

A Models

A1 Geometric Brownian Motion (GBM) GBM can bewritten as a generalisation of a Wiener (continuous time-stochasticMarkov process with independent increments andwhose changes over any infinite interval of time are normallydistributed with a variance that increases linearly with thetime interval [22]) process

119889119909 = 120572119909119889119905 + 120590119909119889119911 (A1)

where according to Marathe and Ryan [30] the first termis known as the Expected Value whereas the second is theStochastic component with 120572 being the drift parameter and120590 the volatility of the process Also dz is the Wiener processwhich induces the abovementioned stochastic behaviour inthe model

119889119911 = isin119905radic119889119905 997888rarr isin119905 sim 119873 (0 1) (A2)

TheGBMmodel can be expressed in discrete terms accordingto

119909 = 119909119905 minus 119909119905minus1 = 120572119909119905minus1119905 + 120590119909119905minus1120598radic119905 (A3)

In GBM percentage changes in x 119909119909 are normally dis-tributed thus absolute changes in x 119909 are lognormallydistributed Also the expected value and variance for 119909(119905) are

E [119909 (119905)] = 1199090119890120572119905 (A4)

var [119909 (119905)] = 119909201198902120572119905 (1198901205902119905 minus 1) (A5)

A2 Orstein-Uhlenbeck Process (OUP) TheOUPprocess wasfirst defined by Uhlenbeck and Orstein [31] as an alternativeto the regular Brownian Motion to model the velocity ofthe diffusion movement of a particle that accounts for itslosses due to friction with other particles The OUP processcan be regarded as a modification of Brownian Motion incontinuous time where its properties have been changed

(Stationary Gaussian Markov and stochastic process) [32]These modifications cause the process to move towards acentral position with a stronger attraction the further it isfrom this position As mentioned above the OUP is usuallyemployed to model commodity prices and is the simplestversion of amean-reverting process [22]

119889119878 = 120582 (120583 minus 119878) 119889119905 + 120590119889119882119905 (A6)

where S is the level of prices 120583 the long-term average towhich prices tend to revert and 120582 the speed of reversionAdditionally in a similar fashion to that of the GBM 120590 isthe volatility of the process and 119889119882119905 is a Wiener processwith an identical definition However in contrast to GBMtime intervals in OUP are not independent since differencesbetween current levels of prices S and long-term averageprices 120583 make the expected change in prices dS more likelyeither positive or negative

The discrete version of the model can be expressed asfollows

119878119905 = 120583 (1 minus 119890minus120582Δ119905) + 119890minus120582Δ119905119878119905minus1 + 120590radic1 minus 119890minus2120582Δ1199052120582 119889119882119905 (A7)

where the expected value for 119909(119905) and the variance for (119909(119905) minus120583) areE [119909 (119905)] = 120583 + (1199090 minus 120583) 119890minus120582119905 (A8)

var [119909 (119905) minus 120583] = 12059022120582 (1 minus 119890minus2120582119905) (A9)

It can be derived from previous equations that as timeincreases prices will tend to long-term average levels 120583 Inaddition with large time spans if the speed of reversion 120582becomes high variance tends to 0 On the other hand if thespeed of reversion is 0 then var[119909(119905)] 997888rarr 1205902119905 making theprocess a simple Brownian Motion

A3Holtrsquos Linear Exponential Smoothing (LES) Linear Expo-nential Smoothing models are capable of considering bothlevels and trends at every instant assigning higher weights inthe overall calculation to values closer to the present than toolder ones LES models carry that out by constantly updatinglocal estimations of levels and trends with the intervention ofone or two smoothing constants which enable the models todampen older value effects Although it is possible to employa single smoothing constant for both the level and the trendknown as Brownrsquos LES to use two one for each known asHoltrsquos LES is usually preferred since Brownrsquos LES tends torender estimations of the trend ldquounstablerdquo as suggested byauthors such as Nau [33] Holtrsquos LES model comes definedby the level trend and forecast updating equations each ofthese expressed as follows respectively

119871 119905 = 120572119884119905 + (1 minus 120572) (119871 119905minus1 + 119879119905minus1) (A10)

119879119905 = 120573 (119871 119905 minus 119871 119905minus1) + (1 minus 120573) 119879119905minus1 (A11)

119905+119896 = 119871 119905 + 119896119879119905 (A12)

Complexity 13

With 120572 being the first smoothing constant for the levelsand 120573 the second smoothing constant for the trend Highervalues for the smoothing constants imply that either levels ortrends are changing rapidly over time whereas lower valuesimply the contrary Hence the higher the constant the moreuncertain the future is believed to be

B Model Calibration

Each model has been calibrated individually using two setsof data containing TC and RC levels respectively from 2004to 2012 The available dataset is comprised of data from 2004to 2017 which make the calibration data approximately 23 ofthe total

B1 GBM Increments in the logarithm of variable x aredistributed as follows

(ln119909) sim 119873((120572 minus 12059022 ) 119905 120590119905) (B1)

Hence if m is defined as the sample mean of the differenceof the natural logarithm of the time series for TCRC levelsconsidered for the calibration and n as the number ofincrements of the series considered with n=9

119898 = 1119899119899sum119905=1

(ln119909119905 minus ln119909119905minus1) (B2)

119904 = radic 1119899 minus 1119899sum119894=1

(ln119909119905 minus ln119909119905minus1 minus 119898)2 (B3)

119898 = 120572 minus 12059022 (B4)

119904 = 120590 (B5)

B2 OUP The OUP process is an AR1 process [22] whoseresolution is well presented by Woolridge [34] using OLStechniques fitting the following

119910119905 = 119886 + 119887119910119905minus1 + 120576119905 (B6)

Hence the estimators for the parameters of the OUP modelare obtained by OLS for both TC and RC levels indepen-dently

= minus ln 119887Δ119905 (B7)

120583 = 1198861 minus 119887 (B8)

= 120576119905radic 2 ln (1 + 119887)(1 + 119887)2 minus 1 (B9)

B3 LES A linear regression is conducted on the inputdataset available to find the starting parameters for the LES

model the initial Level 1198710 and the initial value of thetrend 1198790 irrespective of TC values and RC values Hereas recommended by Gardner [35] the use of OLS is highlyadvisable due to the erratic behaviour shown by trends inthe historic data so the obtaining of negative values of 1198780 isprevented Linear regression fulfils the following

119884119905 = 119886119905 + 119887 (B10)

1198710 = 119887 (B11)

1198790 = 119886 (B12)

By fixing the two smoothing constants the values for theforecasts 119905+119896 can be calculated at each step using the modelequations There are multiple references in the literature onwhat the optimum range for each smoothing constant isGardner [35] speaks of setting moderate values for bothparameter less than 03 to obtain the best results Examplespointing out the same may be found in Brown [36] Coutie[37] Harrison [38] and Montgomery and Johnson [39]Also for many applications Makridakis and Hibon [20] andChatfield [40] found that parameter values should fall withinthe range of 03-1 On the other hand McClain and Thomas[41] provided a condition of stability for the nonseasonal LESmodel given by

0 lt 120572 lt 20 lt 120573 lt 4 minus 2120572120572

(B13)

Also the largest possible value of 120572 that allows the avoidanceof areas of oscillation is proposed by McClain and Thomas[41] and McClain [42]

120572 lt 4120573(1 + 120573)2 (B14)

However according to Gardner there is no tangible proofthat this value improves accuracy in any form Nonethelesswe have opted to follow what Nau [33] refers to as ldquotheusual wayrdquo namely minimising the Mean Squared Error(MSE) of the one-step-ahead forecast of TCRC for eachinput data series previously mentioned To do so Matlabrsquosfminsearch function has been used with function and variabletolerance levels of 1x10minus4 as well as a set maximum numberof function iterations and function evaluations of 1x106 tolimit computing resources In Table 18 the actual number ofnecessary iterations to obtain optimum values for smoothingconstants is shown As can be seen the criteria are wellbeyond the final results which ensured that an optimumsolution was reached with assumable computing usage (thesimulation required less than one minute) and with a highdegree of certainty

14 Complexity

Table 18 Iterations on LES smoothing constants optimisation

Iterations Function EvaluationsTC 1-Step 36 40TC 2-Steps 249945 454211TC 5-Steps 40 77RC 1-Step 32 62RC 2-Steps 254388 462226RC 5-Steps 34 66

C Error Calculations

C1 Mean Square Error (MSE)

119872119878119864 = 1119899119899sum119894=1

(119894 minus 119884119894)2 (C1)

where 119894 are the forecasted values and 119884119894 those observedC2 Mean Absolute Deviation (MAD)

119872119860119863 = 1119899119899sum119894=1

10038161003816100381610038161003816119894 minus 11988410038161003816100381610038161003816 (C2)

where 119894 are the forecasted values and 119884 the average value ofall the observations

C3 Mean Absolute Percentage Error (MAPE)

119872119860119875119864 = 1119899119899sum119894=1

10038161003816100381610038161003816119894 minus 11988411989410038161003816100381610038161003816119884119894 (C3)

The above formula is expressed in parts-per-one and is theone used in the calculations conducted here Hence multi-plying the result by 100 would deliver percentage outcomes

C4 Root Mean Square Error (RMSE)

119877119872119878119864 = radic119872119878119864 = radic 1119899119899sum119894=1

(119894 minus 119884119894)2 (C4)

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Supplementary Materials

The Matlab codes developed to carry out the simulationsfor each of the models proposed as well as the datasetsused and the Matlab workspaces with the simulationsoutcomes as they have been reflected in this paper areprovided in separate folders for each model GBM foldercontains the code to make the simulations for the Geo-metric Brownian Motion model ldquoGBMmrdquo as well as aseparate script which allows determining the total numberof monthly step simulations for which errors levels havebeen lower than annual-step simulations ldquoErrCoutermrdquoThe datasets used are included in two excel files ldquoTCxlsxrdquoand ldquoRCxlsxrdquo ldquoGBM OK MonthlyStepsmatrdquo is the Matlabworkspace containing the outcome of monthly step simu-lations whereas ldquoGBM OK AnnualStepsmatrdquo contains theoutcome for annual-step simulations Also ldquoGBM OKmatrdquois the workspace to be loaded prior to performing GBMsimulations MR folder contains the code file to carryout the simulations for the Orstein-Uhlenbeck ProcessldquoMRmrdquo as well as the separate script to compare errorsof monthly step simulations with annual-step simula-tions ldquoErrCountermrdquo ldquoTCxlsxrdquo and ldquoRCxlsxrdquo containthe TCRC benchmark levels from 2004 to 2017 Finallymonthly step simulationsrsquo outcome has been saved inthe workspace ldquoMR OK MonthlyStepsmatrdquo while annual-step simulationsrsquo outcome has been saved in the Matlabworkspace ldquoMR OK AnnualStepsmatrdquo Also ldquoMR OKmatrdquois the workspace to be loaded prior to performing MRsimulations LES folder contains the code file to carry outthe calculations necessary to obtain the LES model forecastsldquoLESmrdquo Three separate Matlab functions are includedldquomapeLESmrdquo ldquomapeLES1mrdquo and ldquomapeLES2mrdquo whichdefine the calculation of the MAPE for the LESrsquo five-year-ahead one-year-ahead and two-year-ahead forecastsAlso ldquoLES OKmatrdquo is the workspace to be loaded prior toperforming LES simulations Finally graphs of each modelhave been included in separate files as well in a subfoldernamed ldquoGraphsrdquo within each of the modelsrsquo folder Figuresare included in Matlabrsquos format (fig) and TIFF format TIFFfigures have been generated in both colours and black andwhite for easier visualization (Supplementary Materials)

References

[1] United Nations International Trade Statistics Yearbook vol 22016

[2] International Copper StudyGroupTheWorld Copper Factbook2017

[3] K M Johnson J M Hammarstrom M L Zientek and CL Dicken ldquoEstimate of undiscovered copper resources of theworld 2013rdquo US Geological Survey 2017

[4] M E Schlesinger and W G Davenport Extractive Metallurgyof Copper Elsevier 5th edition 2011

[5] S Gloser M Soulier and L A T Espinoza ldquoDynamic analysisof global copper flows Global stocks postconsumer materialflows recycling indicators and uncertainty evaluationrdquo Envi-ronmental Science amp Technology vol 47 no 12 pp 6564ndash65722013

Complexity 5

GBM RC First 20 pathsGBM TC First 20 paths

0

50

100

150

200

250

300

TC (U

SDM

T)

2013 2014 2015 2016 20172012

GBM TC First 20 paths - Year Averaged

05

101520253035

RC (U

Sclb

)

0

5

10

15

20

25

30

RC (U

Slb

)

0

50

100

150

200

250

TC (U

SDM

T)

2013 2014 2015 2016 20172012

Observed TC Benchmark Levels

2013 2014 2015 2016 20172012

Observed RC Benchmark Levels

2013 2014 2015 2016 20172012

GBM RC First 20 paths - Year Averaged

Figure 2 The upper illustrations show the first 20 Monte Carlo paths for either TC or RC levels using monthly steps The illustrations belowshow the annual averages of monthly step forecasts for TC and RC levels

MR TC First 20 paths

2012 2013 2014 2015 2016 2017

MR RC First 20 paths

MR RC First 20 paths - Year Averaged

2013 2014 2015 2016 20172012

Observed

5

6

7

8

9

10

11

RC (U

Slb

)

2468

10121416

RC (U

Sclb

)

0

50

100

150

200

TC (U

SDM

T)

2013 2014 2015 2016 20172012

MR TC First 20 paths - Year Averaged

2013 2014 2015 2016 20172012

Observed

50

60

70

80

90

100

110

TC (U

SDM

T)

Figure 3 The first 20 Monte Carlo paths following the OUP model using monthly steps are shown on the upper illustrations for either TCor RC The annual averaged simulated paths every 12 values are shown below

dollars per metric tonne and RC in US cents per pound ofpayable copper content in concentrates In fact benchmarkdata show that of the last 14 years it was only in 2010that values of TC and RC did not conform to this relation

though it did remain close to it (46547) Nonetheless thisrelation has not been observed in the methodology hereinthus treating TC and RC independently developing separateunrelated forecasts for TC and RC to understand whether

6 Complexity

Table 2 TCRC year benchmark levels (Data available free ofcharge Source Johansson [50] Svante [51] Teck [52] Shaw [53]Willbrandt and Faust [54 55] Aurubis AG [56] Drouven and Faust[57] Aurubis AG [58] EY [59] Schachler [60] Nakazato [61])

YEAR TC (USDMT) RC (USclb)2004 45 452005 85 852006 95 952007 60 602008 45 452009 75 752010 465 472011 56 562012 635 6352013 70 702014 92 922015 107 1072016 9735 97352017 925 925

these individual forecasts for TC and RC would render betterresults than a single joint forecast for both TCRC levels thatwould take into account the 101 relation existent in the data

34 Models Comparison Method The works referred to inthe literature usually resort to different measures of errorsto conduct the testing of a modelrsquos forecasting capacityregardless of the specific nature of the forecasted value be itcotton prices or macroeconomic parameters Thus the mostwidely errors used includeMean Squared Error (MSE) MeanAbsolute Deviation (MAD) Mean Absolute Percentage Error(MAPE) and Root Mean Square Error (RMSE) [14 16 17 2125ndash29] In this work Geometric Brownian Motion (GBM)Ornstein-Uhlenbeck Process (OUP) and Linear ExponentialSmoothing (LES) models have been used to forecast annualTCRC benchmark levels using all the four main measuresof error mentioned above to test the predictive accuracy ofthese three models The GBM and OUP models have beensimulated and tested with different step sizes while the LESmodel has been analysed solely using annual time steps

The GBM and OUPmodels were treated separately to theLESmodel thus GBM andOUPwere simulated usingMonteCarlo (MC) simulations whereas LES forecasts were simplecalculations In a preliminary stage the models were firstcalibrated to forecast values from 2013 to 2017 using availabledata from 2004 to 2012 for TC and RC separately hencethe disregarding of the well-known inherent 101 relation (seeAppendix B for Models Calibration) The GBM and OUPmodels were calibrated for two different step sizes monthlysteps and annual steps in order to compare forecastingaccuracy with each step size in each model Following thecalibration of the models Monte Carlo simulations (MC)were carried out usingMatlab software to render the pursuedforecasts of GBM andOUPmodels obtaining 1000 simulatedpaths for each step size MC simulations using monthly stepsfor the 2013ndash2017 timespan were averaged every 12 steps to

deliver year forecasts of TCRC benchmark levels On theother hand forecasts obtained by MC simulations takingannual steps for the same period were considered as yearforecasts for TCRC annual benchmark levels without theneed for extra transformation Besides LES model forecastswere calculated at different time horizons to be able tocompare the short-term and long-term predictive accuracyLES forecasts were obtained for one-year-ahead hence usingknown values of TCRC from 2004 to 2016 for two yearsahead stretching the calibrating dataset from 2004 to 2015for five-year-ahead thus using the same input dataset as forthe GBM and OUP models from 2004 to 2012

Finally for every forecast path obtained we have calcu-lated the average of the squares of the errors with respect tothe observed valuesMSE the average distance of a forecast tothe observed meanMAD the average deviation of a forecastfrom observed values MAPE and the square root of MSERMSE (see Appendix C for error calculations) The results ofMSE MAD MAPE and RMSE calculated for each forecastpathwere averaged by the total number of simulations carriedout for each case Averaged values of error measures of allsimulated paths MSE MAD MAPE and RMSE for bothannual-step forecasts and monthly step forecasts have beenused for cross-comparison between models to test predictiveability at every step size possible

Also to test each modelrsquos short-term forecasting capacityagainst its long-term forecasting capacity one-year-aheadforecast errors of the LES model were compared with theerrors from the last year of the GBM and OUP simulatedpaths Two-year-ahead forecast errors of LES models werecomparedwith the average errors of the last two years ofGBMand OUP and five-year-ahead forecast errors of LES modelswere compared with the overall average of errors of the GBMand OUP forecasts

4 Analysis of Results

Monte Carlo simulations of GBM and OUP models render1000 different possible paths for TC and RC respectivelyat each time step size considered Accuracy errors for bothannual time steps and averaged monthly time steps for bothGBM and OUP forecasts were first compared to determinethe most accurate time step size for each model In additionthe LES model outcome for both TC and RC at differenttimeframes was also calculated and measures of errors for allthe three alternative models proposed at an optimum timestep were finally compared

41 GBM Forecast The first 20 of 1000 Monte Carlo pathsfor the GBM model with a monthly step size using Matlabsoftware may be seen in Figure 2 for both the TC and the RClevels compared to their averaged paths for every 12 valuesobtainedThe tendency for GBM forecasts to steeply increaseover time is easily observable in the nonaveraged monthlystep paths shown

The average values of error of all 1000MC paths obtainedthrough simulation for averaged monthly step and annual-step forecasts are shown in Table 3 for both TC and RC

Complexity 7

Table 3 Average of main measures of error for GBM after 1000 MC simulations from 2013 to 2017

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC Averaged Monthly Steps 560x103 4698 050 5638TC Annual Steps 520x103 4902 052 5857RC Averaged Monthly Steps 5874 455 048 546RC Annual Steps 5085 491 052 584

Table 4 Number of paths for which measures of error are higher for monthly steps than for annual steps in GBM

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC 4571000 4551000 4531000 4531000RC 4391000 4301000 4291000 4291000

discounts over the period from 2013 to 2017 which maylead to preliminary conclusions in terms of an accuracycomparison between averaged monthly steps and annualsteps

However a more exhaustive analysis is shown in Table 4where the number of times the values of each error measureare higher for monthly steps is expressed over the totalnumber of MC simulations carried out The results indicatethat better values of error are reached the majority of timesfor averagedmonthly step simulations rather than for straightannual ones

In contrast short-term accuracy was also evaluated byanalysing the error measures of one-year-ahead forecasts(2013) in Table 5 and of two-year-ahead forecasts (2013ndash2014)in Table 6 The results indicate as one may expect thataccuracy decreases as the forecasted horizon is widened withthe accuracy of averaged monthly step forecasts remaininghigher than annual ones as found above for the five-yearforecasting horizon

42 OUP Forecast Long-term levels for TC and RC 120583 thespeed of reversion 120582 and the volatility of the process 120590 arethe parameters determined at themodel calibration stage (seeAppendix B for the model calibration explanation) whichdefine the behaviour of the OUP shown in Table 7 Thecalibration was done prior to the Monte Carlo simulationfor both TC and RC with each step size using availablehistorical data from 2004 to 2012 The OUP model was fittedwith the corresponding parameters for each case upon MCsimulation

Figure 3 shows theMeanReversionMCestimations of theTCRC benchmark values from 2013 to 2017 using monthlystepsThemonthly forecastswere rendered from January 2012through to December 2016 and averaged every twelve valuesto deliver a benchmark forecast for each year The averagedresults can be comparable to actual data as well as to theannual Monte Carlo simulations following Mean ReversionThe lower-side figures show these yearly averaged monthlystep simulation outcomes which clearly move around a dash-dotted red line indicating the long-term run levels for TCRCto which they tend to revert

The accuracy of monthly steps against annual steps forthe TCRC benchmark levels forecast was also tested by

determining the number of simulations for which averageerror measures became higher Table 8 shows the numberof times monthly simulations have been less accurate thanannual simulations for five-year-ahead OUP forecasting bycomparing the four measures of errors proposed The resultsindicate that only 25-32 of the simulations drew a higheraverage error which clearly results in a better predictiveaccuracy for monthly step forecasting of TCRC annualbenchmark levels

The averaged measures of errors obtained after the MCsimulations of the OUP model for both averaged monthlysteps and annual steps giving TCRC benchmark forecastsfrom 2013 to 2017 are shown in Table 9

The error levels of the MC simulations shown in Table 9point towards a higher prediction accuracy of averagedmonthly step forecasts of the OUP Model yielding anaveraged MAPE value that is 129 lower for TC and RC 5-step-ahead forecasts In regard to MAPE values for monthlysteps only 266 of the simulations rise above annual MCsimulations for TC and 25 for RC 5-step-ahead forecastswhich further underpins the greater accuracy of thisOUP set-up for TCRC level forecasts A significant lower probabilityof higher error levels for TCRC forecasts with monthly MCOUP simulations is reached for the other measures providedIn addition short-term and long-term prediction accuracywere tested by comparing errors of forecasts for five-year-ahead error measures in Table 10 one-year-ahead in Table 11and two-year-ahead in Table 12

With a closer forecasting horizon errormeasures show animprovement of forecasting accuracy when average monthlysteps are used rather than annual ones For instance theMAPE values for 2013 forecast for TC are 68 lower foraveraged monthly steps than for annual steps and alsoMAPE for 2013ndash2014 were 30 lower for both TC and RCforecasts Similarly better values of error are achieved for theother measures for averaged monthly short-term forecaststhan in other scenarios In addition as expected accuracyis increased for closer forecasting horizons where the levelof errors shown above becomes lower as the deviation offorecasts is trimmed with short-term predictions

43 LES Forecast In contrast to GBM and OUP the LESmodel lacks any stochastic component so nondeterministic

8 Complexity

Table 5 Average for main measures of error for GBM after 1000 MC simulations for 2013

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC Averaged Monthly Steps 33658 1455 021 1455TC Annual Steps 69381 2081 030 2081RC Averaged Monthly Steps 297 138 020 138RC Annual Steps 657 205 029 205

Table 6 Average for main measures of error for GBM after 1000 MC simulations for 2013-2014

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC Averaged Monthly Steps 99492 2187 031 2435TC Annual Steps 133x103 2833 040 3104RC Averaged Monthly Steps 1035 240 034 271RC Annual Steps 1313 284 034 311

Table 7 OUP parameters obtained in the calibration process

120583 120582 120590TCMonthly 6345 4792x105 2534TC Annual 6345 2974 1308RC Monthly 635 4763x105 2519RC Annual 635 2972 1305

methods such as theMonte Carlo are not required to obtain aforecast Nonetheless the LES model relies on two smoothingconstants which must be properly set in order to deliveraccurate predictions hence the values of the smoothingconstants were first optimised (see Appendix B for LESModel fitting and smoothing constants optimisation) Theoptimisation was carried out for one-year-ahead forecaststwo-year-ahead forecasts and five-year-ahead forecasts byminimising the values of MSE for both TC and RC Thedifferent values used for smoothing constants as well asthe initial values for level and trend obtained by the linearregression of the available dataset from 2004 through to 2012are shown in Table 12

Compared one-year-ahead two-year-ahead and five-year-ahead LES forecasts for TC and RC are shown inFigure 4 clearly indicating a stronger accuracy for shorter-term forecasts as the observed and forecasted plotted linesoverlap

Minimumvalues for errormeasures achieved through theLESmodel parameter optimisation are shown in Table 13Thevalues obtained confirm the strong accuracy for shorter-termforecasts of TCRC seen in Figure 4

5 Discussion and Model Comparison

The forecasted values of TCRC benchmark levels couldeventually be applied to broader valuation models for copperconcentrates and their trading activities as well as to thecopper smeltersrsquo revenue stream thus the importance ofdelivering as accurate a prediction as possible in relation tothese discounts to make any future application possible Eachof the models presented may be a feasible method on its own

with eventual later adaptations to forecasting future valuesof benchmark TCRC Nonetheless the accuracy of thesemodels as they have been used in this work requires firstlya comparison to determine whether any of them could bea good standalone technique and secondly to test whethera combination of two or more of them would deliver moreprecise results

When comparing the different error measures obtainedfor all the three models it is clearly established that resultsfor a randomly chosen simulation of GBM or OUP wouldbe more likely to be more precise had a monthly step beenused to deliver annual forecasts instead of an annual-step sizeIn contrast average error measures for the entire populationof simulations with each step size employed showing thatmonthly step simulations for GBM and OUP models arealways more accurate than straight annual-step forecastswhen a shorter time horizon one or two-year-ahead is takeninto consideration However GBM presents a higher levelof forecasting accuracy when the average for error measuresof all simulations is analysed employing annual steps forlong-term horizons whereas OUP averaged monthly stepforecasts remain more accurate when predicting long-termhorizons Table 14 shows the error improvement for the aver-aged monthly step forecasts of each model Negative valuesindicate that better levels of error averages have been found instraight annual forecasts than for monthly step simulations

Considering the best results for each model and com-paring their corresponding error measures we can opt forthe best technique to employ among the three proposed inthis paper Hence GBM delivers the most accurate one-yearforecast when averaging the next twelve-month predictionsfor TCRC values as does the MR-OUP model Table 15shows best error measures for one-year-ahead forecasts forGBM OUP and LES models

Unarguably the LES model generates minimal errormeasures for one-year-ahead forecasts significantly less thanthe other models employed A similar situation is found fortwo-year-ahead forecasts whereminimumerrormeasures arealso delivered by the LES model shown in Table 16

Finally accuracy measures for five-year-ahead forecastsof the GBM model might result in somewhat contradictory

Complexity 9

Table 8 Number of paths for whichmeasures of error are higher for monthly steps than for annual steps inMR-OUP 5-StepsMC simulation

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC 3111000 2831000 2661000 2661000RC 3161000 2811000 2501000 2501000

Table 9 Average for main measures of error for MR-OUP after 1000 MC simulations 2013-2017

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC Averaged Monthly Steps 84254 2614 027 2895TC Annual Steps 114x103 2931 031 3325RC Averaged Monthly Steps 840 261 027 289RC Annual Steps 1148 294 031 333

Table 10 Average for main measures of error for MR-OUP after 1000 MC simulations 2013

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC Averaged Monthly Steps 3914 522 007 522TC Annual Steps 36643 1555 022 1555RC Averaged Monthly Steps 039 051 007 051RC Annual Steps 363 154 022 154

Table 11 Average for main measures of error for MR-OUP after 1000 MC simulations 2013-2014

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC Averaged Monthly Steps 37165 1567 018 1900TC Annual Steps 68244 2185 026 2324RC Averaged Monthly Steps 376 157 018 191RC Annual Steps 689 219 026 243

Table 12 LES Model Optimised Parameters

L0 T0 120572 120573TC 1 year 713611 -15833 -02372 01598RC 1 year 71333 -01567 -02368 01591TC 2 years 713611 -15833 -1477x10minus4 7774226RC 2 years 71333 -01567 -1448x10minus4 7898336TC 5 years 713611 -15833 -02813 01880RC 5 years 71333 -01567 -02808 01880

Table 13 LES error measures for different steps-ahead forecasts

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC 1 year 10746x10minus5 00033 46831x10minus5 00033RC 1 year 51462x10minus8 22685x10minus4 32408x10minus5 22685x10minus4

TC 2 years 189247 97275 01057 137567RC 2 years 18977 09741 01059 13776TC 5 years 1775531 78881 00951 108422RC 5 years 11759 07889 00952 10844

terms for MSE reaching better values for annual steps thanfor averaged monthly steps while the other figures do betteron averagedmonthly steps Addressing the definition ofMSEthis includes the variance of the estimator as well as its biasbeing equal to its variance in the case of unbiased estimators

Therefore MSE measures the quality of the estimator butalso magnifies estimator deviations from actual values sinceboth positive and negative values are squared and averagedIn contrast RMSE is calculated as the square root of MSEand following the previous analogy stands for the standard

