portfolio investment: are commodities...

Portfolio Investment Are Commodities Useful

by

Lei Yan and Philip Garcia

Suggested citation format

Yan L P Garcia 2014 ldquoPortfolio Investment Are Commodities Usefulrdquo Proceedings of the NCCC-134 Conference on Applied Commodity Price Analysis Forecasting and Market Risk Management St Louis MO [httpwwwfarmdocillinoisedunccc134]


Lei Yan

and

Philip Garcia lowast

Paper presented at the NCR-134 Conference on Applied Commodity PriceAnalysis Forecasting and Market Risk Management

St Louis Missouri April 21-22 2014

Copyright 2014 by Lei Yan and Philip Garcia All rights reserved Readers may makeverbatim copies of this document for non-commercial purposes by any means

provided that this copyright notice appears on all such copies

lowastLei Yan is a PhD student and Philip Garcia is the TA Hieronymus Distinguished Chair in FuturesMarkets both in the Department of Agricultural and Consumer Economics at the University of Illinois atUrbana-Champaign Correspondence may be sent to leiyan2illinoisedu We thank Anton BekkermanJeffrey Dorfman Randall Fortenbery Scott Irwin Berna Karali Paul Peterson Dwight Sanders and par-ticipants at the NCCC-134 conference for helpful discussions and comments Any remaining errors are ourresponsibility alone


This paper investigates the usefulness of commodities in investorsrsquo portfolios within a mean-variance optimization framework The analysis differs from previous research by consideringmultiple investment tools including individual commodity futures contracts three generationsof commodity indices and by controlling for estimation error in portfolio optimization pro-cess Rather generally the results demonstrate that including individual commodities or thefirst- and second-generation commodity indices do little to enhance portfolio performanceSimilarly when an initial portfolio is diversified the risk-reducing ability of agriculturalcommodities is much weaker than identified by previous research In contrast includingthe third-generation indices substantially improves the portfoliorsquos Sharpe ratio by generatinghigher returns and lower risk

Keywords commodity index investments portfolio diversification mean-variance opti-mization estimation error

Introduction

Investments in commodities have grown rapidly in the past decade through commodity indexfunds The total value of commodity index investments was about $210 billion by the endof 20121 This large inflow of investment funds is mainly based on the perception that in-vestors can obtain diversification benefits by including commodity futures since commoditiesshow equity-like returns and low correlation with traditional assets (Gorton and Rouwen-horst 2006) The question of interest is whether including commodity futures can improvea portfoliorsquos performance The literature provides mixed evidence Early studies show thatinvestors are better off in terms of reducing risk by including commodities in portfolios (egBodie and Rosansky 1980 Fortenbery and Hauser 1990 Ankrim and Hensel 1993) How-ever recent research fails to identify consistent diversification benefits in an out-of-samplesetting (Daskalaki and Skiadopoulos 2011 You and Daigler 2013) Where do commoditiesand particularly agricultural commodities fit into investorsrsquo portfolios and does their use-fulness provide insights into future non-traditional investors in these markets

We analyze that most academic research on the usefulness of commodities in portfolios suffersfrom two shortcomings First most studies examine the role of commodities in a portfolioby employing either a few individual commodity futures contracts or widely used indices(eg SampP Goldman Sachs Commodity Index) Few investigate the roles that different com-modity indices and multiple agricultural commodities can play in a portfolio Miffre (2012)argues that returns to commodity indices can greatly depend on imbedded strategies whichsuggests that the usefulness of commodities may vary by the type of index considered Inaddition Commodity Index Traders (CITs) have become large participants in twelve agri-cultural futures markets with an average percentage of positions rising from 7 to 34 for2000-2009 (Aulerich et al 2013) Despite the increasing exposure to agricultural commoditymarkets formal assessments on whether including agricultural commodity futures benefits a

1See CFTC Index Investment Data httpwwwcftcgovMarketReportsIndexInvestmentDataindexhtm

1

portfolio are scarce Second studies that apply portfolio theory to commodity futures oftenignore estimation error Estimation error occurs whenever sample moments are used to esti-mate population values Standard portfolio optimization that replaces expected returns bysample estimates often leads to concentrated and unstable asset allocations (Kan and Zhou2009) For example You and Daigler (2013) find that the ex-post optimal portfolios arequite unstable and suggest that reducing bias in the expected returns would make optimalportfolio models more useful Failure to account for these shortcomings can influence ourunderstanding of the role that commodities play particularly during periods of high volatilitywhich have occurred recently

The paper re-examines the usefulness of commodities in a portfolio and contributes the lit-erature in two dimensions First we evaluate the impacts on investor portfolios of twelveagricultural commodities monitored by the CFTC and three commodity indices (SampP Gold-man Sachs Commodity Index (SPGSCI) Deutsche Bank Optimum Yield Commodity Index(DBOYCI) and Morningstar Long-Short Commodity Index (MSLSCI)) which reflect theclassification scheme developed by Miffre (2012) The first two indices represent passivelong-only exposures to commodities while MSLSCI allows for both long and short positionsand more aggressive momentum strategies2 The different proxies for commodities help usdetermine whether the usefulness of commodities varies by the type of investment toolsSecond to control for estimation error we use Black and Littermanrsquos (1992) procedure thatconducts portfolio optimization with shrinkage estimates Historical returns for equities andbonds are shrunk towards a prior which is derived from reversing the Capital Asset PricingModel (CAPM) with equal weights For commodities the prior returns are assumed to bezero consistent with findings in the literature (Sanders and Irwin 2012) and our calculationsthat returns to commodities are not significantly different from zero

The data consist of monthly returns of multiple asset types for 1991-2012 We constructbenchmark portfolios using US equities US bonds global equities and global bondsand assess the effect of adding commodity indices and individual commodities The mean-variance optimization with sample estimates and shrinkage estimates for expected returnsis implemented in both in- and out-of-sample settings The results suggest that includingindividual commodities does not significantly improve the portfolio performance as measuredby the Sharpe ratio Similarly including the first- or second-generation commodity indicesfails to increase the portfolio Sharpe ratio but evidence does emerge that the third-generationcommodity indices can either increase returns or decrease risk (or both) which leads toenhanced portfolio performance The results also confirm previous findings that standardmean-variance optimization leads to concentrated and unstable asset allocations throughtime and controlling for estimation error produces more diversified and balanced portfoliosHowever we find that commodities play a much smaller role when the remainder of portfoliois diversified Our results are consistent over multiple robustness analyses

2Added description of the indices is provided in the text below

2

Related Literature

In this section we first review studies on commodity futures returns with focus on the dif-ference in performance between individual futures and commodity indices Next we discussthe correlation between equity and commodity returns since low correlation is generallyviewed as a necessary condition for commodity to be part of a portfolio While return andcorrelation are two important factors that influence the commoditiesrsquo investment value theycannot assure portfolio benefits Consequently we discuss research that applies a portfolioframework to evaluate the usefulness of commodities

