market risk and the cattle feeding margin: an application of value-at-risk

Market Risk and the Cattle Feeding Margin:An Application of Value-at-Risk

Mark R. ManfredoMorrison School of Agribusiness and Resource Management,Arizona State University East, Mail Code 0180, 7001 E. Williams Field Rd.,Bldg. 20, Mesa, AZ 85212. E-mail: [email protected]

Raymond M. LeutholdOffice for Futures and Options Research, Department of Agricultural andConsumer Economics, University of Illinois at Urbana-Champaign,305 Mumford Hall, 1301 West Gregory Drive, Urbana, IL 61801.E-mail: [email protected]

ABSTRACT

Value-at-Risk, known as VaR, gives a prediction with a certain level of confidence of potentialportfolio losses that may be encountered over a specified time period due to adverse price move-ments in the portfolio’s assets+ For example, a VaR of 1 million dollars at the 95% level of confi-dence implies that overall portfolio losses should not exceed 1 million dollars more than 5% of thetime over a given holding period+ This research examines the effectiveness of VaR measures, de-veloped using alternative estimation techniques, in predicting large losses in the cattle-feeding mar-gin+ Results show that several estimation techniques, both parametric and nonparametric, providewell-calibrated estimates of VaR such that violations~losses exceeding the VaR estimate! are com-mensurate with the desired level of confidence+ In particular, estimates developed using theRisk-MetricsTM method appear robust for instruments that have linear payoff structures such as cashcommodity prices+ © 2001 John Wiley & Sons, Inc+

1. INTRODUCTION

Value-at-Risk~VaR! is considered by many to be a “state-of-the-art” tool in risk mea-surement+ VaR is receiving considerable attention in the financial literature and, morerecently, in the agricultural economics literature~Boehlje & Lins, 1998; Manfredo &Leuthold, 1999!+ Specifically, VaR gives a prediction with a certain level of confidence ofpotential portfolio losses that may be encountered over a specified time period due toadverse price movements in the portfolio’s assets+ For example, a VaR estimate of 1 mil-lion dollars at the 95% level of confidence implies that portfolio losses should not exceed1 million dollars more than 5% of the time over the given holding period~Jorion, 1997!+

VaR is being embraced by corporate risk managers as an important tool in the overallrisk management process+ In particular, upper level managers who may or may not bewell versed in statistical analysis view VaR as an intuitive measure of risk since it con-centrates only on adverse outcomes and is usually reported in dollars or returns+ Because

Agribusiness, Vol. 17 (3) 333–353 (2001)© 2001 John Wiley & Sons, Inc.

333

of this, VaR is commonly used for internal risk management purposes~e+g+, setting trad-ing limits among traders of financial assets! and is further being touted for use in riskmanagement decision making by nonfinancial firms~Ho, Chen, & Eng, 1996; Jorion,1997; RiskMetricsTM, 1996!+ In the case of a nonfinancial firm, such as an agribusinessenterprise, a risk manager may examine the VaR associated with a portfolio of risky as-sets~e+g+, cash positions! prior to and after risk management strategies~e+g+, use of fu-tures, options, forward contracts, or combination of the above! are implemented+ Thisallows the risk manager to assess the potentially large losses to the portfolio in the pres-ence of various risk management strategies, thus allowing this manager to more effi-ciently implement these strategies+ However, initial interest in VaR stemmed from itspotential applications as a regulatory tool+ In the wake of several financial disasters in-volving the trading of derivatives products, such as the Barrings Bank collapse~see Jo-rion, 1997!, regulatory agencies such as the Securities and Exchange Commission embracedVaR as a transparent measure of downside market risk that could be useful in reportingrisks associated with portfolios of highly market-sensitive assets such as derivatives~Lins-meier & Pearson, 1997!+ Again, since VaR focuses on downside risk, regulators as well aspotential investors view it as being a more intuitive and transparent measure of risk+

VaR is estimated using either parametric or full-valuation procedures+ Parametricprocedures rely on estimates of volatility and correlations in creating portfolio volatilityforecasts, which are scaled by a factor corresponding to the desired confidence level+Full-valuation procedures model the entire return distribution, with the VaR measure be-ing the percentile associated with the desired confidence level~e+g+, 5% percentile for the95% confidence level!+1 Much of the literature related to VaR focuses on the properties ofvarious procedures for estimating the risk measure that fall within these two broad cat-egories+ Specifically, parametric and full-valuation procedures are evaluated on their abil-ity to generate VaR estimates that are consistent with the desired predetermined confidencelevel+ Empirical studies to date~Mahoney, 1996; Hendricks, 1996; Jackson, Maude, &Perraudin, 1997! find that the performance of either parametric or full-valuation proce-dures is sensitive to the data and portfolio composition examined as well as the predeter-mined factors of the VaR model itself~e+g+, confidence level and time horizon!+

Several studies regarding livestock risk management strategies have examined the cat-tle feeding process in a multiproduct or portfolio framework~Leuthold & Mokler, 1979;Peterson & Leuthold, 1987!+ The major market risks to cattle feeding are the variabilityof fed cattle prices~output price! and the variability of corn and feeder cattle prices~inputprices!+ In fact, studies focussing on the major factors affecting the profitability of cattlefeeding operations isolate the variability of these market prices as being particularly in-fluential, especially relative to production risk factors such as feed efficiency~Schroeder,Albright, Langemeier, & Mintert, 1993; Langemeier, Schroeder, & Mintert, 1992; Jones,Mintert, Langemeier, Schroeder, & Albright , 1996!+ The difference between fed cattleprices and the prices of corn and feeder cattle, under assumed production technology, isreferred to as the cattle feeding margin+ Thus, the cattle feeding margin serves as a port-folio of assets~fed cattle, feeder cattle, and corn prices! similar to that of a portfolio offinancial assets~Peterson & Leuthold, 1987!+

Given the general interest and adoption of VaR in the financial risk management com-munity, VaR has the potential of being a valuable risk measure for agricultural risk man-

1Full-valuationprocedures are often referred to as nonparametric procedures in the finance literature+ How-ever, the term full-valuation is used throughout this paper to avoid confusion with traditional nonparametricstatistical methods+

334 MANFREDO AND LEUTHOLD

agement as well+ Therefore, it is important to gain an understanding of how standard VaRestimation techniques perform in the context of agricultural prices+ Cattle feeding enter-prises are exposed to a variety of volatile market prices that can be and have been exam-ined in a portfolio context+ Thus, the cattle feeding margin provides a practical case fortesting VaR methodologies in the context of agricultural prices, helping to establish theappropriateness of VaR as a viable and important risk management tool for agribusinessrisk managers+

Considering the recent interest in VaR and the variability of the market risk factors ofthe cattle feeding margin, the overall objective of this article is to examine VaR measuresin the context of the cattle feeding margin+ In particular, this article develops and testsVaR measures estimated using several alternative procedures~both parametric and full-valuation! in predicting large losses in the cattle feeding margin~e+g+, the number of timesthe VaR is exceeded relative to its predetermined confidence level!+ This research is im-portant and unique since it provides insight into the performance of procedures, oftensuggested for use in creating VaR estimates for portfolios of financial assets, in the con-text of agricultural prices+ Given initial evidence of the sensitivity of VaR measures to theprocedures and data set used, as well as the increasing interest in VaR as a tool in riskmanagement, this research makes advances in understanding VaR estimation techniquesand their performance in the context of agricultural prices—particularly prices importantto the cattle feeding industry+

2. THEORETICAL CONSTRUCTS OF VALUE-AT-RISK

As a downside risk measure, VaR concentrates on low probability events that occur in thelower tail of a distribution+ In establishing a theoretical construct for VaR, Jorion~1996,1997! first defines end-of-period portfolio value as W5W0~11R! where W0 is the initialvalue of the portfolio and R is portfolio returns+ Therefore, for a general distribution offuture portfolio value f~W!, Jorion~1996, 1997! defines VaR as:

12 c 5 E2`

W *

f ~W!dw ~1!

where W* is the end-of-period portfolio value when worst possible returns~R* ! occurand c is a predetermined confidence level associated with W*+ By associating W* with thepredetermined confidence level c, W* should not be encountered more than 12c% of thetime ~Figure 1!+ Since portfolio value is a direct function of portfolio returns, equation 1may also be written in terms of returns such that W* is replaced by26R* 6+2

Full-valuation methods rely on procedures for modeling the entire distribution of port-folio value or returns and defining the VaR estimate as the percentile associated with 12cin equation 1+ However, parametric estimation of VaR relies on the properties of the nor-mal distribution+ Assuming that the general distribution in equation 1 is the standard nor-mal distribution f~v! where v; ~0,1!, Jorion~1996, 1997! associates R* with the standardnormal deviate~a! by normalizing R* 2 m and setting

2a 526R* 6 2 m

s~2!