10 Complexity

Table 14 Error Average Improvement for averaged monthly steps before annual steps

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864GBM TCRC 1 Year 53265653 32723587 33333548 32723587GBM TCRC 2 Years 26952635 25751597 2619286 24341369GBM TCRC 5 Years -1173-1104 873709 926769 747565OUP TCRC 1 Year 88088708 63716250 62916319 63686224OUP TCRC 2 Years 46284421 27712617 29993075 22162075OUP TCRC 5 Years 26022553 1093997 13121218 13061269

Table 15 TCRC best error measures for 2013 forecasts after 1000 MC simulations

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864GBM TCRC 1 Yearlowast 33111336 1425143 020020 1425143OUP TCRC 1 Yearlowast 4105042 536054 0076600777 536054LES TCRC 1 Year 107x10minus5514x10minus8 0003226x10minus4 468x10minus5324x10minus5 0003226x10minus4

lowastAveraged monthly steps

Table 16 TCRC Best error measures for 2013ndash2014 forecasts after 1000 MC simulations

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864GBM TCRC 2 Stepslowast 103x1031006 2200242 031034 2447271OUP TCRC 2 Stepslowast 37331376 1573158 0180201813 1900191LES TCRC 2 Steps 18924189 972097 0105701059 1375137lowastAveraged monthly steps

deviation of the estimator if MSE were considered to be thevarianceThough RMSE overreacts when high values of MSEare reached it is less prone to this than MSE since it iscalculated as its squared root thus not accounting for largeerrors as disproportionately as MSE does Furthermore aswe have compared an average of 1000 measures of errorscorresponding to each MC simulation performed the valuesobtained for average RMSE stay below the square root ofaverage MSE which indicates that some of these dispropor-tionate error measures are to some extent distorting thelatter Hence RMSE average values point towards a higheraccuracy for GBM five-year forecasts with averaged monthlysteps which is further endorsed by the average values ofMAD and MAPE thus being the one used for comparisonwith the other two models as shown in Table 17

The final comparison clearly shows how the LES modeloutperforms the other two at all average measures providedfollowed by the OUP model in accuracy although the lattermore than doubles the average MAPE value for LES

The results of simulations indicate that measures of errorstend to either differ slightly or not at all for either forecastsof any timeframe A coherent value with the 101 relation canthen be given with close to the same level of accuracy bymultiplying RC forecasts or dividing TC ones by 10

6 Conclusions

Copper TCRC are a keystone for pricing copper concen-trates which are the actual feedstock for copper smeltersThe potential evolution of TCRC is a question of botheconomic and technical significance forminers as their value

decreases the potential final selling price of concentratesAdditionally copper minersrsquo revenues are more narrowlyrelated to the market price of copper as well as to othertechnical factors such as ore dilution or the grade of theconcentrates produced Smelters on the contrary are hugelyaffected by the discount which they succeed in getting whenpurchasing the concentrates since that makes up the largestpart of their gross revenue besides other secondary sourcesIn addition eventual differences between TCRC may givecommodity traders ludicrous arbitrage opportunities Alsodifferences between short- and long-termTCRC agreementsoffer arbitrage opportunities for traders hence comprisinga part of their revenue in the copper concentrate tradingbusiness as well copper price fluctuations and the capacityto make economically optimum copper blends

As far as we are aware no rigorous research has beencarried out on the behaviour of these discounts Based onhistorical data on TCRC agreed upon in the LME CopperWeek from 2004 to 2017 three potentially suitable forecastingmodels for TCRC annual benchmark values have beencompared through four measures of forecasting accuracyat different horizons These models were chosen firstlydue to their broad implementation and proven capacityin commodity prices forecasting that they all share andsecondly because of the core differences in terms of pricebehaviour with each other

Focusing on the MAPE values achieved those obtainedby the LES model when TC and RC are treated indepen-dently have been significantly lower than for the rest ofthe models Indeed one-year-ahead MAPE measures for TCvalues for the GBM model (20) almost triple those of

Complexity 11

2004 2006 2008 2010 2012 2014405060708090

100

TC (U

SDM

T)

LES TC 1 Step Ahead 2013

2004 2006 2008 2010 2012 2014

LES TC 2 Steps Ahead 2013-2014

2004 2006 2008 2010 2012 2014 2016 2018

LES TC 5 Steps Ahead 2013-2017

2004 2006 2008 2010 2012 2014

LES RC 1 Step Ahead 2013

2004 2006 2008 2010 2012 2014456789

10LES RC 2 Steps Ahead 2013-2014

Observed Forecast

2004 2006 2008 2010 2012 2014 2016 2018

LES RC 5 Steps Ahead 2013-2017

405060708090

100110

405060708090

100

456789

10

RC (U

Sclb

)

456789

1011

Figure 4 One-step-ahead forecasts (2013) two-step-ahead forecasts (2013-2014) and five-step-ahead forecasts (2013-2017) for TC and RCusing the Linear Exponential Smoothing model (LES)

Table 17 TCRC Best error measures for 2013ndash2017 forecasts after 1000 MC simulations

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864GBM TCRC 5 Stepslowast 600x1036165 4651459 049048 5571551OUP TCRC 5 Stepslowast 84334843 2616262 0271502719 2896289LES TCRC 5 Steps 17755117 788078 0095100952 1084108lowastAveraged monthly steps

the OUP model (766) in contrast with the significantlylower values from the LES model (00046) This gaptends to be narrowed when TCRC values are forecastedat longer horizons when most measures of error becomemore even The GBM and OUP models have proven todeliver better accuracy performance when the TCRC valuesare projected monthly and then averaged to obtain annualbenchmark forecasts Even so the LES model remains themost accurate of all with MAPE values of 10 at two-year-ahead forecasts with 18 and 31 for TC for OUP and GBMrespectively

Finally despite TC and RC being two independentdiscounts applied to copper concentrates they are bothset jointly with an often 101 relation as our data revealsThis relation also transcends to simulation results and errormeasures hence showing a negligible discrepancy betweenthe independent forecasting of TC and RC or the jointforecasting of both values keeping the 101 relationThis is forinstance the case of the five-year-ahead OUP MAPE values(0271502719) which were obtained without observing the101 relation in the data A similar level of discrepancy wasobtained at any horizon with any model which indicates thatboth values could be forecasted with the same accuracy usingthe selected model with any of them and then applying the101 relation

Our findings thus suggest that both at short and atlong-term horizons TCRC annual benchmark levels tendto exhibit a pattern which is best fit by an LES model Thisindicates that these critical discounts for the copper tradingbusiness do maintain a certain dependency on past valuesThis would also suggest the existence of cyclical patterns incopper TCRC possibly driven by many of the same marketfactors that move the price of copper

This work contributes by delivering a formal tool forsmelters or miners to make accurate forecasts of TCRCbenchmark levels The level of errors attained indicatesthe LES model may be a valid model to forecast thesecrucial discounts for the copper market In addition ourwork further contributes to helping market participants toproject the price of concentrates with an acceptable degree ofuncertainty as now they may include a fundamental elementfor their estimation This would enable them to optimise theway they produce or process these copper concentrates Alsothe precise knowledge of these discountsrsquo expected behaviourcontributes to letting miners traders and smelters aliketake the maximum advantage from the copper concentratetrading agreements that they are part of As an additionalcontribution this work may well be applied to gold or silverRC which are relevant deduction concentrates when thesehave a significant amount of gold or silver

12 Complexity

Once TCRC annual benchmark levels are able to beforecasted with a certain level of accuracy future researchshould go further into this research through the explorationof the potential impact that other market factors may have onthese discounts

As a limitation of this research we should point out thetimespan of the data considered compared to those of otherforecasting works on commodity prices for example whichuse broader timespans For our case we have consideredthe maximum available sufficiently reliable data on TCRCbenchmark levels starting back in 2004 as there is noreliable data beyond this year Also as another limitationwe have used four measures of error which are among themost frequently used to compare the accuracy of differentmodels However other measures could have been used at anindividual level to test each modelrsquos accuracy

Appendix

A Models

A1 Geometric Brownian Motion (GBM) GBM can bewritten as a generalisation of a Wiener (continuous time-stochasticMarkov process with independent increments andwhose changes over any infinite interval of time are normallydistributed with a variance that increases linearly with thetime interval [22]) process

119889119909 = 120572119909119889119905 + 120590119909119889119911 (A1)

where according to Marathe and Ryan [30] the first termis known as the Expected Value whereas the second is theStochastic component with 120572 being the drift parameter and120590 the volatility of the process Also dz is the Wiener processwhich induces the abovementioned stochastic behaviour inthe model

119889119911 = isin119905radic119889119905 997888rarr isin119905 sim 119873 (0 1) (A2)

TheGBMmodel can be expressed in discrete terms accordingto

119909 = 119909119905 minus 119909119905minus1 = 120572119909119905minus1119905 + 120590119909119905minus1120598radic119905 (A3)

In GBM percentage changes in x 119909119909 are normally dis-tributed thus absolute changes in x 119909 are lognormallydistributed Also the expected value and variance for 119909(119905) are

E [119909 (119905)] = 1199090119890120572119905 (A4)

var [119909 (119905)] = 119909201198902120572119905 (1198901205902119905 minus 1) (A5)

A2 Orstein-Uhlenbeck Process (OUP) TheOUPprocess wasfirst defined by Uhlenbeck and Orstein [31] as an alternativeto the regular Brownian Motion to model the velocity ofthe diffusion movement of a particle that accounts for itslosses due to friction with other particles The OUP processcan be regarded as a modification of Brownian Motion incontinuous time where its properties have been changed

(Stationary Gaussian Markov and stochastic process) [32]These modifications cause the process to move towards acentral position with a stronger attraction the further it isfrom this position As mentioned above the OUP is usuallyemployed to model commodity prices and is the simplestversion of amean-reverting process [22]

119889119878 = 120582 (120583 minus 119878) 119889119905 + 120590119889119882119905 (A6)

where S is the level of prices 120583 the long-term average towhich prices tend to revert and 120582 the speed of reversionAdditionally in a similar fashion to that of the GBM 120590 isthe volatility of the process and 119889119882119905 is a Wiener processwith an identical definition However in contrast to GBMtime intervals in OUP are not independent since differencesbetween current levels of prices S and long-term averageprices 120583 make the expected change in prices dS more likelyeither positive or negative

The discrete version of the model can be expressed asfollows

119878119905 = 120583 (1 minus 119890minus120582Δ119905) + 119890minus120582Δ119905119878119905minus1 + 120590radic1 minus 119890minus2120582Δ1199052120582 119889119882119905 (A7)

where the expected value for 119909(119905) and the variance for (119909(119905) minus120583) areE [119909 (119905)] = 120583 + (1199090 minus 120583) 119890minus120582119905 (A8)

var [119909 (119905) minus 120583] = 12059022120582 (1 minus 119890minus2120582119905) (A9)

It can be derived from previous equations that as timeincreases prices will tend to long-term average levels 120583 Inaddition with large time spans if the speed of reversion 120582becomes high variance tends to 0 On the other hand if thespeed of reversion is 0 then var[119909(119905)] 997888rarr 1205902119905 making theprocess a simple Brownian Motion

A3Holtrsquos Linear Exponential Smoothing (LES) Linear Expo-nential Smoothing models are capable of considering bothlevels and trends at every instant assigning higher weights inthe overall calculation to values closer to the present than toolder ones LES models carry that out by constantly updatinglocal estimations of levels and trends with the intervention ofone or two smoothing constants which enable the models todampen older value effects Although it is possible to employa single smoothing constant for both the level and the trendknown as Brownrsquos LES to use two one for each known asHoltrsquos LES is usually preferred since Brownrsquos LES tends torender estimations of the trend ldquounstablerdquo as suggested byauthors such as Nau [33] Holtrsquos LES model comes definedby the level trend and forecast updating equations each ofthese expressed as follows respectively

119871 119905 = 120572119884119905 + (1 minus 120572) (119871 119905minus1 + 119879119905minus1) (A10)

119879119905 = 120573 (119871 119905 minus 119871 119905minus1) + (1 minus 120573) 119879119905minus1 (A11)

119905+119896 = 119871 119905 + 119896119879119905 (A12)

Complexity 13

With 120572 being the first smoothing constant for the levelsand 120573 the second smoothing constant for the trend Highervalues for the smoothing constants imply that either levels ortrends are changing rapidly over time whereas lower valuesimply the contrary Hence the higher the constant the moreuncertain the future is believed to be

B Model Calibration

Each model has been calibrated individually using two setsof data containing TC and RC levels respectively from 2004to 2012 The available dataset is comprised of data from 2004to 2017 which make the calibration data approximately 23 ofthe total

B1 GBM Increments in the logarithm of variable x aredistributed as follows

(ln119909) sim 119873((120572 minus 12059022 ) 119905 120590119905) (B1)

Hence if m is defined as the sample mean of the differenceof the natural logarithm of the time series for TCRC levelsconsidered for the calibration and n as the number ofincrements of the series considered with n=9

119898 = 1119899119899sum119905=1

(ln119909119905 minus ln119909119905minus1) (B2)

119904 = radic 1119899 minus 1119899sum119894=1

(ln119909119905 minus ln119909119905minus1 minus 119898)2 (B3)

119898 = 120572 minus 12059022 (B4)

119904 = 120590 (B5)

B2 OUP The OUP process is an AR1 process [22] whoseresolution is well presented by Woolridge [34] using OLStechniques fitting the following

119910119905 = 119886 + 119887119910119905minus1 + 120576119905 (B6)

Hence the estimators for the parameters of the OUP modelare obtained by OLS for both TC and RC levels indepen-dently

= minus ln 119887Δ119905 (B7)

120583 = 1198861 minus 119887 (B8)

= 120576119905radic 2 ln (1 + 119887)(1 + 119887)2 minus 1 (B9)

B3 LES A linear regression is conducted on the inputdataset available to find the starting parameters for the LES

model the initial Level 1198710 and the initial value of thetrend 1198790 irrespective of TC values and RC values Hereas recommended by Gardner [35] the use of OLS is highlyadvisable due to the erratic behaviour shown by trends inthe historic data so the obtaining of negative values of 1198780 isprevented Linear regression fulfils the following

119884119905 = 119886119905 + 119887 (B10)

1198710 = 119887 (B11)

1198790 = 119886 (B12)

By fixing the two smoothing constants the values for theforecasts 119905+119896 can be calculated at each step using the modelequations There are multiple references in the literature onwhat the optimum range for each smoothing constant isGardner [35] speaks of setting moderate values for bothparameter less than 03 to obtain the best results Examplespointing out the same may be found in Brown [36] Coutie[37] Harrison [38] and Montgomery and Johnson [39]Also for many applications Makridakis and Hibon [20] andChatfield [40] found that parameter values should fall withinthe range of 03-1 On the other hand McClain and Thomas[41] provided a condition of stability for the nonseasonal LESmodel given by

0 lt 120572 lt 20 lt 120573 lt 4 minus 2120572120572

(B13)

Also the largest possible value of 120572 that allows the avoidanceof areas of oscillation is proposed by McClain and Thomas[41] and McClain [42]

120572 lt 4120573(1 + 120573)2 (B14)

However according to Gardner there is no tangible proofthat this value improves accuracy in any form Nonethelesswe have opted to follow what Nau [33] refers to as ldquotheusual wayrdquo namely minimising the Mean Squared Error(MSE) of the one-step-ahead forecast of TCRC for eachinput data series previously mentioned To do so Matlabrsquosfminsearch function has been used with function and variabletolerance levels of 1x10minus4 as well as a set maximum numberof function iterations and function evaluations of 1x106 tolimit computing resources In Table 18 the actual number ofnecessary iterations to obtain optimum values for smoothingconstants is shown As can be seen the criteria are wellbeyond the final results which ensured that an optimumsolution was reached with assumable computing usage (thesimulation required less than one minute) and with a highdegree of certainty

14 Complexity

Table 18 Iterations on LES smoothing constants optimisation

Iterations Function EvaluationsTC 1-Step 36 40TC 2-Steps 249945 454211TC 5-Steps 40 77RC 1-Step 32 62RC 2-Steps 254388 462226RC 5-Steps 34 66

C Error Calculations

C1 Mean Square Error (MSE)

119872119878119864 = 1119899119899sum119894=1

(119894 minus 119884119894)2 (C1)

where 119894 are the forecasted values and 119884119894 those observedC2 Mean Absolute Deviation (MAD)

119872119860119863 = 1119899119899sum119894=1

10038161003816100381610038161003816119894 minus 11988410038161003816100381610038161003816 (C2)

where 119894 are the forecasted values and 119884 the average value ofall the observations

C3 Mean Absolute Percentage Error (MAPE)

119872119860119875119864 = 1119899119899sum119894=1

10038161003816100381610038161003816119894 minus 11988411989410038161003816100381610038161003816119884119894 (C3)

The above formula is expressed in parts-per-one and is theone used in the calculations conducted here Hence multi-plying the result by 100 would deliver percentage outcomes

C4 Root Mean Square Error (RMSE)

119877119872119878119864 = radic119872119878119864 = radic 1119899119899sum119894=1

(119894 minus 119884119894)2 (C4)

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Supplementary Materials

The Matlab codes developed to carry out the simulationsfor each of the models proposed as well as the datasetsused and the Matlab workspaces with the simulationsoutcomes as they have been reflected in this paper areprovided in separate folders for each model GBM foldercontains the code to make the simulations for the Geo-metric Brownian Motion model ldquoGBMmrdquo as well as aseparate script which allows determining the total numberof monthly step simulations for which errors levels havebeen lower than annual-step simulations ldquoErrCoutermrdquoThe datasets used are included in two excel files ldquoTCxlsxrdquoand ldquoRCxlsxrdquo ldquoGBM OK MonthlyStepsmatrdquo is the Matlabworkspace containing the outcome of monthly step simu-lations whereas ldquoGBM OK AnnualStepsmatrdquo contains theoutcome for annual-step simulations Also ldquoGBM OKmatrdquois the workspace to be loaded prior to performing GBMsimulations MR folder contains the code file to carryout the simulations for the Orstein-Uhlenbeck ProcessldquoMRmrdquo as well as the separate script to compare errorsof monthly step simulations with annual-step simula-tions ldquoErrCountermrdquo ldquoTCxlsxrdquo and ldquoRCxlsxrdquo containthe TCRC benchmark levels from 2004 to 2017 Finallymonthly step simulationsrsquo outcome has been saved inthe workspace ldquoMR OK MonthlyStepsmatrdquo while annual-step simulationsrsquo outcome has been saved in the Matlabworkspace ldquoMR OK AnnualStepsmatrdquo Also ldquoMR OKmatrdquois the workspace to be loaded prior to performing MRsimulations LES folder contains the code file to carry outthe calculations necessary to obtain the LES model forecastsldquoLESmrdquo Three separate Matlab functions are includedldquomapeLESmrdquo ldquomapeLES1mrdquo and ldquomapeLES2mrdquo whichdefine the calculation of the MAPE for the LESrsquo five-year-ahead one-year-ahead and two-year-ahead forecastsAlso ldquoLES OKmatrdquo is the workspace to be loaded prior toperforming LES simulations Finally graphs of each modelhave been included in separate files as well in a subfoldernamed ldquoGraphsrdquo within each of the modelsrsquo folder Figuresare included in Matlabrsquos format (fig) and TIFF format TIFFfigures have been generated in both colours and black andwhite for easier visualization (Supplementary Materials)

References

[1] United Nations International Trade Statistics Yearbook vol 22016

[2] International Copper StudyGroupTheWorld Copper Factbook2017

[3] K M Johnson J M Hammarstrom M L Zientek and CL Dicken ldquoEstimate of undiscovered copper resources of theworld 2013rdquo US Geological Survey 2017

[4] M E Schlesinger and W G Davenport Extractive Metallurgyof Copper Elsevier 5th edition 2011

[5] S Gloser M Soulier and L A T Espinoza ldquoDynamic analysisof global copper flows Global stocks postconsumer materialflows recycling indicators and uncertainty evaluationrdquo Envi-ronmental Science amp Technology vol 47 no 12 pp 6564ndash65722013

6 Complexity

Table 2 TCRC year benchmark levels (Data available free ofcharge Source Johansson [50] Svante [51] Teck [52] Shaw [53]Willbrandt and Faust [54 55] Aurubis AG [56] Drouven and Faust[57] Aurubis AG [58] EY [59] Schachler [60] Nakazato [61])

YEAR TC (USDMT) RC (USclb)2004 45 452005 85 852006 95 952007 60 602008 45 452009 75 752010 465 472011 56 562012 635 6352013 70 702014 92 922015 107 1072016 9735 97352017 925 925

these individual forecasts for TC and RC would render betterresults than a single joint forecast for both TCRC levels thatwould take into account the 101 relation existent in the data

34 Models Comparison Method The works referred to inthe literature usually resort to different measures of errorsto conduct the testing of a modelrsquos forecasting capacityregardless of the specific nature of the forecasted value be itcotton prices or macroeconomic parameters Thus the mostwidely errors used includeMean Squared Error (MSE) MeanAbsolute Deviation (MAD) Mean Absolute Percentage Error(MAPE) and Root Mean Square Error (RMSE) [14 16 17 2125ndash29] In this work Geometric Brownian Motion (GBM)Ornstein-Uhlenbeck Process (OUP) and Linear ExponentialSmoothing (LES) models have been used to forecast annualTCRC benchmark levels using all the four main measuresof error mentioned above to test the predictive accuracy ofthese three models The GBM and OUP models have beensimulated and tested with different step sizes while the LESmodel has been analysed solely using annual time steps

The GBM and OUPmodels were treated separately to theLESmodel thus GBM andOUPwere simulated usingMonteCarlo (MC) simulations whereas LES forecasts were simplecalculations In a preliminary stage the models were firstcalibrated to forecast values from 2013 to 2017 using availabledata from 2004 to 2012 for TC and RC separately hencethe disregarding of the well-known inherent 101 relation (seeAppendix B for Models Calibration) The GBM and OUPmodels were calibrated for two different step sizes monthlysteps and annual steps in order to compare forecastingaccuracy with each step size in each model Following thecalibration of the models Monte Carlo simulations (MC)were carried out usingMatlab software to render the pursuedforecasts of GBM andOUPmodels obtaining 1000 simulatedpaths for each step size MC simulations using monthly stepsfor the 2013ndash2017 timespan were averaged every 12 steps to

deliver year forecasts of TCRC benchmark levels On theother hand forecasts obtained by MC simulations takingannual steps for the same period were considered as yearforecasts for TCRC annual benchmark levels without theneed for extra transformation Besides LES model forecastswere calculated at different time horizons to be able tocompare the short-term and long-term predictive accuracyLES forecasts were obtained for one-year-ahead hence usingknown values of TCRC from 2004 to 2016 for two yearsahead stretching the calibrating dataset from 2004 to 2015for five-year-ahead thus using the same input dataset as forthe GBM and OUP models from 2004 to 2012

Finally for every forecast path obtained we have calcu-lated the average of the squares of the errors with respect tothe observed valuesMSE the average distance of a forecast tothe observed meanMAD the average deviation of a forecastfrom observed values MAPE and the square root of MSERMSE (see Appendix C for error calculations) The results ofMSE MAD MAPE and RMSE calculated for each forecastpathwere averaged by the total number of simulations carriedout for each case Averaged values of error measures of allsimulated paths MSE MAD MAPE and RMSE for bothannual-step forecasts and monthly step forecasts have beenused for cross-comparison between models to test predictiveability at every step size possible

Also to test each modelrsquos short-term forecasting capacityagainst its long-term forecasting capacity one-year-aheadforecast errors of the LES model were compared with theerrors from the last year of the GBM and OUP simulatedpaths Two-year-ahead forecast errors of LES models werecomparedwith the average errors of the last two years ofGBMand OUP and five-year-ahead forecast errors of LES modelswere compared with the overall average of errors of the GBMand OUP forecasts

4 Analysis of Results

Monte Carlo simulations of GBM and OUP models render1000 different possible paths for TC and RC respectivelyat each time step size considered Accuracy errors for bothannual time steps and averaged monthly time steps for bothGBM and OUP forecasts were first compared to determinethe most accurate time step size for each model In additionthe LES model outcome for both TC and RC at differenttimeframes was also calculated and measures of errors for allthe three alternative models proposed at an optimum timestep were finally compared

41 GBM Forecast The first 20 of 1000 Monte Carlo pathsfor the GBM model with a monthly step size using Matlabsoftware may be seen in Figure 2 for both the TC and the RClevels compared to their averaged paths for every 12 valuesobtainedThe tendency for GBM forecasts to steeply increaseover time is easily observable in the nonaveraged monthlystep paths shown

The average values of error of all 1000MC paths obtainedthrough simulation for averaged monthly step and annual-step forecasts are shown in Table 3 for both TC and RC

Complexity 7

Table 3 Average of main measures of error for GBM after 1000 MC simulations from 2013 to 2017

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC Averaged Monthly Steps 560x103 4698 050 5638TC Annual Steps 520x103 4902 052 5857RC Averaged Monthly Steps 5874 455 048 546RC Annual Steps 5085 491 052 584

Table 4 Number of paths for which measures of error are higher for monthly steps than for annual steps in GBM

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC 4571000 4551000 4531000 4531000RC 4391000 4301000 4291000 4291000

discounts over the period from 2013 to 2017 which maylead to preliminary conclusions in terms of an accuracycomparison between averaged monthly steps and annualsteps

However a more exhaustive analysis is shown in Table 4where the number of times the values of each error measureare higher for monthly steps is expressed over the totalnumber of MC simulations carried out The results indicatethat better values of error are reached the majority of timesfor averagedmonthly step simulations rather than for straightannual ones

In contrast short-term accuracy was also evaluated byanalysing the error measures of one-year-ahead forecasts(2013) in Table 5 and of two-year-ahead forecasts (2013ndash2014)in Table 6 The results indicate as one may expect thataccuracy decreases as the forecasted horizon is widened withthe accuracy of averaged monthly step forecasts remaininghigher than annual ones as found above for the five-yearforecasting horizon

42 OUP Forecast Long-term levels for TC and RC 120583 thespeed of reversion 120582 and the volatility of the process 120590 arethe parameters determined at themodel calibration stage (seeAppendix B for the model calibration explanation) whichdefine the behaviour of the OUP shown in Table 7 Thecalibration was done prior to the Monte Carlo simulationfor both TC and RC with each step size using availablehistorical data from 2004 to 2012 The OUP model was fittedwith the corresponding parameters for each case upon MCsimulation

Figure 3 shows theMeanReversionMCestimations of theTCRC benchmark values from 2013 to 2017 using monthlystepsThemonthly forecastswere rendered from January 2012through to December 2016 and averaged every twelve valuesto deliver a benchmark forecast for each year The averagedresults can be comparable to actual data as well as to theannual Monte Carlo simulations following Mean ReversionThe lower-side figures show these yearly averaged monthlystep simulation outcomes which clearly move around a dash-dotted red line indicating the long-term run levels for TCRCto which they tend to revert

The accuracy of monthly steps against annual steps forthe TCRC benchmark levels forecast was also tested by

determining the number of simulations for which averageerror measures became higher Table 8 shows the numberof times monthly simulations have been less accurate thanannual simulations for five-year-ahead OUP forecasting bycomparing the four measures of errors proposed The resultsindicate that only 25-32 of the simulations drew a higheraverage error which clearly results in a better predictiveaccuracy for monthly step forecasting of TCRC annualbenchmark levels

The averaged measures of errors obtained after the MCsimulations of the OUP model for both averaged monthlysteps and annual steps giving TCRC benchmark forecastsfrom 2013 to 2017 are shown in Table 9

The error levels of the MC simulations shown in Table 9point towards a higher prediction accuracy of averagedmonthly step forecasts of the OUP Model yielding anaveraged MAPE value that is 129 lower for TC and RC 5-step-ahead forecasts In regard to MAPE values for monthlysteps only 266 of the simulations rise above annual MCsimulations for TC and 25 for RC 5-step-ahead forecastswhich further underpins the greater accuracy of thisOUP set-up for TCRC level forecasts A significant lower probabilityof higher error levels for TCRC forecasts with monthly MCOUP simulations is reached for the other measures providedIn addition short-term and long-term prediction accuracywere tested by comparing errors of forecasts for five-year-ahead error measures in Table 10 one-year-ahead in Table 11and two-year-ahead in Table 12

With a closer forecasting horizon errormeasures show animprovement of forecasting accuracy when average monthlysteps are used rather than annual ones For instance theMAPE values for 2013 forecast for TC are 68 lower foraveraged monthly steps than for annual steps and alsoMAPE for 2013ndash2014 were 30 lower for both TC and RCforecasts Similarly better values of error are achieved for theother measures for averaged monthly short-term forecaststhan in other scenarios In addition as expected accuracyis increased for closer forecasting horizons where the levelof errors shown above becomes lower as the deviation offorecasts is trimmed with short-term predictions

43 LES Forecast In contrast to GBM and OUP the LESmodel lacks any stochastic component so nondeterministic

8 Complexity

Table 5 Average for main measures of error for GBM after 1000 MC simulations for 2013

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC Averaged Monthly Steps 33658 1455 021 1455TC Annual Steps 69381 2081 030 2081RC Averaged Monthly Steps 297 138 020 138RC Annual Steps 657 205 029 205