Literature on commodity futures returns falls into two categories The first line of researchdating back to Keynes (1930) seeks evidence on risk premium for individual commodityfutures markets For instance Dusak (1973) and Fama and French (1987) find only limitedevidence of a constant risk premium However Bessembinder and Chan (1992) and Bjornsonand Carter (1997) when allowing for a time-varying risk premium do find support for a riskpremium in several commodity futures markets Despite these mixed findings the resultsidentify a few factors such as hedging pressure that appear to be related to risk premiumThe other line of literature examines returns to a portfolio of commodities or commodityindices For example in a direct assessment Gorton and Rouwenhorst (2006) documentthat an equally-weighted commodity portfolio can achieve ldquoequity-likerdquo returns In contrastSanders and Irwin (2012) show that returns to individual commodity futures contracts donot differ from zero question the source of value in an equally-weighted commodity port-folio and suggest that the superiority of commodity indices may be largely due to theirimbedded strategies The notion that differences in strategies may influence commodityportfolio performance is informative for investors and analysts seeking to understand com-modity market behavior Miffre (2012) classifies commodity indices into three generationsFirst-generation indices provide passive long-only exposure to broad range of commoditiesand include widely used ones (eg SPGSCI) Second-generation indices also are long-onlybut improve on the first generation indices by mitigating the impact of negative roll yieldsfrom rolling positions when the market is in contango Third-generation indices differ fromthe former two by allowing both long and short positions based on selected strategies Miffre(2012) further shows that third-generation commodity indices perform best for 2008-2012This is consistent with recent research (Fuertes et al 2010 Szymanowska et al 2013) thatmomentum and term structure based strategies can work well in commodity futures markets

In addition to returns correlation also plays an important role in determining portfolio per-formance Commodity futures have long been considered as an isolated asset class whosereturns are uncorrelated or even negatively correlated with returns of traditional assets (Gor-ton and Rouwenhorst 2006) Recently Buyukahin et al (2010) investigate the time-varyingcorrelation between equity and commodity returns and find the correlation generally remainslow for 1991-2008 despite a temporary modest increase during the financial turmoil How-ever low correlation only implies diversification opportunity not real diversification benefitsMoreover recent evidence points to increasing correlation between commodities (Tang andXiong 2012) and between commodities and other financial assets (Silvennoinen and Thorp2013) which calls into question the diversification benefits of commodity futures

3

The investment potential of commodities also has been investigated in a portfolio choiceframework Bodie and Rosansky (1980) and Georgiev (2001) show that blending commod-ity futures with common stocks can reduce risk without sacrificing return for the periods1950-1976 and 1990-2001 They form portfolios by considering a range of proportions be-tween assets instead of using an optimization procedure Other researchers examine thisissue within the Markowitzrsquos (1952) mean-variance optimization framework Fortenbery andHauser (1990) find that the addition of agricultural futures contracts to a stock portfoliorarely increases portfolio return but does reduce portfolio risk Ankrim and Hensel (1993)use SPGSCI to represent exposure to commodities and find similar improvement on return-risk profile Satyanarayan and Varangis (1996) augment Ankrim and Henselrsquos (1993) workby considering global investors with broader asset classes and find that the inclusion ofcommodities shifts the efficient frontier upwards While most prior research agrees on thediversification benefits of commodities contradictory results also exist Daskalaki and Ski-adopoulos (2011) extend analysis to an out-of-sample setting and find that the in-samplebenefits if any by including commodities do not persist out-of-sample3 You and Daigler(2013) also compare the in-sample and out-of-sample performance of portfolios with andwithout futures contracts using mean-variance optimization They find that portfolios withfutures outperform the traditional portfolio in-sample but out-of-sample gains in perfor-mance are negligible

The identified research on the role of commodities in investorsrsquo portfolios is subject to twoshortcomings First studies on asset allocation with commodities examines either a fewindividual commodities or a passive long-only index (eg SPGSCI) which fails to reflect thevariety of commodity indices (Miffre 2012) Also few examine the usefulness of includingthe twelve agricultural commodity futures in portfolios in light of rapid growth of CITpositions in those markets (Aulerich et al 2013) You and Daigler (2013) examine 39futures contracts which include these agricultural commodity futures within a classical mean-variance optimization framework but their results may be susceptible to estimation errorsSecond previous research ignores estimation error that can arise when the sample estimatesof mean and variance of returns are treated as ldquotruerdquo values in optimization You andDaigler (2013) is the only study that recognizes the potential impact of estimation errorThey find that the optimal portfolio weights are highly volatile which is mainly driven byerrors in expected return estimates With imprecise mean and variance as inputs the derivedoptimal portfolios tend to be concentrated and extremely sensitive to the mean estimate (egMichaud 1989) The problem is worse in a more realistic out-of-sample setting becausein-sample distributions of returns often do not persist out-of-sample (Kan and Zhou 2009)Failure to control estimation error can lead to misallocated portfolios and incorrect evaluationon the role of commodities To address these issues we consider alternate ways of investingin commodities including twelve agricultural commodity futures and three generations ofcommodity indices The use of alternative indices allows us to test whether the usefulness of

3Daskalaki and Skiadopoulos (2011) use different approaches to evaluate the role of commodities un-der in-sample and out-of-sample settings They conduct spanning tests in-sample and implement portfoliooptimization out-of-sample

4

commodities depends on specific investment instruments Employing agricultural commodityfutures enables us to explore whether they can provide diversification benefits to a portfoliogiven increasing CIT exposures With regards to the estimation error different methodshave been proposed (Fabozzi et al 2007) Here we follow Black and Littermanrsquos (1992)approach which allows us to incorporate the priors that many economists have about thezero expected returns to agricultural futures markets

Data

The dataset consists of monthly returns of multiple types of assets for 1991-2012 To investi-gate the role of commodities in a portfolio we construct a benchmark portfolio that includesfour asset types - US equities US bonds global equities and global bonds The USequity market is represented by the SampP 500 Index - a widely used investment benchmarkThe Barclays US Aggregate Bond Index is chosen to reflect the performance of investmentgrade bonds as it covers treasury bonds government agency bonds mortgage-backed bondscorporate bonds and a small amount of foreign bonds traded in US Global equity perfor-mance is measured by the MSCI World Index (ex US) which includes securities from 23countries and is often used as a common benchmark for global stock funds The JPMorganGlobal Aggregate Bond Index is considered as a representative of global bonds which tracksinstruments from over 60 countries

For commodities we consider both commodity indices and individual commodity futurescontracts In particular we use three commodity indices - SampP Goldman Sachs Commod-ity Index (SPGSCI) Deutsche Bank Optimum Yield Commodities Index (DBOYCI) andMorningstar Long-Short Commodity Index (MSLSCI) which belong to the three genera-tions classified by Miffre (2012)4 The first-generation indices are designed to provide broadexposure to commodities among which SPGSCI is widely used as a benchmark These in-dices hold liquid contracts with the shortest maturity which can lead to poor performancewhen prices of distant futures contracts are higher than near-by contract prices The second-generation indices differ in that they take into account the shape of futures price curve andalso are willing to invest in distant contracts DBOYCI as its name suggests choose con-tracts with the maximum implied roll yield56 The first two generations indices are bothpassive long-only exposures to commodities In contrast the third-generation indices cantake long and short positions based on strategies they identify For example strategies basedon momentum and term structure signal have been shown to work well (Fuertes et al 2010Szymanowska et al 2013) MSLSCI uses a momentum rule to determine its long or short

4These three indices are chosen because they have historical data back to 1991 which is uncommon Wealso use another set of indices for a robustness check

5Implied roll yields are measured as price differences between near-by and distant futures contractsscaled by maturity differences For a market in contango negative roll yield arises by rolling from thenear-by contract to the distant contract that has higher price DBOYCI is designed to invest into aparticular distant contract that generates the smallest negative roll yield See DBOYCI index guidehttpsindexdbcomdbiqweb2dataguidesDBLCI-OY v15pdf