2In most cases, the critical return R* associated with large portfolio losses is negative+

VaR IN CATTLE FEEDING 335

wherem is the mean return+ Through equation 2, Jorion~1996, 1997! subsequently showsthat

12 c 5 E2`

W *

f ~W!dw5E2`

26 R* 6

f ~R!dR5E2`

2a

f ~n!dv ~3!

which allows VaR to be expressed in terms of critical portfolio value~W* !, critical re-turns~R* !, or normal deviate~a!+ Therefore, VaR can be defined as:

VaR5 W0as ~4!

where W0 is initial portfolio value, ~a! is the normal deviate associated with 12c, andsis portfolio standard deviation+ Thus, the critical element of parametric VaR~equation 4!is the estimate of portfolio standard deviation~s!, also referred to as portfolio volatility+

Several studies have examined the properties as well as the pros and cons of using bothfull-valuation and parametric procedures~Linsmeier & Pearson, 1996, 1997; Duffie &Pan, 1997; Jorion, 1996; Manfredo & Leuthold, 1999!+ For instance, full-valuation pro-cedures are often praised for their flexibility but are often criticized for not being able tocapture time-varying volatility often found in financial return series+ Parametric proce-dures, however, are able to capture time-varying volatility through the incorporation ofconditional volatility forecasts+ In fact, much of the impetus for using parametric VaRstems from the existing and growing literature on volatility forecasting~Bollerslev, Chou,

Figure 1 Illustration of Value-at-Risk+


& Kroner, 1992; Figlewski, 1997!, as well as from the publicity and popularity of theRiskMetricsTM ~1996! method, which advocates the use of an exponentially weightedmoving average technique for estimating volatility and correlations+3

Despite these issues, the central concern with any VaR estimation technique is its abil-ity to adequately capture portfolio values that occur in the lower tail of a distributioncommensurate with the predetermined confidence level+ This is especially true since bothfull-valuation and parametric procedures inherently provide estimates of the total vari-ance of a distribution and do not explicitly model distribution tails+ This is often a majorcriticism of using VaR in general due to leptokurtosis observed with financial as well asagricultural price returns~Yang & Brorsen, 1992!+ Regardless, empirical studies to date~e+g+, Mahoney, 1996; Hendricks, 1996; Jackson et al+, 1997! have found that both para-metric and full-valuation procedures adequately cover large portfolio losses, especially atconfidence levels greater than or equal to 95% for the portfolios and data they tested+However, these initial empirical findings also suggest that the performance of any VaRestimation technique is sensitive to the data set used in developing and evaluating theestimates, the predetermined confidence level, forecast horizon, and portfolio composition+

3. VALUE-AT-RISK IN THE CONTEXT OF CATTLE FEEDING

In order to illustrate the linkage between the theoretical constructs of VaR and its prac-tical application and value as a tool for risk management, the following hypothetical ex-ample is presented+ This example parallels descriptions of VaR often found in the financeliterature where portfolios of equities or currencies are the norm+ This example also helpsin understanding the methods used in determining the performance of alternative VaRestimation procedures in the context of the cattle feeding margin which are developedlater in the article+

Formally defined in section 5, the cattle feeding margin is the difference between therevenue generated from the sale of fed cattle less the costs incurred for the purchase offeeder cattle, feed~e+g+, corn!, as well as other variable costs+ Using the definition of VaRin ~4!, suppose that at a given time period, a cattle feeder observes the value of the cattlefeeding margin~W0! to be $1100head+ The cattle feeder also estimates the weekly vol-atility or portfolio standard deviation~s! of the feeding margin to be 0+16 or 16%+ Sub-sequently, the VaR at the 95% level of confidence~a 5 1+65! over the upcoming 1-weekperiod would be $1100head3 1+65 3 0+16 5 $29+040head+ Therefore, with 95% confi-dence, the cattle feeder should expect to realize no greater than a $29+040head decrease inthe value of the feeding margin over a 1-week time frame+ In other words, the loss of$29+040head is one that a cattle feeder should realize only 5% of the time+ If the actualloss in the cattle feeding margin exceeds the VaR estimate, then this would be considereda violation of the estimate+

From a risk management perspective, the VaR estimate of $29+04 is a valuable piece ofinformation+ Since every cattle feeding business has different financial characteristicsand tolerances toward risk, the cattle feeder or risk manager must examine the VaR esti-mate relative to the financial position of the entire business+ Simply put, can the firm

3In 1994, JP Morgan published theirRiskMetricsTM system for developing VaR estimates on the World WideWeb+ Their goal was to establish a market standard for estimating VaR+ JP Morgan’s marketing of theRiskMet-ricsTM method was a catalyst for the growth and adoption of VaR as a downside risk measure throughout thefinancial community+ In 1998, the RiskMetricsTM group was spun off from JP Morgan and is now a separateentity devoted to market and credit risk research+


tolerate or even survive such a rare event—a loss of $29+04 to the margin? This questionis not only important to the cattle feeding operation, but also to banks and other financialinstitutions who lend money to these firms+ The inability of a cattle feeding operation toabsorb large losses in the feeding margin may jeopardize their ability to make principaland interest payments+ Therefore, various risk management strategies could be examinedin the context of how they might affect the VaR estimate+ Presumably, risk managementstrategies, such as the use of futures and options contracts in hedging fed cattle, feedercattle, and0or corn prices, should reduce the VaR estimate+ In other words, those extremelosses in the cattle feeding margin that would normally occur only 5% of the time shouldbe smaller with the incorporation of some type of risk management strategy+

Since implementing a risk management strategy is not without costs~e+g+, commis-sions, option premiums, margin deposits, and opportunity costs!, a risk manger can ex-amine the VaR of the cattle feeding margin with or without the incorporation of variousrisk management strategies, thus tailoring a strategy to a unique operation+ For instance,a liquid, well-capitalized firm might be able to absorb a large loss in the margin~e+g+,$29+04! and thus opt not to engage in any risk management strategy or selectively hedgecertain components of the cattle feeding margin~e+g+, fed cattle price!, or utilize inex-pensive out-of-the-money call options+ A firm that has less tolerance for such a large lossin the feeding margin is likely to be more aggressive in its risk management decisions~e+g+, through the use of more expensive at- or in-the-money options!+ Either way, a cattlefeeder is able to use this information in tailoring a risk management strategy that keepsthe VaR at a level consistent with the firm’s tolerance for large losses, and its lender’srequirements+ Ultimately this leads to a more focussed, successful, and cost-effective riskmanagement strategy that makes the best use of valuable working capital+4

4. DATA

To examine various VaR estimation techniques for the cattle feeding margin, price returnseries are needed+ Returns are constructed from Wednesday cash prices of fed cattle, feedercattle, and corn+ Cash prices are used, since it is the variability of cash prices that causethe cattle feeding margin to fluctuate over time+ Returns are defined as Ri,t 5 ln~Pi,t! 2ln~Pi,t21! where Ri,t is the weekly return of commodity i; ln is the natural logarithm; Pi,t

is the current Wednesday price of commodity i; and Pi,t21 is the previous Wednesdayprice+ Wednesday price data are used since fed cattle and feeder cattle are actively tradedonly 1 day per week, with that day typically in midweek~J+ Rob, Livestock MarketingInformation Center, December 1997, personal communication!+ If a Wednesday price isnot available, then a Tuesday price is used+ The three data series span from January 1984through December 1997, providing 14 years~729 observations! of returns for estimationand out-of-sample testing+

Fed cattle prices~$0cwt! are for the Texas-Oklahoma direct market~1,100- to 1,300-pound steers!, feeder cattle~$0cwt! are for the Oklahoma City terminal market~650 to700 pounds!, and corn prices~$0bu! are for Central Illinois~#2 yellow!+ These data arereported daily in theWall Street Journal+ Furthermore, these prices serve as proxies forlocal cash market prices since each cattle feeding operation is exposed to specific prices

4The AgRiskTM program developed by the University of Illinois at Urbana-Champaign and The Ohio StateUniversity incorporates VaR analysis in assessing revenue losses from implementing or not implementing var-ious risk management strategies for corn, soybeans, and wheat~see http:00www-agecon+ag+ohio-state+edu0agrisk0!+


in its particular region that may or may not have different volatility from the prices ex-amined in this study+ However, due to the liquidity of these cash markets as well as theirfrequency and reliability of reporting, these data are assumed robust for examining theperformance of alternative VaR estimation methods for the cattle feeding margin+

5. METHODS

In defining the cattle feeding margin, Leuthold and Mokler~1979!, Peterson and Leuthold~1987!, and Schroeder and Hayenga~1988! describe similar cattle feeding scenarios thatincorporate fixed feeding technology+ It is assumed that cattle are placed on feed at 650pounds and fed to 1,100 pounds, consuming 45 bushels of corn in the process+5 Based onthis technology, the cattle feeding margin is defined as:

margin~$0head! 5 ~fed cattle price!112 ~feeder cattle price!6+5 2 ~corn price!45+6 ~5!