Table 6 Average for main measures of error for GBM after 1000 MC simulations for 2013-2014

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC Averaged Monthly Steps 99492 2187 031 2435TC Annual Steps 133x103 2833 040 3104RC Averaged Monthly Steps 1035 240 034 271RC Annual Steps 1313 284 034 311

Table 7 OUP parameters obtained in the calibration process

120583 120582 120590TCMonthly 6345 4792x105 2534TC Annual 6345 2974 1308RC Monthly 635 4763x105 2519RC Annual 635 2972 1305

methods such as theMonte Carlo are not required to obtain aforecast Nonetheless the LES model relies on two smoothingconstants which must be properly set in order to deliveraccurate predictions hence the values of the smoothingconstants were first optimised (see Appendix B for LESModel fitting and smoothing constants optimisation) Theoptimisation was carried out for one-year-ahead forecaststwo-year-ahead forecasts and five-year-ahead forecasts byminimising the values of MSE for both TC and RC Thedifferent values used for smoothing constants as well asthe initial values for level and trend obtained by the linearregression of the available dataset from 2004 through to 2012are shown in Table 12

Compared one-year-ahead two-year-ahead and five-year-ahead LES forecasts for TC and RC are shown inFigure 4 clearly indicating a stronger accuracy for shorter-term forecasts as the observed and forecasted plotted linesoverlap

Minimumvalues for errormeasures achieved through theLESmodel parameter optimisation are shown in Table 13Thevalues obtained confirm the strong accuracy for shorter-termforecasts of TCRC seen in Figure 4

5 Discussion and Model Comparison

The forecasted values of TCRC benchmark levels couldeventually be applied to broader valuation models for copperconcentrates and their trading activities as well as to thecopper smeltersrsquo revenue stream thus the importance ofdelivering as accurate a prediction as possible in relation tothese discounts to make any future application possible Eachof the models presented may be a feasible method on its own

with eventual later adaptations to forecasting future valuesof benchmark TCRC Nonetheless the accuracy of thesemodels as they have been used in this work requires firstlya comparison to determine whether any of them could bea good standalone technique and secondly to test whethera combination of two or more of them would deliver moreprecise results

When comparing the different error measures obtainedfor all the three models it is clearly established that resultsfor a randomly chosen simulation of GBM or OUP wouldbe more likely to be more precise had a monthly step beenused to deliver annual forecasts instead of an annual-step sizeIn contrast average error measures for the entire populationof simulations with each step size employed showing thatmonthly step simulations for GBM and OUP models arealways more accurate than straight annual-step forecastswhen a shorter time horizon one or two-year-ahead is takeninto consideration However GBM presents a higher levelof forecasting accuracy when the average for error measuresof all simulations is analysed employing annual steps forlong-term horizons whereas OUP averaged monthly stepforecasts remain more accurate when predicting long-termhorizons Table 14 shows the error improvement for the aver-aged monthly step forecasts of each model Negative valuesindicate that better levels of error averages have been found instraight annual forecasts than for monthly step simulations

Considering the best results for each model and com-paring their corresponding error measures we can opt forthe best technique to employ among the three proposed inthis paper Hence GBM delivers the most accurate one-yearforecast when averaging the next twelve-month predictionsfor TCRC values as does the MR-OUP model Table 15shows best error measures for one-year-ahead forecasts forGBM OUP and LES models

Unarguably the LES model generates minimal errormeasures for one-year-ahead forecasts significantly less thanthe other models employed A similar situation is found fortwo-year-ahead forecasts whereminimumerrormeasures arealso delivered by the LES model shown in Table 16

Finally accuracy measures for five-year-ahead forecastsof the GBM model might result in somewhat contradictory

Complexity 9

Table 8 Number of paths for whichmeasures of error are higher for monthly steps than for annual steps inMR-OUP 5-StepsMC simulation

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC 3111000 2831000 2661000 2661000RC 3161000 2811000 2501000 2501000

Table 9 Average for main measures of error for MR-OUP after 1000 MC simulations 2013-2017

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC Averaged Monthly Steps 84254 2614 027 2895TC Annual Steps 114x103 2931 031 3325RC Averaged Monthly Steps 840 261 027 289RC Annual Steps 1148 294 031 333

Table 10 Average for main measures of error for MR-OUP after 1000 MC simulations 2013

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC Averaged Monthly Steps 3914 522 007 522TC Annual Steps 36643 1555 022 1555RC Averaged Monthly Steps 039 051 007 051RC Annual Steps 363 154 022 154

Table 11 Average for main measures of error for MR-OUP after 1000 MC simulations 2013-2014

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC Averaged Monthly Steps 37165 1567 018 1900TC Annual Steps 68244 2185 026 2324RC Averaged Monthly Steps 376 157 018 191RC Annual Steps 689 219 026 243

Table 12 LES Model Optimised Parameters

L0 T0 120572 120573TC 1 year 713611 -15833 -02372 01598RC 1 year 71333 -01567 -02368 01591TC 2 years 713611 -15833 -1477x10minus4 7774226RC 2 years 71333 -01567 -1448x10minus4 7898336TC 5 years 713611 -15833 -02813 01880RC 5 years 71333 -01567 -02808 01880

Table 13 LES error measures for different steps-ahead forecasts

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC 1 year 10746x10minus5 00033 46831x10minus5 00033RC 1 year 51462x10minus8 22685x10minus4 32408x10minus5 22685x10minus4

TC 2 years 189247 97275 01057 137567RC 2 years 18977 09741 01059 13776TC 5 years 1775531 78881 00951 108422RC 5 years 11759 07889 00952 10844

terms for MSE reaching better values for annual steps thanfor averaged monthly steps while the other figures do betteron averagedmonthly steps Addressing the definition ofMSEthis includes the variance of the estimator as well as its biasbeing equal to its variance in the case of unbiased estimators

Therefore MSE measures the quality of the estimator butalso magnifies estimator deviations from actual values sinceboth positive and negative values are squared and averagedIn contrast RMSE is calculated as the square root of MSEand following the previous analogy stands for the standard

10 Complexity

Table 14 Error Average Improvement for averaged monthly steps before annual steps

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864GBM TCRC 1 Year 53265653 32723587 33333548 32723587GBM TCRC 2 Years 26952635 25751597 2619286 24341369GBM TCRC 5 Years -1173-1104 873709 926769 747565OUP TCRC 1 Year 88088708 63716250 62916319 63686224OUP TCRC 2 Years 46284421 27712617 29993075 22162075OUP TCRC 5 Years 26022553 1093997 13121218 13061269

Table 15 TCRC best error measures for 2013 forecasts after 1000 MC simulations

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864GBM TCRC 1 Yearlowast 33111336 1425143 020020 1425143OUP TCRC 1 Yearlowast 4105042 536054 0076600777 536054LES TCRC 1 Year 107x10minus5514x10minus8 0003226x10minus4 468x10minus5324x10minus5 0003226x10minus4

lowastAveraged monthly steps

Table 16 TCRC Best error measures for 2013ndash2014 forecasts after 1000 MC simulations

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864GBM TCRC 2 Stepslowast 103x1031006 2200242 031034 2447271OUP TCRC 2 Stepslowast 37331376 1573158 0180201813 1900191LES TCRC 2 Steps 18924189 972097 0105701059 1375137lowastAveraged monthly steps

deviation of the estimator if MSE were considered to be thevarianceThough RMSE overreacts when high values of MSEare reached it is less prone to this than MSE since it iscalculated as its squared root thus not accounting for largeerrors as disproportionately as MSE does Furthermore aswe have compared an average of 1000 measures of errorscorresponding to each MC simulation performed the valuesobtained for average RMSE stay below the square root ofaverage MSE which indicates that some of these dispropor-tionate error measures are to some extent distorting thelatter Hence RMSE average values point towards a higheraccuracy for GBM five-year forecasts with averaged monthlysteps which is further endorsed by the average values ofMAD and MAPE thus being the one used for comparisonwith the other two models as shown in Table 17

The final comparison clearly shows how the LES modeloutperforms the other two at all average measures providedfollowed by the OUP model in accuracy although the lattermore than doubles the average MAPE value for LES

The results of simulations indicate that measures of errorstend to either differ slightly or not at all for either forecastsof any timeframe A coherent value with the 101 relation canthen be given with close to the same level of accuracy bymultiplying RC forecasts or dividing TC ones by 10

6 Conclusions

Copper TCRC are a keystone for pricing copper concen-trates which are the actual feedstock for copper smeltersThe potential evolution of TCRC is a question of botheconomic and technical significance forminers as their value

decreases the potential final selling price of concentratesAdditionally copper minersrsquo revenues are more narrowlyrelated to the market price of copper as well as to othertechnical factors such as ore dilution or the grade of theconcentrates produced Smelters on the contrary are hugelyaffected by the discount which they succeed in getting whenpurchasing the concentrates since that makes up the largestpart of their gross revenue besides other secondary sourcesIn addition eventual differences between TCRC may givecommodity traders ludicrous arbitrage opportunities Alsodifferences between short- and long-termTCRC agreementsoffer arbitrage opportunities for traders hence comprisinga part of their revenue in the copper concentrate tradingbusiness as well copper price fluctuations and the capacityto make economically optimum copper blends

As far as we are aware no rigorous research has beencarried out on the behaviour of these discounts Based onhistorical data on TCRC agreed upon in the LME CopperWeek from 2004 to 2017 three potentially suitable forecastingmodels for TCRC annual benchmark values have beencompared through four measures of forecasting accuracyat different horizons These models were chosen firstlydue to their broad implementation and proven capacityin commodity prices forecasting that they all share andsecondly because of the core differences in terms of pricebehaviour with each other

Focusing on the MAPE values achieved those obtainedby the LES model when TC and RC are treated indepen-dently have been significantly lower than for the rest ofthe models Indeed one-year-ahead MAPE measures for TCvalues for the GBM model (20) almost triple those of

Complexity 11

2004 2006 2008 2010 2012 2014405060708090

100

TC (U

SDM

T)

LES TC 1 Step Ahead 2013

2004 2006 2008 2010 2012 2014

LES TC 2 Steps Ahead 2013-2014

2004 2006 2008 2010 2012 2014 2016 2018

LES TC 5 Steps Ahead 2013-2017

2004 2006 2008 2010 2012 2014

LES RC 1 Step Ahead 2013

2004 2006 2008 2010 2012 2014456789

10LES RC 2 Steps Ahead 2013-2014

Observed Forecast

2004 2006 2008 2010 2012 2014 2016 2018

LES RC 5 Steps Ahead 2013-2017

405060708090

100110

405060708090

100

456789

10

RC (U

Sclb

)

456789

1011

Figure 4 One-step-ahead forecasts (2013) two-step-ahead forecasts (2013-2014) and five-step-ahead forecasts (2013-2017) for TC and RCusing the Linear Exponential Smoothing model (LES)

Table 17 TCRC Best error measures for 2013ndash2017 forecasts after 1000 MC simulations

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864GBM TCRC 5 Stepslowast 600x1036165 4651459 049048 5571551OUP TCRC 5 Stepslowast 84334843 2616262 0271502719 2896289LES TCRC 5 Steps 17755117 788078 0095100952 1084108lowastAveraged monthly steps

the OUP model (766) in contrast with the significantlylower values from the LES model (00046) This gaptends to be narrowed when TCRC values are forecastedat longer horizons when most measures of error becomemore even The GBM and OUP models have proven todeliver better accuracy performance when the TCRC valuesare projected monthly and then averaged to obtain annualbenchmark forecasts Even so the LES model remains themost accurate of all with MAPE values of 10 at two-year-ahead forecasts with 18 and 31 for TC for OUP and GBMrespectively

Finally despite TC and RC being two independentdiscounts applied to copper concentrates they are bothset jointly with an often 101 relation as our data revealsThis relation also transcends to simulation results and errormeasures hence showing a negligible discrepancy betweenthe independent forecasting of TC and RC or the jointforecasting of both values keeping the 101 relationThis is forinstance the case of the five-year-ahead OUP MAPE values(0271502719) which were obtained without observing the101 relation in the data A similar level of discrepancy wasobtained at any horizon with any model which indicates thatboth values could be forecasted with the same accuracy usingthe selected model with any of them and then applying the101 relation

Our findings thus suggest that both at short and atlong-term horizons TCRC annual benchmark levels tendto exhibit a pattern which is best fit by an LES model Thisindicates that these critical discounts for the copper tradingbusiness do maintain a certain dependency on past valuesThis would also suggest the existence of cyclical patterns incopper TCRC possibly driven by many of the same marketfactors that move the price of copper

This work contributes by delivering a formal tool forsmelters or miners to make accurate forecasts of TCRCbenchmark levels The level of errors attained indicatesthe LES model may be a valid model to forecast thesecrucial discounts for the copper market In addition ourwork further contributes to helping market participants toproject the price of concentrates with an acceptable degree ofuncertainty as now they may include a fundamental elementfor their estimation This would enable them to optimise theway they produce or process these copper concentrates Alsothe precise knowledge of these discountsrsquo expected behaviourcontributes to letting miners traders and smelters aliketake the maximum advantage from the copper concentratetrading agreements that they are part of As an additionalcontribution this work may well be applied to gold or silverRC which are relevant deduction concentrates when thesehave a significant amount of gold or silver

12 Complexity

Once TCRC annual benchmark levels are able to beforecasted with a certain level of accuracy future researchshould go further into this research through the explorationof the potential impact that other market factors may have onthese discounts

As a limitation of this research we should point out thetimespan of the data considered compared to those of otherforecasting works on commodity prices for example whichuse broader timespans For our case we have consideredthe maximum available sufficiently reliable data on TCRCbenchmark levels starting back in 2004 as there is noreliable data beyond this year Also as another limitationwe have used four measures of error which are among themost frequently used to compare the accuracy of differentmodels However other measures could have been used at anindividual level to test each modelrsquos accuracy

Appendix

A Models

A1 Geometric Brownian Motion (GBM) GBM can bewritten as a generalisation of a Wiener (continuous time-stochasticMarkov process with independent increments andwhose changes over any infinite interval of time are normallydistributed with a variance that increases linearly with thetime interval [22]) process

119889119909 = 120572119909119889119905 + 120590119909119889119911 (A1)

where according to Marathe and Ryan [30] the first termis known as the Expected Value whereas the second is theStochastic component with 120572 being the drift parameter and120590 the volatility of the process Also dz is the Wiener processwhich induces the abovementioned stochastic behaviour inthe model

119889119911 = isin119905radic119889119905 997888rarr isin119905 sim 119873 (0 1) (A2)

TheGBMmodel can be expressed in discrete terms accordingto

119909 = 119909119905 minus 119909119905minus1 = 120572119909119905minus1119905 + 120590119909119905minus1120598radic119905 (A3)

In GBM percentage changes in x 119909119909 are normally dis-tributed thus absolute changes in x 119909 are lognormallydistributed Also the expected value and variance for 119909(119905) are

E [119909 (119905)] = 1199090119890120572119905 (A4)

var [119909 (119905)] = 119909201198902120572119905 (1198901205902119905 minus 1) (A5)

A2 Orstein-Uhlenbeck Process (OUP) TheOUPprocess wasfirst defined by Uhlenbeck and Orstein [31] as an alternativeto the regular Brownian Motion to model the velocity ofthe diffusion movement of a particle that accounts for itslosses due to friction with other particles The OUP processcan be regarded as a modification of Brownian Motion incontinuous time where its properties have been changed

(Stationary Gaussian Markov and stochastic process) [32]These modifications cause the process to move towards acentral position with a stronger attraction the further it isfrom this position As mentioned above the OUP is usuallyemployed to model commodity prices and is the simplestversion of amean-reverting process [22]

119889119878 = 120582 (120583 minus 119878) 119889119905 + 120590119889119882119905 (A6)

where S is the level of prices 120583 the long-term average towhich prices tend to revert and 120582 the speed of reversionAdditionally in a similar fashion to that of the GBM 120590 isthe volatility of the process and 119889119882119905 is a Wiener processwith an identical definition However in contrast to GBMtime intervals in OUP are not independent since differencesbetween current levels of prices S and long-term averageprices 120583 make the expected change in prices dS more likelyeither positive or negative

The discrete version of the model can be expressed asfollows

119878119905 = 120583 (1 minus 119890minus120582Δ119905) + 119890minus120582Δ119905119878119905minus1 + 120590radic1 minus 119890minus2120582Δ1199052120582 119889119882119905 (A7)

where the expected value for 119909(119905) and the variance for (119909(119905) minus120583) areE [119909 (119905)] = 120583 + (1199090 minus 120583) 119890minus120582119905 (A8)

var [119909 (119905) minus 120583] = 12059022120582 (1 minus 119890minus2120582119905) (A9)

It can be derived from previous equations that as timeincreases prices will tend to long-term average levels 120583 Inaddition with large time spans if the speed of reversion 120582becomes high variance tends to 0 On the other hand if thespeed of reversion is 0 then var[119909(119905)] 997888rarr 1205902119905 making theprocess a simple Brownian Motion

A3Holtrsquos Linear Exponential Smoothing (LES) Linear Expo-nential Smoothing models are capable of considering bothlevels and trends at every instant assigning higher weights inthe overall calculation to values closer to the present than toolder ones LES models carry that out by constantly updatinglocal estimations of levels and trends with the intervention ofone or two smoothing constants which enable the models todampen older value effects Although it is possible to employa single smoothing constant for both the level and the trendknown as Brownrsquos LES to use two one for each known asHoltrsquos LES is usually preferred since Brownrsquos LES tends torender estimations of the trend ldquounstablerdquo as suggested byauthors such as Nau [33] Holtrsquos LES model comes definedby the level trend and forecast updating equations each ofthese expressed as follows respectively

119871 119905 = 120572119884119905 + (1 minus 120572) (119871 119905minus1 + 119879119905minus1) (A10)

119879119905 = 120573 (119871 119905 minus 119871 119905minus1) + (1 minus 120573) 119879119905minus1 (A11)

119905+119896 = 119871 119905 + 119896119879119905 (A12)

Complexity 13

With 120572 being the first smoothing constant for the levelsand 120573 the second smoothing constant for the trend Highervalues for the smoothing constants imply that either levels ortrends are changing rapidly over time whereas lower valuesimply the contrary Hence the higher the constant the moreuncertain the future is believed to be

B Model Calibration

Each model has been calibrated individually using two setsof data containing TC and RC levels respectively from 2004to 2012 The available dataset is comprised of data from 2004to 2017 which make the calibration data approximately 23 ofthe total

B1 GBM Increments in the logarithm of variable x aredistributed as follows

(ln119909) sim 119873((120572 minus 12059022 ) 119905 120590119905) (B1)

Hence if m is defined as the sample mean of the differenceof the natural logarithm of the time series for TCRC levelsconsidered for the calibration and n as the number ofincrements of the series considered with n=9

119898 = 1119899119899sum119905=1

(ln119909119905 minus ln119909119905minus1) (B2)

119904 = radic 1119899 minus 1119899sum119894=1

(ln119909119905 minus ln119909119905minus1 minus 119898)2 (B3)

119898 = 120572 minus 12059022 (B4)

119904 = 120590 (B5)

B2 OUP The OUP process is an AR1 process [22] whoseresolution is well presented by Woolridge [34] using OLStechniques fitting the following

119910119905 = 119886 + 119887119910119905minus1 + 120576119905 (B6)

Hence the estimators for the parameters of the OUP modelare obtained by OLS for both TC and RC levels indepen-dently

= minus ln 119887Δ119905 (B7)

120583 = 1198861 minus 119887 (B8)

= 120576119905radic 2 ln (1 + 119887)(1 + 119887)2 minus 1 (B9)

B3 LES A linear regression is conducted on the inputdataset available to find the starting parameters for the LES

model the initial Level 1198710 and the initial value of thetrend 1198790 irrespective of TC values and RC values Hereas recommended by Gardner [35] the use of OLS is highlyadvisable due to the erratic behaviour shown by trends inthe historic data so the obtaining of negative values of 1198780 isprevented Linear regression fulfils the following

119884119905 = 119886119905 + 119887 (B10)

1198710 = 119887 (B11)

1198790 = 119886 (B12)

By fixing the two smoothing constants the values for theforecasts 119905+119896 can be calculated at each step using the modelequations There are multiple references in the literature onwhat the optimum range for each smoothing constant isGardner [35] speaks of setting moderate values for bothparameter less than 03 to obtain the best results Examplespointing out the same may be found in Brown [36] Coutie[37] Harrison [38] and Montgomery and Johnson [39]Also for many applications Makridakis and Hibon [20] andChatfield [40] found that parameter values should fall withinthe range of 03-1 On the other hand McClain and Thomas[41] provided a condition of stability for the nonseasonal LESmodel given by

0 lt 120572 lt 20 lt 120573 lt 4 minus 2120572120572

(B13)

Also the largest possible value of 120572 that allows the avoidanceof areas of oscillation is proposed by McClain and Thomas[41] and McClain [42]

120572 lt 4120573(1 + 120573)2 (B14)

However according to Gardner there is no tangible proofthat this value improves accuracy in any form Nonethelesswe have opted to follow what Nau [33] refers to as ldquotheusual wayrdquo namely minimising the Mean Squared Error(MSE) of the one-step-ahead forecast of TCRC for eachinput data series previously mentioned To do so Matlabrsquosfminsearch function has been used with function and variabletolerance levels of 1x10minus4 as well as a set maximum numberof function iterations and function evaluations of 1x106 tolimit computing resources In Table 18 the actual number ofnecessary iterations to obtain optimum values for smoothingconstants is shown As can be seen the criteria are wellbeyond the final results which ensured that an optimumsolution was reached with assumable computing usage (thesimulation required less than one minute) and with a highdegree of certainty

14 Complexity

Table 18 Iterations on LES smoothing constants optimisation

Iterations Function EvaluationsTC 1-Step 36 40TC 2-Steps 249945 454211TC 5-Steps 40 77RC 1-Step 32 62RC 2-Steps 254388 462226RC 5-Steps 34 66

C Error Calculations

C1 Mean Square Error (MSE)

119872119878119864 = 1119899119899sum119894=1

(119894 minus 119884119894)2 (C1)

where 119894 are the forecasted values and 119884119894 those observedC2 Mean Absolute Deviation (MAD)

119872119860119863 = 1119899119899sum119894=1

10038161003816100381610038161003816119894 minus 11988410038161003816100381610038161003816 (C2)

where 119894 are the forecasted values and 119884 the average value ofall the observations

C3 Mean Absolute Percentage Error (MAPE)

119872119860119875119864 = 1119899119899sum119894=1

10038161003816100381610038161003816119894 minus 11988411989410038161003816100381610038161003816119884119894 (C3)

The above formula is expressed in parts-per-one and is theone used in the calculations conducted here Hence multi-plying the result by 100 would deliver percentage outcomes

C4 Root Mean Square Error (RMSE)

119877119872119878119864 = radic119872119878119864 = radic 1119899119899sum119894=1

(119894 minus 119884119894)2 (C4)

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Supplementary Materials

The Matlab codes developed to carry out the simulationsfor each of the models proposed as well as the datasetsused and the Matlab workspaces with the simulationsoutcomes as they have been reflected in this paper areprovided in separate folders for each model GBM foldercontains the code to make the simulations for the Geo-metric Brownian Motion model ldquoGBMmrdquo as well as aseparate script which allows determining the total numberof monthly step simulations for which errors levels havebeen lower than annual-step simulations ldquoErrCoutermrdquoThe datasets used are included in two excel files ldquoTCxlsxrdquoand ldquoRCxlsxrdquo ldquoGBM OK MonthlyStepsmatrdquo is the Matlabworkspace containing the outcome of monthly step simu-lations whereas ldquoGBM OK AnnualStepsmatrdquo contains theoutcome for annual-step simulations Also ldquoGBM OKmatrdquois the workspace to be loaded prior to performing GBMsimulations MR folder contains the code file to carryout the simulations for the Orstein-Uhlenbeck ProcessldquoMRmrdquo as well as the separate script to compare errorsof monthly step simulations with annual-step simula-tions ldquoErrCountermrdquo ldquoTCxlsxrdquo and ldquoRCxlsxrdquo containthe TCRC benchmark levels from 2004 to 2017 Finallymonthly step simulationsrsquo outcome has been saved inthe workspace ldquoMR OK MonthlyStepsmatrdquo while annual-step simulationsrsquo outcome has been saved in the Matlabworkspace ldquoMR OK AnnualStepsmatrdquo Also ldquoMR OKmatrdquois the workspace to be loaded prior to performing MRsimulations LES folder contains the code file to carry outthe calculations necessary to obtain the LES model forecastsldquoLESmrdquo Three separate Matlab functions are includedldquomapeLESmrdquo ldquomapeLES1mrdquo and ldquomapeLES2mrdquo whichdefine the calculation of the MAPE for the LESrsquo five-year-ahead one-year-ahead and two-year-ahead forecastsAlso ldquoLES OKmatrdquo is the workspace to be loaded prior toperforming LES simulations Finally graphs of each modelhave been included in separate files as well in a subfoldernamed ldquoGraphsrdquo within each of the modelsrsquo folder Figuresare included in Matlabrsquos format (fig) and TIFF format TIFFfigures have been generated in both colours and black andwhite for easier visualization (Supplementary Materials)

References

[1] United Nations International Trade Statistics Yearbook vol 22016

[2] International Copper StudyGroupTheWorld Copper Factbook2017

[3] K M Johnson J M Hammarstrom M L Zientek and CL Dicken ldquoEstimate of undiscovered copper resources of theworld 2013rdquo US Geological Survey 2017

[4] M E Schlesinger and W G Davenport Extractive Metallurgyof Copper Elsevier 5th edition 2011

[5] S Gloser M Soulier and L A T Espinoza ldquoDynamic analysisof global copper flows Global stocks postconsumer materialflows recycling indicators and uncertainty evaluationrdquo Envi-ronmental Science amp Technology vol 47 no 12 pp 6564ndash65722013

Complexity 7

Table 3 Average of main measures of error for GBM after 1000 MC simulations from 2013 to 2017

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC Averaged Monthly Steps 560x103 4698 050 5638TC Annual Steps 520x103 4902 052 5857RC Averaged Monthly Steps 5874 455 048 546RC Annual Steps 5085 491 052 584

Table 4 Number of paths for which measures of error are higher for monthly steps than for annual steps in GBM

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC 4571000 4551000 4531000 4531000RC 4391000 4301000 4291000 4291000

discounts over the period from 2013 to 2017 which maylead to preliminary conclusions in terms of an accuracycomparison between averaged monthly steps and annualsteps

However a more exhaustive analysis is shown in Table 4where the number of times the values of each error measureare higher for monthly steps is expressed over the totalnumber of MC simulations carried out The results indicatethat better values of error are reached the majority of timesfor averagedmonthly step simulations rather than for straightannual ones

In contrast short-term accuracy was also evaluated byanalysing the error measures of one-year-ahead forecasts(2013) in Table 5 and of two-year-ahead forecasts (2013ndash2014)in Table 6 The results indicate as one may expect thataccuracy decreases as the forecasted horizon is widened withthe accuracy of averaged monthly step forecasts remaininghigher than annual ones as found above for the five-yearforecasting horizon

42 OUP Forecast Long-term levels for TC and RC 120583 thespeed of reversion 120582 and the volatility of the process 120590 arethe parameters determined at themodel calibration stage (seeAppendix B for the model calibration explanation) whichdefine the behaviour of the OUP shown in Table 7 Thecalibration was done prior to the Monte Carlo simulationfor both TC and RC with each step size using availablehistorical data from 2004 to 2012 The OUP model was fittedwith the corresponding parameters for each case upon MCsimulation

Figure 3 shows theMeanReversionMCestimations of theTCRC benchmark values from 2013 to 2017 using monthlystepsThemonthly forecastswere rendered from January 2012through to December 2016 and averaged every twelve valuesto deliver a benchmark forecast for each year The averagedresults can be comparable to actual data as well as to theannual Monte Carlo simulations following Mean ReversionThe lower-side figures show these yearly averaged monthlystep simulation outcomes which clearly move around a dash-dotted red line indicating the long-term run levels for TCRCto which they tend to revert

The accuracy of monthly steps against annual steps forthe TCRC benchmark levels forecast was also tested by

determining the number of simulations for which averageerror measures became higher Table 8 shows the numberof times monthly simulations have been less accurate thanannual simulations for five-year-ahead OUP forecasting bycomparing the four measures of errors proposed The resultsindicate that only 25-32 of the simulations drew a higheraverage error which clearly results in a better predictiveaccuracy for monthly step forecasting of TCRC annualbenchmark levels

The averaged measures of errors obtained after the MCsimulations of the OUP model for both averaged monthlysteps and annual steps giving TCRC benchmark forecastsfrom 2013 to 2017 are shown in Table 9

The error levels of the MC simulations shown in Table 9point towards a higher prediction accuracy of averagedmonthly step forecasts of the OUP Model yielding anaveraged MAPE value that is 129 lower for TC and RC 5-step-ahead forecasts In regard to MAPE values for monthlysteps only 266 of the simulations rise above annual MCsimulations for TC and 25 for RC 5-step-ahead forecastswhich further underpins the greater accuracy of thisOUP set-up for TCRC level forecasts A significant lower probabilityof higher error levels for TCRC forecasts with monthly MCOUP simulations is reached for the other measures providedIn addition short-term and long-term prediction accuracywere tested by comparing errors of forecasts for five-year-ahead error measures in Table 10 one-year-ahead in Table 11and two-year-ahead in Table 12