6Recently researchers have begun to explore the sources of commodity index returns and cast doubton the role of roll yields (Willenbrock 2011 Sanders and Irwin 2012) Here we abstract away from thosequestions and simply use the indices as commonly done in the industry

5

positions every month - taking long in the subsequent month if current price exceeds its 12-month moving average and taking short otherwise7 Note these three commodity indices areall broad and cover major commodity sectors including energy precious metals industrialmetals and agriculture The data on equities bonds and commodity indices come from theBloomberg database8 We also consider twelve agricultural commodities including cocoacoffee cotton sugar feeder cattle lean hogs live cattle corn soybeans soybean oil Chicagowheat and Kansas wheat These twelve commodities are monitored by the CFTC in theirweekly Supplemental Commitment of Traders reports and research has shown that the CITpositions in those markets have increased pronounced since 2004 The prices of individualcommodity futures come from the Commodity Research Bureau

Monthly returns are calculated as the percentage changes of asset prices

Rt = ln(ptptminus1

)times 100 (1)

where pt is the closing price on the last trading day in month t9 Several points require at-tention in excess return construction First total return indices are used for equities bondsand commodity indices since they are target indices tracked by many funds Excess returnsare obtained by subtracting total returns from equation 1 by risk-free rate which is repre-sented by the one-month Libor rate Second investing in futures different from investing inequities or bonds requires no principal except margins Futures margins are typically muchsmaller than the contractsrsquo nominal values that is futures investments involve substantialleverage To draw a meaningful comparison between the performance of futures and otherasset classes a full-collateralized assumption is often made (Gorton and Rouwenhorst 2006Sanders and Irwin 2012) Following this convention the excess returns to futures positionsare just percentage changes in futures prices as shown in equation 110 Third commodityfutures returns are calculated using near-by contracts and adjusted for roll dates (see ap-pendix of Singleton 2014)

Table 1 provides annualized mean and standard deviation of monthly excess returns for allassets for different periods For 1991-2012 investing in US equities and bonds achievedaverage returns about 57 and 323 per year while allocating funds to global equitiesand bonds obtained 04 and 328 per year The average returns for three commodityindices are 025 612 and 528 Returns for individual commodities vary considerablyranging from 638 (soybeans) to -624 (cotton) Compared with equities bonds tend

7See MSLSCI Fact SheethttpcorporatemorningstarcomUSdocumentsIndexesCommodityFactsheetpdf

8Second- and third-generation commodity indices are all launched after 2005 but some provide historicaldata based on backward computation using the same strategies Use of these returns to make comparisonshere assumes that the relative effectiveness of the strategies has not changed

9pt denotes the settlement price for futures contract10Leverage only matters in calculating return for individual commodity futures commodity indices are

designed to be fully collateralized Investing in a particular index is typically performed through an indexfund which ties to mimic the index performance by holding positions in both commodity futures and risk-freebonds markets

6

to display much lower standard deviations and most commodities (except feeder cattle andlive cattle) exhibit higher standard deviations as expected The second- and third-generationcommodity indices (DBOYCI and MSLSCI) show equity-like returns and moderate standarddeviations We also report test results for the hypothesis that the mean excess returns areequal to zero Five markets show significantly positive returns at the 10 significance levelincluding US equities US bonds global bonds the third-generation commodity index andlive cattle futures The strong performance of MSLSCI is probably related to its embeddedstrategies In contrast all individual commodity futures returns do not differ from zeroexcept live cattle which is consistent with findings by Erb and Harvey (2006) and Sandersand Irwin (2012) that the historical returns for individual commodities are almost zero11

Table 1 also reports the mean and standard deviation of monthly excess returns for temporalsubsamples To reflect increases in commodity prices beginning in 2004 and their collapsefollowing financial crisis in mid-2008 we consider three subsamples - 1991-2003 2004-2007and 2008-2012 Wide differences in asset performance can be identified across subsamplesFor example US equities performed best before 2004 global equities did best for 2004-2007 US and global bonds have become more valuable since 2008 For the commodityindices DBOYCI provides consistently higher excess return and lower standard deviationthan SPGSCI Interestingly MSLSCI shows moderate return or loss but consistently lowerstandard deviation compared to the other two indices Seven of twelve individual commodi-ties show higher excess returns in the commodity boom period 2004-2007 Three commodities(cotton sugar and soybeans) perform better after 2008 and the two cattle contracts showsimilar performance across subsamples The standard deviations tend to increase for mostof assets post 2008 The p-values are not reported for subsamples due to limited observations

Table 2 provides correlations between asset returns for different periods For the entire pe-riod US equities are highly correlated with global equities (078) but are weakly correlatedwith US and global bonds (002 and 009) The correlation between US equities and threegenerations of commodity indices are 02 023 and -01 For the subsamples we find thatUS equities are nearly uncorrelated or negatively correlated with commodities before crisisbut the correlation increases considerably in 2008-2012 This is consistent with Buyukahinet al (2010) and Silvennoinen and Thorp (2013) who find a recent increase in correlation be-tween equity and commodity markets Informatively the third-generation commodity indexMSLSCI maintains negative correlation with equities even in the post-crisis period whichis a reflection of its ability to take short positions in the downside market The last columnprovides the average correlation between equities or bonds return and twelve agriculturalcommodity futures returns12 Despite an increase after 2008 average correlations betweentraditional assets and twelve individual commodity futures are generally low US bonds arehighly correlated with global bonds and almost uncorrelated with commodities throughoutthe sample and global equities and bonds seem to have higher correlations with commodities

11Erb and Harvey (2006) also find that the live cattle shows marginally significant positive returns in1982-2004 In part this may be related to the marginally significant time-varying risk premium found byFrank and Garcia (2009) in the live cattle market

12Since we add only one commodity to the traditional portfolio at time the pairwise correlations betweenindividual commodities do not matter So we report average values to save space

7

than US equities and bonds in all periods13

Methods

In this section we first briefly review the classical mean-variance optimization frameworkWe then discuss the importance of estimation error and introduce Black and Littermanrsquos(1992) approach to mitigate its effect Finally we describe in-sample and out-of-sampleimplementation of portfolio optimization

Mean-variance optimization

Markowitzrsquos (1952) mean-variance paradigm is by far the most common framework to studyportfolio choice Consider N risky assets with an Ntimes1 random return vector r and a risk-freeasset with known return rf Define excess returns as re = rminusrf ι ι = [1 1 1]prime and denotethe expected mean and variance-covariance matrix by micro = E[re] and Σ = E[(reminusmicro)(reminusmicro)prime]Given a weight vector w = [w1 w2 wN ]prime the portfoliorsquos expected return and variance arewprimemicro and wprimeΣw respectively

When there is a risk-free asset the two-fund separation theorem implies that all investorsshould hold a combination of the risk-free asset and the same portfolio of risky assets - thetangent portfolio (Tobin 1958) The tangent portfolio is achieved by maximizing the Sharperatio which is defined as the ratio of excess return and standard deviation

maxw

wprimemicroradicwprimeΣw

(2)

st wprimeι = 1 ιprime = [1 1 1]

Optimal portfolio weights can be solved as

wlowast =1

ιprimeΣminus1microΣminus1micro (3)

In the analysis we also impose non-negative constraints on asset weights consistent with thecommon practice that most investors only hold long asset positions