Typically, the cattle feeding margin specified in~5! is examined on a pen-by-pen basiswith a planning period established prior to the purchase of the corn and feeder cattle inwhich risk management decisions are made+ For most large feedlot operations, however,there is routine and continuous~weekly! buying and placing of feeder cattle on feed,purchasing of corn, and marketing of fed cattle+ For example, feedlots with a capacity of30,000 head or more typically do not maintain corn inventories for more than two weeks~Davies & Widawsky, 1995!+ Consistent with these observations for large feedlots, thisstudy takes a global approach to cattle feeding and does not examine the cattle feedingmargin on a pen-by-pen basis+ Therefore, the specification of the cattle feeding margin in~5! assumes cattle feeding is a continuous process with decision makers routinely~weekly!evaluating the variability of fed cattle, feeder cattle, and corn prices in a portfolio frame-work+ Because of this, VaR measures are estimated and evaluated for weekly horizonsconsistent with the periodicity of the three price return series+ This assumption is not onlyrealistic for large feeding operations, but is also practical from an empirical standpointsince it allows for many observations of VaR and realized portfolio value critical fordetermining the ability of VaR in forecasting large losses for the cattle feeding margin+

For parametric VaR estimation, the variance of the cattle feeding margin~portfoliovariance! is defined as:

sfm2 5 wfc

2 sfc2 1 wfdr

2 sfdr2 1 wc

2 sc2 1 2wfcwfdr rfc, fdr sfcsfdr 1 2wfcwc rfc, csfcsc

1 2wfdr wc rfdr, csfdr sc ~6!

wheresfc2 , sfdr

2 , andsc2 are the variances of fed cattle, feeder cattle, and corn returns and

rfc, fdr , rfc,c, and rfdr,c are the respective correlation coefficients between returns+ Theportfolio weights~wfc , wfdr , and wc! are defined as PiQi where Pi is the price of com-modity i and Qi is quantity of commodity i based on the assumed production technologyallowing equation 6 to be expressed in dollar terms~see Jorion, 1997, p+ 156!+ Consider-

5Studies examining the hedging of the cattle feeding margin typically assume the consumption of corn to bein the range of 42 to 49 bushels+

6Typically, other variable costs are also subtracted from the margin presented in~5!+ Since this study focuseson market risk, other variable costs~e+g+, vet costs! are held constant+


ing equation 6 in a forecasting framework such that the individual variances~volatility!and correlation coefficients are forecasts, VaR at any given week t is:

VaRfm, t 5 a [sfm, t11 ~7!

where [sfm, t11 is the portfolio volatility forecast of the cattle feeding margin~$0head! anda is the scaling factor corresponding to the desired confidence level+ Several forecastingprocedures are employed in estimating the individual volatilities and correlations used inequation 6, all of which have been advocated for use in developing VaR measures and0orused in previous empirical studies related to VaR+ These methods include a long-run his-torical average, a 150-week historical moving average, a GARCH ~1,1! ; t, exponen-tially weighted moving averages advocated byRiskMetricsTM, and implied volatilitiesfrom options on futures contracts+

For both the long-run historic and 150-week moving averages, the volatility forecast isdefined as:

[si, t11 5 ! 1

T (m50

T21

Ri, t2m2 ~8!

where [si, t11 is the volatility forecast for commodity i, T is the number of past squaredreturns used in developing the forecast, and Ri,t

2 is the realized squared return for com-modity i in week t where the mean return of the series is constrained to zero+7 Similarly,covariance forecasts, which are needed to calculate correlations between the three com-modity price series, take the form:

[sij , t11 51

T (m50

T21

Ri, t2m{Rj, t2m ~9!

where [sij , t11 is the forecasted covariance between commodity i and commodity j and Ri,t

and Rj,t are returns for commodity i and j, respectively+ In the case of the long-run his-torical average, the sample size is anchored to the first return observation~growing sam-ple size!+ For the 150-week historical moving average, T is equal to 150+8 Moving averages~or moving windows! are very similar to long-run historical averages but are thought tobe more sensitive to structural change and observed time variation than models that use agrowing sample size+

Due to the popularity of GARCH models in the volatility forecasting literature~Bol-lerslev et al+, 1992!, as well as discussion of their potential use in VaR modeling~Jorion,

7In the volatility forecasting literature, it is standard practice that the mean return of a series be constrainedto zero when developing volatility forecasts+ In addition, Figlewski provides empirical evidence that setting themean to zero provides more accurate volatility forecasts+ Therefore, throughout the remainder of this paper, themean return is set to zero+

8The existing literature provides very little guidance into the number of past observations to use in creatingthese forecasts+ Setting T5150 corresponds with approximately 3 years of past return data, which was deemedadequate in picking up the long-term price variability while being sensitive to time-variation and structuralchange versus other values of T that were examined+ In related work, it was also found that there was little if nosignificant difference in forecast accuracy for historical moving average volatility forecasts of the examinedcash return series using various sample sizes~e+g+, T 5 50, T 5 100, and T5 150!+


1997; Hopper, 1996; Duffie & Pan, 1997!, a GARCH ~1,1! ; t model is also used anddefined as:

[si, t11 5 !a0 1 a1 Ri, t2 1 b1si, t

2 ~10!

wherea1, a1, andb1 are maximum likelihood GARCH estimates~MLE ! generated usingthe BHHH~Berndt, Hall, Hall, & Hausman, 1974! algorithm+ Unlike the common GARCH~1,1!, which assumes the normal distribution, the GARCH~1,1! ; t uses the Student’s-tdistribution in the maximum likelihood estimation which better handles leptokurtotic re-turns and has been found to adequately fit various agricultural price returns~Yang &Brorsen, 1992!+

TheRiskMetricsTM method of estimating volatilities and covariances~correlations! isalso used+ The methodology incorporates an exponentially weighted average that relieson a fixed decay factorl+ It has been touted for its ease of estimation and its ability torepresent time-varying volatility without resorting to GARCH estimation~Mahoney, 1996!+Because of this, the RiskMetricsTM methodology for estimating volatilities and covari-ances~correlations! has been a major impetus for the use of parametric methods in theVaR literature+ TheRiskMetricsTM volatility forecast is:

[si, t11 5 ! [si, t2 1 ~12 l!Ri, t

2 ~11!

wherel is the predetermined decay factor and[si, t2 is the t-period variance forecast+ Time

varying covariances are forecasted in a similar fashion such that:

[sij , t11 5 l [sij , t 1 ~12 l!Ri, t{Rj, t + ~12!