With a closer forecasting horizon errormeasures show animprovement of forecasting accuracy when average monthlysteps are used rather than annual ones For instance theMAPE values for 2013 forecast for TC are 68 lower foraveraged monthly steps than for annual steps and alsoMAPE for 2013ndash2014 were 30 lower for both TC and RCforecasts Similarly better values of error are achieved for theother measures for averaged monthly short-term forecaststhan in other scenarios In addition as expected accuracyis increased for closer forecasting horizons where the levelof errors shown above becomes lower as the deviation offorecasts is trimmed with short-term predictions

43 LES Forecast In contrast to GBM and OUP the LESmodel lacks any stochastic component so nondeterministic

8 Complexity

Table 5 Average for main measures of error for GBM after 1000 MC simulations for 2013

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC Averaged Monthly Steps 33658 1455 021 1455TC Annual Steps 69381 2081 030 2081RC Averaged Monthly Steps 297 138 020 138RC Annual Steps 657 205 029 205

Table 6 Average for main measures of error for GBM after 1000 MC simulations for 2013-2014

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC Averaged Monthly Steps 99492 2187 031 2435TC Annual Steps 133x103 2833 040 3104RC Averaged Monthly Steps 1035 240 034 271RC Annual Steps 1313 284 034 311

Table 7 OUP parameters obtained in the calibration process

120583 120582 120590TCMonthly 6345 4792x105 2534TC Annual 6345 2974 1308RC Monthly 635 4763x105 2519RC Annual 635 2972 1305

methods such as theMonte Carlo are not required to obtain aforecast Nonetheless the LES model relies on two smoothingconstants which must be properly set in order to deliveraccurate predictions hence the values of the smoothingconstants were first optimised (see Appendix B for LESModel fitting and smoothing constants optimisation) Theoptimisation was carried out for one-year-ahead forecaststwo-year-ahead forecasts and five-year-ahead forecasts byminimising the values of MSE for both TC and RC Thedifferent values used for smoothing constants as well asthe initial values for level and trend obtained by the linearregression of the available dataset from 2004 through to 2012are shown in Table 12

Compared one-year-ahead two-year-ahead and five-year-ahead LES forecasts for TC and RC are shown inFigure 4 clearly indicating a stronger accuracy for shorter-term forecasts as the observed and forecasted plotted linesoverlap

Minimumvalues for errormeasures achieved through theLESmodel parameter optimisation are shown in Table 13Thevalues obtained confirm the strong accuracy for shorter-termforecasts of TCRC seen in Figure 4

5 Discussion and Model Comparison

The forecasted values of TCRC benchmark levels couldeventually be applied to broader valuation models for copperconcentrates and their trading activities as well as to thecopper smeltersrsquo revenue stream thus the importance ofdelivering as accurate a prediction as possible in relation tothese discounts to make any future application possible Eachof the models presented may be a feasible method on its own

with eventual later adaptations to forecasting future valuesof benchmark TCRC Nonetheless the accuracy of thesemodels as they have been used in this work requires firstlya comparison to determine whether any of them could bea good standalone technique and secondly to test whethera combination of two or more of them would deliver moreprecise results

When comparing the different error measures obtainedfor all the three models it is clearly established that resultsfor a randomly chosen simulation of GBM or OUP wouldbe more likely to be more precise had a monthly step beenused to deliver annual forecasts instead of an annual-step sizeIn contrast average error measures for the entire populationof simulations with each step size employed showing thatmonthly step simulations for GBM and OUP models arealways more accurate than straight annual-step forecastswhen a shorter time horizon one or two-year-ahead is takeninto consideration However GBM presents a higher levelof forecasting accuracy when the average for error measuresof all simulations is analysed employing annual steps forlong-term horizons whereas OUP averaged monthly stepforecasts remain more accurate when predicting long-termhorizons Table 14 shows the error improvement for the aver-aged monthly step forecasts of each model Negative valuesindicate that better levels of error averages have been found instraight annual forecasts than for monthly step simulations

Considering the best results for each model and com-paring their corresponding error measures we can opt forthe best technique to employ among the three proposed inthis paper Hence GBM delivers the most accurate one-yearforecast when averaging the next twelve-month predictionsfor TCRC values as does the MR-OUP model Table 15shows best error measures for one-year-ahead forecasts forGBM OUP and LES models

Unarguably the LES model generates minimal errormeasures for one-year-ahead forecasts significantly less thanthe other models employed A similar situation is found fortwo-year-ahead forecasts whereminimumerrormeasures arealso delivered by the LES model shown in Table 16

Finally accuracy measures for five-year-ahead forecastsof the GBM model might result in somewhat contradictory

Complexity 9

Table 8 Number of paths for whichmeasures of error are higher for monthly steps than for annual steps inMR-OUP 5-StepsMC simulation

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC 3111000 2831000 2661000 2661000RC 3161000 2811000 2501000 2501000

Table 9 Average for main measures of error for MR-OUP after 1000 MC simulations 2013-2017

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC Averaged Monthly Steps 84254 2614 027 2895TC Annual Steps 114x103 2931 031 3325RC Averaged Monthly Steps 840 261 027 289RC Annual Steps 1148 294 031 333

Table 10 Average for main measures of error for MR-OUP after 1000 MC simulations 2013

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC Averaged Monthly Steps 3914 522 007 522TC Annual Steps 36643 1555 022 1555RC Averaged Monthly Steps 039 051 007 051RC Annual Steps 363 154 022 154

Table 11 Average for main measures of error for MR-OUP after 1000 MC simulations 2013-2014

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC Averaged Monthly Steps 37165 1567 018 1900TC Annual Steps 68244 2185 026 2324RC Averaged Monthly Steps 376 157 018 191RC Annual Steps 689 219 026 243

Table 12 LES Model Optimised Parameters

L0 T0 120572 120573TC 1 year 713611 -15833 -02372 01598RC 1 year 71333 -01567 -02368 01591TC 2 years 713611 -15833 -1477x10minus4 7774226RC 2 years 71333 -01567 -1448x10minus4 7898336TC 5 years 713611 -15833 -02813 01880RC 5 years 71333 -01567 -02808 01880

Table 13 LES error measures for different steps-ahead forecasts

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC 1 year 10746x10minus5 00033 46831x10minus5 00033RC 1 year 51462x10minus8 22685x10minus4 32408x10minus5 22685x10minus4

TC 2 years 189247 97275 01057 137567RC 2 years 18977 09741 01059 13776TC 5 years 1775531 78881 00951 108422RC 5 years 11759 07889 00952 10844

terms for MSE reaching better values for annual steps thanfor averaged monthly steps while the other figures do betteron averagedmonthly steps Addressing the definition ofMSEthis includes the variance of the estimator as well as its biasbeing equal to its variance in the case of unbiased estimators

Therefore MSE measures the quality of the estimator butalso magnifies estimator deviations from actual values sinceboth positive and negative values are squared and averagedIn contrast RMSE is calculated as the square root of MSEand following the previous analogy stands for the standard

10 Complexity

Table 14 Error Average Improvement for averaged monthly steps before annual steps

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864GBM TCRC 1 Year 53265653 32723587 33333548 32723587GBM TCRC 2 Years 26952635 25751597 2619286 24341369GBM TCRC 5 Years -1173-1104 873709 926769 747565OUP TCRC 1 Year 88088708 63716250 62916319 63686224OUP TCRC 2 Years 46284421 27712617 29993075 22162075OUP TCRC 5 Years 26022553 1093997 13121218 13061269

Table 15 TCRC best error measures for 2013 forecasts after 1000 MC simulations

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864GBM TCRC 1 Yearlowast 33111336 1425143 020020 1425143OUP TCRC 1 Yearlowast 4105042 536054 0076600777 536054LES TCRC 1 Year 107x10minus5514x10minus8 0003226x10minus4 468x10minus5324x10minus5 0003226x10minus4

lowastAveraged monthly steps

Table 16 TCRC Best error measures for 2013ndash2014 forecasts after 1000 MC simulations

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864GBM TCRC 2 Stepslowast 103x1031006 2200242 031034 2447271OUP TCRC 2 Stepslowast 37331376 1573158 0180201813 1900191LES TCRC 2 Steps 18924189 972097 0105701059 1375137lowastAveraged monthly steps

deviation of the estimator if MSE were considered to be thevarianceThough RMSE overreacts when high values of MSEare reached it is less prone to this than MSE since it iscalculated as its squared root thus not accounting for largeerrors as disproportionately as MSE does Furthermore aswe have compared an average of 1000 measures of errorscorresponding to each MC simulation performed the valuesobtained for average RMSE stay below the square root ofaverage MSE which indicates that some of these dispropor-tionate error measures are to some extent distorting thelatter Hence RMSE average values point towards a higheraccuracy for GBM five-year forecasts with averaged monthlysteps which is further endorsed by the average values ofMAD and MAPE thus being the one used for comparisonwith the other two models as shown in Table 17

The final comparison clearly shows how the LES modeloutperforms the other two at all average measures providedfollowed by the OUP model in accuracy although the lattermore than doubles the average MAPE value for LES

The results of simulations indicate that measures of errorstend to either differ slightly or not at all for either forecastsof any timeframe A coherent value with the 101 relation canthen be given with close to the same level of accuracy bymultiplying RC forecasts or dividing TC ones by 10

6 Conclusions

Copper TCRC are a keystone for pricing copper concen-trates which are the actual feedstock for copper smeltersThe potential evolution of TCRC is a question of botheconomic and technical significance forminers as their value

decreases the potential final selling price of concentratesAdditionally copper minersrsquo revenues are more narrowlyrelated to the market price of copper as well as to othertechnical factors such as ore dilution or the grade of theconcentrates produced Smelters on the contrary are hugelyaffected by the discount which they succeed in getting whenpurchasing the concentrates since that makes up the largestpart of their gross revenue besides other secondary sourcesIn addition eventual differences between TCRC may givecommodity traders ludicrous arbitrage opportunities Alsodifferences between short- and long-termTCRC agreementsoffer arbitrage opportunities for traders hence comprisinga part of their revenue in the copper concentrate tradingbusiness as well copper price fluctuations and the capacityto make economically optimum copper blends

As far as we are aware no rigorous research has beencarried out on the behaviour of these discounts Based onhistorical data on TCRC agreed upon in the LME CopperWeek from 2004 to 2017 three potentially suitable forecastingmodels for TCRC annual benchmark values have beencompared through four measures of forecasting accuracyat different horizons These models were chosen firstlydue to their broad implementation and proven capacityin commodity prices forecasting that they all share andsecondly because of the core differences in terms of pricebehaviour with each other

Focusing on the MAPE values achieved those obtainedby the LES model when TC and RC are treated indepen-dently have been significantly lower than for the rest ofthe models Indeed one-year-ahead MAPE measures for TCvalues for the GBM model (20) almost triple those of

Complexity 11

2004 2006 2008 2010 2012 2014405060708090

100

TC (U

SDM

T)

LES TC 1 Step Ahead 2013

2004 2006 2008 2010 2012 2014

LES TC 2 Steps Ahead 2013-2014

2004 2006 2008 2010 2012 2014 2016 2018

LES TC 5 Steps Ahead 2013-2017

2004 2006 2008 2010 2012 2014

LES RC 1 Step Ahead 2013

2004 2006 2008 2010 2012 2014456789

10LES RC 2 Steps Ahead 2013-2014

Observed Forecast

2004 2006 2008 2010 2012 2014 2016 2018

LES RC 5 Steps Ahead 2013-2017

405060708090

100110

405060708090

100

456789

10

RC (U

Sclb

)

456789

1011

Figure 4 One-step-ahead forecasts (2013) two-step-ahead forecasts (2013-2014) and five-step-ahead forecasts (2013-2017) for TC and RCusing the Linear Exponential Smoothing model (LES)

Table 17 TCRC Best error measures for 2013ndash2017 forecasts after 1000 MC simulations

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864GBM TCRC 5 Stepslowast 600x1036165 4651459 049048 5571551OUP TCRC 5 Stepslowast 84334843 2616262 0271502719 2896289LES TCRC 5 Steps 17755117 788078 0095100952 1084108lowastAveraged monthly steps

the OUP model (766) in contrast with the significantlylower values from the LES model (00046) This gaptends to be narrowed when TCRC values are forecastedat longer horizons when most measures of error becomemore even The GBM and OUP models have proven todeliver better accuracy performance when the TCRC valuesare projected monthly and then averaged to obtain annualbenchmark forecasts Even so the LES model remains themost accurate of all with MAPE values of 10 at two-year-ahead forecasts with 18 and 31 for TC for OUP and GBMrespectively

Finally despite TC and RC being two independentdiscounts applied to copper concentrates they are bothset jointly with an often 101 relation as our data revealsThis relation also transcends to simulation results and errormeasures hence showing a negligible discrepancy betweenthe independent forecasting of TC and RC or the jointforecasting of both values keeping the 101 relationThis is forinstance the case of the five-year-ahead OUP MAPE values(0271502719) which were obtained without observing the101 relation in the data A similar level of discrepancy wasobtained at any horizon with any model which indicates thatboth values could be forecasted with the same accuracy usingthe selected model with any of them and then applying the101 relation

Our findings thus suggest that both at short and atlong-term horizons TCRC annual benchmark levels tendto exhibit a pattern which is best fit by an LES model Thisindicates that these critical discounts for the copper tradingbusiness do maintain a certain dependency on past valuesThis would also suggest the existence of cyclical patterns incopper TCRC possibly driven by many of the same marketfactors that move the price of copper

This work contributes by delivering a formal tool forsmelters or miners to make accurate forecasts of TCRCbenchmark levels The level of errors attained indicatesthe LES model may be a valid model to forecast thesecrucial discounts for the copper market In addition ourwork further contributes to helping market participants toproject the price of concentrates with an acceptable degree ofuncertainty as now they may include a fundamental elementfor their estimation This would enable them to optimise theway they produce or process these copper concentrates Alsothe precise knowledge of these discountsrsquo expected behaviourcontributes to letting miners traders and smelters aliketake the maximum advantage from the copper concentratetrading agreements that they are part of As an additionalcontribution this work may well be applied to gold or silverRC which are relevant deduction concentrates when thesehave a significant amount of gold or silver

12 Complexity

Once TCRC annual benchmark levels are able to beforecasted with a certain level of accuracy future researchshould go further into this research through the explorationof the potential impact that other market factors may have onthese discounts

As a limitation of this research we should point out thetimespan of the data considered compared to those of otherforecasting works on commodity prices for example whichuse broader timespans For our case we have consideredthe maximum available sufficiently reliable data on TCRCbenchmark levels starting back in 2004 as there is noreliable data beyond this year Also as another limitationwe have used four measures of error which are among themost frequently used to compare the accuracy of differentmodels However other measures could have been used at anindividual level to test each modelrsquos accuracy

Appendix

A Models

A1 Geometric Brownian Motion (GBM) GBM can bewritten as a generalisation of a Wiener (continuous time-stochasticMarkov process with independent increments andwhose changes over any infinite interval of time are normallydistributed with a variance that increases linearly with thetime interval [22]) process

119889119909 = 120572119909119889119905 + 120590119909119889119911 (A1)

where according to Marathe and Ryan [30] the first termis known as the Expected Value whereas the second is theStochastic component with 120572 being the drift parameter and120590 the volatility of the process Also dz is the Wiener processwhich induces the abovementioned stochastic behaviour inthe model

119889119911 = isin119905radic119889119905 997888rarr isin119905 sim 119873 (0 1) (A2)

TheGBMmodel can be expressed in discrete terms accordingto

119909 = 119909119905 minus 119909119905minus1 = 120572119909119905minus1119905 + 120590119909119905minus1120598radic119905 (A3)

In GBM percentage changes in x 119909119909 are normally dis-tributed thus absolute changes in x 119909 are lognormallydistributed Also the expected value and variance for 119909(119905) are

E [119909 (119905)] = 1199090119890120572119905 (A4)

var [119909 (119905)] = 119909201198902120572119905 (1198901205902119905 minus 1) (A5)

A2 Orstein-Uhlenbeck Process (OUP) TheOUPprocess wasfirst defined by Uhlenbeck and Orstein [31] as an alternativeto the regular Brownian Motion to model the velocity ofthe diffusion movement of a particle that accounts for itslosses due to friction with other particles The OUP processcan be regarded as a modification of Brownian Motion incontinuous time where its properties have been changed

(Stationary Gaussian Markov and stochastic process) [32]These modifications cause the process to move towards acentral position with a stronger attraction the further it isfrom this position As mentioned above the OUP is usuallyemployed to model commodity prices and is the simplestversion of amean-reverting process [22]

119889119878 = 120582 (120583 minus 119878) 119889119905 + 120590119889119882119905 (A6)

where S is the level of prices 120583 the long-term average towhich prices tend to revert and 120582 the speed of reversionAdditionally in a similar fashion to that of the GBM 120590 isthe volatility of the process and 119889119882119905 is a Wiener processwith an identical definition However in contrast to GBMtime intervals in OUP are not independent since differencesbetween current levels of prices S and long-term averageprices 120583 make the expected change in prices dS more likelyeither positive or negative

The discrete version of the model can be expressed asfollows

119878119905 = 120583 (1 minus 119890minus120582Δ119905) + 119890minus120582Δ119905119878119905minus1 + 120590radic1 minus 119890minus2120582Δ1199052120582 119889119882119905 (A7)

where the expected value for 119909(119905) and the variance for (119909(119905) minus120583) areE [119909 (119905)] = 120583 + (1199090 minus 120583) 119890minus120582119905 (A8)

var [119909 (119905) minus 120583] = 12059022120582 (1 minus 119890minus2120582119905) (A9)

It can be derived from previous equations that as timeincreases prices will tend to long-term average levels 120583 Inaddition with large time spans if the speed of reversion 120582becomes high variance tends to 0 On the other hand if thespeed of reversion is 0 then var[119909(119905)] 997888rarr 1205902119905 making theprocess a simple Brownian Motion

A3Holtrsquos Linear Exponential Smoothing (LES) Linear Expo-nential Smoothing models are capable of considering bothlevels and trends at every instant assigning higher weights inthe overall calculation to values closer to the present than toolder ones LES models carry that out by constantly updatinglocal estimations of levels and trends with the intervention ofone or two smoothing constants which enable the models todampen older value effects Although it is possible to employa single smoothing constant for both the level and the trendknown as Brownrsquos LES to use two one for each known asHoltrsquos LES is usually preferred since Brownrsquos LES tends torender estimations of the trend ldquounstablerdquo as suggested byauthors such as Nau [33] Holtrsquos LES model comes definedby the level trend and forecast updating equations each ofthese expressed as follows respectively

119871 119905 = 120572119884119905 + (1 minus 120572) (119871 119905minus1 + 119879119905minus1) (A10)

119879119905 = 120573 (119871 119905 minus 119871 119905minus1) + (1 minus 120573) 119879119905minus1 (A11)

119905+119896 = 119871 119905 + 119896119879119905 (A12)

Complexity 13

With 120572 being the first smoothing constant for the levelsand 120573 the second smoothing constant for the trend Highervalues for the smoothing constants imply that either levels ortrends are changing rapidly over time whereas lower valuesimply the contrary Hence the higher the constant the moreuncertain the future is believed to be

B Model Calibration

Each model has been calibrated individually using two setsof data containing TC and RC levels respectively from 2004to 2012 The available dataset is comprised of data from 2004to 2017 which make the calibration data approximately 23 ofthe total

B1 GBM Increments in the logarithm of variable x aredistributed as follows

(ln119909) sim 119873((120572 minus 12059022 ) 119905 120590119905) (B1)

Hence if m is defined as the sample mean of the differenceof the natural logarithm of the time series for TCRC levelsconsidered for the calibration and n as the number ofincrements of the series considered with n=9

119898 = 1119899119899sum119905=1

(ln119909119905 minus ln119909119905minus1) (B2)

119904 = radic 1119899 minus 1119899sum119894=1

(ln119909119905 minus ln119909119905minus1 minus 119898)2 (B3)

119898 = 120572 minus 12059022 (B4)

119904 = 120590 (B5)

B2 OUP The OUP process is an AR1 process [22] whoseresolution is well presented by Woolridge [34] using OLStechniques fitting the following

119910119905 = 119886 + 119887119910119905minus1 + 120576119905 (B6)

Hence the estimators for the parameters of the OUP modelare obtained by OLS for both TC and RC levels indepen-dently

= minus ln 119887Δ119905 (B7)

120583 = 1198861 minus 119887 (B8)

= 120576119905radic 2 ln (1 + 119887)(1 + 119887)2 minus 1 (B9)

B3 LES A linear regression is conducted on the inputdataset available to find the starting parameters for the LES

model the initial Level 1198710 and the initial value of thetrend 1198790 irrespective of TC values and RC values Hereas recommended by Gardner [35] the use of OLS is highlyadvisable due to the erratic behaviour shown by trends inthe historic data so the obtaining of negative values of 1198780 isprevented Linear regression fulfils the following

119884119905 = 119886119905 + 119887 (B10)

1198710 = 119887 (B11)

1198790 = 119886 (B12)

By fixing the two smoothing constants the values for theforecasts 119905+119896 can be calculated at each step using the modelequations There are multiple references in the literature onwhat the optimum range for each smoothing constant isGardner [35] speaks of setting moderate values for bothparameter less than 03 to obtain the best results Examplespointing out the same may be found in Brown [36] Coutie[37] Harrison [38] and Montgomery and Johnson [39]Also for many applications Makridakis and Hibon [20] andChatfield [40] found that parameter values should fall withinthe range of 03-1 On the other hand McClain and Thomas[41] provided a condition of stability for the nonseasonal LESmodel given by

0 lt 120572 lt 20 lt 120573 lt 4 minus 2120572120572

(B13)

Also the largest possible value of 120572 that allows the avoidanceof areas of oscillation is proposed by McClain and Thomas[41] and McClain [42]

120572 lt 4120573(1 + 120573)2 (B14)

However according to Gardner there is no tangible proofthat this value improves accuracy in any form Nonethelesswe have opted to follow what Nau [33] refers to as ldquotheusual wayrdquo namely minimising the Mean Squared Error(MSE) of the one-step-ahead forecast of TCRC for eachinput data series previously mentioned To do so Matlabrsquosfminsearch function has been used with function and variabletolerance levels of 1x10minus4 as well as a set maximum numberof function iterations and function evaluations of 1x106 tolimit computing resources In Table 18 the actual number ofnecessary iterations to obtain optimum values for smoothingconstants is shown As can be seen the criteria are wellbeyond the final results which ensured that an optimumsolution was reached with assumable computing usage (thesimulation required less than one minute) and with a highdegree of certainty

14 Complexity

Table 18 Iterations on LES smoothing constants optimisation

Iterations Function EvaluationsTC 1-Step 36 40TC 2-Steps 249945 454211TC 5-Steps 40 77RC 1-Step 32 62RC 2-Steps 254388 462226RC 5-Steps 34 66

C Error Calculations

C1 Mean Square Error (MSE)

119872119878119864 = 1119899119899sum119894=1

(119894 minus 119884119894)2 (C1)

where 119894 are the forecasted values and 119884119894 those observedC2 Mean Absolute Deviation (MAD)

119872119860119863 = 1119899119899sum119894=1

10038161003816100381610038161003816119894 minus 11988410038161003816100381610038161003816 (C2)

where 119894 are the forecasted values and 119884 the average value ofall the observations

C3 Mean Absolute Percentage Error (MAPE)

119872119860119875119864 = 1119899119899sum119894=1

10038161003816100381610038161003816119894 minus 11988411989410038161003816100381610038161003816119884119894 (C3)

The above formula is expressed in parts-per-one and is theone used in the calculations conducted here Hence multi-plying the result by 100 would deliver percentage outcomes

C4 Root Mean Square Error (RMSE)

119877119872119878119864 = radic119872119878119864 = radic 1119899119899sum119894=1

(119894 minus 119884119894)2 (C4)

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Supplementary Materials

The Matlab codes developed to carry out the simulationsfor each of the models proposed as well as the datasetsused and the Matlab workspaces with the simulationsoutcomes as they have been reflected in this paper areprovided in separate folders for each model GBM foldercontains the code to make the simulations for the Geo-metric Brownian Motion model ldquoGBMmrdquo as well as aseparate script which allows determining the total numberof monthly step simulations for which errors levels havebeen lower than annual-step simulations ldquoErrCoutermrdquoThe datasets used are included in two excel files ldquoTCxlsxrdquoand ldquoRCxlsxrdquo ldquoGBM OK MonthlyStepsmatrdquo is the Matlabworkspace containing the outcome of monthly step simu-lations whereas ldquoGBM OK AnnualStepsmatrdquo contains theoutcome for annual-step simulations Also ldquoGBM OKmatrdquois the workspace to be loaded prior to performing GBMsimulations MR folder contains the code file to carryout the simulations for the Orstein-Uhlenbeck ProcessldquoMRmrdquo as well as the separate script to compare errorsof monthly step simulations with annual-step simula-tions ldquoErrCountermrdquo ldquoTCxlsxrdquo and ldquoRCxlsxrdquo containthe TCRC benchmark levels from 2004 to 2017 Finallymonthly step simulationsrsquo outcome has been saved inthe workspace ldquoMR OK MonthlyStepsmatrdquo while annual-step simulationsrsquo outcome has been saved in the Matlabworkspace ldquoMR OK AnnualStepsmatrdquo Also ldquoMR OKmatrdquois the workspace to be loaded prior to performing MRsimulations LES folder contains the code file to carry outthe calculations necessary to obtain the LES model forecastsldquoLESmrdquo Three separate Matlab functions are includedldquomapeLESmrdquo ldquomapeLES1mrdquo and ldquomapeLES2mrdquo whichdefine the calculation of the MAPE for the LESrsquo five-year-ahead one-year-ahead and two-year-ahead forecastsAlso ldquoLES OKmatrdquo is the workspace to be loaded prior toperforming LES simulations Finally graphs of each modelhave been included in separate files as well in a subfoldernamed ldquoGraphsrdquo within each of the modelsrsquo folder Figuresare included in Matlabrsquos format (fig) and TIFF format TIFFfigures have been generated in both colours and black andwhite for easier visualization (Supplementary Materials)

References

[1] United Nations International Trade Statistics Yearbook vol 22016

[2] International Copper StudyGroupTheWorld Copper Factbook2017

[3] K M Johnson J M Hammarstrom M L Zientek and CL Dicken ldquoEstimate of undiscovered copper resources of theworld 2013rdquo US Geological Survey 2017

[4] M E Schlesinger and W G Davenport Extractive Metallurgyof Copper Elsevier 5th edition 2011

[5] S Gloser M Soulier and L A T Espinoza ldquoDynamic analysisof global copper flows Global stocks postconsumer materialflows recycling indicators and uncertainty evaluationrdquo Envi-ronmental Science amp Technology vol 47 no 12 pp 6564ndash65722013

8 Complexity

Table 5 Average for main measures of error for GBM after 1000 MC simulations for 2013

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC Averaged Monthly Steps 33658 1455 021 1455TC Annual Steps 69381 2081 030 2081RC Averaged Monthly Steps 297 138 020 138RC Annual Steps 657 205 029 205

Table 6 Average for main measures of error for GBM after 1000 MC simulations for 2013-2014

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC Averaged Monthly Steps 99492 2187 031 2435TC Annual Steps 133x103 2833 040 3104RC Averaged Monthly Steps 1035 240 034 271RC Annual Steps 1313 284 034 311

Table 7 OUP parameters obtained in the calibration process

120583 120582 120590TCMonthly 6345 4792x105 2534TC Annual 6345 2974 1308RC Monthly 635 4763x105 2519RC Annual 635 2972 1305

methods such as theMonte Carlo are not required to obtain aforecast Nonetheless the LES model relies on two smoothingconstants which must be properly set in order to deliveraccurate predictions hence the values of the smoothingconstants were first optimised (see Appendix B for LESModel fitting and smoothing constants optimisation) Theoptimisation was carried out for one-year-ahead forecaststwo-year-ahead forecasts and five-year-ahead forecasts byminimising the values of MSE for both TC and RC Thedifferent values used for smoothing constants as well asthe initial values for level and trend obtained by the linearregression of the available dataset from 2004 through to 2012are shown in Table 12

Compared one-year-ahead two-year-ahead and five-year-ahead LES forecasts for TC and RC are shown inFigure 4 clearly indicating a stronger accuracy for shorter-term forecasts as the observed and forecasted plotted linesoverlap

Minimumvalues for errormeasures achieved through theLESmodel parameter optimisation are shown in Table 13Thevalues obtained confirm the strong accuracy for shorter-termforecasts of TCRC seen in Figure 4

5 Discussion and Model Comparison

The forecasted values of TCRC benchmark levels couldeventually be applied to broader valuation models for copperconcentrates and their trading activities as well as to thecopper smeltersrsquo revenue stream thus the importance ofdelivering as accurate a prediction as possible in relation tothese discounts to make any future application possible Eachof the models presented may be a feasible method on its own

with eventual later adaptations to forecasting future valuesof benchmark TCRC Nonetheless the accuracy of thesemodels as they have been used in this work requires firstlya comparison to determine whether any of them could bea good standalone technique and secondly to test whethera combination of two or more of them would deliver moreprecise results