Estimation error and shrinkage estimation

In practice classical mean-variance optimization does not always perform well To imple-ment it sample mean and covariance are used as inputs (micro and Σ) to solve for optimalweights based on equation (3) There is considerable research documenting the imprecisionof sample estimates (eg Michaud 1989 Kan and Zhou 2009) The general conclusionsare that sample estimates involve large estimation error and the derived optimal weightsinherit those errors resulting in unreliable portfolios In addition to being imprecise sample

13The reason why commodities are more correlated with global financial markets than US markets isunclear and warrants further research

8

estimates tend to produce concentrated allocations which seems to contradict the notion ofdiversification

The literature has suggested different ways to control estimation error14 Shrinkage estima-tion is attractive because of its ability to incorporate investorsrsquo own beliefs Developed byJames and Stein (1961) and applied to portfolio choice by (eg Jorion 1986 Ledoit andWolf 2003) the idea is to shrink the sample mean towards a prior 15

micros = (1minus δ)micro0 + δmicro (4)

where micros micro0 and micro denote the shrinkage estimate the prior and the sample estimate forexpected excess return The shrinkage factor δ isin [0 1] reflects the relative precision betweensample and prior estimates When δ rarr 0 the shrinkage estimate converges to the prior andwhen δ rarr 1 it converges to the sample estimate leading to the traditional mean-varianceoptimization As illustrated in table 1 the sample estimate micro varies considerably across sub-samples but the prior micro0 by specification will be more stable In this sense the shrinkageestimates provide a more conservative framework in which investors underreact to both goodand bad past performance in optimizing their portfolios There is no fixed rule to select δThe literature often assume that the prior is more precise than the sample estimate (egDrobetz 2001) Here we use δ = 02 and check the sensitivity of the results to alternatevalues in robustness analyses

While appealing the shrinkage estimation requires an informative prior Black and Litter-man (1992) lend an intuitive way of specifying the prior micro0 Originally Black and Litterman(1992) propose a framework to combine investorrsquos subjective views and implied returns whichderived from the CAPM model To make correspondence our prior returns refer to the im-plied returns and the sample returns are used to represent investorrsquos subjective views Theuse of sample returns as a proxy of investorrsquos views reflects the fact that many investorsmake decisions based only on past prices The implied returns are derived using the reverseoptimization process described in Sharpe (1974) The intuition is that the market itself is inequilibrium and investors should only deviate from the CAPM market equilibrium returnsif they have reliable information on future returns Start with the CAPM model

E(ri)minus rf = βi(E(rm)minus rf ) βi =cov(ri rm)

σ2m

(5)

where E(ri) E(rm) and rf are the expected return on asset i the expected return on marketportfolio and the risk-free rate respectively cov(ri rm) and σ2

m denote the covariance be-tween returns of asset i and market portfolio and the variance of returns of market portfolioLet wm = (wm1 wmN)prime be market capitalization then rm =

sumNj=1wmj rj Substitute rm

14Fabozzi et al (2007) provide a review on recent developments in robust optimization and its applicationsin portfolio management

15Shrinkage estimation can also be applied to the covariance matrix We focus on the mean because theprior on covariance is more difficult to establish and because estimation errors in mean have a much largerinfluence (Best and Grauer 1991)

9

in equation (5) and put it in matrix form

micro0 equiv E(r)minus rf ι = γΣwm γ =E(rm)minus rf

σ2m

ι = [1 1 1]prime (6)

where micro0 equiv E(r) minus rf ι is defined as the CAPM market equilibrium return vector Σ is thecovariance matrix wm is the weight vector measured by market capitalization and γ can beexplained as the coefficient of risk aversion (He and Litterman 1999) In the same way op-tions traders imply volatility from option prices using the Black-Scholes model the impliedreturns are implied from market capitalization weights and covariance matrix

To implement equation (6) we need to specify each term on the right-hand side Σ γand wm The sample covariance (Σ) can be used as an estimate for Σ Since E(rm) isunobservable Black and Litterman (1992) suggest calibrating γ such that the resultingimplied return micro0 (or Sharpe ratio) for particular asset looks reasonable In the originalpaper they solve γ to make sure that the Sharpe ratio of US equities is equal to 05 If theannualized standard deviation of US equities ranges between 16 and 20 fixing Sharperatio at 05 corresponds to a total return from 8 to 10 per year Both these numbers arereasonable and widely accepted by industry and academia and their procedure is used tocalibrate γ To develop a structure for wm Black and Litterman (1992) argue for the useof market capitalization but the difficulty is to measure the scale of any asset market Forcommodity futures market capitalization is not a meaningful concept since the net positionis always equal to zero Moreover it is unclear whether commodity futures can be pricedusing the CAPM model To circumvent these issues we directly specify the prior returnsof commodities to be zero which is consistent with Sanders and Irwin (2012) and our ownfindings that the average returns to commodity futures are almost zero16 For the other assets(US equities US bonds global equities and global bonds) we simply use equal weights(wm = [1

414

14

14]prime) to derive their prior returns which may capture real world constraints

often placed by investors to ensure a more diversified portfolio (Brentani 2004) Once weobtain the prior (micro0) based on equation (6) the shrinkage estimates micros are then calculatedas the weighted average between micro0 and micro from equation (4)

Implementation

The impact of including commodities in a portfolio is evaluated in three steps First weconstruct sample estimates for the mean (micro) and covariance (Σ) of expected returns Tocontrol for estimation error the shrinkage estimate for the mean (micro0) is also established fol-lowing procedure just described

Second we implement the mean-variance optimization for two sets of assets - benchmarkand expanded The benchmark consists of US equities US bonds global equities andglobal bonds and the expanded portfolio includes these assets as well as commodities Onecommodity (either an index or an individual commodity futures contract) is introduced at

16This may be inappropriate for MSLSCI since it shows significantly positive returns Shrinking towardszero provides a conservative assessment of the impact of MSLSCI on a portfolio

10

a time We explore the role of commodity under both in-sample and out-of-sample settingsMost of the literature falls into the in-sample category - deriving optimal portfolios and as-sessing their performance based on the same sample period Out-of-sample analysis may bemore meaningful to investors who are concerned more about future performance We followDaskalaki and Skiadopoulos (2011) in assessing out-of-sample portfolio performance with amonthly rebalancing scheme At the end of each month the previous K-months returns areused to generate the mean and covariance estimates which are then used for mean-variancemaximization to derive optimal weights The portfoliorsquos return in the next month is productof optimal weights and the month K + 1 return vector This process is repeated until theend of the sample is reached By doing so we obtain a series of monthly portfolio returnswhich are used for evaluation Note we use K observations for estimation and the portfolioevaluation is actually for 1996-2012 instead the whole period 1991-2012 Estimation windowK is set at 60 months and alternative sizes of K = 36 48 72 are checked for robustness17

Finally we compare the performance between benchmark and expanded portfolios to assessif including commodities is beneficial In particular we calculate the portfoliorsquos annualizedexcess return (Mean) annualized standard deviation (SD) and Sharpe ratio (Sharpe) If theexpanded portfolio achieves a higher Sharpe ratio we conclude that including commoditiesis beneficial to portfolios Optimal weights are also examined to fully reveal the role ofcommodities in a portfolio