Three fixed decay factors are used, including l 5 +94 andl 5 +97, which are recom-mended byRiskMetricsTM for daily and monthly data, respectively, as well as al opti-mized for weekly data over the respective sample period of the three return series viaMLE techniques using the BHHH algorithm in the S-Plus statistical package~l 5 +96!+

Finally, implied volatilities from observed options prices are used for developing para-metric VaR estimates+ Since exchange-traded options contracts written specifically oncash prices do not exist, it is assumed that volatilities implied from options written on livecattle, feeder cattle, and corn futures provide a reasonable proxy for the option market’sassessment of future price volatility for these cash prices+ Implied volatilities are derivedusing the Black-1976 model~Black, 1976! for options on futures contracts using the Fi-nancial CAD software+ Since the Black-1976 model is a European model, the impliedvolatilities used are computed as the simple average of the implied volatility from nearby,at-the-money, put and call options+ This is done in order to reduce potential bias associ-ated with using a European model for American style options~Mayhew, 1995; Jorion,1995!+ Since implied volatilities yield annualized estimates, it is necessary to convertthese annualized estimates to weekly estimates such that:

I ZVi, t11 5IV~annual!, i, t

!52~13!


whereI ZVi, t11 is the volatility forecast for commodity i at week t11, andIV~annual!, i, t is theannualized implied volatility estimate+

Utilizing the above procedures for forecasting volatilities and covariances, correlationforecasts used in equation 6 are computed as:

[rij , t11 5[sij , t11

[si, t11 [sj, t11~14!

where [si, t11 and [sj, t11 are the volatility forecasts for commodities i and j and[sij , t11 is theforecasted covariance between commodities i and j+ It is important to note that both theGARCH ~1,1! ; t and the implied volatilities, equations 10 and 13, respectively, do nothave corresponding methods to explicitly develop covariances needed for developing cor-relation forecasts in equation 14+ Because of this, the portfolio variance~volatility! fore-cast in equation 6 is created using either GARCH~1,1! ; t or implied volatilities withcorrelations estimated from the long-run historical average, 150-week moving average,and theRiskMetricsTM method, respectively+ These are referred to as “mixed” VaR mod-els throughout the remainder of the paper~Table 1!+

Since the use ofRiskMetricsTM volatilities and correlations is advocated as a simplealternative to multivariate GARCH procedures, a constant correlation MGARCH proce-dure is also used+ Specifically the constant correlation MGARCH model presented byBollerslev~1990! assumes that conditional correlations between commodity price returnsare constant over time and that individual commodity price return variances follow aunivariate GARCH~1,1! process~Campbell, Lo, & MacKinlay, 1997; Bera, Garcia, &Roh, 1997!+ Thus, the model can be shown as:

[sii , t112 5 cii 1 aii Rii , t

2 1 bii sii , t2

[sij , t11 5 [rij , t11 [sii , t11 [sjj , t11 ~15!

TABLE 1+ Value-at-Risk Measures

Abbreviation Description

MGARCH Multivariate GARCH~constant correlation!MGARCH-t Multivariate GARCH using the Student’s t-distribution in the estimation

~constant correlation!H150-VaR 150-day moving average volatilities and correlationsHISTAVG-VaR Long-run historical average volatilities and correlationsRM97-VaR RiskMetricsTM volatilities and correlations usingl 5 +97RM94-VaR RiskMetricsTM volatilities and correlations usingl 5 +94RM96-VaR RiskMetricsTM volatilities and correlations using optimizedl 5 +96GRM97-VaR GARCH-t volatilities and RM97 correlationsGH150-VaR GARCH-t volatilities and H150 correlationsGHIST-VaR GARCH-t volatilities and HISTAVG correlationsIVRM97-VaR Implied volatilities~IV ! and RM97 correlationsIVH150-VaR Implied volatilities~IV ! and H150 correlationsIVHIST-VaR Implied volatilities~IV ! and HISTAVG correlationsHISTSIM-VAR Historical simulationCOMP-VaR Simple average composite of RM97-VaR and HISTSIM-VaR


where [sii , t112 is the conditional variance of asset i for t11, [sij , t11 is the conditional co-

variance, [rij , t11 is the constant correlation between assets i and j, and [sii , t11 is the con-ditional standard deviation of asset i at t11+ In all cases, iÞj+ This model provides apositive definite covariance matrix if all parameters are positive, aii 1 bii , 1, and [rij isbetween21 and 1+ The MGARCH estimates, both assuming a normal distribution as wellas a Student’s-t distribution in the MLE estimation, are then compared to the more sim-plistic procedure~s! where volatilities and covariances~correlations! are estimated inde-pendently of each other+9

Thus using the described parametric methods, weekly VaR measures are estimated fromJanuary 1987 through October 1997 providing 564 weekly forecasts of volatility and cor-relations among the three return series+ Value-at-Risk estimates are calculated for the90% ~a 5 1+28!, 95% ~a 5 1+65!, and 99%~a 5 2+33! levels of confidence+ Each of theparametric VaR measures developed and tested are described and outlined in Table 1+

In addition to the parametric VaR estimates, a simple full-valuation procedure~histor-ical simulation! is also developed for the 90%, 95%, and 99% confidence levels and com-pared with the parametric methods described~Table 1!+ The historical simulation methodmodels the entire return distribution with the VaR designated as the percentile associatedwith the desired level of confidence+ The historical simulation method thus allows thehistorical return data to determine the distribution of the cattle feeding margin unlikeparametric methods, which assume normality+

The historical simulation procedure used follows the methods of Linsmeier and Pear-son~1996!+ First, at time period t, the cattle feeding margin is calculated as in equation 5+Second, hypothetical future values~prices! of fed cattle, feeder cattle, and corn are cal-culated+ This is done by taking the prices observed at time t and exposing them to theirrespective previous 150 weeks of returns such that Pt

* 5 Pt~11 Rt2T! for all T 51+ + +150+Third, hypothetical future values of the cattle feeding margin are calculated using theseprices and the respective technology outlined in~5!, creating 150 simulated values of thecattle feeding margin+ Next, each of the simulated values of the cattle feeding margin issubtracted from the actual feeding margin realized at time t, yielding 150 differencesbetween the cattle feeding margin at week t and the simulated future values of the feedingmargin previously generated+ Finally, from the distribution of these differences, the per-centile associated with the desired confidence level~e+g+, 5% for the 95% level of con-fidence! becomes the VaR estimate~see Appendix A!+

6. EVALUATION

The specific parametric and historical simulation VaR estimates in Table 1 are evaluatedon their ability to predict large losses~decreases! in the cattle feeding margin resultingfrom fluctuations in fed cattle, feeder cattle, and corn prices+ All evaluation is conductedon an out-of-sample basis+ If actual portfolio losses over the desired horizon~e+g+, 1 week!exceeds the VaR estimate made for that particular period, a violation occurs+ Hence, ifviolations are in excess to that implied by a particular confidence level, the VaR measure

9Bera, Garcia, and Roh note that assuming a constant correlation structure is a very strong proposition+Several other MGARCH techniques have been suggested that also limit the number of parameters and attemptto ensure a positive definite covariance matrix such as the diagonal vech and BEEK models~Bera, Garcia, &Roh, 1997; Campbell, Lo, & MacKinlay, 1997!+ These specifications, among others, were tried for the cattlefeeding margin portfolio+ However, model convergence and0or failure to produce a positive definite covariancematrix were consistent problems+


is considered inadequate in measuring large losses of the cattle feeding margin+ To de-termine if violations are commensurate with the designated confidence level of VaR, alikelihood ratio test is developed following the procedures of Lopez~1997!+ The nullhypothesis isd 5 d* whered is the desired coverage level~e+g+, 5%! corresponding to thegiven confidence level~e+g+, 95%!, d* is X0N where X is the number of realized viola-tions and N is the number of out-of-sample observations+ The probability of realizing Xviolations of VaR for a sample of N is:

Pr ~X;d, N! 5 SNXDdX~12 d!N2X ~16!

and the likelihood ratio test statistic is~Lopez, 1997, p+ 7!:

LR~d! 5 2{ ln~d *X~12 d * !N2X! 2 ln~dX~12 d!N2X!} ~17!

which has an asymptoticx2 distribution with 1 degree of freedom+Similarly, a test of bias in VaR estimates is conducted consistent with the procedures of

Mahoney~1996, pp+ 206 and 207!, which is based on a binomial probability distribution+The expectation of the number of violations of a VaR estimate is N~12c! where N is thenumber of out-of-sample observations and c is the confidence level expressed in decimalform ~e+g+, 0+95 for 95% level of confidence!+ The variance of this estimate is Nc~12c!+Thus, the test for bias is defined as a Z test, which in large samples is distributed nor-mally, such that:

Zc 5Lrealized2 N~12 c!