When comparing the different error measures obtainedfor all the three models it is clearly established that resultsfor a randomly chosen simulation of GBM or OUP wouldbe more likely to be more precise had a monthly step beenused to deliver annual forecasts instead of an annual-step sizeIn contrast average error measures for the entire populationof simulations with each step size employed showing thatmonthly step simulations for GBM and OUP models arealways more accurate than straight annual-step forecastswhen a shorter time horizon one or two-year-ahead is takeninto consideration However GBM presents a higher levelof forecasting accuracy when the average for error measuresof all simulations is analysed employing annual steps forlong-term horizons whereas OUP averaged monthly stepforecasts remain more accurate when predicting long-termhorizons Table 14 shows the error improvement for the aver-aged monthly step forecasts of each model Negative valuesindicate that better levels of error averages have been found instraight annual forecasts than for monthly step simulations

Considering the best results for each model and com-paring their corresponding error measures we can opt forthe best technique to employ among the three proposed inthis paper Hence GBM delivers the most accurate one-yearforecast when averaging the next twelve-month predictionsfor TCRC values as does the MR-OUP model Table 15shows best error measures for one-year-ahead forecasts forGBM OUP and LES models

Unarguably the LES model generates minimal errormeasures for one-year-ahead forecasts significantly less thanthe other models employed A similar situation is found fortwo-year-ahead forecasts whereminimumerrormeasures arealso delivered by the LES model shown in Table 16

Finally accuracy measures for five-year-ahead forecastsof the GBM model might result in somewhat contradictory

Complexity 9

Table 8 Number of paths for whichmeasures of error are higher for monthly steps than for annual steps inMR-OUP 5-StepsMC simulation

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC 3111000 2831000 2661000 2661000RC 3161000 2811000 2501000 2501000

Table 9 Average for main measures of error for MR-OUP after 1000 MC simulations 2013-2017

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC Averaged Monthly Steps 84254 2614 027 2895TC Annual Steps 114x103 2931 031 3325RC Averaged Monthly Steps 840 261 027 289RC Annual Steps 1148 294 031 333

Table 10 Average for main measures of error for MR-OUP after 1000 MC simulations 2013

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC Averaged Monthly Steps 3914 522 007 522TC Annual Steps 36643 1555 022 1555RC Averaged Monthly Steps 039 051 007 051RC Annual Steps 363 154 022 154

Table 11 Average for main measures of error for MR-OUP after 1000 MC simulations 2013-2014

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC Averaged Monthly Steps 37165 1567 018 1900TC Annual Steps 68244 2185 026 2324RC Averaged Monthly Steps 376 157 018 191RC Annual Steps 689 219 026 243

Table 12 LES Model Optimised Parameters

L0 T0 120572 120573TC 1 year 713611 -15833 -02372 01598RC 1 year 71333 -01567 -02368 01591TC 2 years 713611 -15833 -1477x10minus4 7774226RC 2 years 71333 -01567 -1448x10minus4 7898336TC 5 years 713611 -15833 -02813 01880RC 5 years 71333 -01567 -02808 01880

Table 13 LES error measures for different steps-ahead forecasts

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC 1 year 10746x10minus5 00033 46831x10minus5 00033RC 1 year 51462x10minus8 22685x10minus4 32408x10minus5 22685x10minus4

TC 2 years 189247 97275 01057 137567RC 2 years 18977 09741 01059 13776TC 5 years 1775531 78881 00951 108422RC 5 years 11759 07889 00952 10844

terms for MSE reaching better values for annual steps thanfor averaged monthly steps while the other figures do betteron averagedmonthly steps Addressing the definition ofMSEthis includes the variance of the estimator as well as its biasbeing equal to its variance in the case of unbiased estimators

Therefore MSE measures the quality of the estimator butalso magnifies estimator deviations from actual values sinceboth positive and negative values are squared and averagedIn contrast RMSE is calculated as the square root of MSEand following the previous analogy stands for the standard

10 Complexity

Table 14 Error Average Improvement for averaged monthly steps before annual steps

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864GBM TCRC 1 Year 53265653 32723587 33333548 32723587GBM TCRC 2 Years 26952635 25751597 2619286 24341369GBM TCRC 5 Years -1173-1104 873709 926769 747565OUP TCRC 1 Year 88088708 63716250 62916319 63686224OUP TCRC 2 Years 46284421 27712617 29993075 22162075OUP TCRC 5 Years 26022553 1093997 13121218 13061269

Table 15 TCRC best error measures for 2013 forecasts after 1000 MC simulations

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864GBM TCRC 1 Yearlowast 33111336 1425143 020020 1425143OUP TCRC 1 Yearlowast 4105042 536054 0076600777 536054LES TCRC 1 Year 107x10minus5514x10minus8 0003226x10minus4 468x10minus5324x10minus5 0003226x10minus4

lowastAveraged monthly steps

Table 16 TCRC Best error measures for 2013ndash2014 forecasts after 1000 MC simulations

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864GBM TCRC 2 Stepslowast 103x1031006 2200242 031034 2447271OUP TCRC 2 Stepslowast 37331376 1573158 0180201813 1900191LES TCRC 2 Steps 18924189 972097 0105701059 1375137lowastAveraged monthly steps

deviation of the estimator if MSE were considered to be thevarianceThough RMSE overreacts when high values of MSEare reached it is less prone to this than MSE since it iscalculated as its squared root thus not accounting for largeerrors as disproportionately as MSE does Furthermore aswe have compared an average of 1000 measures of errorscorresponding to each MC simulation performed the valuesobtained for average RMSE stay below the square root ofaverage MSE which indicates that some of these dispropor-tionate error measures are to some extent distorting thelatter Hence RMSE average values point towards a higheraccuracy for GBM five-year forecasts with averaged monthlysteps which is further endorsed by the average values ofMAD and MAPE thus being the one used for comparisonwith the other two models as shown in Table 17

The final comparison clearly shows how the LES modeloutperforms the other two at all average measures providedfollowed by the OUP model in accuracy although the lattermore than doubles the average MAPE value for LES

The results of simulations indicate that measures of errorstend to either differ slightly or not at all for either forecastsof any timeframe A coherent value with the 101 relation canthen be given with close to the same level of accuracy bymultiplying RC forecasts or dividing TC ones by 10

6 Conclusions

Copper TCRC are a keystone for pricing copper concen-trates which are the actual feedstock for copper smeltersThe potential evolution of TCRC is a question of botheconomic and technical significance forminers as their value

decreases the potential final selling price of concentratesAdditionally copper minersrsquo revenues are more narrowlyrelated to the market price of copper as well as to othertechnical factors such as ore dilution or the grade of theconcentrates produced Smelters on the contrary are hugelyaffected by the discount which they succeed in getting whenpurchasing the concentrates since that makes up the largestpart of their gross revenue besides other secondary sourcesIn addition eventual differences between TCRC may givecommodity traders ludicrous arbitrage opportunities Alsodifferences between short- and long-termTCRC agreementsoffer arbitrage opportunities for traders hence comprisinga part of their revenue in the copper concentrate tradingbusiness as well copper price fluctuations and the capacityto make economically optimum copper blends

As far as we are aware no rigorous research has beencarried out on the behaviour of these discounts Based onhistorical data on TCRC agreed upon in the LME CopperWeek from 2004 to 2017 three potentially suitable forecastingmodels for TCRC annual benchmark values have beencompared through four measures of forecasting accuracyat different horizons These models were chosen firstlydue to their broad implementation and proven capacityin commodity prices forecasting that they all share andsecondly because of the core differences in terms of pricebehaviour with each other

Focusing on the MAPE values achieved those obtainedby the LES model when TC and RC are treated indepen-dently have been significantly lower than for the rest ofthe models Indeed one-year-ahead MAPE measures for TCvalues for the GBM model (20) almost triple those of

Complexity 11

2004 2006 2008 2010 2012 2014405060708090

100

TC (U

SDM

T)

LES TC 1 Step Ahead 2013

2004 2006 2008 2010 2012 2014

LES TC 2 Steps Ahead 2013-2014

2004 2006 2008 2010 2012 2014 2016 2018

LES TC 5 Steps Ahead 2013-2017

2004 2006 2008 2010 2012 2014

LES RC 1 Step Ahead 2013

2004 2006 2008 2010 2012 2014456789

10LES RC 2 Steps Ahead 2013-2014

Observed Forecast

2004 2006 2008 2010 2012 2014 2016 2018

LES RC 5 Steps Ahead 2013-2017

405060708090

100110

405060708090

100

456789

10

RC (U

Sclb

)

456789

1011

Figure 4 One-step-ahead forecasts (2013) two-step-ahead forecasts (2013-2014) and five-step-ahead forecasts (2013-2017) for TC and RCusing the Linear Exponential Smoothing model (LES)

Table 17 TCRC Best error measures for 2013ndash2017 forecasts after 1000 MC simulations

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864GBM TCRC 5 Stepslowast 600x1036165 4651459 049048 5571551OUP TCRC 5 Stepslowast 84334843 2616262 0271502719 2896289LES TCRC 5 Steps 17755117 788078 0095100952 1084108lowastAveraged monthly steps

the OUP model (766) in contrast with the significantlylower values from the LES model (00046) This gaptends to be narrowed when TCRC values are forecastedat longer horizons when most measures of error becomemore even The GBM and OUP models have proven todeliver better accuracy performance when the TCRC valuesare projected monthly and then averaged to obtain annualbenchmark forecasts Even so the LES model remains themost accurate of all with MAPE values of 10 at two-year-ahead forecasts with 18 and 31 for TC for OUP and GBMrespectively

Finally despite TC and RC being two independentdiscounts applied to copper concentrates they are bothset jointly with an often 101 relation as our data revealsThis relation also transcends to simulation results and errormeasures hence showing a negligible discrepancy betweenthe independent forecasting of TC and RC or the jointforecasting of both values keeping the 101 relationThis is forinstance the case of the five-year-ahead OUP MAPE values(0271502719) which were obtained without observing the101 relation in the data A similar level of discrepancy wasobtained at any horizon with any model which indicates thatboth values could be forecasted with the same accuracy usingthe selected model with any of them and then applying the101 relation

Our findings thus suggest that both at short and atlong-term horizons TCRC annual benchmark levels tendto exhibit a pattern which is best fit by an LES model Thisindicates that these critical discounts for the copper tradingbusiness do maintain a certain dependency on past valuesThis would also suggest the existence of cyclical patterns incopper TCRC possibly driven by many of the same marketfactors that move the price of copper

This work contributes by delivering a formal tool forsmelters or miners to make accurate forecasts of TCRCbenchmark levels The level of errors attained indicatesthe LES model may be a valid model to forecast thesecrucial discounts for the copper market In addition ourwork further contributes to helping market participants toproject the price of concentrates with an acceptable degree ofuncertainty as now they may include a fundamental elementfor their estimation This would enable them to optimise theway they produce or process these copper concentrates Alsothe precise knowledge of these discountsrsquo expected behaviourcontributes to letting miners traders and smelters aliketake the maximum advantage from the copper concentratetrading agreements that they are part of As an additionalcontribution this work may well be applied to gold or silverRC which are relevant deduction concentrates when thesehave a significant amount of gold or silver

12 Complexity

Once TCRC annual benchmark levels are able to beforecasted with a certain level of accuracy future researchshould go further into this research through the explorationof the potential impact that other market factors may have onthese discounts

As a limitation of this research we should point out thetimespan of the data considered compared to those of otherforecasting works on commodity prices for example whichuse broader timespans For our case we have consideredthe maximum available sufficiently reliable data on TCRCbenchmark levels starting back in 2004 as there is noreliable data beyond this year Also as another limitationwe have used four measures of error which are among themost frequently used to compare the accuracy of differentmodels However other measures could have been used at anindividual level to test each modelrsquos accuracy

Appendix

A Models

A1 Geometric Brownian Motion (GBM) GBM can bewritten as a generalisation of a Wiener (continuous time-stochasticMarkov process with independent increments andwhose changes over any infinite interval of time are normallydistributed with a variance that increases linearly with thetime interval [22]) process

119889119909 = 120572119909119889119905 + 120590119909119889119911 (A1)

where according to Marathe and Ryan [30] the first termis known as the Expected Value whereas the second is theStochastic component with 120572 being the drift parameter and120590 the volatility of the process Also dz is the Wiener processwhich induces the abovementioned stochastic behaviour inthe model

119889119911 = isin119905radic119889119905 997888rarr isin119905 sim 119873 (0 1) (A2)

TheGBMmodel can be expressed in discrete terms accordingto

119909 = 119909119905 minus 119909119905minus1 = 120572119909119905minus1119905 + 120590119909119905minus1120598radic119905 (A3)

In GBM percentage changes in x 119909119909 are normally dis-tributed thus absolute changes in x 119909 are lognormallydistributed Also the expected value and variance for 119909(119905) are

E [119909 (119905)] = 1199090119890120572119905 (A4)

var [119909 (119905)] = 119909201198902120572119905 (1198901205902119905 minus 1) (A5)

A2 Orstein-Uhlenbeck Process (OUP) TheOUPprocess wasfirst defined by Uhlenbeck and Orstein [31] as an alternativeto the regular Brownian Motion to model the velocity ofthe diffusion movement of a particle that accounts for itslosses due to friction with other particles The OUP processcan be regarded as a modification of Brownian Motion incontinuous time where its properties have been changed

(Stationary Gaussian Markov and stochastic process) [32]These modifications cause the process to move towards acentral position with a stronger attraction the further it isfrom this position As mentioned above the OUP is usuallyemployed to model commodity prices and is the simplestversion of amean-reverting process [22]

119889119878 = 120582 (120583 minus 119878) 119889119905 + 120590119889119882119905 (A6)

where S is the level of prices 120583 the long-term average towhich prices tend to revert and 120582 the speed of reversionAdditionally in a similar fashion to that of the GBM 120590 isthe volatility of the process and 119889119882119905 is a Wiener processwith an identical definition However in contrast to GBMtime intervals in OUP are not independent since differencesbetween current levels of prices S and long-term averageprices 120583 make the expected change in prices dS more likelyeither positive or negative

The discrete version of the model can be expressed asfollows

119878119905 = 120583 (1 minus 119890minus120582Δ119905) + 119890minus120582Δ119905119878119905minus1 + 120590radic1 minus 119890minus2120582Δ1199052120582 119889119882119905 (A7)

where the expected value for 119909(119905) and the variance for (119909(119905) minus120583) areE [119909 (119905)] = 120583 + (1199090 minus 120583) 119890minus120582119905 (A8)

var [119909 (119905) minus 120583] = 12059022120582 (1 minus 119890minus2120582119905) (A9)

It can be derived from previous equations that as timeincreases prices will tend to long-term average levels 120583 Inaddition with large time spans if the speed of reversion 120582becomes high variance tends to 0 On the other hand if thespeed of reversion is 0 then var[119909(119905)] 997888rarr 1205902119905 making theprocess a simple Brownian Motion

A3Holtrsquos Linear Exponential Smoothing (LES) Linear Expo-nential Smoothing models are capable of considering bothlevels and trends at every instant assigning higher weights inthe overall calculation to values closer to the present than toolder ones LES models carry that out by constantly updatinglocal estimations of levels and trends with the intervention ofone or two smoothing constants which enable the models todampen older value effects Although it is possible to employa single smoothing constant for both the level and the trendknown as Brownrsquos LES to use two one for each known asHoltrsquos LES is usually preferred since Brownrsquos LES tends torender estimations of the trend ldquounstablerdquo as suggested byauthors such as Nau [33] Holtrsquos LES model comes definedby the level trend and forecast updating equations each ofthese expressed as follows respectively

119871 119905 = 120572119884119905 + (1 minus 120572) (119871 119905minus1 + 119879119905minus1) (A10)

119879119905 = 120573 (119871 119905 minus 119871 119905minus1) + (1 minus 120573) 119879119905minus1 (A11)

119905+119896 = 119871 119905 + 119896119879119905 (A12)

Complexity 13

With 120572 being the first smoothing constant for the levelsand 120573 the second smoothing constant for the trend Highervalues for the smoothing constants imply that either levels ortrends are changing rapidly over time whereas lower valuesimply the contrary Hence the higher the constant the moreuncertain the future is believed to be

B Model Calibration

Each model has been calibrated individually using two setsof data containing TC and RC levels respectively from 2004to 2012 The available dataset is comprised of data from 2004to 2017 which make the calibration data approximately 23 ofthe total

B1 GBM Increments in the logarithm of variable x aredistributed as follows

(ln119909) sim 119873((120572 minus 12059022 ) 119905 120590119905) (B1)

Hence if m is defined as the sample mean of the differenceof the natural logarithm of the time series for TCRC levelsconsidered for the calibration and n as the number ofincrements of the series considered with n=9

119898 = 1119899119899sum119905=1

(ln119909119905 minus ln119909119905minus1) (B2)

119904 = radic 1119899 minus 1119899sum119894=1

(ln119909119905 minus ln119909119905minus1 minus 119898)2 (B3)

119898 = 120572 minus 12059022 (B4)

119904 = 120590 (B5)

B2 OUP The OUP process is an AR1 process [22] whoseresolution is well presented by Woolridge [34] using OLStechniques fitting the following

119910119905 = 119886 + 119887119910119905minus1 + 120576119905 (B6)

Hence the estimators for the parameters of the OUP modelare obtained by OLS for both TC and RC levels indepen-dently

= minus ln 119887Δ119905 (B7)

120583 = 1198861 minus 119887 (B8)

= 120576119905radic 2 ln (1 + 119887)(1 + 119887)2 minus 1 (B9)

B3 LES A linear regression is conducted on the inputdataset available to find the starting parameters for the LES

model the initial Level 1198710 and the initial value of thetrend 1198790 irrespective of TC values and RC values Hereas recommended by Gardner [35] the use of OLS is highlyadvisable due to the erratic behaviour shown by trends inthe historic data so the obtaining of negative values of 1198780 isprevented Linear regression fulfils the following

119884119905 = 119886119905 + 119887 (B10)

1198710 = 119887 (B11)

1198790 = 119886 (B12)

By fixing the two smoothing constants the values for theforecasts 119905+119896 can be calculated at each step using the modelequations There are multiple references in the literature onwhat the optimum range for each smoothing constant isGardner [35] speaks of setting moderate values for bothparameter less than 03 to obtain the best results Examplespointing out the same may be found in Brown [36] Coutie[37] Harrison [38] and Montgomery and Johnson [39]Also for many applications Makridakis and Hibon [20] andChatfield [40] found that parameter values should fall withinthe range of 03-1 On the other hand McClain and Thomas[41] provided a condition of stability for the nonseasonal LESmodel given by

0 lt 120572 lt 20 lt 120573 lt 4 minus 2120572120572

(B13)

Also the largest possible value of 120572 that allows the avoidanceof areas of oscillation is proposed by McClain and Thomas[41] and McClain [42]

120572 lt 4120573(1 + 120573)2 (B14)

However according to Gardner there is no tangible proofthat this value improves accuracy in any form Nonethelesswe have opted to follow what Nau [33] refers to as ldquotheusual wayrdquo namely minimising the Mean Squared Error(MSE) of the one-step-ahead forecast of TCRC for eachinput data series previously mentioned To do so Matlabrsquosfminsearch function has been used with function and variabletolerance levels of 1x10minus4 as well as a set maximum numberof function iterations and function evaluations of 1x106 tolimit computing resources In Table 18 the actual number ofnecessary iterations to obtain optimum values for smoothingconstants is shown As can be seen the criteria are wellbeyond the final results which ensured that an optimumsolution was reached with assumable computing usage (thesimulation required less than one minute) and with a highdegree of certainty

14 Complexity

Table 18 Iterations on LES smoothing constants optimisation

Iterations Function EvaluationsTC 1-Step 36 40TC 2-Steps 249945 454211TC 5-Steps 40 77RC 1-Step 32 62RC 2-Steps 254388 462226RC 5-Steps 34 66

C Error Calculations

C1 Mean Square Error (MSE)

119872119878119864 = 1119899119899sum119894=1

(119894 minus 119884119894)2 (C1)

where 119894 are the forecasted values and 119884119894 those observedC2 Mean Absolute Deviation (MAD)

119872119860119863 = 1119899119899sum119894=1

10038161003816100381610038161003816119894 minus 11988410038161003816100381610038161003816 (C2)

where 119894 are the forecasted values and 119884 the average value ofall the observations

C3 Mean Absolute Percentage Error (MAPE)

119872119860119875119864 = 1119899119899sum119894=1

10038161003816100381610038161003816119894 minus 11988411989410038161003816100381610038161003816119884119894 (C3)

The above formula is expressed in parts-per-one and is theone used in the calculations conducted here Hence multi-plying the result by 100 would deliver percentage outcomes

C4 Root Mean Square Error (RMSE)

119877119872119878119864 = radic119872119878119864 = radic 1119899119899sum119894=1

(119894 minus 119884119894)2 (C4)

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Supplementary Materials

The Matlab codes developed to carry out the simulationsfor each of the models proposed as well as the datasetsused and the Matlab workspaces with the simulationsoutcomes as they have been reflected in this paper areprovided in separate folders for each model GBM foldercontains the code to make the simulations for the Geo-metric Brownian Motion model ldquoGBMmrdquo as well as aseparate script which allows determining the total numberof monthly step simulations for which errors levels havebeen lower than annual-step simulations ldquoErrCoutermrdquoThe datasets used are included in two excel files ldquoTCxlsxrdquoand ldquoRCxlsxrdquo ldquoGBM OK MonthlyStepsmatrdquo is the Matlabworkspace containing the outcome of monthly step simu-lations whereas ldquoGBM OK AnnualStepsmatrdquo contains theoutcome for annual-step simulations Also ldquoGBM OKmatrdquois the workspace to be loaded prior to performing GBMsimulations MR folder contains the code file to carryout the simulations for the Orstein-Uhlenbeck ProcessldquoMRmrdquo as well as the separate script to compare errorsof monthly step simulations with annual-step simula-tions ldquoErrCountermrdquo ldquoTCxlsxrdquo and ldquoRCxlsxrdquo containthe TCRC benchmark levels from 2004 to 2017 Finallymonthly step simulationsrsquo outcome has been saved inthe workspace ldquoMR OK MonthlyStepsmatrdquo while annual-step simulationsrsquo outcome has been saved in the Matlabworkspace ldquoMR OK AnnualStepsmatrdquo Also ldquoMR OKmatrdquois the workspace to be loaded prior to performing MRsimulations LES folder contains the code file to carry outthe calculations necessary to obtain the LES model forecastsldquoLESmrdquo Three separate Matlab functions are includedldquomapeLESmrdquo ldquomapeLES1mrdquo and ldquomapeLES2mrdquo whichdefine the calculation of the MAPE for the LESrsquo five-year-ahead one-year-ahead and two-year-ahead forecastsAlso ldquoLES OKmatrdquo is the workspace to be loaded prior toperforming LES simulations Finally graphs of each modelhave been included in separate files as well in a subfoldernamed ldquoGraphsrdquo within each of the modelsrsquo folder Figuresare included in Matlabrsquos format (fig) and TIFF format TIFFfigures have been generated in both colours and black andwhite for easier visualization (Supplementary Materials)

References

[1] United Nations International Trade Statistics Yearbook vol 22016

[2] International Copper StudyGroupTheWorld Copper Factbook2017

[3] K M Johnson J M Hammarstrom M L Zientek and CL Dicken ldquoEstimate of undiscovered copper resources of theworld 2013rdquo US Geological Survey 2017

[4] M E Schlesinger and W G Davenport Extractive Metallurgyof Copper Elsevier 5th edition 2011

[5] S Gloser M Soulier and L A T Espinoza ldquoDynamic analysisof global copper flows Global stocks postconsumer materialflows recycling indicators and uncertainty evaluationrdquo Envi-ronmental Science amp Technology vol 47 no 12 pp 6564ndash65722013

Complexity 9

Table 8 Number of paths for whichmeasures of error are higher for monthly steps than for annual steps inMR-OUP 5-StepsMC simulation

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC 3111000 2831000 2661000 2661000RC 3161000 2811000 2501000 2501000

Table 9 Average for main measures of error for MR-OUP after 1000 MC simulations 2013-2017

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC Averaged Monthly Steps 84254 2614 027 2895TC Annual Steps 114x103 2931 031 3325RC Averaged Monthly Steps 840 261 027 289RC Annual Steps 1148 294 031 333

Table 10 Average for main measures of error for MR-OUP after 1000 MC simulations 2013

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC Averaged Monthly Steps 3914 522 007 522TC Annual Steps 36643 1555 022 1555RC Averaged Monthly Steps 039 051 007 051RC Annual Steps 363 154 022 154

Table 11 Average for main measures of error for MR-OUP after 1000 MC simulations 2013-2014

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC Averaged Monthly Steps 37165 1567 018 1900TC Annual Steps 68244 2185 026 2324RC Averaged Monthly Steps 376 157 018 191RC Annual Steps 689 219 026 243

Table 12 LES Model Optimised Parameters

L0 T0 120572 120573TC 1 year 713611 -15833 -02372 01598RC 1 year 71333 -01567 -02368 01591TC 2 years 713611 -15833 -1477x10minus4 7774226RC 2 years 71333 -01567 -1448x10minus4 7898336TC 5 years 713611 -15833 -02813 01880RC 5 years 71333 -01567 -02808 01880

Table 13 LES error measures for different steps-ahead forecasts

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864TC 1 year 10746x10minus5 00033 46831x10minus5 00033RC 1 year 51462x10minus8 22685x10minus4 32408x10minus5 22685x10minus4

TC 2 years 189247 97275 01057 137567RC 2 years 18977 09741 01059 13776TC 5 years 1775531 78881 00951 108422RC 5 years 11759 07889 00952 10844

terms for MSE reaching better values for annual steps thanfor averaged monthly steps while the other figures do betteron averagedmonthly steps Addressing the definition ofMSEthis includes the variance of the estimator as well as its biasbeing equal to its variance in the case of unbiased estimators

Therefore MSE measures the quality of the estimator butalso magnifies estimator deviations from actual values sinceboth positive and negative values are squared and averagedIn contrast RMSE is calculated as the square root of MSEand following the previous analogy stands for the standard

10 Complexity

Table 14 Error Average Improvement for averaged monthly steps before annual steps

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864GBM TCRC 1 Year 53265653 32723587 33333548 32723587GBM TCRC 2 Years 26952635 25751597 2619286 24341369GBM TCRC 5 Years -1173-1104 873709 926769 747565OUP TCRC 1 Year 88088708 63716250 62916319 63686224OUP TCRC 2 Years 46284421 27712617 29993075 22162075OUP TCRC 5 Years 26022553 1093997 13121218 13061269

Table 15 TCRC best error measures for 2013 forecasts after 1000 MC simulations

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864GBM TCRC 1 Yearlowast 33111336 1425143 020020 1425143OUP TCRC 1 Yearlowast 4105042 536054 0076600777 536054LES TCRC 1 Year 107x10minus5514x10minus8 0003226x10minus4 468x10minus5324x10minus5 0003226x10minus4

lowastAveraged monthly steps

Table 16 TCRC Best error measures for 2013ndash2014 forecasts after 1000 MC simulations

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864GBM TCRC 2 Stepslowast 103x1031006 2200242 031034 2447271OUP TCRC 2 Stepslowast 37331376 1573158 0180201813 1900191LES TCRC 2 Steps 18924189 972097 0105701059 1375137lowastAveraged monthly steps

deviation of the estimator if MSE were considered to be thevarianceThough RMSE overreacts when high values of MSEare reached it is less prone to this than MSE since it iscalculated as its squared root thus not accounting for largeerrors as disproportionately as MSE does Furthermore aswe have compared an average of 1000 measures of errorscorresponding to each MC simulation performed the valuesobtained for average RMSE stay below the square root ofaverage MSE which indicates that some of these dispropor-tionate error measures are to some extent distorting thelatter Hence RMSE average values point towards a higheraccuracy for GBM five-year forecasts with averaged monthlysteps which is further endorsed by the average values ofMAD and MAPE thus being the one used for comparisonwith the other two models as shown in Table 17

The final comparison clearly shows how the LES modeloutperforms the other two at all average measures providedfollowed by the OUP model in accuracy although the lattermore than doubles the average MAPE value for LES

The results of simulations indicate that measures of errorstend to either differ slightly or not at all for either forecastsof any timeframe A coherent value with the 101 relation canthen be given with close to the same level of accuracy bymultiplying RC forecasts or dividing TC ones by 10

6 Conclusions

Copper TCRC are a keystone for pricing copper concen-trates which are the actual feedstock for copper smeltersThe potential evolution of TCRC is a question of botheconomic and technical significance forminers as their value

decreases the potential final selling price of concentratesAdditionally copper minersrsquo revenues are more narrowlyrelated to the market price of copper as well as to othertechnical factors such as ore dilution or the grade of theconcentrates produced Smelters on the contrary are hugelyaffected by the discount which they succeed in getting whenpurchasing the concentrates since that makes up the largestpart of their gross revenue besides other secondary sourcesIn addition eventual differences between TCRC may givecommodity traders ludicrous arbitrage opportunities Alsodifferences between short- and long-termTCRC agreementsoffer arbitrage opportunities for traders hence comprisinga part of their revenue in the copper concentrate tradingbusiness as well copper price fluctuations and the capacityto make economically optimum copper blends