Results

In-sample performance

Table 3 compares in-sample performance of the benchmark and expanded portfolios PanelA is based on mean-variance optimization with sample estimates for expected returns Thebenchmark portfolio achieves a Sharpe ratio 098 with annualized excess return and standarddeviation 345 and 354 The first two generations of commodity indices represented bySPGSCI and DBOYCI have limited influence on the portfolio while including the third-generation commodity index MSLSCI improves return-risk profile resulting in a substantialenhancement in Sharpe ratio For many individual commodities (cocoa coffee cottonleanhogs corn soybean oil wheat CBOT wheat KBOT) Sharpe ratios of expanded portfolios donot change from those of the benchmark because these commodities do not enter portfolioThe exceptions are feeder cattle and live cattle which is partly explained by their marginallysignificant positive returns identified earlier Since including twelve individual agriculturalcommodity futures fail to improve portfolio performance except for feeder cattle and livecattle we only report those two in table 3 for conciseness

Panel B shows the results from mean-variance maximization with shrinkage estimates Com-pared with results in panel A the means are similar but the standard deviations have in-creased substantially leading to a drop in the Sharpe ratios Including the third-generation

17In a forecasting context one may be interested in the optimal selection of K in terms of trade-off betweenreliability and flexibility especially in the presence of structural breaks Selecting optimal estimation windowis not considered in the paper Instead we show the effect of different lengths on the results

11

commodity index MSLSCI still improves the portfolio performance relative to the bench-mark but the effects of individual commodities on the portfolio are negligible

These results can be further understood by examining the optimal portfolio weights Table4 panel A shows the optimal weights based on in-sample mean-variance maximization withsample and shrinkage estimates With sample estimates the optimal portfolios are dom-inated by US bonds with a weight around 90 Global equities and bonds are entirelyexcluded This confirms the argument that classical mean-variance tends to generate con-centrated asset allocations (eg Michaud 1989) Of the three commodity indices SPGSCIplays no role in the portfolio DBOYCI takes 59 while MSLSCI has the largest proportion164 On average the individual commodities have a small part in the portfolio (17)In contrast the optimal weights based on optimization with shrinkage estimates are morebalanced The proportion of US bonds declines to about 37 and both US equities andglobal bonds play larger roles in the portfolio The increased share of equities raises bothreturn and standard deviation of the whole portfolio which was seen in table 3 Weightson commodities are reduced suggesting commodities are even less useful in more diversi-fied portfolio These small weights are consistent with findings in table 3 that individualcommodities have limited impacts on the portfolios

Out-of-sample performance

Table 5 shows the out-of-sample performance of benchmark and expanded portfolios us-ing both sample and shrinkage estimates Recall the estimates are generated using past60-months observations and the portfolio is rebalanced monthly To test whether the dif-ferences between Sharpe ratios from the benchmark and expanded portfolios are significantwe use the test proposed by Ledoit and Wolf (2008) which has demonstrated robust finitesample properties when returns are non-iid

Panel A shows the out-of-sample performance based on mean-variance optimization withsample estimates The Sharpe ratio for the benchmark portfolio is 031 which does notdiffer statistically from those of the expanded portfolios except for the third-generation com-modity index Including MSLSCI achieves higher returns lower risk and a significantlyhigher Sharpe ratio (067) With regards to the individual commodities ten of twelve re-duce the portfoliorsquos standard deviation although they also decrease the returns Again wereport expanded portfolios by feeder cattle and live cattle as they improve the Sharpe ratiosthough insignificantly The standard deviation of benchmark portfolio is 678 and the av-erage standard deviation of portfolios expanded by those ten commodities is 656 a 022average reduction in portfolio risk This risk reduction result is consistent with findings byearly literature that commodities contribute to the portfolio by reducing the volatility (egFortenbery and Hauser 1990)

Panel B shows results based on mean-variance optimization with shrinkage estimatorsMSLSCI still improves the portfoliorsquos Sharpe ratio (from 029 to 04) reflecting higher re-turns lower risk and negative correlation with traditional assets identified earlier Com-pared with panel A commodities appear to play a smaller role in portfolios First the

12

third-generation commodity index MSLSCI raises the portfolio Sharpe ratio from 029 to04 which is smaller in magnitude than the Sharpe ratio increase (from 031 to 067) whensample estimates are used However this Sharpe ratio increase becomes insignificant withp-value of 021 Second the second-generation commodity index DBOYCI and two agricul-tural commodity futures (feeder cattle and live cattle) have no impact on the Sharpe ratioThird the risk-reducing ability of agricultural commodity futures is also weaker Althoughten of twelve expanded portfolios show some degree of reduction in standard deviation themagnitude decreases from 022 to 005 when shrinkage estimates are used

These findings can be further understood by inspecting the optimal portfolio weights Table4 panel B reports the out-of-sample optimal weights and their standard deviations Eachvalue represents the average optimal weights over time for a particular asset and values inthe last column denote an average over twelve individual commodities for the period Withsample estimates US bonds has a weight close to 54 much larger than weights estimatedfor the other assets SPGSCI assumes only 31 but the other two commodity indices areweighted heavily reaching 128 and 227 The individual commodities are on average35 With shrinkage estimates the portfolios are more diversified The weight of USbonds is less than 34 and other assets contribute more in the portfolio Note the weights ofthe commodity indices and composites are all reduced suggesting weaker impacts Standarddeviations of optimal weights are shown in parentheses The variability of optimal weightsis uniformly smaller when shrinkage estimates are used which again confirms that shrinkageestimation generate stable portfolios in an out-of-sample environment

Robust analysis

We provide several robustness checks18 First we use different indices to represent the threegenerations of commodity indices Second we consider including all twelve agricultural com-modity futures in the portfolio instead of just one at a time Third we examine alternativevalues of the shrinkage factor δ Fourth we examine how the usefulness of commoditieschanges over time by repeating analysis for different periods Furthermore we investigatethe impact of futures leverage different sizes of estimation window and transaction costs

Alternate commodity indices

We consider a different set of commodity indices - Dow Jones UBS Commodity Index(DJUBS) Merrill Lynch Commodity Index (MLCI) and CYD Long-Short Commodity In-dex (CYDLS) which enables examining whether the usefulness of commodity indices variesby the index type DJUBS is a first generation index similar to SPGSCI but differs by beingless highly concentrated on energy and by imposing upper bounds on individual commodity(15) and sector (33) to ensure diversification MLCI is a second generation index similarto DBOYCI which reduces the negative effects of rolling contracts when the market is in

18Due to space limit we only report a portion of the results The complete set of findings are availablefrom the authors All robust tests are conducted in an out-of-sample setting since the in-sample portfolio isdominated by bonds and is less useful

13

contango19 Similar to MLSCI the CYDLS index also allows for long and short positionsdepending on the term structural signals20

Table 6 shows the out-of-sample portfolio performance with sample and shrinkage estimatesFor both the sample and shrinkage estimates the DJUBS and MLCI indices fail to improvethe Sharpe ratio but including CYDLS does increase the Sharpe ratio though insignificantlyfrom 031 to 058 with the sample estimates Again the benefit from Sharpe ratio increasedisappears when shrinkage estimates are used

Multiple commodities

In table 7 we include all twelve individual commodities in the portfolio which enables us tocheck whether there exists any subset of commodities that jointly benefit the portfolio PanelA shows the performance measures for benchmark and expanded portfolios Based on mean-variance optimization with sample estimates the expanded portfolio has a lower Shapreratio (020) compared to the benchmark portfolio (031) Although including commoditiescan reduce standard deviation from 678 to 633 it decreases the return even more from213 to 127 Similar results arise using the shrinkage estimates Panel B provides theaverage optimal weights for each asset With sample estimates the US bonds are 55of the portfolio followed by global bonds (21) US equities (19) and global equities(5) Once commodities are included the weights of benchmark assets drop and eightcommodities enter the portfolio with a total weight about 21 If the shrinkage estimatesare used the portfolios become more diversified The total weight of commodities is reducedto 9 indicating a less important role in the portfolio On balance including multiplecommodities does not improve the portfolio performance