!Nc~12 c!~18!

where Lrealized5 the number of observed violations of VaR at a given confidence level c~Mahoney, 1996!+ Hence, if the Z statistic is significantly positive~negative! then VaRregularly underestimates~overestimates! actual downside risks+10

Summary statistics of the VaR violations are also used to evaluate the VaR modelsincluding the number of violations realized~X !, the percentage of violations that oc-curred over the sample period~X 0N!*100, the average size violation~“sum of violations”0X !, the maximum violation, and the minimum violation+ Therefore, if several VaR measuresare determined to be “well calibrated”~e+g+, d 5 d* ! the preferred VaR model amongalternatives is the one with the smallest of these summary statistics+

7. EMPIRICAL RESULTS

Over the sample period from January 1987 to October 1997, the average feeding marginas defined in equation 5 is $118+37 per head+ The largest feeding margin over this timeperiod is $258+45 per head, realized in the week of 1002096, while the smallest feedingmargin is2$44+57 per head in week 709097+ The largest single weekly loss~decrease! in

10The likelihood ratio statistic and Z statistic are commonly used in the literature as well as by industryprofessionals; however, Lopez notes that these tests are designed to determine unconditional coverage, and donot take into consideration potential serial dependence of violations+ The development of VaR evaluation mea-sures that examine conditional coverage is a topic of current research~see Lopez, 1997; Crnkovic & Drachman,1996!+


the cattle feeding margin~portfolio! over the sample period, 2$53+62, occurred from6016093 to 6023093+ For all of the VaR measures tested in Table 1, this large changeresulted in the largest~maximum! violation of VaR for all confidence levels tested; a rareevent since losses like these are expected to occur less than 1% of the time+ Using thisevent to illustrate a violation of VaR, the VaR estimate on 6016093 is $39+58 at the 99%confidence level for RM97-VaR+ Based on this VaR estimate, it is predicted that the cattlefeeding margin would not decrease from its current level of $124+05 by more than $39+58with 99% confidence+ However, over the next week, the cattle feeding margin decreasesby $53+62 to $70+43, a violation of the VaR estimate by the size of $14+05+

The results of the likelihood ratio test, Z test, and summary statistics of the VaR vio-lations are presented in Tables 2 through 4+ Based on the results of the likelihood ratio andZ tests~equations 17 and 18!, the RiskMetricsTM models~RM97-VaR, RM94-VaR, andRMOPT-VaR!, the historical moving average model~H150-VaR!, and the full-valuationhistorical simulation~HISTSIM-VaR! provide coverage consistent with all three confi-dence levels~90%, 95%, and 99%!+ In other words, in each case, these VaR specificationsfail to reject the null hypothesis that the number of violations realized over the sampleperiod equals the number implied by the predetermined confidence level~d 5 d* !+ All ofthe other VaR models, barring IVRM97-VaR at the 90% level, have more violations oc-curring than accepted by the predetermined confidence level+ For those VaR measuresthat have more violations than allowed, the associated Z statistic is both positive andsignificant, suggesting that those specifications produce estimates that consistently un-derestimate the true downside risk of the cattle feeding margin+

Overall, among the well-calibrated VaR measures, it is difficult to deem one measureto be the best+ However, theRiskMetricsTM specifications, especially RM97-VaR wherethe decay factorl 5 +97, appears to provide robust VaR estimates for each of the threeconfidence levels tested using a wide array of evaluation criteria+ Despite this, any im-provement provided by RM97-VaR relative to the other well-calibrated VaR models isfairly minimal and most likely not economically meaningful+ In other words, among thewell-calibrated models, the performance of the VaR model actually implemented willlikely not effect the firms’ bottom line+ Furthermore, evaluation based on the summarystatistics presented, beyond the results of the LR test and Z test, is somewhat subjective+However, RM97-VaR does have the smallest maximum violation among all VaR modelsfor each of the three confidence levels examined+ Hence, for the largest violation thatoccurs from 6016093 to 6023093, RM97-VaR has the most conservative VaR estimateamong all tested+

Examining each of the confidence levels individually, at the 90% confidence level~Table 2!, several VaR measures fail to reject the null hypothesis ofd 5 d* includingH150-VaR, RMOPT-VaR, RM94-VaR, RM97-VaR, HISTSIM-VaR, and IVRM97-VaR+Of these, RMOPT-VaR has the smallest values of both the likelihood ratio and Z statisticswhile IVRM97-VaR and HISTSIM-VaR have the largest+ However, RM97-VaR, has thesmallest average size violation, maximum violation, and minimum violation at $8+081,$31+887, and $0+002, respectively+ Of the well-calibrated VaR measures at the 95% con-fidence level~Table 3!, again RMOPT-VaR has the smallest values of the test statisticsand percent violations~5+5%! that are closest to what would be expected by the predeter-mined confidence level of 95%+ Both RM97-VaR and RM94-VaR are next with likeli-hood ratio statistics and Z statistics at 0+518 and 0+734, respectively+ Again, in this case,RM97-VaR has the smaller average size violation, maximum violation, and minimumviolation, with the average size violation being approximately $0+68 per head smallerthan that for RMOPT-VaR~$6+838 vs+ $7+519! and $1+075 per head smaller RM94-VaR