As far as we are aware no rigorous research has beencarried out on the behaviour of these discounts Based onhistorical data on TCRC agreed upon in the LME CopperWeek from 2004 to 2017 three potentially suitable forecastingmodels for TCRC annual benchmark values have beencompared through four measures of forecasting accuracyat different horizons These models were chosen firstlydue to their broad implementation and proven capacityin commodity prices forecasting that they all share andsecondly because of the core differences in terms of pricebehaviour with each other

Focusing on the MAPE values achieved those obtainedby the LES model when TC and RC are treated indepen-dently have been significantly lower than for the rest ofthe models Indeed one-year-ahead MAPE measures for TCvalues for the GBM model (20) almost triple those of

Complexity 11

2004 2006 2008 2010 2012 2014405060708090

100

TC (U

SDM

T)

LES TC 1 Step Ahead 2013

2004 2006 2008 2010 2012 2014

LES TC 2 Steps Ahead 2013-2014

2004 2006 2008 2010 2012 2014 2016 2018

LES TC 5 Steps Ahead 2013-2017

2004 2006 2008 2010 2012 2014

LES RC 1 Step Ahead 2013

2004 2006 2008 2010 2012 2014456789

10LES RC 2 Steps Ahead 2013-2014

Observed Forecast

2004 2006 2008 2010 2012 2014 2016 2018

LES RC 5 Steps Ahead 2013-2017

405060708090

100110

405060708090

100

456789

10

RC (U

Sclb

)

456789

1011

Figure 4 One-step-ahead forecasts (2013) two-step-ahead forecasts (2013-2014) and five-step-ahead forecasts (2013-2017) for TC and RCusing the Linear Exponential Smoothing model (LES)

Table 17 TCRC Best error measures for 2013ndash2017 forecasts after 1000 MC simulations

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864GBM TCRC 5 Stepslowast 600x1036165 4651459 049048 5571551OUP TCRC 5 Stepslowast 84334843 2616262 0271502719 2896289LES TCRC 5 Steps 17755117 788078 0095100952 1084108lowastAveraged monthly steps

the OUP model (766) in contrast with the significantlylower values from the LES model (00046) This gaptends to be narrowed when TCRC values are forecastedat longer horizons when most measures of error becomemore even The GBM and OUP models have proven todeliver better accuracy performance when the TCRC valuesare projected monthly and then averaged to obtain annualbenchmark forecasts Even so the LES model remains themost accurate of all with MAPE values of 10 at two-year-ahead forecasts with 18 and 31 for TC for OUP and GBMrespectively

Finally despite TC and RC being two independentdiscounts applied to copper concentrates they are bothset jointly with an often 101 relation as our data revealsThis relation also transcends to simulation results and errormeasures hence showing a negligible discrepancy betweenthe independent forecasting of TC and RC or the jointforecasting of both values keeping the 101 relationThis is forinstance the case of the five-year-ahead OUP MAPE values(0271502719) which were obtained without observing the101 relation in the data A similar level of discrepancy wasobtained at any horizon with any model which indicates thatboth values could be forecasted with the same accuracy usingthe selected model with any of them and then applying the101 relation

Our findings thus suggest that both at short and atlong-term horizons TCRC annual benchmark levels tendto exhibit a pattern which is best fit by an LES model Thisindicates that these critical discounts for the copper tradingbusiness do maintain a certain dependency on past valuesThis would also suggest the existence of cyclical patterns incopper TCRC possibly driven by many of the same marketfactors that move the price of copper

This work contributes by delivering a formal tool forsmelters or miners to make accurate forecasts of TCRCbenchmark levels The level of errors attained indicatesthe LES model may be a valid model to forecast thesecrucial discounts for the copper market In addition ourwork further contributes to helping market participants toproject the price of concentrates with an acceptable degree ofuncertainty as now they may include a fundamental elementfor their estimation This would enable them to optimise theway they produce or process these copper concentrates Alsothe precise knowledge of these discountsrsquo expected behaviourcontributes to letting miners traders and smelters aliketake the maximum advantage from the copper concentratetrading agreements that they are part of As an additionalcontribution this work may well be applied to gold or silverRC which are relevant deduction concentrates when thesehave a significant amount of gold or silver

12 Complexity

Once TCRC annual benchmark levels are able to beforecasted with a certain level of accuracy future researchshould go further into this research through the explorationof the potential impact that other market factors may have onthese discounts

As a limitation of this research we should point out thetimespan of the data considered compared to those of otherforecasting works on commodity prices for example whichuse broader timespans For our case we have consideredthe maximum available sufficiently reliable data on TCRCbenchmark levels starting back in 2004 as there is noreliable data beyond this year Also as another limitationwe have used four measures of error which are among themost frequently used to compare the accuracy of differentmodels However other measures could have been used at anindividual level to test each modelrsquos accuracy

Appendix

A Models

A1 Geometric Brownian Motion (GBM) GBM can bewritten as a generalisation of a Wiener (continuous time-stochasticMarkov process with independent increments andwhose changes over any infinite interval of time are normallydistributed with a variance that increases linearly with thetime interval [22]) process

119889119909 = 120572119909119889119905 + 120590119909119889119911 (A1)

where according to Marathe and Ryan [30] the first termis known as the Expected Value whereas the second is theStochastic component with 120572 being the drift parameter and120590 the volatility of the process Also dz is the Wiener processwhich induces the abovementioned stochastic behaviour inthe model

119889119911 = isin119905radic119889119905 997888rarr isin119905 sim 119873 (0 1) (A2)

TheGBMmodel can be expressed in discrete terms accordingto

119909 = 119909119905 minus 119909119905minus1 = 120572119909119905minus1119905 + 120590119909119905minus1120598radic119905 (A3)

In GBM percentage changes in x 119909119909 are normally dis-tributed thus absolute changes in x 119909 are lognormallydistributed Also the expected value and variance for 119909(119905) are

E [119909 (119905)] = 1199090119890120572119905 (A4)

var [119909 (119905)] = 119909201198902120572119905 (1198901205902119905 minus 1) (A5)

A2 Orstein-Uhlenbeck Process (OUP) TheOUPprocess wasfirst defined by Uhlenbeck and Orstein [31] as an alternativeto the regular Brownian Motion to model the velocity ofthe diffusion movement of a particle that accounts for itslosses due to friction with other particles The OUP processcan be regarded as a modification of Brownian Motion incontinuous time where its properties have been changed

(Stationary Gaussian Markov and stochastic process) [32]These modifications cause the process to move towards acentral position with a stronger attraction the further it isfrom this position As mentioned above the OUP is usuallyemployed to model commodity prices and is the simplestversion of amean-reverting process [22]

119889119878 = 120582 (120583 minus 119878) 119889119905 + 120590119889119882119905 (A6)

where S is the level of prices 120583 the long-term average towhich prices tend to revert and 120582 the speed of reversionAdditionally in a similar fashion to that of the GBM 120590 isthe volatility of the process and 119889119882119905 is a Wiener processwith an identical definition However in contrast to GBMtime intervals in OUP are not independent since differencesbetween current levels of prices S and long-term averageprices 120583 make the expected change in prices dS more likelyeither positive or negative

The discrete version of the model can be expressed asfollows

119878119905 = 120583 (1 minus 119890minus120582Δ119905) + 119890minus120582Δ119905119878119905minus1 + 120590radic1 minus 119890minus2120582Δ1199052120582 119889119882119905 (A7)

where the expected value for 119909(119905) and the variance for (119909(119905) minus120583) areE [119909 (119905)] = 120583 + (1199090 minus 120583) 119890minus120582119905 (A8)

var [119909 (119905) minus 120583] = 12059022120582 (1 minus 119890minus2120582119905) (A9)

It can be derived from previous equations that as timeincreases prices will tend to long-term average levels 120583 Inaddition with large time spans if the speed of reversion 120582becomes high variance tends to 0 On the other hand if thespeed of reversion is 0 then var[119909(119905)] 997888rarr 1205902119905 making theprocess a simple Brownian Motion

A3Holtrsquos Linear Exponential Smoothing (LES) Linear Expo-nential Smoothing models are capable of considering bothlevels and trends at every instant assigning higher weights inthe overall calculation to values closer to the present than toolder ones LES models carry that out by constantly updatinglocal estimations of levels and trends with the intervention ofone or two smoothing constants which enable the models todampen older value effects Although it is possible to employa single smoothing constant for both the level and the trendknown as Brownrsquos LES to use two one for each known asHoltrsquos LES is usually preferred since Brownrsquos LES tends torender estimations of the trend ldquounstablerdquo as suggested byauthors such as Nau [33] Holtrsquos LES model comes definedby the level trend and forecast updating equations each ofthese expressed as follows respectively

119871 119905 = 120572119884119905 + (1 minus 120572) (119871 119905minus1 + 119879119905minus1) (A10)

119879119905 = 120573 (119871 119905 minus 119871 119905minus1) + (1 minus 120573) 119879119905minus1 (A11)

119905+119896 = 119871 119905 + 119896119879119905 (A12)

Complexity 13

With 120572 being the first smoothing constant for the levelsand 120573 the second smoothing constant for the trend Highervalues for the smoothing constants imply that either levels ortrends are changing rapidly over time whereas lower valuesimply the contrary Hence the higher the constant the moreuncertain the future is believed to be

B Model Calibration

Each model has been calibrated individually using two setsof data containing TC and RC levels respectively from 2004to 2012 The available dataset is comprised of data from 2004to 2017 which make the calibration data approximately 23 ofthe total

B1 GBM Increments in the logarithm of variable x aredistributed as follows

(ln119909) sim 119873((120572 minus 12059022 ) 119905 120590119905) (B1)

Hence if m is defined as the sample mean of the differenceof the natural logarithm of the time series for TCRC levelsconsidered for the calibration and n as the number ofincrements of the series considered with n=9

119898 = 1119899119899sum119905=1

(ln119909119905 minus ln119909119905minus1) (B2)

119904 = radic 1119899 minus 1119899sum119894=1

(ln119909119905 minus ln119909119905minus1 minus 119898)2 (B3)

119898 = 120572 minus 12059022 (B4)

119904 = 120590 (B5)

B2 OUP The OUP process is an AR1 process [22] whoseresolution is well presented by Woolridge [34] using OLStechniques fitting the following

119910119905 = 119886 + 119887119910119905minus1 + 120576119905 (B6)

Hence the estimators for the parameters of the OUP modelare obtained by OLS for both TC and RC levels indepen-dently

= minus ln 119887Δ119905 (B7)

120583 = 1198861 minus 119887 (B8)

= 120576119905radic 2 ln (1 + 119887)(1 + 119887)2 minus 1 (B9)

B3 LES A linear regression is conducted on the inputdataset available to find the starting parameters for the LES

model the initial Level 1198710 and the initial value of thetrend 1198790 irrespective of TC values and RC values Hereas recommended by Gardner [35] the use of OLS is highlyadvisable due to the erratic behaviour shown by trends inthe historic data so the obtaining of negative values of 1198780 isprevented Linear regression fulfils the following

119884119905 = 119886119905 + 119887 (B10)

1198710 = 119887 (B11)

1198790 = 119886 (B12)

By fixing the two smoothing constants the values for theforecasts 119905+119896 can be calculated at each step using the modelequations There are multiple references in the literature onwhat the optimum range for each smoothing constant isGardner [35] speaks of setting moderate values for bothparameter less than 03 to obtain the best results Examplespointing out the same may be found in Brown [36] Coutie[37] Harrison [38] and Montgomery and Johnson [39]Also for many applications Makridakis and Hibon [20] andChatfield [40] found that parameter values should fall withinthe range of 03-1 On the other hand McClain and Thomas[41] provided a condition of stability for the nonseasonal LESmodel given by

0 lt 120572 lt 20 lt 120573 lt 4 minus 2120572120572

(B13)

Also the largest possible value of 120572 that allows the avoidanceof areas of oscillation is proposed by McClain and Thomas[41] and McClain [42]

120572 lt 4120573(1 + 120573)2 (B14)

However according to Gardner there is no tangible proofthat this value improves accuracy in any form Nonethelesswe have opted to follow what Nau [33] refers to as ldquotheusual wayrdquo namely minimising the Mean Squared Error(MSE) of the one-step-ahead forecast of TCRC for eachinput data series previously mentioned To do so Matlabrsquosfminsearch function has been used with function and variabletolerance levels of 1x10minus4 as well as a set maximum numberof function iterations and function evaluations of 1x106 tolimit computing resources In Table 18 the actual number ofnecessary iterations to obtain optimum values for smoothingconstants is shown As can be seen the criteria are wellbeyond the final results which ensured that an optimumsolution was reached with assumable computing usage (thesimulation required less than one minute) and with a highdegree of certainty

14 Complexity

Table 18 Iterations on LES smoothing constants optimisation

Iterations Function EvaluationsTC 1-Step 36 40TC 2-Steps 249945 454211TC 5-Steps 40 77RC 1-Step 32 62RC 2-Steps 254388 462226RC 5-Steps 34 66

C Error Calculations

C1 Mean Square Error (MSE)

119872119878119864 = 1119899119899sum119894=1

(119894 minus 119884119894)2 (C1)

where 119894 are the forecasted values and 119884119894 those observedC2 Mean Absolute Deviation (MAD)

119872119860119863 = 1119899119899sum119894=1

10038161003816100381610038161003816119894 minus 11988410038161003816100381610038161003816 (C2)

where 119894 are the forecasted values and 119884 the average value ofall the observations

C3 Mean Absolute Percentage Error (MAPE)

119872119860119875119864 = 1119899119899sum119894=1

10038161003816100381610038161003816119894 minus 11988411989410038161003816100381610038161003816119884119894 (C3)

The above formula is expressed in parts-per-one and is theone used in the calculations conducted here Hence multi-plying the result by 100 would deliver percentage outcomes

C4 Root Mean Square Error (RMSE)

119877119872119878119864 = radic119872119878119864 = radic 1119899119899sum119894=1

(119894 minus 119884119894)2 (C4)

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Supplementary Materials

The Matlab codes developed to carry out the simulationsfor each of the models proposed as well as the datasetsused and the Matlab workspaces with the simulationsoutcomes as they have been reflected in this paper areprovided in separate folders for each model GBM foldercontains the code to make the simulations for the Geo-metric Brownian Motion model ldquoGBMmrdquo as well as aseparate script which allows determining the total numberof monthly step simulations for which errors levels havebeen lower than annual-step simulations ldquoErrCoutermrdquoThe datasets used are included in two excel files ldquoTCxlsxrdquoand ldquoRCxlsxrdquo ldquoGBM OK MonthlyStepsmatrdquo is the Matlabworkspace containing the outcome of monthly step simu-lations whereas ldquoGBM OK AnnualStepsmatrdquo contains theoutcome for annual-step simulations Also ldquoGBM OKmatrdquois the workspace to be loaded prior to performing GBMsimulations MR folder contains the code file to carryout the simulations for the Orstein-Uhlenbeck ProcessldquoMRmrdquo as well as the separate script to compare errorsof monthly step simulations with annual-step simula-tions ldquoErrCountermrdquo ldquoTCxlsxrdquo and ldquoRCxlsxrdquo containthe TCRC benchmark levels from 2004 to 2017 Finallymonthly step simulationsrsquo outcome has been saved inthe workspace ldquoMR OK MonthlyStepsmatrdquo while annual-step simulationsrsquo outcome has been saved in the Matlabworkspace ldquoMR OK AnnualStepsmatrdquo Also ldquoMR OKmatrdquois the workspace to be loaded prior to performing MRsimulations LES folder contains the code file to carry outthe calculations necessary to obtain the LES model forecastsldquoLESmrdquo Three separate Matlab functions are includedldquomapeLESmrdquo ldquomapeLES1mrdquo and ldquomapeLES2mrdquo whichdefine the calculation of the MAPE for the LESrsquo five-year-ahead one-year-ahead and two-year-ahead forecastsAlso ldquoLES OKmatrdquo is the workspace to be loaded prior toperforming LES simulations Finally graphs of each modelhave been included in separate files as well in a subfoldernamed ldquoGraphsrdquo within each of the modelsrsquo folder Figuresare included in Matlabrsquos format (fig) and TIFF format TIFFfigures have been generated in both colours and black andwhite for easier visualization (Supplementary Materials)

References

[1] United Nations International Trade Statistics Yearbook vol 22016

[2] International Copper StudyGroupTheWorld Copper Factbook2017

[3] K M Johnson J M Hammarstrom M L Zientek and CL Dicken ldquoEstimate of undiscovered copper resources of theworld 2013rdquo US Geological Survey 2017

[4] M E Schlesinger and W G Davenport Extractive Metallurgyof Copper Elsevier 5th edition 2011

[5] S Gloser M Soulier and L A T Espinoza ldquoDynamic analysisof global copper flows Global stocks postconsumer materialflows recycling indicators and uncertainty evaluationrdquo Envi-ronmental Science amp Technology vol 47 no 12 pp 6564ndash65722013

10 Complexity

Table 14 Error Average Improvement for averaged monthly steps before annual steps

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864GBM TCRC 1 Year 53265653 32723587 33333548 32723587GBM TCRC 2 Years 26952635 25751597 2619286 24341369GBM TCRC 5 Years -1173-1104 873709 926769 747565OUP TCRC 1 Year 88088708 63716250 62916319 63686224OUP TCRC 2 Years 46284421 27712617 29993075 22162075OUP TCRC 5 Years 26022553 1093997 13121218 13061269

Table 15 TCRC best error measures for 2013 forecasts after 1000 MC simulations

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864GBM TCRC 1 Yearlowast 33111336 1425143 020020 1425143OUP TCRC 1 Yearlowast 4105042 536054 0076600777 536054LES TCRC 1 Year 107x10minus5514x10minus8 0003226x10minus4 468x10minus5324x10minus5 0003226x10minus4

lowastAveraged monthly steps

Table 16 TCRC Best error measures for 2013ndash2014 forecasts after 1000 MC simulations

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864GBM TCRC 2 Stepslowast 103x1031006 2200242 031034 2447271OUP TCRC 2 Stepslowast 37331376 1573158 0180201813 1900191LES TCRC 2 Steps 18924189 972097 0105701059 1375137lowastAveraged monthly steps

deviation of the estimator if MSE were considered to be thevarianceThough RMSE overreacts when high values of MSEare reached it is less prone to this than MSE since it iscalculated as its squared root thus not accounting for largeerrors as disproportionately as MSE does Furthermore aswe have compared an average of 1000 measures of errorscorresponding to each MC simulation performed the valuesobtained for average RMSE stay below the square root ofaverage MSE which indicates that some of these dispropor-tionate error measures are to some extent distorting thelatter Hence RMSE average values point towards a higheraccuracy for GBM five-year forecasts with averaged monthlysteps which is further endorsed by the average values ofMAD and MAPE thus being the one used for comparisonwith the other two models as shown in Table 17

The final comparison clearly shows how the LES modeloutperforms the other two at all average measures providedfollowed by the OUP model in accuracy although the lattermore than doubles the average MAPE value for LES

The results of simulations indicate that measures of errorstend to either differ slightly or not at all for either forecastsof any timeframe A coherent value with the 101 relation canthen be given with close to the same level of accuracy bymultiplying RC forecasts or dividing TC ones by 10

6 Conclusions

Copper TCRC are a keystone for pricing copper concen-trates which are the actual feedstock for copper smeltersThe potential evolution of TCRC is a question of botheconomic and technical significance forminers as their value

decreases the potential final selling price of concentratesAdditionally copper minersrsquo revenues are more narrowlyrelated to the market price of copper as well as to othertechnical factors such as ore dilution or the grade of theconcentrates produced Smelters on the contrary are hugelyaffected by the discount which they succeed in getting whenpurchasing the concentrates since that makes up the largestpart of their gross revenue besides other secondary sourcesIn addition eventual differences between TCRC may givecommodity traders ludicrous arbitrage opportunities Alsodifferences between short- and long-termTCRC agreementsoffer arbitrage opportunities for traders hence comprisinga part of their revenue in the copper concentrate tradingbusiness as well copper price fluctuations and the capacityto make economically optimum copper blends

As far as we are aware no rigorous research has beencarried out on the behaviour of these discounts Based onhistorical data on TCRC agreed upon in the LME CopperWeek from 2004 to 2017 three potentially suitable forecastingmodels for TCRC annual benchmark values have beencompared through four measures of forecasting accuracyat different horizons These models were chosen firstlydue to their broad implementation and proven capacityin commodity prices forecasting that they all share andsecondly because of the core differences in terms of pricebehaviour with each other

Focusing on the MAPE values achieved those obtainedby the LES model when TC and RC are treated indepen-dently have been significantly lower than for the rest ofthe models Indeed one-year-ahead MAPE measures for TCvalues for the GBM model (20) almost triple those of

Complexity 11

2004 2006 2008 2010 2012 2014405060708090

100

TC (U

SDM

T)

LES TC 1 Step Ahead 2013

2004 2006 2008 2010 2012 2014

LES TC 2 Steps Ahead 2013-2014

2004 2006 2008 2010 2012 2014 2016 2018

LES TC 5 Steps Ahead 2013-2017

2004 2006 2008 2010 2012 2014

LES RC 1 Step Ahead 2013

2004 2006 2008 2010 2012 2014456789

10LES RC 2 Steps Ahead 2013-2014

Observed Forecast

2004 2006 2008 2010 2012 2014 2016 2018

LES RC 5 Steps Ahead 2013-2017

405060708090

100110

405060708090

100

456789

10

RC (U

Sclb

)

456789

1011

Figure 4 One-step-ahead forecasts (2013) two-step-ahead forecasts (2013-2014) and five-step-ahead forecasts (2013-2017) for TC and RCusing the Linear Exponential Smoothing model (LES)

Table 17 TCRC Best error measures for 2013ndash2017 forecasts after 1000 MC simulations

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864GBM TCRC 5 Stepslowast 600x1036165 4651459 049048 5571551OUP TCRC 5 Stepslowast 84334843 2616262 0271502719 2896289LES TCRC 5 Steps 17755117 788078 0095100952 1084108lowastAveraged monthly steps

the OUP model (766) in contrast with the significantlylower values from the LES model (00046) This gaptends to be narrowed when TCRC values are forecastedat longer horizons when most measures of error becomemore even The GBM and OUP models have proven todeliver better accuracy performance when the TCRC valuesare projected monthly and then averaged to obtain annualbenchmark forecasts Even so the LES model remains themost accurate of all with MAPE values of 10 at two-year-ahead forecasts with 18 and 31 for TC for OUP and GBMrespectively

Finally despite TC and RC being two independentdiscounts applied to copper concentrates they are bothset jointly with an often 101 relation as our data revealsThis relation also transcends to simulation results and errormeasures hence showing a negligible discrepancy betweenthe independent forecasting of TC and RC or the jointforecasting of both values keeping the 101 relationThis is forinstance the case of the five-year-ahead OUP MAPE values(0271502719) which were obtained without observing the101 relation in the data A similar level of discrepancy wasobtained at any horizon with any model which indicates thatboth values could be forecasted with the same accuracy usingthe selected model with any of them and then applying the101 relation

Our findings thus suggest that both at short and atlong-term horizons TCRC annual benchmark levels tendto exhibit a pattern which is best fit by an LES model Thisindicates that these critical discounts for the copper tradingbusiness do maintain a certain dependency on past valuesThis would also suggest the existence of cyclical patterns incopper TCRC possibly driven by many of the same marketfactors that move the price of copper

This work contributes by delivering a formal tool forsmelters or miners to make accurate forecasts of TCRCbenchmark levels The level of errors attained indicatesthe LES model may be a valid model to forecast thesecrucial discounts for the copper market In addition ourwork further contributes to helping market participants toproject the price of concentrates with an acceptable degree ofuncertainty as now they may include a fundamental elementfor their estimation This would enable them to optimise theway they produce or process these copper concentrates Alsothe precise knowledge of these discountsrsquo expected behaviourcontributes to letting miners traders and smelters aliketake the maximum advantage from the copper concentratetrading agreements that they are part of As an additionalcontribution this work may well be applied to gold or silverRC which are relevant deduction concentrates when thesehave a significant amount of gold or silver

12 Complexity

Once TCRC annual benchmark levels are able to beforecasted with a certain level of accuracy future researchshould go further into this research through the explorationof the potential impact that other market factors may have onthese discounts

As a limitation of this research we should point out thetimespan of the data considered compared to those of otherforecasting works on commodity prices for example whichuse broader timespans For our case we have consideredthe maximum available sufficiently reliable data on TCRCbenchmark levels starting back in 2004 as there is noreliable data beyond this year Also as another limitationwe have used four measures of error which are among themost frequently used to compare the accuracy of differentmodels However other measures could have been used at anindividual level to test each modelrsquos accuracy

Appendix

A Models

A1 Geometric Brownian Motion (GBM) GBM can bewritten as a generalisation of a Wiener (continuous time-stochasticMarkov process with independent increments andwhose changes over any infinite interval of time are normallydistributed with a variance that increases linearly with thetime interval [22]) process

119889119909 = 120572119909119889119905 + 120590119909119889119911 (A1)

where according to Marathe and Ryan [30] the first termis known as the Expected Value whereas the second is theStochastic component with 120572 being the drift parameter and120590 the volatility of the process Also dz is the Wiener processwhich induces the abovementioned stochastic behaviour inthe model

119889119911 = isin119905radic119889119905 997888rarr isin119905 sim 119873 (0 1) (A2)

TheGBMmodel can be expressed in discrete terms accordingto

119909 = 119909119905 minus 119909119905minus1 = 120572119909119905minus1119905 + 120590119909119905minus1120598radic119905 (A3)

In GBM percentage changes in x 119909119909 are normally dis-tributed thus absolute changes in x 119909 are lognormallydistributed Also the expected value and variance for 119909(119905) are

E [119909 (119905)] = 1199090119890120572119905 (A4)

var [119909 (119905)] = 119909201198902120572119905 (1198901205902119905 minus 1) (A5)

A2 Orstein-Uhlenbeck Process (OUP) TheOUPprocess wasfirst defined by Uhlenbeck and Orstein [31] as an alternativeto the regular Brownian Motion to model the velocity ofthe diffusion movement of a particle that accounts for itslosses due to friction with other particles The OUP processcan be regarded as a modification of Brownian Motion incontinuous time where its properties have been changed

(Stationary Gaussian Markov and stochastic process) [32]These modifications cause the process to move towards acentral position with a stronger attraction the further it isfrom this position As mentioned above the OUP is usuallyemployed to model commodity prices and is the simplestversion of amean-reverting process [22]

119889119878 = 120582 (120583 minus 119878) 119889119905 + 120590119889119882119905 (A6)

where S is the level of prices 120583 the long-term average towhich prices tend to revert and 120582 the speed of reversionAdditionally in a similar fashion to that of the GBM 120590 isthe volatility of the process and 119889119882119905 is a Wiener processwith an identical definition However in contrast to GBMtime intervals in OUP are not independent since differencesbetween current levels of prices S and long-term averageprices 120583 make the expected change in prices dS more likelyeither positive or negative

The discrete version of the model can be expressed asfollows

119878119905 = 120583 (1 minus 119890minus120582Δ119905) + 119890minus120582Δ119905119878119905minus1 + 120590radic1 minus 119890minus2120582Δ1199052120582 119889119882119905 (A7)

where the expected value for 119909(119905) and the variance for (119909(119905) minus120583) areE [119909 (119905)] = 120583 + (1199090 minus 120583) 119890minus120582119905 (A8)

var [119909 (119905) minus 120583] = 12059022120582 (1 minus 119890minus2120582119905) (A9)

It can be derived from previous equations that as timeincreases prices will tend to long-term average levels 120583 Inaddition with large time spans if the speed of reversion 120582becomes high variance tends to 0 On the other hand if thespeed of reversion is 0 then var[119909(119905)] 997888rarr 1205902119905 making theprocess a simple Brownian Motion

A3Holtrsquos Linear Exponential Smoothing (LES) Linear Expo-nential Smoothing models are capable of considering bothlevels and trends at every instant assigning higher weights inthe overall calculation to values closer to the present than toolder ones LES models carry that out by constantly updatinglocal estimations of levels and trends with the intervention ofone or two smoothing constants which enable the models todampen older value effects Although it is possible to employa single smoothing constant for both the level and the trendknown as Brownrsquos LES to use two one for each known asHoltrsquos LES is usually preferred since Brownrsquos LES tends torender estimations of the trend ldquounstablerdquo as suggested byauthors such as Nau [33] Holtrsquos LES model comes definedby the level trend and forecast updating equations each ofthese expressed as follows respectively

119871 119905 = 120572119884119905 + (1 minus 120572) (119871 119905minus1 + 119879119905minus1) (A10)

119879119905 = 120573 (119871 119905 minus 119871 119905minus1) + (1 minus 120573) 119879119905minus1 (A11)

119905+119896 = 119871 119905 + 119896119879119905 (A12)