Shrinkage factor

We examine the impact of shrinkage factor δ by changing it from 02 to 05 in which casethe sample estimates and the prior are equally weighted Since larger δ represents a moreconfident for the sample estimates it is expected that the results using shrinkage estimateswill look closer to those using sample estimates The general conclusions are similar Allcommodities fail to improve the portfoliorsquos Sharpe ratio except MSLSCI Including individualcommodities seems to reduce the portfoliorsquos standard deviation but the magnitude is smallerwith shrinkage estimates

Subsamples

To investigate the temporal usefulness of commodities we split the whole period into threesubsamples 1991-2003 2004-2007 and 2008-2012 Period 2004-2007 represents the com-modity price boom period while 2008-2012 includes the market meltdown and subsequentperiod in which cross-market correlation is strengthened Recall that the out-of-sampleoptimization needs 60-month observations for parameter estimation We use returns for

19See Merrill Lynch commodity paper httpwwwmlcommedia67354pdf20See CYD indices overview httpwwwcyd-researchcomenindiceslongshort tr indexphp

14

1991-1995 1999-2003 and 2003-2007 to start the optimization for each subsamples Table8 panel A reports the Sharpe ratios derived from optimization with sample estimates Asexpected the first- and second-generation commodity indices (SPGSCI and DBOYCI) in-creases portfolio Sharpe ratio only in the commodity boom period 2004-2007 In contrastthe third-generation commodity index MSLSCI improves the portfolio performance in theother periods Including agricultural commodity futures barely enhances portfolio perfor-mance in any periods except feeder cattle and live cattle When shrinkage estimates are used(table 8 panel B) benefits to commodities almost disappear except for the third-generationcommodity index in 1991-2003

Other robustness tests

In addition we consider the impacts of full-collateralization length of estimation windowand transaction costs First we relax the full-collateralized assumption since trading in-dividual commodity futures does not require principal except a small margin FollowingEgelkraut et al (2005) we assume a futures margin as 10 of the contract value21 Giventhe leverage the futures return is ten times as the usual return measured by percentageprice changes Using these levered returns we re-examine the role of twelve individual com-modity futures and find similar results The reason is that the leverage amplifies return andvolatility simultaneously leading to no difference in portfolio performance We also consideralternative sizes of estimation window (K) in the out-of-sample optimization The initial Kis specified as 60 months We check for the cases K = 36 48 72 and the results are ratherrobust

To include transaction costs we define the net-of-transaction-cost returns as

rnct+1 = (1 + rct+1)[1minus ctimesNsumj=1

(|wjt+1 minus wjt|)] (7)

where rct+1 and rnct+1 are portfolio returns before and after the transaction costs in period

t+ 1 c denotes the transaction cost vector andsumN

j=1(|wjt+1 minus wjt|) measures the amountof portfolio rebalanced Since the transaction costs may differ between asset types wefollow Daskalaki and Skiadopoulos (2011) by setting c equal to 50 basis points (05) pertransaction for equities and bonds and 35 basis points (035) for individual commodityfutures contracts22 For commodity indices we set it at 100 basis point (1) since the feecharged by most commodity index funds ranges from 075 to 15 We obtain qualitativelysimilar conclusions when allowing for transaction costs

21The actual margins are not available for all commodities Analyzing several commodities includingcorn soybeans wheat live cattle and lean hogs we find that the historical margins fluctuate about 5-10 See historical margins from CME httpwwwcmegroupcomclearingrisk-managementhistorical-marginshtml

22Daskalaki and Skiadopoulos (2011) establish the transaction costs levels based on discussion with prac-titioners Trading commodities may induce much lower transaction costs (Fuertes et al 2010) To ensurethe robustness we have also considered lower cost values for trading commodities and the results remainsimilar

15

Concluding remarks

This paper examines whether investors benefit by including commodities in their portfoliosWe differ from previous literature in two aspects First we evaluate whether the usefulnessof commodities varies by the type of investment tool Specifically we use the twelve agricul-tural commodities monitored by the CFTC and three commodity indices which allow fora broader range of commodities and represent the three generations of commodity indicesdeveloped by Miffre (2012) Second we explicitly control for estimation error in the processof optimization which has been ignored in prior research on this topic Estimation error hasbeen shown to induce concentrated and unstable portfolios which may mislead the evalu-ation on the role of commodities To reduce estimation error we shrink the sample meanestimates to a prior following Black and Littermanrsquos (1992) approach For commodities theprior returns are assumed to be zero consistent with findings that returns to commoditiesdo not differ from zero (Sanders and Irwin 2012) while the prior returns of traditional com-ponents - US and global equities and bonds - are derived from reversing the CAPM withequal weights

Overall the findings indicate that including individual agricultural commodity futures or thefirst- or second-generation commodity indices fail to significantly improve portfolio Sharperatios in both in-sample and out-of-sample analyses In contrast the third-generation com-modity indices improve the portfolio performance due to their imbedding strategies relatedto momentum and term structure signals Accounting for estimation error with shrinkageestimates we find that the optimal asset allocations are more diversified and stable and thatcommodities play a smaller role in a more diversified portfolio Specifically while includ-ing commodities can reduce volatility in highly concentrated portfolios their risk-reducingeffects almost disappear in the more diversified portfolios considered here In general theresults are robust to alternative commodity indices degree of shrinkage and including multi-ple commodities together We also relax the full-collateralized assumption take into accounttransaction costs change sizes of estimation window and perform the analysis over subsam-ples and find similar results

Our findings are largely consistent with recent studies on the role of commodities in investorrsquosportfolios Using out-of-sample utility maximization Daskalaki and Skiadopoulos (2011)find no significant improvements in Sharpe ratios by including popular commodity indices(SPGSCI and DJUBS) and five individual commodities (cotton crude oil copper gold andlive cattle) in a portfolio Similarly You and Daigler (2013) examine a number of commodityand financial futures in a mean-variance framework and find that the portfolio with futurescontracts outperforms the traditional portfolio in-sample but improved performance doesnot continue in an out-of-sample context Here we identify limited improvement in Sharperatios for some commodities (eg feeder and live cattle) in the in-sample analysis but noout-of-sample evidence of statistically significant portfolio improvement for both individualcommodities and the first-generation indices emerges We expand the literature by consid-ering the second- and third-generation commodity indices and find that the third-generationcommodity indices can improve portfolio Sharpe ratios significantly in most cases Thisfinding highlights the importance of commodity index categorization (Miffre 2012) when

16

studying portfolio investment by researchers as well as investors

Our findings contrast to some degree with the view that including commodities in portfolioswill be beneficial to investors by reducing volatility (eg Bodie and Rosansky 1980 Forten-bery and Hauser 1990 Ankrim and Hensel 1993) While commodities can marginally reducerisk in concentrated portfolios we show that in more balanced portfolios the risk reducingability of commodities is negligible Here the more balanced portfolios resulted from usingshrinkage estimates to reduce estimation error You and Daiglerrsquos (2013) conjecture that thefailure to account for estimation error which can lead to concentrated portfolio allocationsmight influence our understanding of the role commodities play in portfolios was perceptiveOur analysis demonstrates that in terms of risk mitigation their role is reduced