TAB

LE

2+

Eva

lua

tion

ofV

aR

Me

asu

res

for

the

90

%C

on

fide

nce

Le

vel

Nu

mb

er

of

vio

latio

ns

Va

Rm

ea

sure

~N5

56

4!a

Pe

rce

nt

vio

latio

ns

Avg

+siz

evi

ola

tionb

Ma

xim

um

vio

latio

nM

inim

um

vio

latio

nL

Rst

atis

ticZ

sta

tistic

Ave

rag

eV

aR

est

ima

te

MG

AR

CH

-t8

41

4+89

0$

9+73

7$

36+1

05

$0+0

63

13+2

51

*3+

87

4**

$1

7+8

62

MG

AR

CH

73

12+9

40

$9+0

13

$3

4+42

6$

0+22

65+

01

5*

2+3

30

**$

20+

10

7H

15

0-V

aR

61

10+8

20

$8+2

06

$3

2+34

9$

0+04

40+

40

70+

64

6$

21+6

32

HIS

TAV

G-V

aR

72

12+7

70

$8+4

25

$3

2+02

7$

0+00

94+

44

9*

2+1

90

**$

20+

28

0R

MO

PT-

Va

R5

91

0+46

0$

8+62

1$

32+1

19

$0+4

51

0+1

31

0+3

65

$2

1+95

9R

M9

4-V

aR

60

10+6

40

$8+7

26

$3

2+59

7$

0+02

30+

25

10+

50

5$

21+9

31

RM

97

-Va

R6

21

0+99

0$

8+08

1$

31+8

87

$0+0

02

0+6

00

0+7

86

$2

1+92

5G

RM

97

-Va

R7

51

3+30

0$

8+89

8$

32+2

80

$0+0

38

6+2

43

*2+

61

1**

$1

9+9

83

GH

15

0-V

aR

75

13+3

00

$9+5

34

$3

4+57

2$

0+27

96+

24

3*

2+6

11

**$

19+

44

4G

HIS

T-V

aR

87

15+4

30

$9+5

30

$3

6+64

9$

0+06

21

6+10

1*

4+2

95

**$

17+

87

1IV

RM

97

-Va

R6

91

2+23

0$

9+72

6$

35+8

15

$0+3

06

2+9

41

1+7

69

$1

9+35

1IV

H1

50

-Va

R7

81

3+83

0$

9+04

1$

37+4

24

$0+0

70

8+3

14

*3+

03

2**

$1

8+8

80

IVH

IST-

Va

R8

31

4+72

0$

9+87

9$

39+1

20

$0+2

61

12+3

57

*3+

73

4**

$1

7+5

15

HIS

TS

IM-V

aR

69

12+2

30

$8+3

67

$3

3+28

3$

0+05

22+

94

11+

76

9$

20+8

54

CO

MP

-Va

R6

01

0+64

0$

8+73

8$

33+4

29

$0+1

26

0+2

51

0+5

05

$2

1+51

5a N

isth

en

um

be

ro

fw

ee

kly

Va

Re

stim

ate

sa

nd

sub

seq

ue

nt

cha

ng

es

inp

ort

folio

valu

e+

b Avg

+siz

evi

ola

tion,

ma

xim

um

vio

latio

n,m

inim

um

vio

latio

n,a

nd

ave

rag

eV

aR

est

ima

tea

rein

do

llars

pe

rh

ea

d+

*Sig

nifi

can

ta

tth

e5

%le

vel+T

he

Ch

i-sq

ua

red

criti

calv

alu

eis

3+84

1+**

Sig

nifi

can

ta

tth

e5

%le

vel+T

he

criti

calZ

valu

eis

1+96+


TAB

LE

3+

Eva

lua

tion

ofV

aR

Me

asu

res

for

the

95

%C

on

fide

nce

Le

vel

Nu

mb

er

of

Vio

latio

ns

Va

Rm

ea

sure

~N5

56

4!a

Pe

rce

nt

vio

latio

ns

Avg

+siz

evi

ola

tionb

Ma

xim

um

vio

latio

nM

inim

um

vio

latio

nL

Rst

atis

ticZ

sta

tistic

Ave

rag

eV

aR

est

ima

te

MG

AR

CH

-t5

69+

93

0$

8+51

9$

31+0

40

$0+5

39

22+0

73

*5+

37

1**

$2

3+0

25

MG

AR

CH

44

7+8

00

$7+7

71

$2

8+87

5$

0+39

48+

01

9*

3+0

53

**$

25+

92

0H

15

0-V

aR

34

6+03

0$

6+58

0$

26+1

97

$0+0

10

1+1

82

1+1

21

$2

7+88

5H

ISTA

VG

-Va

R3

96+9

10

$7+5

83

$2

5+79

6$

0+09

73+

91

0*

2+0

87

**$

26+

14

3R

MO

PT-

Va

R3

15+5

00

$7+5

18

$2

5+90

1$

0+38

80+

28

40+

54

1$

28+3

06

RM

94

-Va

R3

25+6

70

$7+9

13

$2

6+51

7$

0+23

30+

51

80+

73

4$

28+2

71

RM

97

-Va

R3

25+6

70

$6+8

38

$2

5+06

1$

0+03

20+

51

80+

73

4$

28+2

62

GR

M9

7-V

aR

46

8+16

0$

7+54

1$

26+1

08

$0+7

51

10+0

15

*3+

43

9**

$2

5+7

59

GH

15

0-V

aR

51

9+04

0$

7+66

7$

29+0

63

$0+0

33

15+8

20

*4+

40

5**

$2

5+0

89

GH

IST-

Va

R5

81

0+28

0$

8+41

5$

31+7

41

$0+2

06

25+7

39

*5+

75

7**

$2

3+0

48

IVR

M9

7-V

aR

44

7+80

0$

8+33

9$

30+6

66

$0+8

74

8+0

19

*3+

05

3**

$2

4+9

45

IVH

15

0-V

aR

45

7+98

0$

8+71

2$

32+7

40

$0+7

45

8+9

93

*3+

24

6**

$2

4+3

38

IVH

IST-

Va

R5

49+

57

0$

9+05

7$

34+9

26

$0+3

33

19+8

26

*4+

98

5**

$2

2+5

78

HIS

TS

IM-V

aR

36

6+38

0$

6+70

5$

26+6

24

$0+0

16

2+0

96

1+5

07

$2

7+92

9C

OM

P-V

aR

34

6+03

0$

6+61

4$

27+2

00

$0+2

62

1+1

82

1+1

21

$2

8+25

8a N

isth

en

um

be

ro

fw

ee

kly

Va

Re

stim

ate

sa

nd

sub

seq

ue

nt

cha

ng

es

inp

ort

folio

valu

e+

b Avg

+siz

evi

ola

tion,

ma

xim

um

vio

latio

n,m

inim

um

vio

latio

n,a

nd

ave

rag

eV

aR

est

ima

tea

rein

do

llars

pe

rh

ea

d+

*Sig

nifi

can

ta

tth

e5

%le

vel+T

he

Ch

i-sq

ua

red

criti

calv

alu

eis

3+84

1+**

Sig

nifi

can

ta

tth

e5

%le

vel+T

he

criti

calZ

valu

eis

1+96+


TAB

LE

4+

Eva

lua

tion

ofV

aR

Me

asu

res

for

the

99

%C

on

fide

nce

Le

vel

Nu

mb

er

of

Vio

latio

ns

Va

Rm

ea

sure

~N5

56

4!a

Pe

rce

nt

vio

latio

ns

Avg

+siz

evi

ola

tionb

Ma

xim

um

vio

latio

nM

inim

um

vio

latio

nL

Rst

atis

ticZ

sta

tistic

Ave

rag

eV

aR

est

ima

te

MG

AR

CH

-t2

23+

90

0$

6+37

0$

21+7

30

$0+4

00

27+6

55

*6+

92

4**

$3

2+5

14

MG

AR

CH

14

2+4

80

$5+7

95

$1

8+67

4$

1+30

18+

86

3*

3+5

38

**$

36+

60

2H

15

0-V

aR

61+0

60

$6+0

01

$1

4+89

1$

0+58

50+

02

30+

15

2$

39+3

77

HIS

TAV

G-V

aR

10

1+77

0$

6+06

2$

16+0

36

$0+6

07

2+7

68

1+8

45

$3

6+91

7R

MO

PT-

Va

R1

01+7

70

$4+4

78

$1

4+47

3$

0+42

72+

76

81+

84

5$

39+9

72

RM

94

-Va

R1

11+9

50

$5+0

21

$1

5+34

4$

0+42

04+

02

8*

2+2

68

**$

39+

92

2R

M9

7-V

aR

61+0

60

$5+6

68

$1

4+05

0$

0+59

60+

02

30+

15

2$

39+9

10

GR

M9

7-V

aR

15

2+66

0$

5+33

0$

14+7

66

$1+2

17

10+7

83

*3+

96

1**

$3

6+3

74

GH

15

0-V

aR

15

2+66

0$

6+38

8$

18+9

39

$1+2

57

10+7

83

*3+

96

1**

$3

5+3

94

GH

IST-

Va

R2

34+0

80

$6+4

45

$2

2+72

0$

0+14

73

0+48

3*

7+3

47

**$

32+

53

0IV

RM

97

-Va

R1

62+8

40

$4+4

42

$2

1+20

2$

0+53

31

2+84

0*

4+3

84

**$

35+

22

5IV

H1

50

-Va

R1

73+0

10

$4+8

58

$2

4+13

1$

0+36

81

5+02

6*

4+8

08

**$

34+

36

8IV

HIS

T-V

aR

25

4+4

30

$5+8

77

$2

7+21

8$

0+66

63

6+40

9*

8+1

93

**$

31+

88

3H

IST

SIM

-Va

R1

01+7

70

$5+3

81

$1

7+60

2$

0+27

32+

76

81+

84

5$

37+7

59

CO

MP

-Va

R6

1+06

0$

6+16

8$

17+3

62

$1+6

48

0+0

23

0+1

52

$3

9+06

3a N

isth

en

um

be

ro

fw

ee

kly

Va

Re

stim

ate

sa

nd

sub

seq

ue

nt

cha

ng

es

inp

ort

folio

valu

e+

b Avg

+siz

evi

ola

tion,

ma

xim

um

vio

latio

n,m

inim

um

vio

latio

n,a

nd

ave

rag

eV

aR

est

ima

tea

rein

do

llars

pe

rh

ea

d+

*Sig

nifi

can

ta

tth

e5

%le

vel+T

he

Ch

i-sq

ua

red

criti

calv

alu

eis

3+84

1+**

Sig

nifi

can

ta

t5

%le

vel+T

he

criti

calZ

valu

eis

1+96+


~$6+838 vs+ $7+913!+ It is difficult to judge whether these differences in the size of VaRviolations on average over the sample period are economically meaningful+ Interestingly,at the 99% level of confidence, where violations of VaR are expected to occur no greaterthan 1% of the time~Table 4!, both RM97-VaR and H150-VaR have the same values oflikelihood ratio and Z statistics at 0+023 and 0+152, respectively+ As with the 90% and95% confidence levels, RM97-VaR again has the smallest maximum violation+