Complexity 13

With 120572 being the first smoothing constant for the levelsand 120573 the second smoothing constant for the trend Highervalues for the smoothing constants imply that either levels ortrends are changing rapidly over time whereas lower valuesimply the contrary Hence the higher the constant the moreuncertain the future is believed to be

B Model Calibration

Each model has been calibrated individually using two setsof data containing TC and RC levels respectively from 2004to 2012 The available dataset is comprised of data from 2004to 2017 which make the calibration data approximately 23 ofthe total

B1 GBM Increments in the logarithm of variable x aredistributed as follows

(ln119909) sim 119873((120572 minus 12059022 ) 119905 120590119905) (B1)

Hence if m is defined as the sample mean of the differenceof the natural logarithm of the time series for TCRC levelsconsidered for the calibration and n as the number ofincrements of the series considered with n=9

119898 = 1119899119899sum119905=1

(ln119909119905 minus ln119909119905minus1) (B2)

119904 = radic 1119899 minus 1119899sum119894=1

(ln119909119905 minus ln119909119905minus1 minus 119898)2 (B3)

119898 = 120572 minus 12059022 (B4)

119904 = 120590 (B5)

B2 OUP The OUP process is an AR1 process [22] whoseresolution is well presented by Woolridge [34] using OLStechniques fitting the following

119910119905 = 119886 + 119887119910119905minus1 + 120576119905 (B6)

Hence the estimators for the parameters of the OUP modelare obtained by OLS for both TC and RC levels indepen-dently

= minus ln 119887Δ119905 (B7)

120583 = 1198861 minus 119887 (B8)

= 120576119905radic 2 ln (1 + 119887)(1 + 119887)2 minus 1 (B9)

B3 LES A linear regression is conducted on the inputdataset available to find the starting parameters for the LES

model the initial Level 1198710 and the initial value of thetrend 1198790 irrespective of TC values and RC values Hereas recommended by Gardner [35] the use of OLS is highlyadvisable due to the erratic behaviour shown by trends inthe historic data so the obtaining of negative values of 1198780 isprevented Linear regression fulfils the following

119884119905 = 119886119905 + 119887 (B10)

1198710 = 119887 (B11)

1198790 = 119886 (B12)

By fixing the two smoothing constants the values for theforecasts 119905+119896 can be calculated at each step using the modelequations There are multiple references in the literature onwhat the optimum range for each smoothing constant isGardner [35] speaks of setting moderate values for bothparameter less than 03 to obtain the best results Examplespointing out the same may be found in Brown [36] Coutie[37] Harrison [38] and Montgomery and Johnson [39]Also for many applications Makridakis and Hibon [20] andChatfield [40] found that parameter values should fall withinthe range of 03-1 On the other hand McClain and Thomas[41] provided a condition of stability for the nonseasonal LESmodel given by

0 lt 120572 lt 20 lt 120573 lt 4 minus 2120572120572

(B13)

Also the largest possible value of 120572 that allows the avoidanceof areas of oscillation is proposed by McClain and Thomas[41] and McClain [42]

120572 lt 4120573(1 + 120573)2 (B14)

However according to Gardner there is no tangible proofthat this value improves accuracy in any form Nonethelesswe have opted to follow what Nau [33] refers to as ldquotheusual wayrdquo namely minimising the Mean Squared Error(MSE) of the one-step-ahead forecast of TCRC for eachinput data series previously mentioned To do so Matlabrsquosfminsearch function has been used with function and variabletolerance levels of 1x10minus4 as well as a set maximum numberof function iterations and function evaluations of 1x106 tolimit computing resources In Table 18 the actual number ofnecessary iterations to obtain optimum values for smoothingconstants is shown As can be seen the criteria are wellbeyond the final results which ensured that an optimumsolution was reached with assumable computing usage (thesimulation required less than one minute) and with a highdegree of certainty

14 Complexity

Table 18 Iterations on LES smoothing constants optimisation

Iterations Function EvaluationsTC 1-Step 36 40TC 2-Steps 249945 454211TC 5-Steps 40 77RC 1-Step 32 62RC 2-Steps 254388 462226RC 5-Steps 34 66

C Error Calculations

C1 Mean Square Error (MSE)

119872119878119864 = 1119899119899sum119894=1

(119894 minus 119884119894)2 (C1)

where 119894 are the forecasted values and 119884119894 those observedC2 Mean Absolute Deviation (MAD)

119872119860119863 = 1119899119899sum119894=1

10038161003816100381610038161003816119894 minus 11988410038161003816100381610038161003816 (C2)

where 119894 are the forecasted values and 119884 the average value ofall the observations

C3 Mean Absolute Percentage Error (MAPE)

119872119860119875119864 = 1119899119899sum119894=1

10038161003816100381610038161003816119894 minus 11988411989410038161003816100381610038161003816119884119894 (C3)

The above formula is expressed in parts-per-one and is theone used in the calculations conducted here Hence multi-plying the result by 100 would deliver percentage outcomes

C4 Root Mean Square Error (RMSE)

119877119872119878119864 = radic119872119878119864 = radic 1119899119899sum119894=1

(119894 minus 119884119894)2 (C4)

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Supplementary Materials

The Matlab codes developed to carry out the simulationsfor each of the models proposed as well as the datasetsused and the Matlab workspaces with the simulationsoutcomes as they have been reflected in this paper areprovided in separate folders for each model GBM foldercontains the code to make the simulations for the Geo-metric Brownian Motion model ldquoGBMmrdquo as well as aseparate script which allows determining the total numberof monthly step simulations for which errors levels havebeen lower than annual-step simulations ldquoErrCoutermrdquoThe datasets used are included in two excel files ldquoTCxlsxrdquoand ldquoRCxlsxrdquo ldquoGBM OK MonthlyStepsmatrdquo is the Matlabworkspace containing the outcome of monthly step simu-lations whereas ldquoGBM OK AnnualStepsmatrdquo contains theoutcome for annual-step simulations Also ldquoGBM OKmatrdquois the workspace to be loaded prior to performing GBMsimulations MR folder contains the code file to carryout the simulations for the Orstein-Uhlenbeck ProcessldquoMRmrdquo as well as the separate script to compare errorsof monthly step simulations with annual-step simula-tions ldquoErrCountermrdquo ldquoTCxlsxrdquo and ldquoRCxlsxrdquo containthe TCRC benchmark levels from 2004 to 2017 Finallymonthly step simulationsrsquo outcome has been saved inthe workspace ldquoMR OK MonthlyStepsmatrdquo while annual-step simulationsrsquo outcome has been saved in the Matlabworkspace ldquoMR OK AnnualStepsmatrdquo Also ldquoMR OKmatrdquois the workspace to be loaded prior to performing MRsimulations LES folder contains the code file to carry outthe calculations necessary to obtain the LES model forecastsldquoLESmrdquo Three separate Matlab functions are includedldquomapeLESmrdquo ldquomapeLES1mrdquo and ldquomapeLES2mrdquo whichdefine the calculation of the MAPE for the LESrsquo five-year-ahead one-year-ahead and two-year-ahead forecastsAlso ldquoLES OKmatrdquo is the workspace to be loaded prior toperforming LES simulations Finally graphs of each modelhave been included in separate files as well in a subfoldernamed ldquoGraphsrdquo within each of the modelsrsquo folder Figuresare included in Matlabrsquos format (fig) and TIFF format TIFFfigures have been generated in both colours and black andwhite for easier visualization (Supplementary Materials)

References

[1] United Nations International Trade Statistics Yearbook vol 22016

[2] International Copper StudyGroupTheWorld Copper Factbook2017

[3] K M Johnson J M Hammarstrom M L Zientek and CL Dicken ldquoEstimate of undiscovered copper resources of theworld 2013rdquo US Geological Survey 2017

[4] M E Schlesinger and W G Davenport Extractive Metallurgyof Copper Elsevier 5th edition 2011

[5] S Gloser M Soulier and L A T Espinoza ldquoDynamic analysisof global copper flows Global stocks postconsumer materialflows recycling indicators and uncertainty evaluationrdquo Envi-ronmental Science amp Technology vol 47 no 12 pp 6564ndash65722013

Complexity 11

2004 2006 2008 2010 2012 2014405060708090

100

TC (U

SDM

T)

LES TC 1 Step Ahead 2013

2004 2006 2008 2010 2012 2014

LES TC 2 Steps Ahead 2013-2014

2004 2006 2008 2010 2012 2014 2016 2018

LES TC 5 Steps Ahead 2013-2017

2004 2006 2008 2010 2012 2014

LES RC 1 Step Ahead 2013

2004 2006 2008 2010 2012 2014456789

10LES RC 2 Steps Ahead 2013-2014

Observed Forecast

2004 2006 2008 2010 2012 2014 2016 2018

LES RC 5 Steps Ahead 2013-2017

405060708090

100110

405060708090

100

456789

10

RC (U

Sclb

)

456789

1011

Figure 4 One-step-ahead forecasts (2013) two-step-ahead forecasts (2013-2014) and five-step-ahead forecasts (2013-2017) for TC and RCusing the Linear Exponential Smoothing model (LES)

Table 17 TCRC Best error measures for 2013ndash2017 forecasts after 1000 MC simulations

119872119878119864 119872119860119863 119872119860119875119864 119877119872119878119864GBM TCRC 5 Stepslowast 600x1036165 4651459 049048 5571551OUP TCRC 5 Stepslowast 84334843 2616262 0271502719 2896289LES TCRC 5 Steps 17755117 788078 0095100952 1084108lowastAveraged monthly steps

the OUP model (766) in contrast with the significantlylower values from the LES model (00046) This gaptends to be narrowed when TCRC values are forecastedat longer horizons when most measures of error becomemore even The GBM and OUP models have proven todeliver better accuracy performance when the TCRC valuesare projected monthly and then averaged to obtain annualbenchmark forecasts Even so the LES model remains themost accurate of all with MAPE values of 10 at two-year-ahead forecasts with 18 and 31 for TC for OUP and GBMrespectively

Finally despite TC and RC being two independentdiscounts applied to copper concentrates they are bothset jointly with an often 101 relation as our data revealsThis relation also transcends to simulation results and errormeasures hence showing a negligible discrepancy betweenthe independent forecasting of TC and RC or the jointforecasting of both values keeping the 101 relationThis is forinstance the case of the five-year-ahead OUP MAPE values(0271502719) which were obtained without observing the101 relation in the data A similar level of discrepancy wasobtained at any horizon with any model which indicates thatboth values could be forecasted with the same accuracy usingthe selected model with any of them and then applying the101 relation

Our findings thus suggest that both at short and atlong-term horizons TCRC annual benchmark levels tendto exhibit a pattern which is best fit by an LES model Thisindicates that these critical discounts for the copper tradingbusiness do maintain a certain dependency on past valuesThis would also suggest the existence of cyclical patterns incopper TCRC possibly driven by many of the same marketfactors that move the price of copper

This work contributes by delivering a formal tool forsmelters or miners to make accurate forecasts of TCRCbenchmark levels The level of errors attained indicatesthe LES model may be a valid model to forecast thesecrucial discounts for the copper market In addition ourwork further contributes to helping market participants toproject the price of concentrates with an acceptable degree ofuncertainty as now they may include a fundamental elementfor their estimation This would enable them to optimise theway they produce or process these copper concentrates Alsothe precise knowledge of these discountsrsquo expected behaviourcontributes to letting miners traders and smelters aliketake the maximum advantage from the copper concentratetrading agreements that they are part of As an additionalcontribution this work may well be applied to gold or silverRC which are relevant deduction concentrates when thesehave a significant amount of gold or silver

12 Complexity

Once TCRC annual benchmark levels are able to beforecasted with a certain level of accuracy future researchshould go further into this research through the explorationof the potential impact that other market factors may have onthese discounts

As a limitation of this research we should point out thetimespan of the data considered compared to those of otherforecasting works on commodity prices for example whichuse broader timespans For our case we have consideredthe maximum available sufficiently reliable data on TCRCbenchmark levels starting back in 2004 as there is noreliable data beyond this year Also as another limitationwe have used four measures of error which are among themost frequently used to compare the accuracy of differentmodels However other measures could have been used at anindividual level to test each modelrsquos accuracy

Appendix

A Models

A1 Geometric Brownian Motion (GBM) GBM can bewritten as a generalisation of a Wiener (continuous time-stochasticMarkov process with independent increments andwhose changes over any infinite interval of time are normallydistributed with a variance that increases linearly with thetime interval [22]) process

119889119909 = 120572119909119889119905 + 120590119909119889119911 (A1)

where according to Marathe and Ryan [30] the first termis known as the Expected Value whereas the second is theStochastic component with 120572 being the drift parameter and120590 the volatility of the process Also dz is the Wiener processwhich induces the abovementioned stochastic behaviour inthe model

119889119911 = isin119905radic119889119905 997888rarr isin119905 sim 119873 (0 1) (A2)

TheGBMmodel can be expressed in discrete terms accordingto

119909 = 119909119905 minus 119909119905minus1 = 120572119909119905minus1119905 + 120590119909119905minus1120598radic119905 (A3)

In GBM percentage changes in x 119909119909 are normally dis-tributed thus absolute changes in x 119909 are lognormallydistributed Also the expected value and variance for 119909(119905) are

E [119909 (119905)] = 1199090119890120572119905 (A4)

var [119909 (119905)] = 119909201198902120572119905 (1198901205902119905 minus 1) (A5)

A2 Orstein-Uhlenbeck Process (OUP) TheOUPprocess wasfirst defined by Uhlenbeck and Orstein [31] as an alternativeto the regular Brownian Motion to model the velocity ofthe diffusion movement of a particle that accounts for itslosses due to friction with other particles The OUP processcan be regarded as a modification of Brownian Motion incontinuous time where its properties have been changed

(Stationary Gaussian Markov and stochastic process) [32]These modifications cause the process to move towards acentral position with a stronger attraction the further it isfrom this position As mentioned above the OUP is usuallyemployed to model commodity prices and is the simplestversion of amean-reverting process [22]

119889119878 = 120582 (120583 minus 119878) 119889119905 + 120590119889119882119905 (A6)

where S is the level of prices 120583 the long-term average towhich prices tend to revert and 120582 the speed of reversionAdditionally in a similar fashion to that of the GBM 120590 isthe volatility of the process and 119889119882119905 is a Wiener processwith an identical definition However in contrast to GBMtime intervals in OUP are not independent since differencesbetween current levels of prices S and long-term averageprices 120583 make the expected change in prices dS more likelyeither positive or negative

The discrete version of the model can be expressed asfollows

119878119905 = 120583 (1 minus 119890minus120582Δ119905) + 119890minus120582Δ119905119878119905minus1 + 120590radic1 minus 119890minus2120582Δ1199052120582 119889119882119905 (A7)

where the expected value for 119909(119905) and the variance for (119909(119905) minus120583) areE [119909 (119905)] = 120583 + (1199090 minus 120583) 119890minus120582119905 (A8)

var [119909 (119905) minus 120583] = 12059022120582 (1 minus 119890minus2120582119905) (A9)

It can be derived from previous equations that as timeincreases prices will tend to long-term average levels 120583 Inaddition with large time spans if the speed of reversion 120582becomes high variance tends to 0 On the other hand if thespeed of reversion is 0 then var[119909(119905)] 997888rarr 1205902119905 making theprocess a simple Brownian Motion

A3Holtrsquos Linear Exponential Smoothing (LES) Linear Expo-nential Smoothing models are capable of considering bothlevels and trends at every instant assigning higher weights inthe overall calculation to values closer to the present than toolder ones LES models carry that out by constantly updatinglocal estimations of levels and trends with the intervention ofone or two smoothing constants which enable the models todampen older value effects Although it is possible to employa single smoothing constant for both the level and the trendknown as Brownrsquos LES to use two one for each known asHoltrsquos LES is usually preferred since Brownrsquos LES tends torender estimations of the trend ldquounstablerdquo as suggested byauthors such as Nau [33] Holtrsquos LES model comes definedby the level trend and forecast updating equations each ofthese expressed as follows respectively

119871 119905 = 120572119884119905 + (1 minus 120572) (119871 119905minus1 + 119879119905minus1) (A10)

119879119905 = 120573 (119871 119905 minus 119871 119905minus1) + (1 minus 120573) 119879119905minus1 (A11)

119905+119896 = 119871 119905 + 119896119879119905 (A12)

Complexity 13

With 120572 being the first smoothing constant for the levelsand 120573 the second smoothing constant for the trend Highervalues for the smoothing constants imply that either levels ortrends are changing rapidly over time whereas lower valuesimply the contrary Hence the higher the constant the moreuncertain the future is believed to be

B Model Calibration

Each model has been calibrated individually using two setsof data containing TC and RC levels respectively from 2004to 2012 The available dataset is comprised of data from 2004to 2017 which make the calibration data approximately 23 ofthe total

B1 GBM Increments in the logarithm of variable x aredistributed as follows

(ln119909) sim 119873((120572 minus 12059022 ) 119905 120590119905) (B1)

Hence if m is defined as the sample mean of the differenceof the natural logarithm of the time series for TCRC levelsconsidered for the calibration and n as the number ofincrements of the series considered with n=9

119898 = 1119899119899sum119905=1

(ln119909119905 minus ln119909119905minus1) (B2)

119904 = radic 1119899 minus 1119899sum119894=1

(ln119909119905 minus ln119909119905minus1 minus 119898)2 (B3)

119898 = 120572 minus 12059022 (B4)

119904 = 120590 (B5)

B2 OUP The OUP process is an AR1 process [22] whoseresolution is well presented by Woolridge [34] using OLStechniques fitting the following

119910119905 = 119886 + 119887119910119905minus1 + 120576119905 (B6)

Hence the estimators for the parameters of the OUP modelare obtained by OLS for both TC and RC levels indepen-dently

= minus ln 119887Δ119905 (B7)

120583 = 1198861 minus 119887 (B8)

= 120576119905radic 2 ln (1 + 119887)(1 + 119887)2 minus 1 (B9)

B3 LES A linear regression is conducted on the inputdataset available to find the starting parameters for the LES

model the initial Level 1198710 and the initial value of thetrend 1198790 irrespective of TC values and RC values Hereas recommended by Gardner [35] the use of OLS is highlyadvisable due to the erratic behaviour shown by trends inthe historic data so the obtaining of negative values of 1198780 isprevented Linear regression fulfils the following

119884119905 = 119886119905 + 119887 (B10)

1198710 = 119887 (B11)

1198790 = 119886 (B12)

By fixing the two smoothing constants the values for theforecasts 119905+119896 can be calculated at each step using the modelequations There are multiple references in the literature onwhat the optimum range for each smoothing constant isGardner [35] speaks of setting moderate values for bothparameter less than 03 to obtain the best results Examplespointing out the same may be found in Brown [36] Coutie[37] Harrison [38] and Montgomery and Johnson [39]Also for many applications Makridakis and Hibon [20] andChatfield [40] found that parameter values should fall withinthe range of 03-1 On the other hand McClain and Thomas[41] provided a condition of stability for the nonseasonal LESmodel given by

0 lt 120572 lt 20 lt 120573 lt 4 minus 2120572120572

(B13)

Also the largest possible value of 120572 that allows the avoidanceof areas of oscillation is proposed by McClain and Thomas[41] and McClain [42]

120572 lt 4120573(1 + 120573)2 (B14)

However according to Gardner there is no tangible proofthat this value improves accuracy in any form Nonethelesswe have opted to follow what Nau [33] refers to as ldquotheusual wayrdquo namely minimising the Mean Squared Error(MSE) of the one-step-ahead forecast of TCRC for eachinput data series previously mentioned To do so Matlabrsquosfminsearch function has been used with function and variabletolerance levels of 1x10minus4 as well as a set maximum numberof function iterations and function evaluations of 1x106 tolimit computing resources In Table 18 the actual number ofnecessary iterations to obtain optimum values for smoothingconstants is shown As can be seen the criteria are wellbeyond the final results which ensured that an optimumsolution was reached with assumable computing usage (thesimulation required less than one minute) and with a highdegree of certainty

14 Complexity

Table 18 Iterations on LES smoothing constants optimisation

Iterations Function EvaluationsTC 1-Step 36 40TC 2-Steps 249945 454211TC 5-Steps 40 77RC 1-Step 32 62RC 2-Steps 254388 462226RC 5-Steps 34 66

C Error Calculations

C1 Mean Square Error (MSE)

119872119878119864 = 1119899119899sum119894=1

(119894 minus 119884119894)2 (C1)

where 119894 are the forecasted values and 119884119894 those observedC2 Mean Absolute Deviation (MAD)

119872119860119863 = 1119899119899sum119894=1

10038161003816100381610038161003816119894 minus 11988410038161003816100381610038161003816 (C2)

where 119894 are the forecasted values and 119884 the average value ofall the observations

C3 Mean Absolute Percentage Error (MAPE)

119872119860119875119864 = 1119899119899sum119894=1

10038161003816100381610038161003816119894 minus 11988411989410038161003816100381610038161003816119884119894 (C3)

The above formula is expressed in parts-per-one and is theone used in the calculations conducted here Hence multi-plying the result by 100 would deliver percentage outcomes

C4 Root Mean Square Error (RMSE)

119877119872119878119864 = radic119872119878119864 = radic 1119899119899sum119894=1

(119894 minus 119884119894)2 (C4)

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Supplementary Materials

The Matlab codes developed to carry out the simulationsfor each of the models proposed as well as the datasetsused and the Matlab workspaces with the simulationsoutcomes as they have been reflected in this paper areprovided in separate folders for each model GBM foldercontains the code to make the simulations for the Geo-metric Brownian Motion model ldquoGBMmrdquo as well as aseparate script which allows determining the total numberof monthly step simulations for which errors levels havebeen lower than annual-step simulations ldquoErrCoutermrdquoThe datasets used are included in two excel files ldquoTCxlsxrdquoand ldquoRCxlsxrdquo ldquoGBM OK MonthlyStepsmatrdquo is the Matlabworkspace containing the outcome of monthly step simu-lations whereas ldquoGBM OK AnnualStepsmatrdquo contains theoutcome for annual-step simulations Also ldquoGBM OKmatrdquois the workspace to be loaded prior to performing GBMsimulations MR folder contains the code file to carryout the simulations for the Orstein-Uhlenbeck ProcessldquoMRmrdquo as well as the separate script to compare errorsof monthly step simulations with annual-step simula-tions ldquoErrCountermrdquo ldquoTCxlsxrdquo and ldquoRCxlsxrdquo containthe TCRC benchmark levels from 2004 to 2017 Finallymonthly step simulationsrsquo outcome has been saved inthe workspace ldquoMR OK MonthlyStepsmatrdquo while annual-step simulationsrsquo outcome has been saved in the Matlabworkspace ldquoMR OK AnnualStepsmatrdquo Also ldquoMR OKmatrdquois the workspace to be loaded prior to performing MRsimulations LES folder contains the code file to carry outthe calculations necessary to obtain the LES model forecastsldquoLESmrdquo Three separate Matlab functions are includedldquomapeLESmrdquo ldquomapeLES1mrdquo and ldquomapeLES2mrdquo whichdefine the calculation of the MAPE for the LESrsquo five-year-ahead one-year-ahead and two-year-ahead forecastsAlso ldquoLES OKmatrdquo is the workspace to be loaded prior toperforming LES simulations Finally graphs of each modelhave been included in separate files as well in a subfoldernamed ldquoGraphsrdquo within each of the modelsrsquo folder Figuresare included in Matlabrsquos format (fig) and TIFF format TIFFfigures have been generated in both colours and black andwhite for easier visualization (Supplementary Materials)

References

[1] United Nations International Trade Statistics Yearbook vol 22016

[2] International Copper StudyGroupTheWorld Copper Factbook2017

[3] K M Johnson J M Hammarstrom M L Zientek and CL Dicken ldquoEstimate of undiscovered copper resources of theworld 2013rdquo US Geological Survey 2017

[4] M E Schlesinger and W G Davenport Extractive Metallurgyof Copper Elsevier 5th edition 2011

[5] S Gloser M Soulier and L A T Espinoza ldquoDynamic analysisof global copper flows Global stocks postconsumer materialflows recycling indicators and uncertainty evaluationrdquo Envi-ronmental Science amp Technology vol 47 no 12 pp 6564ndash65722013

12 Complexity

Once TCRC annual benchmark levels are able to beforecasted with a certain level of accuracy future researchshould go further into this research through the explorationof the potential impact that other market factors may have onthese discounts

As a limitation of this research we should point out thetimespan of the data considered compared to those of otherforecasting works on commodity prices for example whichuse broader timespans For our case we have consideredthe maximum available sufficiently reliable data on TCRCbenchmark levels starting back in 2004 as there is noreliable data beyond this year Also as another limitationwe have used four measures of error which are among themost frequently used to compare the accuracy of differentmodels However other measures could have been used at anindividual level to test each modelrsquos accuracy

Appendix

A Models

A1 Geometric Brownian Motion (GBM) GBM can bewritten as a generalisation of a Wiener (continuous time-stochasticMarkov process with independent increments andwhose changes over any infinite interval of time are normallydistributed with a variance that increases linearly with thetime interval [22]) process

119889119909 = 120572119909119889119905 + 120590119909119889119911 (A1)

where according to Marathe and Ryan [30] the first termis known as the Expected Value whereas the second is theStochastic component with 120572 being the drift parameter and120590 the volatility of the process Also dz is the Wiener processwhich induces the abovementioned stochastic behaviour inthe model

119889119911 = isin119905radic119889119905 997888rarr isin119905 sim 119873 (0 1) (A2)

TheGBMmodel can be expressed in discrete terms accordingto

119909 = 119909119905 minus 119909119905minus1 = 120572119909119905minus1119905 + 120590119909119905minus1120598radic119905 (A3)

In GBM percentage changes in x 119909119909 are normally dis-tributed thus absolute changes in x 119909 are lognormallydistributed Also the expected value and variance for 119909(119905) are

E [119909 (119905)] = 1199090119890120572119905 (A4)

var [119909 (119905)] = 119909201198902120572119905 (1198901205902119905 minus 1) (A5)

A2 Orstein-Uhlenbeck Process (OUP) TheOUPprocess wasfirst defined by Uhlenbeck and Orstein [31] as an alternativeto the regular Brownian Motion to model the velocity ofthe diffusion movement of a particle that accounts for itslosses due to friction with other particles The OUP processcan be regarded as a modification of Brownian Motion incontinuous time where its properties have been changed

(Stationary Gaussian Markov and stochastic process) [32]These modifications cause the process to move towards acentral position with a stronger attraction the further it isfrom this position As mentioned above the OUP is usuallyemployed to model commodity prices and is the simplestversion of amean-reverting process [22]

119889119878 = 120582 (120583 minus 119878) 119889119905 + 120590119889119882119905 (A6)

where S is the level of prices 120583 the long-term average towhich prices tend to revert and 120582 the speed of reversionAdditionally in a similar fashion to that of the GBM 120590 isthe volatility of the process and 119889119882119905 is a Wiener processwith an identical definition However in contrast to GBMtime intervals in OUP are not independent since differencesbetween current levels of prices S and long-term averageprices 120583 make the expected change in prices dS more likelyeither positive or negative

The discrete version of the model can be expressed asfollows

119878119905 = 120583 (1 minus 119890minus120582Δ119905) + 119890minus120582Δ119905119878119905minus1 + 120590radic1 minus 119890minus2120582Δ1199052120582 119889119882119905 (A7)

where the expected value for 119909(119905) and the variance for (119909(119905) minus120583) areE [119909 (119905)] = 120583 + (1199090 minus 120583) 119890minus120582119905 (A8)

var [119909 (119905) minus 120583] = 12059022120582 (1 minus 119890minus2120582119905) (A9)

It can be derived from previous equations that as timeincreases prices will tend to long-term average levels 120583 Inaddition with large time spans if the speed of reversion 120582becomes high variance tends to 0 On the other hand if thespeed of reversion is 0 then var[119909(119905)] 997888rarr 1205902119905 making theprocess a simple Brownian Motion

A3Holtrsquos Linear Exponential Smoothing (LES) Linear Expo-nential Smoothing models are capable of considering bothlevels and trends at every instant assigning higher weights inthe overall calculation to values closer to the present than toolder ones LES models carry that out by constantly updatinglocal estimations of levels and trends with the intervention ofone or two smoothing constants which enable the models todampen older value effects Although it is possible to employa single smoothing constant for both the level and the trendknown as Brownrsquos LES to use two one for each known asHoltrsquos LES is usually preferred since Brownrsquos LES tends torender estimations of the trend ldquounstablerdquo as suggested byauthors such as Nau [33] Holtrsquos LES model comes definedby the level trend and forecast updating equations each ofthese expressed as follows respectively

119871 119905 = 120572119884119905 + (1 minus 120572) (119871 119905minus1 + 119879119905minus1) (A10)

119879119905 = 120573 (119871 119905 minus 119871 119905minus1) + (1 minus 120573) 119879119905minus1 (A11)

119905+119896 = 119871 119905 + 119896119879119905 (A12)

Complexity 13

With 120572 being the first smoothing constant for the levelsand 120573 the second smoothing constant for the trend Highervalues for the smoothing constants imply that either levels ortrends are changing rapidly over time whereas lower valuesimply the contrary Hence the higher the constant the moreuncertain the future is believed to be