These findings have practical implications in a broader investment context and implicationsfor market behavior Investors often place constraints on portfolio allocations to diversifytheir risk and the notion that more balanced portfolios makes commodities less useful mayinfluence allocations and the presence of some financial investors in commodity marketsOur findings seem to be consistent with recent actions by some large pension funds Forexample the California Public Employees Retirement System (CalPERS) the largest USpension fund reduced its commodity allocation from 14 to 06 in October 2012 and to05 in early 2013 The same has been done by the California State Teachersrsquo RetirementSystem and the Ohio Police amp Fire Pension Fund23 On the market level since the top tenlargest exchanged-traded products (ETPs) on broad commodities track the first- or second-generation commodity indices according to the 2013 Q1 report by ETF Securities24 thegrowth of commodity index investments may slow given few benefits to portfolios Howeverinvestors may shift investment tools Since some of the third-generation commodity indicesproduce better performance it is likely that investors may move towards third-generation in-dices and be selective in identifying those indices that provides most attractive performanceIn a market context this suggests that future non-traditional investors in commodity marketsmay be more informed and selective in their market activities

23httpwwwpionlinecomarticle20130218PRINT302189978investors-get-active-over-commodities24httpwwwetfsecuritiescomDocumentsGlobal20Commodity20ETP20Quarterly20-

20Q1202013Europepdf

17

References

Ankrim E M and Hensel C R (1993) Commodities in asset allocation A real-assetalternative to real estate Financial Analysts Journal 49(3)20ndash29

Aulerich N M Irwin S H and Garcia P (2013) Bubbles food prices and speculationevidence from the CFTCs daily large trader data files Working paper

Bessembinder H and Chan K (1992) Time-varying risk premia and forecastable returnsin futures markets Journal of Financial Economics 32(2)149ndash168

Best M and Grauer R (1991) On the sensitivity of mean-variance-efficient portfolios tochanges in asset means Some analytical and computational retults Review of FinancialStudies 4(2)315ndash342

Bjornson B and Carter C (1997) New evidence on agricultural commodity return perfor-mance under time-varying risk American Journal of Agricultural Economics 79(3)918ndash930

Black F and Litterman R (1992) Global portfolio optimization Financial AnalystsJournal 48(5)28ndash43

Bodie Z and Rosansky V (1980) Risk and return in commodity futures Financial AnalystsJournal 36(3)27ndash39

Brentani C (2004) Portfolio Management in Practice Essential Capital Markets ElsevierScience

Buyukahin B Haigh M and Robe M (2010) Commodities and equities Ever a marketof one Journal of Alternative Investments 12(3)76ndash95

Daskalaki C and Skiadopoulos G (2011) Should investors include commodities in theirportfolios after all New evidence Journal of Banking amp Finance 35(10)2606ndash2626

Drobetz W (2001) How to avoid the pitfalls in portfolio optimization Putting the Black-Litterman approach at work Financial Markets and Portfolio Management 15(1)59ndash75

Dusak K (1973) Futures trading and investor returns An investigation of commoditymarket risk premiums Journal of Political Economy 81(6)1387ndash1406

Egelkraut T Woodard J Garcia P and Pennings J M E (2005) Portfolio diver-sification with commodity futures Properties of leveraged futures NCR-134 Conferencepaper

Erb C and Harvey C (2006) The strategic and tactical value of commodity futuresFinancial Analysts Journal 62(2)69ndash98

Fabozzi F J Kolm P N Pachamanova D and Focardi S M (2007) Robust portfoliooptimization Journal of Portfolio Management 33(3)40ndash48

18

Fama E and French K (1987) Commodity futures prices Some evidence on forecastpower premiums and the theory of storage Journal of Business 60(1)55ndash73

Fortenbery T and Hauser R (1990) Investment potentical of agricultural futures contractsAmerican Journal of Agricultural Economics 72(3)721ndash728

Frank J and Garcia P (2009) Time-varying risk premium further evidence in agriculturalfutures markets Applied Economics 41(6)715ndash725

Fuertes A-M Miffre J and Rallis G (2010) Tactical allocation in commodity futuresmarkets Combining momentum and term structure signals Journal of Banking amp Finance34(10)2530ndash2548

Georgiev G (2001) Benefits of commodity investment Journal of Alternative Investments4(1)40ndash48

Gorton G and Rouwenhorst K (2006) Facts and fantasies about commodity futuresFinancial Analysts Journal 62(2)47ndash68

He G and Litterman R (1999) The intuition behind Black-Litterman model portfoliosGoldman Sachs Asset Management working paper

James W and Stein C (1961) Estimation with quadratic loss In Proceedings of the FourthBerkeley Symposium on Mathematical Statistics and Probability Volume 1 Contributionsto the Theory of Statistics pages 361ndash379 University of California Press

Jorion P (1986) Bayes-Stein estimation for portfolio analysis Journal of Financial andQuantitative Analysis 21(3)279ndash292

Kan R and Zhou G (2009) Optimal portfolio choice with parameter uncertainty Journalof Financial and Quantitative Analysis 42(03)621ndash656

Keynes J M (1930) A Treatise on Money London Macmillan

Ledoit O and Wolf M (2003) Improved estimation of the covariance matrix of stockreturns with an application to portfolio selection Journal of Empirical Finance 10(5)603ndash621

Ledoit O and Wolf M (2008) Robust performance hypothesis testing with the Sharperatio Journal of Empirical Finance 15(5)850ndash859

Markowitz H (1952) Portfolio selection Journal of Finance 7(1)77ndash91

Michaud R O (1989) The Markowitz optimization enigma Is the optimized optimalFinancial Analysts Journal 45(1)31ndash42

Miffre J (2012) Comparing first second and third generation commodity indices EDHECworking paper

19

Sanders D R and Irwin S H (2012) A reappraisal of investing in commodity futuresmarkets Applied Economic Perspectives and Policy 34(3)515ndash530

Satyanarayan S and Varangis P (1996) Diversification benefits of commodity assets inglobal portfolios Journal of Investing 5(1)69ndash78

Sharpe W F (1974) Imputing expected security returns from porfolio composition Journalof Financial and Quantitative Analysis 9(3)463ndash472

Silvennoinen A and Thorp S (2013) Financialization crisis and commodity correlationdynamics Journal of International Financial Markets Institutions and Money 24(0)42ndash65

Singleton K (2014) Investor flows and the 2008 boombust in oil prices ManagementScience 60(2)300ndash318

Szymanowska M De Roon F Nijman T and den Goorbergh R V (2013) An anatomyof commodity futures risk premia Journal of Finance forthcoming

Tang K and Xiong W (2012) Index investment and the financialization of commoditiesFinancial Analysts Journal 68(6)54ndash74

Tobin J (1958) Liquidity preference as behavior towards risk Review of Economic Studies25(2)65ndash86

Willenbrock S (2011) Diversification return portfolio rebalancing and the commodityreturn puzzle Financial Analysts Journal 67(4)42ndash49

You L and Daigler R (2013) A Markowitz optimization of commodity futures portfoliosJournal of Futures Markets 33(4)343ndash368