Since it is well known that composite forecasting techniques often yield superior fore-casts~Clemen, 1989!, a composite VaR estimate is also created and examined versus theindividual models+ Based on the findings discussed above, the composite measure is con-structed as a simple average of RM97-VaR and HISTSIM-VaR~COMP-VaR!+ These twoVaR measures are chosen for combining since they are both constructed using differentmethods~parametric vs+ full-valuation!+ COMP-VaR is found to provide coverage con-sistent with all three confidence levels and provides improvement over HISTSIM-VaRindividually at all confidence levels based on the percentage of violations realized, andsubsequently, the values of the test statistics+ In fact, at the 90% level, COMP-VaR pro-vided improvement over both of its component forecasts~RM97-VaR and HISTSIM-VaR! based on the size of the LR and Z statistics+

The results presented in Tables 2 through 4 also suggest that performance of any VaRmeasure is greatly affected by the correlation structure incorporated+ Those VaR modelsthat combine univariate volatilities in conjunction with alternative correlation estimates~mixed models! consistently underestimate the true downside risk of the cattle feedingmargin+ This is evident since the Z statistics for these models are positive and significantand the average VaR estimates are smaller than those of the best performing models~e+g+,RM97-VaR!+ However, performance of the mixed VaR models improve as the correla-tions go from being long-run historical to conditionala la RiskMetricsTM ~e+g+, IVHIST-VaR to IVRM97-VaR!+ The parametric VaR measures found to be well calibrated acrossconfidence levels use volatilities and correlations that are estimated from the same un-derlying method~i+e+, RiskMetricsTM !+ This observation calls into question the computa-tional validity of using implied volatility or another univariate volatility estimationprocedure for VaR that does not have a corresponding way of defining covariances and,subsequently, correlations+ Furthermore, the weak performance of both multivariateGARCH specifications is likely due to the constant correlation assumption incorporatedthat was necessary to provide a positive definite covariance matrix as well as modelconvergence+

Interestingly, the majority of violations occurred during the period from April to Oc-tober+ Using RM97-VaR at the 90% level of confidence, 41 out of 62 violations~66% ofviolations! occur during this period+ During this time, 12 out of the 62 violations~20%!occur in June while 27 violations~43%! occur from June through August+ Similarly, atthe 95% level of confidence, the April through October time periods saw 22 out of the 32violations while 8 out of 22~25%! are realized during the month of June alone+ In addi-tion, at the 99% confidence level, 5 out of the 6 violations were realized during the Mayto October time span+ These observations are consistent with known increased price vari-ability of fed cattle and corn prices during these months+

8. SUMMARY AND CONCLUSIONS

The methods recommended byRiskMetricsTM, in particular usingl 5 +97, provide a sim-plistic, accurate, and robust specification for a covariance matrix in calculating weekly


parametric VaR for the examined portfolio~the cattle feeding margin!+ However, otherspecifications, such as the otherRiskMetricsTM specifications as well as the simple his-torical simulation~full-valuation procedure!, also produce well-calibrated VaR measuresfor all three confidence levels examined+

Therefore, the conclusion as to the superiority of theRiskMetricsTM method usingl 5 +97, relative to the other procedures that produced well-calibrated measures, ismade with caution+11 The fact that both parametric procedures~e+g+, RiskMetricsTM !and full-valuation~e+g+, historical simulation! are found to provide well-calibrated VaRmeasures is most likely due to the fact that the cattle feeding margin, as defined in thisstudy, is a portfolio of linear instruments~cash prices!+12 This conclusion is similar tothat of other studies that examined the performance of both full-valuation and paramet-ric VaR estimation methods for portfolios of financial asset returns~e+g+, Mahoney, 1996;Hendricks, 1996; Jackson et al+, 1997!+ Overall, it is concluded that at least for thisportfolio, correlations are more important to the overall performance of a particularVaR measure than volatilities+ Furthermore, the majority of violations of VaR occurduring times of observed seasonal increases in the volatility of fed cattle and corn re-turns+ Because of this, risk managers that build VaR models in which the portfolio ofinterest contains positions in agricultural commodities or other seasonal commoditiesshould be more cautious of their VaR estimates during these known times of increasedvolatility+13

This study is the first known attempt at empirically examining the performanceof various VaR measures in the context of an agricultural enterprise portfolio~e+g+,cattle feeding!+ To date, all known empirical studies examining the performance ofalternative VaR measures have been conducted in the context of portfolios contain-ing currency, interest rate, or equity data with portfolios often developed randomly~Mahoney, 1996; Hendricks, 1996!+ The cattle feeding margin provides a realistic alter-native portfolio, as well as new data, for studying existing techniques of VaR estima-tion+ The results of this study also provide an impetus for further research in the areaof VaR+ For instance, research is needed that focuses on the applicability and per-formance of VaR in the context of other agricultural prices and portfolios~e+g+, thesoybean crush margin! as well as the performance of alternative parametric and full-valuation procedures when options positions, which have a nonlinear payoff structure,are included in a portfolio+ Therefore, as interest and use of VaR increases among riskmangers, research should focus on models that are robust for a variety of prices andportfolios+

11A reviewer noted that the results and conclusions presented here do not provide scientific evidence thatthese well-calibrated models, in particular RM97, can be relied on for future applications to the cattle feedingmargin+ As with any forecasting experiment, there is no guarantee that the future performance of a forecast willbe the same as its past performance+ However, an experiment such as that conducted in this study is the onlyway to get an indication of how a forecasting procedure might work over future time periods+

12It is important to consider that the cattle feeding margin examined in this study was the unhedged cashmargin+ Based on the results of this study, it is expected that the various VaR models examined would performsimilarly if the feeding margin were hedged using either futures or forward contracts, since both of these in-struments also have linear payoff structures+ However, if options positions were used to hedge the margin, theperformance of the VaR models might be different since options have nonlinear payoffs+

13While volatility clustering is routinely observed, its economic explanations are not clear and “the eco-nomic theory explaining temporal variation in conditional variances is very limited,” ~Bollerslev, Chou, &Kroner, 1992, p+ 7!+


AP

PE

ND

IXA

:C

alc

ula

tion

ofV

aR

Usi

ng

His

tori

calS

imu

latio

nM

eth

od

Ob

serv

ed

valu

eo

fth

eca

ttle

fee

din

gm

arg

ina

ttim

ep

eri

od

t(6

/28

/95

)

Da

teT

ime

pe

rio

dF

ed

catt

lep

rice

~$0c

wt!

Fe

ed

er

catt

lep

rice

~$0c

wt!

Co

rnp

rice

~$0b

u!O

bse

rve

dm

arg

in~$

0hd!

a

602

809

5t

63

71+5

2+6

45

10

9+22

5

Pre

vio

us

15

0w

ee

kso

fh

isto

rica

lpri

ces

an

dre

turn

s

Da

teT

ime

pe

rio

dF

ed

catt

lep

rice

~$0c

wt!

Fe

dca

ttle

retu

rns

Fe

ed

er

catt

lep

rice

~$0c

wt!

Fe

ed

er

catt

lere

turn

sC

orn

pri

ce~$

0bu!

Co

rnre

turn

s

602

109

5t-

16

4+00

0+0

00

07

1+75

20+

02

91

2+75

00+

02

39

601

409

5t-

26

4+00

20+

01

55

73+8

72

0+0

08

52+6

85

0+0

49

660

709

5t-

36

5+00

0+0

31

37

4+50

0+0

23

82+5

55

0+0

03

9+

++

++

++

++

++

++

++

++

++

++

++

+80

260

92

t-1

48

75+0

00+

01

34

91+5

02

0+0

15

02+2

15

0+0

25

180

190

92

t-1

49

74+0

02

0+0

10

19

2+88

20+

00

40

2+16

00+

01

87

801

209

2t-

15

07

4+75

0+0

13

59

3+25

0+0

57

92+1

20

20+

01

87

Hyp

oth

etic

alf

utu

reva

lue

s(p

rice

s)ca

lcu

late

du

sin

gh

isto

rica

lre

turn

da

ta

Da

taT

ime

pe

rio

dF

ed

catt

lep

rice

~$0c

wt!

Fe

ed

er

catt

lep

rice

~$0c

wt!

Co

rnp

rice

~$0b

u!H

ypo

the

tica

lma

rgin

~$0h

d!O

bse

rve

dm

arg

in~$

0hd!

Diff

ere

nce

~$0h

d!