B Model Calibration

Each model has been calibrated individually using two setsof data containing TC and RC levels respectively from 2004to 2012 The available dataset is comprised of data from 2004to 2017 which make the calibration data approximately 23 ofthe total

B1 GBM Increments in the logarithm of variable x aredistributed as follows

(ln119909) sim 119873((120572 minus 12059022 ) 119905 120590119905) (B1)

Hence if m is defined as the sample mean of the differenceof the natural logarithm of the time series for TCRC levelsconsidered for the calibration and n as the number ofincrements of the series considered with n=9

119898 = 1119899119899sum119905=1

(ln119909119905 minus ln119909119905minus1) (B2)

119904 = radic 1119899 minus 1119899sum119894=1

(ln119909119905 minus ln119909119905minus1 minus 119898)2 (B3)

119898 = 120572 minus 12059022 (B4)

119904 = 120590 (B5)

B2 OUP The OUP process is an AR1 process [22] whoseresolution is well presented by Woolridge [34] using OLStechniques fitting the following

119910119905 = 119886 + 119887119910119905minus1 + 120576119905 (B6)

Hence the estimators for the parameters of the OUP modelare obtained by OLS for both TC and RC levels indepen-dently

= minus ln 119887Δ119905 (B7)

120583 = 1198861 minus 119887 (B8)

= 120576119905radic 2 ln (1 + 119887)(1 + 119887)2 minus 1 (B9)

B3 LES A linear regression is conducted on the inputdataset available to find the starting parameters for the LES

model the initial Level 1198710 and the initial value of thetrend 1198790 irrespective of TC values and RC values Hereas recommended by Gardner [35] the use of OLS is highlyadvisable due to the erratic behaviour shown by trends inthe historic data so the obtaining of negative values of 1198780 isprevented Linear regression fulfils the following

119884119905 = 119886119905 + 119887 (B10)

1198710 = 119887 (B11)

1198790 = 119886 (B12)

By fixing the two smoothing constants the values for theforecasts 119905+119896 can be calculated at each step using the modelequations There are multiple references in the literature onwhat the optimum range for each smoothing constant isGardner [35] speaks of setting moderate values for bothparameter less than 03 to obtain the best results Examplespointing out the same may be found in Brown [36] Coutie[37] Harrison [38] and Montgomery and Johnson [39]Also for many applications Makridakis and Hibon [20] andChatfield [40] found that parameter values should fall withinthe range of 03-1 On the other hand McClain and Thomas[41] provided a condition of stability for the nonseasonal LESmodel given by

0 lt 120572 lt 20 lt 120573 lt 4 minus 2120572120572

(B13)

Also the largest possible value of 120572 that allows the avoidanceof areas of oscillation is proposed by McClain and Thomas[41] and McClain [42]

120572 lt 4120573(1 + 120573)2 (B14)

However according to Gardner there is no tangible proofthat this value improves accuracy in any form Nonethelesswe have opted to follow what Nau [33] refers to as ldquotheusual wayrdquo namely minimising the Mean Squared Error(MSE) of the one-step-ahead forecast of TCRC for eachinput data series previously mentioned To do so Matlabrsquosfminsearch function has been used with function and variabletolerance levels of 1x10minus4 as well as a set maximum numberof function iterations and function evaluations of 1x106 tolimit computing resources In Table 18 the actual number ofnecessary iterations to obtain optimum values for smoothingconstants is shown As can be seen the criteria are wellbeyond the final results which ensured that an optimumsolution was reached with assumable computing usage (thesimulation required less than one minute) and with a highdegree of certainty

14 Complexity

Table 18 Iterations on LES smoothing constants optimisation

Iterations Function EvaluationsTC 1-Step 36 40TC 2-Steps 249945 454211TC 5-Steps 40 77RC 1-Step 32 62RC 2-Steps 254388 462226RC 5-Steps 34 66

C Error Calculations

C1 Mean Square Error (MSE)

119872119878119864 = 1119899119899sum119894=1

(119894 minus 119884119894)2 (C1)

where 119894 are the forecasted values and 119884119894 those observedC2 Mean Absolute Deviation (MAD)

119872119860119863 = 1119899119899sum119894=1

10038161003816100381610038161003816119894 minus 11988410038161003816100381610038161003816 (C2)

where 119894 are the forecasted values and 119884 the average value ofall the observations

C3 Mean Absolute Percentage Error (MAPE)

119872119860119875119864 = 1119899119899sum119894=1

10038161003816100381610038161003816119894 minus 11988411989410038161003816100381610038161003816119884119894 (C3)

The above formula is expressed in parts-per-one and is theone used in the calculations conducted here Hence multi-plying the result by 100 would deliver percentage outcomes

C4 Root Mean Square Error (RMSE)

119877119872119878119864 = radic119872119878119864 = radic 1119899119899sum119894=1

(119894 minus 119884119894)2 (C4)

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Supplementary Materials

The Matlab codes developed to carry out the simulationsfor each of the models proposed as well as the datasetsused and the Matlab workspaces with the simulationsoutcomes as they have been reflected in this paper areprovided in separate folders for each model GBM foldercontains the code to make the simulations for the Geo-metric Brownian Motion model ldquoGBMmrdquo as well as aseparate script which allows determining the total numberof monthly step simulations for which errors levels havebeen lower than annual-step simulations ldquoErrCoutermrdquoThe datasets used are included in two excel files ldquoTCxlsxrdquoand ldquoRCxlsxrdquo ldquoGBM OK MonthlyStepsmatrdquo is the Matlabworkspace containing the outcome of monthly step simu-lations whereas ldquoGBM OK AnnualStepsmatrdquo contains theoutcome for annual-step simulations Also ldquoGBM OKmatrdquois the workspace to be loaded prior to performing GBMsimulations MR folder contains the code file to carryout the simulations for the Orstein-Uhlenbeck ProcessldquoMRmrdquo as well as the separate script to compare errorsof monthly step simulations with annual-step simula-tions ldquoErrCountermrdquo ldquoTCxlsxrdquo and ldquoRCxlsxrdquo containthe TCRC benchmark levels from 2004 to 2017 Finallymonthly step simulationsrsquo outcome has been saved inthe workspace ldquoMR OK MonthlyStepsmatrdquo while annual-step simulationsrsquo outcome has been saved in the Matlabworkspace ldquoMR OK AnnualStepsmatrdquo Also ldquoMR OKmatrdquois the workspace to be loaded prior to performing MRsimulations LES folder contains the code file to carry outthe calculations necessary to obtain the LES model forecastsldquoLESmrdquo Three separate Matlab functions are includedldquomapeLESmrdquo ldquomapeLES1mrdquo and ldquomapeLES2mrdquo whichdefine the calculation of the MAPE for the LESrsquo five-year-ahead one-year-ahead and two-year-ahead forecastsAlso ldquoLES OKmatrdquo is the workspace to be loaded prior toperforming LES simulations Finally graphs of each modelhave been included in separate files as well in a subfoldernamed ldquoGraphsrdquo within each of the modelsrsquo folder Figuresare included in Matlabrsquos format (fig) and TIFF format TIFFfigures have been generated in both colours and black andwhite for easier visualization (Supplementary Materials)

References

[1] United Nations International Trade Statistics Yearbook vol 22016

[2] International Copper StudyGroupTheWorld Copper Factbook2017

[3] K M Johnson J M Hammarstrom M L Zientek and CL Dicken ldquoEstimate of undiscovered copper resources of theworld 2013rdquo US Geological Survey 2017

[4] M E Schlesinger and W G Davenport Extractive Metallurgyof Copper Elsevier 5th edition 2011

[5] S Gloser M Soulier and L A T Espinoza ldquoDynamic analysisof global copper flows Global stocks postconsumer materialflows recycling indicators and uncertainty evaluationrdquo Envi-ronmental Science amp Technology vol 47 no 12 pp 6564ndash65722013

Complexity 13

With 120572 being the first smoothing constant for the levelsand 120573 the second smoothing constant for the trend Highervalues for the smoothing constants imply that either levels ortrends are changing rapidly over time whereas lower valuesimply the contrary Hence the higher the constant the moreuncertain the future is believed to be

B Model Calibration

Each model has been calibrated individually using two setsof data containing TC and RC levels respectively from 2004to 2012 The available dataset is comprised of data from 2004to 2017 which make the calibration data approximately 23 ofthe total

B1 GBM Increments in the logarithm of variable x aredistributed as follows

(ln119909) sim 119873((120572 minus 12059022 ) 119905 120590119905) (B1)

Hence if m is defined as the sample mean of the differenceof the natural logarithm of the time series for TCRC levelsconsidered for the calibration and n as the number ofincrements of the series considered with n=9

119898 = 1119899119899sum119905=1

(ln119909119905 minus ln119909119905minus1) (B2)

119904 = radic 1119899 minus 1119899sum119894=1

(ln119909119905 minus ln119909119905minus1 minus 119898)2 (B3)

119898 = 120572 minus 12059022 (B4)

119904 = 120590 (B5)

B2 OUP The OUP process is an AR1 process [22] whoseresolution is well presented by Woolridge [34] using OLStechniques fitting the following

119910119905 = 119886 + 119887119910119905minus1 + 120576119905 (B6)

Hence the estimators for the parameters of the OUP modelare obtained by OLS for both TC and RC levels indepen-dently

= minus ln 119887Δ119905 (B7)

120583 = 1198861 minus 119887 (B8)

= 120576119905radic 2 ln (1 + 119887)(1 + 119887)2 minus 1 (B9)

B3 LES A linear regression is conducted on the inputdataset available to find the starting parameters for the LES

model the initial Level 1198710 and the initial value of thetrend 1198790 irrespective of TC values and RC values Hereas recommended by Gardner [35] the use of OLS is highlyadvisable due to the erratic behaviour shown by trends inthe historic data so the obtaining of negative values of 1198780 isprevented Linear regression fulfils the following

119884119905 = 119886119905 + 119887 (B10)

1198710 = 119887 (B11)

1198790 = 119886 (B12)

By fixing the two smoothing constants the values for theforecasts 119905+119896 can be calculated at each step using the modelequations There are multiple references in the literature onwhat the optimum range for each smoothing constant isGardner [35] speaks of setting moderate values for bothparameter less than 03 to obtain the best results Examplespointing out the same may be found in Brown [36] Coutie[37] Harrison [38] and Montgomery and Johnson [39]Also for many applications Makridakis and Hibon [20] andChatfield [40] found that parameter values should fall withinthe range of 03-1 On the other hand McClain and Thomas[41] provided a condition of stability for the nonseasonal LESmodel given by

0 lt 120572 lt 20 lt 120573 lt 4 minus 2120572120572

(B13)

Also the largest possible value of 120572 that allows the avoidanceof areas of oscillation is proposed by McClain and Thomas[41] and McClain [42]

120572 lt 4120573(1 + 120573)2 (B14)

However according to Gardner there is no tangible proofthat this value improves accuracy in any form Nonethelesswe have opted to follow what Nau [33] refers to as ldquotheusual wayrdquo namely minimising the Mean Squared Error(MSE) of the one-step-ahead forecast of TCRC for eachinput data series previously mentioned To do so Matlabrsquosfminsearch function has been used with function and variabletolerance levels of 1x10minus4 as well as a set maximum numberof function iterations and function evaluations of 1x106 tolimit computing resources In Table 18 the actual number ofnecessary iterations to obtain optimum values for smoothingconstants is shown As can be seen the criteria are wellbeyond the final results which ensured that an optimumsolution was reached with assumable computing usage (thesimulation required less than one minute) and with a highdegree of certainty

14 Complexity

Table 18 Iterations on LES smoothing constants optimisation

Iterations Function EvaluationsTC 1-Step 36 40TC 2-Steps 249945 454211TC 5-Steps 40 77RC 1-Step 32 62RC 2-Steps 254388 462226RC 5-Steps 34 66

C Error Calculations

C1 Mean Square Error (MSE)

119872119878119864 = 1119899119899sum119894=1

(119894 minus 119884119894)2 (C1)

where 119894 are the forecasted values and 119884119894 those observedC2 Mean Absolute Deviation (MAD)

119872119860119863 = 1119899119899sum119894=1

10038161003816100381610038161003816119894 minus 11988410038161003816100381610038161003816 (C2)

where 119894 are the forecasted values and 119884 the average value ofall the observations

C3 Mean Absolute Percentage Error (MAPE)

119872119860119875119864 = 1119899119899sum119894=1

10038161003816100381610038161003816119894 minus 11988411989410038161003816100381610038161003816119884119894 (C3)

The above formula is expressed in parts-per-one and is theone used in the calculations conducted here Hence multi-plying the result by 100 would deliver percentage outcomes

C4 Root Mean Square Error (RMSE)

119877119872119878119864 = radic119872119878119864 = radic 1119899119899sum119894=1

(119894 minus 119884119894)2 (C4)

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Supplementary Materials

The Matlab codes developed to carry out the simulationsfor each of the models proposed as well as the datasetsused and the Matlab workspaces with the simulationsoutcomes as they have been reflected in this paper areprovided in separate folders for each model GBM foldercontains the code to make the simulations for the Geo-metric Brownian Motion model ldquoGBMmrdquo as well as aseparate script which allows determining the total numberof monthly step simulations for which errors levels havebeen lower than annual-step simulations ldquoErrCoutermrdquoThe datasets used are included in two excel files ldquoTCxlsxrdquoand ldquoRCxlsxrdquo ldquoGBM OK MonthlyStepsmatrdquo is the Matlabworkspace containing the outcome of monthly step simu-lations whereas ldquoGBM OK AnnualStepsmatrdquo contains theoutcome for annual-step simulations Also ldquoGBM OKmatrdquois the workspace to be loaded prior to performing GBMsimulations MR folder contains the code file to carryout the simulations for the Orstein-Uhlenbeck ProcessldquoMRmrdquo as well as the separate script to compare errorsof monthly step simulations with annual-step simula-tions ldquoErrCountermrdquo ldquoTCxlsxrdquo and ldquoRCxlsxrdquo containthe TCRC benchmark levels from 2004 to 2017 Finallymonthly step simulationsrsquo outcome has been saved inthe workspace ldquoMR OK MonthlyStepsmatrdquo while annual-step simulationsrsquo outcome has been saved in the Matlabworkspace ldquoMR OK AnnualStepsmatrdquo Also ldquoMR OKmatrdquois the workspace to be loaded prior to performing MRsimulations LES folder contains the code file to carry outthe calculations necessary to obtain the LES model forecastsldquoLESmrdquo Three separate Matlab functions are includedldquomapeLESmrdquo ldquomapeLES1mrdquo and ldquomapeLES2mrdquo whichdefine the calculation of the MAPE for the LESrsquo five-year-ahead one-year-ahead and two-year-ahead forecastsAlso ldquoLES OKmatrdquo is the workspace to be loaded prior toperforming LES simulations Finally graphs of each modelhave been included in separate files as well in a subfoldernamed ldquoGraphsrdquo within each of the modelsrsquo folder Figuresare included in Matlabrsquos format (fig) and TIFF format TIFFfigures have been generated in both colours and black andwhite for easier visualization (Supplementary Materials)

References

[1] United Nations International Trade Statistics Yearbook vol 22016

[2] International Copper StudyGroupTheWorld Copper Factbook2017

[3] K M Johnson J M Hammarstrom M L Zientek and CL Dicken ldquoEstimate of undiscovered copper resources of theworld 2013rdquo US Geological Survey 2017

[4] M E Schlesinger and W G Davenport Extractive Metallurgyof Copper Elsevier 5th edition 2011

[5] S Gloser M Soulier and L A T Espinoza ldquoDynamic analysisof global copper flows Global stocks postconsumer materialflows recycling indicators and uncertainty evaluationrdquo Envi-ronmental Science amp Technology vol 47 no 12 pp 6564ndash65722013

14 Complexity

Table 18 Iterations on LES smoothing constants optimisation

Iterations Function EvaluationsTC 1-Step 36 40TC 2-Steps 249945 454211TC 5-Steps 40 77RC 1-Step 32 62RC 2-Steps 254388 462226RC 5-Steps 34 66

C Error Calculations

C1 Mean Square Error (MSE)

119872119878119864 = 1119899119899sum119894=1

(119894 minus 119884119894)2 (C1)

where 119894 are the forecasted values and 119884119894 those observedC2 Mean Absolute Deviation (MAD)

119872119860119863 = 1119899119899sum119894=1

10038161003816100381610038161003816119894 minus 11988410038161003816100381610038161003816 (C2)

where 119894 are the forecasted values and 119884 the average value ofall the observations

C3 Mean Absolute Percentage Error (MAPE)

119872119860119875119864 = 1119899119899sum119894=1

10038161003816100381610038161003816119894 minus 11988411989410038161003816100381610038161003816119884119894 (C3)

The above formula is expressed in parts-per-one and is theone used in the calculations conducted here Hence multi-plying the result by 100 would deliver percentage outcomes

C4 Root Mean Square Error (RMSE)

119877119872119878119864 = radic119872119878119864 = radic 1119899119899sum119894=1

(119894 minus 119884119894)2 (C4)

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Supplementary Materials

The Matlab codes developed to carry out the simulationsfor each of the models proposed as well as the datasetsused and the Matlab workspaces with the simulationsoutcomes as they have been reflected in this paper areprovided in separate folders for each model GBM foldercontains the code to make the simulations for the Geo-metric Brownian Motion model ldquoGBMmrdquo as well as aseparate script which allows determining the total numberof monthly step simulations for which errors levels havebeen lower than annual-step simulations ldquoErrCoutermrdquoThe datasets used are included in two excel files ldquoTCxlsxrdquoand ldquoRCxlsxrdquo ldquoGBM OK MonthlyStepsmatrdquo is the Matlabworkspace containing the outcome of monthly step simu-lations whereas ldquoGBM OK AnnualStepsmatrdquo contains theoutcome for annual-step simulations Also ldquoGBM OKmatrdquois the workspace to be loaded prior to performing GBMsimulations MR folder contains the code file to carryout the simulations for the Orstein-Uhlenbeck ProcessldquoMRmrdquo as well as the separate script to compare errorsof monthly step simulations with annual-step simula-tions ldquoErrCountermrdquo ldquoTCxlsxrdquo and ldquoRCxlsxrdquo containthe TCRC benchmark levels from 2004 to 2017 Finallymonthly step simulationsrsquo outcome has been saved inthe workspace ldquoMR OK MonthlyStepsmatrdquo while annual-step simulationsrsquo outcome has been saved in the Matlabworkspace ldquoMR OK AnnualStepsmatrdquo Also ldquoMR OKmatrdquois the workspace to be loaded prior to performing MRsimulations LES folder contains the code file to carry outthe calculations necessary to obtain the LES model forecastsldquoLESmrdquo Three separate Matlab functions are includedldquomapeLESmrdquo ldquomapeLES1mrdquo and ldquomapeLES2mrdquo whichdefine the calculation of the MAPE for the LESrsquo five-year-ahead one-year-ahead and two-year-ahead forecastsAlso ldquoLES OKmatrdquo is the workspace to be loaded prior toperforming LES simulations Finally graphs of each modelhave been included in separate files as well in a subfoldernamed ldquoGraphsrdquo within each of the modelsrsquo folder Figuresare included in Matlabrsquos format (fig) and TIFF format TIFFfigures have been generated in both colours and black andwhite for easier visualization (Supplementary Materials)

References

[1] United Nations International Trade Statistics Yearbook vol 22016

[2] International Copper StudyGroupTheWorld Copper Factbook2017

[3] K M Johnson J M Hammarstrom M L Zientek and CL Dicken ldquoEstimate of undiscovered copper resources of theworld 2013rdquo US Geological Survey 2017

[4] M E Schlesinger and W G Davenport Extractive Metallurgyof Copper Elsevier 5th edition 2011

[5] S Gloser M Soulier and L A T Espinoza ldquoDynamic analysisof global copper flows Global stocks postconsumer materialflows recycling indicators and uncertainty evaluationrdquo Envi-ronmental Science amp Technology vol 47 no 12 pp 6564ndash65722013

Complexity 15

[6] The London Metal Exchange ldquoSpecial contract rules for cop-per grade Ardquo httpswwwlmecomen-GBTradingBrandsComposition

[7] U Soderstrom ldquoCopper smelter revenue streamrdquo 2008[8] F K Crundwell Finance for Engineers (Evaluation and Funding

of Capital Projects) 2008[9] Metal Bulletin TCRC Index ldquoMeteoritical Bulletinrdquo https

wwwmetalbulletincomArticle3319574CopperPRICING-NOTICE-Specification-of-Metal-Bulletin-copper-concen-trates-TCRC-indexhtml

[10] J Savolainen ldquoReal options in metal mining project valuationreview of literaturerdquo Resources Policy vol 50 pp 49ndash65 2016

[11] W J Hahn J A DiLellio and J S Dyer ldquoRisk premia incommodity price forecasts and their impact on valuationrdquoEnergy Econ vol 7 p 28 2018

[12] C Alberto C Pinheiro and V Senna ldquoPrice forecastingthrough multivariate spectral analysis evidence for commodi-ties of BMampFbovespardquo Brazilian Business Review vol 13 no 5pp 129ndash157 2016

[13] Z Hlouskova P Zenıskova and M Prasilova ldquoComparison ofagricultural costs prediction approachesrdquo AGRIS on-line Papersin Economics and Informatics vol 10 no 1 pp 3ndash13 2018

[14] T Xiong C Li Y Bao Z Hu and L Zhang ldquoA combinationmethod for interval forecasting of agricultural commodityfutures pricesrdquo Knowledge-Based Systems vol 77 pp 92ndash1022015

[15] Y E Shao and J-T Dai ldquoIntegrated feature selection of ARIMAwith computational intelligence approaches for food crop pricepredictionrdquo Complexity vol 2018 Article ID 1910520 17 pages2018

[16] R Heaney ldquoDoes knowledge of the cost of carrymodel improvecommodity futures price forecasting ability A case study usingthe london metal exchange lead contractrdquo International Journalof Forecasting vol 18 no 1 pp 45ndash65 2002

[17] S Shafiee and E Topal ldquoAn overview of global gold market andgold price forecastingrdquo Resources Policy vol 35 no 3 pp 178ndash189 2010

[18] Y Li N Hu G Li and X Yao ldquoForecastingmineral commodityprices with ARIMA-Markov chainrdquo in Proceedings of the 20124th International Conference on Intelligent Human-MachineSystems andCybernetics (IHMSC) pp 49ndash52Nanchang ChinaAugust 2012

[19] J V Issler C Rodrigues and R Burjack ldquoUsing commonfeatures to understand the behavior of metal-commodity pricesand forecast them at different horizonsrdquo Journal of InternationalMoney and Finance vol 42 pp 310ndash335 2014

[20] S Makridakis M Hibon and C Moser ldquoAccuracy of forecast-ing an empirical investigationrdquo Journal of the Royal StatisticalSociety Series A (General) vol 142 no 2 p 97 1979

[21] S Makridakis A Andersen R Carbone et al ldquoThe accuracyof extrapolation (time series) methods Results of a forecastingcompetitionrdquo Journal of Forecasting vol 1 no 2 pp 111ndash1531982

[22] A K Dixit and R S Pindyck Investment under uncertainty vol256 1994

[23] F Black and M Scholes ldquoThe pricing of options corporateliabilitiesrdquo Journal of Political Economy vol 81 pp 637ndash6591973

[24] N Foo H Bloch and R Salim ldquoThe optimisation rule forinvestment in mining projectsrdquo Resources Policy vol 55 pp123ndash132 2018

[25] P Kalekar ldquoTime series forecasting using Holt-Wintersexponential smoothingrdquo Kanwal Rekhi School of InformationTechnology Article ID 04329008 pp 1ndash13 2004 httpwwwitiitbacinsimprajacadsseminar04329008 Exponential-Smoothingpdf

[26] L Gomez-Valle and J Martınez-Rodrıguez ldquoAdvances in pric-ing commodity futures multifactor modelsrdquoMathematical andComputer Modelling vol 57 no 7-8 pp 1722ndash1731 2013

[27] M Khashei and M Bijari ldquoA novel hybridization of artificialneural networks and ARIMA models for time series forecast-ingrdquo Applied Soft Computing vol 11 no 2 pp 2664ndash2675 2011

[28] M Khashei and M Bijari ldquoAn artificial neural network (pd q) model for timeseries forecastingrdquo Expert Systems withApplications vol 37 no 1 pp 479ndash489 2010

[29] A Choudhury and J Jones ldquoCrop yield prediction using timeseries modelsrdquo The Journal of Economic Education vol 15 pp53ndash68 2014

[30] R R Marathe and S M Ryan ldquoOn the validity of the geometricbrownian motion assumptionrdquoThe Engineering Economist vol50 no 2 pp 159ndash192 2005

[31] G E Uhlenbeck and L S Ornstein ldquoOn the theory of theBrownian motionrdquo Physical Review A Atomic Molecular andOptical Physics vol 36 no 5 pp 823ndash841 1930

[32] A Tresierra Tanaka andCM CarrascoMontero ldquoValorizacionde opciones reales modelo ornstein-uhlenbeckrdquo Journal ofEconomics Finance and Administrative Science vol 21 no 41pp 56ndash62 2016

[33] R Nau Forecasting with Moving Averages Dukersquos FuquaSchool of Business Duke University Durham NC USA 2014httppeopledukeedu128simrnauNotes on forecasting withmoving averagesndashRobert Naupdf

[34] J M Woolridge Introductory Econometrics 2011[35] E S Gardner Jr ldquoExponential smoothing the state of the art-

part IIrdquo International Journal of Forecasting vol 22 no 4 pp637ndash666 2006

[36] R G Brown Smoothing Forecasting and Prediction of DiscreteTime Series Dover Phoenix Editions Dover Publications 2ndedition 1963

[37] G Coutie et al Short-Term Forecasting ICT Monogr No 2Oliver Boyd Edinburgh Scotland 1964

[38] P J Harrison ldquoExponential smoothing and short-term salesforecastingrdquo Management Science vol 13 no 11 pp 821ndash8421967

[39] D C Montgomery and L A Johnson Forecasting and TimeSeries Analysis New York NY USA 1976

[40] C Chatfield ldquoThe holt-winters forecasting procedurerdquo Journalof Applied Statistics vol 27 no 3 p 264 1978

[41] J O McClain and L JThomas ldquoResponse-variance tradeoffs inadaptive forecastingrdquo Operations Research vol 21 pp 554ndash5681973

[42] J O McClain ldquoDynamics of exponential smoothing with trendand seasonal termsrdquo Management Science vol 20 no 9 pp1300ndash1304 1974

[43] J V Issler and L R Lima ldquoA panel data approach to economicforecasting the bias-corrected average forecastrdquo Journal ofEconometrics vol 152 no 2 pp 153ndash164 2009

[44] M F Hamid and A Shabri ldquoPalm oil price forecasting modelan autoregressive distributed lag (ARDL) approachrdquo in Pro-ceedings of the 3rd ISM International Statistical Conference 2016(ISM-III) Bringing Professionalism and Prestige in StatisticsKuala Lumpur Malaysia 2017

16 Complexity

[45] H Duan G R Lei and K Shao ldquoForecasting crude oilconsumption in China using a grey prediction model with anoptimal fractional-order accumulating operatorrdquo Complexityvol 2018 12 pages 2018

[46] M J Brennan and E S Schwartz ldquoEvaluating natural resourceinvestmentsrdquoThe Journal of Business vol 58 no 2 pp 135ndash1571985

[47] RMcDonald and D Siegel ldquoThe value of waiting to investrdquoTheQuarterly Journal of Economics vol 101 no 4 pp 707ndash727 1986

[48] K Zhang A Nieto and A N Kleit ldquoThe real option valueof mining operations using mean-reverting commodity pricesrdquoMineral Economics vol 28 no 1-2 pp 11ndash22 2015

[49] R K Sharma ldquoForecasting gold price with box jenkins autore-gressive integratedmoving average methodrdquo Journal of Interna-tional Economics vol 7 pp 32ndash60 2016

[50] J Johansson ldquoBoliden capital markets day 2007rdquo 2007[51] N Svante ldquoBoliden capital markets day September 2011rdquo 2011[52] Teck ldquoTeck modelling workshop 2012rdquo 2012[53] A Shaw ldquoAngloAmerican copper market outlookrdquo in Proceed-

ings of the Copper Conference Brisbane vol 24 Nanning China2015 httpwwwminexconsultingcompublicationsCRU

[54] P Willbrandt and E Faust ldquoAurubis DVFA analyst conferencereport 2013rdquo 2013

[55] P Willbrandt and E Faust ldquoAurubis DVFA analyst conferencereport 2014rdquo 2014

[56] A G Aurubis ldquoAurubis copper mail December 2015rdquo 2015[57] D B Drouven and E Faust ldquoAurubis DVFA analyst conference

report 2015rdquo 2015[58] A G Aurubis ldquoAurubis copper mail February 2016rdquo 2016[59] EY ldquoEY mining and metals commodity briefcaserdquo 2017[60] J Schachler Aurubis AG interim report 2017 Frankfurt Ger-

many 2017[61] Y Nakazato ldquoAurubis AG interim reportrdquo 2017

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