602

109

5t-

16

3+00

69+

42

2+7

08

11

9+91

10

9+23

10+

69

601

409

5t-

26

2+02

70+

89

2+7

76

96+5

21

09+2

32

12+

70

6070

95

t-3

64+9

77

3+2

02+

65

51

19+3

71

09+2

31

0+1

4+

++

++

++

++

++

++

++

++

++

++

++

+80

260

92

t-1

48

63+8

57

0+4

32+

71

21

22+4

91

09+2

31

3+2

780

190

92

t-1

49

62+3

67

1+2

22+

69

41

01+8

61

09+2

32

7+3

780

120

92

t-1

50

63+8

57

5+6

42+

59

69

3+85

10

9+23

21

5+3

7a T

he

catt

lefe

ed

ing

ma

rgin

isd

efin

ed

ine

qu

atio

n5

such

tha

t:m

arg

in~$

0he

ad!

5~f

ed

catt

lep

rice

!11

2~f

ee

de

rca

ttle

pri

ce!6+

52

~co

rnp

rice

!45+

b Hyp

oth

etic

alf

utu

reva

lue

s~pri

ces!

of

fed

catt

le,fe

ed

er

catt

le,a

nd

corn

are

calc

ula

ted

as

:Pt*

5P

t~1

1R

t-T!

for

all

T5

1++

+15

0+c F

rom

the

dis

trib

utio

no

fth

ese

diff

ere

nce

s,V

aR

isca

lcu

late

da

sth

ep

erc

en

tile

ass

oci

ate

dw

ithth

ed

esi

red

leve

lof

con

fide

nce

~e+g

+,5

%fo

rth

e9

5%

leve

lof

con

fide

nce!+


REFERENCES

Bera, A+K+, Garcia, P+, & Roh, J+S+ ~1997!+ Estimation of time-varying hedge ratios for corn andsoybeans: BGARCH and random coefficient approaches+ Sankhya: The Indian Journal of Sta-tistics, Series B, 59, 346–368+

Berndt, E+K+, Hall, B+H+, Hall, R+E+, & Hausman, J+A+ ~1974!+ Estimation inference in nonlinearstructural models+ Annals of Economic and Social Measurement, 304, 653–665+

Black, F+ ~1976!+ The pricing of commodity contracts+ Journal of Financial Economics, 3, 167–179+Boehlje, M+D+, & Lins, D+A+ ~1998!+ Risks and risk management in an industrialized agriculture+

Agricultural Finance Review, 58, 1–16+Bollerslev, T+ ~1990!+ Modeling the coherence in short-run nominal exchange rates: A multivariate

generalized ARCH approach+ Review of Economics and Statistics, 72, 498–505+Bollerslev, T+, Chou, R+Y+, & Kroner, K+F+ ~1992!+ ARCH modeling in finance: A review of the

theory and empirical evidence+ Journal of Econometrics, 52, 5–59+Campbell, J+Y+, Lo, A+W+, & MacKinlay, A+C+ ~1997!+ The econometrics of financial markets+ Prince-

ton, NJ: Princeton University Press+Clemen, R+T+ ~1989!+ Combining forecasts: A review and annotated bibliography+ International Jour-

nal of Forecasting, 59, 559–583+Crnkovic, C+, & Drachman, J+ ~1996!+ Quality control+ Risk, 9, 138–143+Davies, S+, & Widawsky, D+ ~1995!+ The impact of the changing size of feedlots on corn marketing

in Colorado+ Journal of the American Society of Farm Managers and Rural Appraisers, 59,120–127+

Duffie, D+, & Pan, J+ ~1997!+ An overview of value at risk+ The Journal of Derivatives, 4, 7–49+Figlewski, S+ ~1997!+ Forecasting volatility+ Financial Markets, Institutions, and Instruments, 6,

2–87+Hendricks, D+ ~1996!+ Evaluation of value-at-risk models using historical data+ Federal Reserve

Bank of New York Policy Review, 2, 39–69+Ho, T+S+Y+, Chen, M+Z+H+, & Eng, F+H+T+ ~1996!+ VAR analytics: Portfolio structure, key rate con-

vexities, and VAR betas+ The Journal of Portfolio Management, 23, 90–98+Hopper, G+P+ ~1996!+ Value-at-risk: A new methodology for measuring portfolio risk+ Business Re-

view Federal Reserve Bank of Philadelphia, July0August, 19–31+Jackson, P+, Maude, D+J+, & Perraudin, W+ ~1997!+ Bank capital and value at risk+ The Journal of

Derivatives, 4, 73–89+Jones, R+, Mintert, J+, Langemeier, M+, Schroeder, T+, & Albright , M+ ~1996!+ Sources of economic

variability in cattle feeding+ Proceedings of the NCR-134 Conference on Applied CommodityForecasting and Risk Management, Chicago, IL , April , 336–348+

Jorion, P+ ~1995!+ Predicting volatility in the foreign exchange market+ The Journal of Finance, 50,507–528+

Jorion, P+ ~1996!+ Risk2: Measuring the risk in value at risk+ Financial Analysts Journal, 52, 47–56+Jorion, P+ ~1997!+ Value at risk: The new benchmark for controlling derivatives risk+ Chicago, IL :

Irwin Publishing+RiskMetricsTM Technical Document 4th Edition, Morgan Guaranty Trust Company, New York+

Available: http:00www+RiskMetrics+com0research0, 1996Langemeier, M+R+, Schroeder, T+, & Mintert, J+ ~1992!+ Determinants of cattle finishing profitabil-

ity+ Southern Journal of Agricultural Economics, 24, 41–48+Leuthold, R+M+, & Mokler, R+S+ ~1979!+ Feeding margin hedging in the cattle industry+ Proceedings

of the International Futures Trading Seminar of the Chicago Board of Trade, 6, 56–68+Linsmeier, T+J+, & Pearson, N+D+ ~1996!+ Risk measurement: An introduction to value at risk+ Office

for Futures and Options Research Working Paper396–04, University of Illinois atUrbana-Champaign+

Linsmeier, T+J+, & Pearson, N+D+ ~1997!+ Quantitative disclosures of market risk in the SEC release+Accounting Horizons, 11, 107–135+

Lopez, J+A+ ~1997!+ Regulatory evaluation of value-at-risk models+ Staff Report Number 33, Fed-eral Reserve Bank of New York+

Mahoney, J+M+ ~1996!+ Empirical-based versus model-based approaches to value-at-risk: An exam-ination of foreign exchange and global equity portfolios+ Proceedings of a Joint Central BankResearch Conference, Board of Governors of the Federal Reserve System, pp+ 199–217+


Manfredo, M+R+, & R+M+ Leuthold ~1999!+ Value-at-risk analysis: A review and the potential foragricultural applications+ Review of Agricultural Economics, 21, 99–111+

Mayhew, S+ ~1995!+ Implied volatility+ Financial Analysts Journal, 51, 8–19+Peterson, P+E+, & Leuthold, R+M+ ~1987!+ A portfolio approach to optimal hedging for a commercial

cattle feedlot+ The Journal of Futures Markets, 7, 443–457+Schroeder, T+C+, Albright, M+L+, Langemeier, M+R+, & Mintert, J+ ~1993!+ Factors affecting cattle

feeding profitability+ Journal of the American Society of Farm Managers and Rural Appraisers,57, 48–54+

Schroeder, T+C+, & Hayenga, M+L+ ~1988!+ Comparison of selective hedging and options strategiesin cattle feedlot risk management+ The Journal of Futures Markets, 8, 141–156+

Yang, S+, & Brorsen, B+W+ ~1992!+ Nonlinear dynamics of daily cash prices+ American Journal ofAgricultural Economics, 74, 707–715+

Mark R. Manfredo is an assistant professor in the Morrison School of Agribusiness and ResourceManagement at Arizona State University East. He received a PhD in Agricultural Economics fromthe University of Illinois at Urbana-Champaign in 1999. His current research interests are in theareas of agribusiness finance, futures and options markets, and commodity marketing.

Raymond M. Leuthold is professor emeritus and director of the Office for Futures and OptionsResearch, Department of Agricultural and Consumer Economics, University of Illinois at Urbana-Champaign. He received a PhD in Agricultural Economics from the University of Wisconsin–Madison in 1968. His research interests include futures and options markets, agricultural marketing,and agricultural price forecasting and analysis.


market risk and the cattle feeding margin: an application of value-at-risk

Documents