20

Table 1 Summary statistics on asset excess returns

1991 - 2012 1991 - 2003 2004 - 2007 2008 - 2012Mean SD p Mean SD Mean SD Mean SD

US equities 570 1476 007 730 1386 545 856 183 2009US bonds 323 357 000 332 358 079 312 489 383Global equities 040 1693 091 -036 1472 1105 1019 -605 2467Global bonds 328 592 001 328 589 165 496 457 670SPGSCI 025 2153 096 089 1795 886 2149 -817 2890DBOYCI 612 1741 010 429 1435 2247 1442 -208 2479MSLSCI 528 1067 002 630 882 680 1149 148 1397Cocoa -238 2888 070 -790 2738 953 2505 234 3498Coffee -294 3539 070 -510 3885 646 2610 -481 3271Cotton -624 2907 031 -601 2584 -2182 2515 544 3835Sugar 441 3227 052 479 2906 -121 2732 787 4257Feeder cattle 343 1169 017 314 1120 865 1188 004 1280Lean hogs -189 2742 075 -266 2930 298 2458 -376 2482Live cattle 552 1409 007 516 1490 773 1319 468 1281Corn -431 2548 043 -556 1965 -710 2960 109 3418Soybeans 638 2390 021 515 1893 357 2990 1171 2975Soybean oil 005 2456 099 026 2024 722 2714 -612 3182Wheat CBOT -313 2819 060 -010 2476 170 2504 -1467 3751Wheat KBOT 371 2678 052 471 2267 1773 2314 -989 3716

Note Excess returns are defined as the percentage changes of monthly prices in excess of risk-freerate Mean and SD denote the average annualized excess returns and standard deviations inpercentage formats The p-value is for a two-tailed t-test that the mean equals to zero

Table 2 Correlation between asset excess returns

USbonds

Globalequities

Globalbonds

SPGSCI DBOYCI MSLSCI Comm-odities

US equities 1991-2012 002 078 009 020 023 -010 0141991-2003 003 069 -001 -002 000 -014 0062004-2007 -014 070 -009 -024 -011 -024 0082008-2012 006 091 030 057 055 -003 029

US bonds 1991-2012 006 068 -001 -002 -008 0041991-2003 002 066 004 -004 -002 0022004-2007 002 068 002 012 012 0022008-2012 015 074 -007 001 -028 010

Global equities 1991-2012 032 037 040 003 0161991-2003 023 018 017 001 0042004-2007 027 009 022 012 0072008-2012 048 063 062 001 033

Global bonds 1991-2012 014 016 005 0071991-2003 012 007 004 0002004-2007 020 030 023 0042008-2012 016 026 -002 020

Note The last column reports averaged correlations across twelve individual commodity futures

21

Table 3 Portfolio performance comparison In-sample 1991-2012

Benchmarkportfolio

Expanded portfolioSPGSCI DBOYCI MSLSCI Feeder cattle Live cattle

Panel A Mean-variance optimization with sample estimates

Mean 345 345 357 377 344 361SD 354 354 351 327 341 347Sharpe 098 098 102 115 101 104

Panel B Mean-variance optimization with shrinkage estimates

Mean 344 344 344 362 344 345SD 670 671 671 612 671 671Sharpe 051 051 051 059 051 051

Note Benchmark portfolio consists of US equities US bonds global equities and global bonds andis expanded by adding one commodity at a time Mean and SD are based on the annualized portfolioexcess returns Sharpe ratio (Sharpe) is defined as the ratio between Mean and SD The shrinkagefactor δ is assumed to be 02 when shrinkage estimates are used

22

Table 4 Optimal portfolio weights In-sample and out-of-sample

Sample estimates Shrinkage estimatesBenchmark SPGSCI DBOYCI MSLSCI Commodities Benchmark SPGSCI DBOYCI MSLSCI Commodities

Panel A In-sample

US equities 0090 0090 0070 0082 0087 0254 0254 0254 0244 0254US bonds 0910 0910 0872 0753 0896 0374 0374 0374 0375 0373Global equities 0000 0000 0000 0000 0000 0149 0150 0150 0127 0150Global bonds 0000 0000 0000 0000 0000 0222 0222 0222 0186 0223Commodities 0000 0059 0164 0017 0000 0000 0069 0000

Panel B Out-of-sample

US equities 0189 0195 0188 0138 0177 0266 0268 0267 0258 0263(0276) (0273) (0239) (0184) (0257) (0070) (0068) (0068) (0045) (0069)

US bonds 0546 0542 0601 0513 0564 0338 0340 0339 0341 0332(0332) (0322) (0342) (0297) (0360) (0128) (0127) (0127) (0124) (0129)

Global equities 0050 0038 0009 0032 0047 0167 0164 0163 0126 0166(0128) (0098) (0028) (0087) (0120) (0071) (0067) (0068) (0069) (0070)

Global bonds 0215 0194 0075 0090 0178 0230 0226 0227 0184 0231(0199) (0192) (0131) (0136) (0204) (0080) (0079) (0079) (0078) (0081)

Commodities 0031 0128 0227 0035 0003 0004 0091 0009(0041) (0108) (0114) (0044) (0009) (0010) (0057) (0013)

Note Benchmark portfolio consists of US equities US bonds global equities and global bonds and is expanded by adding one commodity at atime For twelve individual commodities the weights are further averaged across commodities and reported in the last column Standard deviationsof weights over periods are reported in the parenthesis The out-of-sample optimization is based on 60-months estimation window (K = 60) andmonthly rebalancing and the weights are reported as averages over periods The shrinkage factor δ is assumed to be 02 when shrinkage estimatesare used

23

Table 5 Portfolio performance comparison Out-sample 1991-2012

Benchmarkportfolio



Mean 213 190 232 362 240 274SD 678 686 736 540 660 648Sharpe 031 028 032 067 036 042p-value 067 099 005 050 017



Note Benchmark portfolio consists of US equities US bonds global equities and global bonds andis expanded by adding one commodity at a time Mean and SD are based on the annualized portfolioexcess returns Sharpe ratio (Sharpe) is defined as the ratio between Mean and SD The p-values arecomputed based on Ledoit and Wolf (2008) with the null hypothesis that the Sharpe ratio obtained frombenchmark portfolio is equal to that derived from expanded portfolios The out-of-sample optimizationis based on a 60-month estimation window (K = 60) and monthly rebalancing The shrinkage factor δ isassumed to be 02 when shrinkage estimates are used

Table 6 Portfolio performance comparison Alternate commodity indices 1991-2012

Sample estimates Shrinkage estimatesBenchmark DJUBS MLCI CYDLS Benchmark DJUBS MLCI CYDLS

Mean 213 154 234 301 224 224 227 192SD 678 652 698 519 776 776 776 700Sharpe 031 024 034 058 029 029 029 027p-value 038 092 017 079 057 076

Note Benchmark portfolio consists of US equities US bonds global equities and global bondsand is expanded by including a different set of commodity indices - Dow Jones UBS Commodity index(DJUBS) Merrill Lynch Commodity index (MLCI) and CYD Long-Short Commodity index (CYDLS)Mean and SD are based on the annualized portfolio excess returns Sharpe ratio (Sharpe) is definedas the ratio between Mean and SD The p-values are computed based on Ledoit and Wolf (2008) withthe null hypothesis that the Sharpe ratio obtained from benchmark portfolio is equal to that derivedfrom expanded portfolios The out-of-sample optimization is based on a 60-month estimation window(K = 60) and monthly rebalancing The shrinkage factor δ is assumed to be 02 when shrinkage estimatesare used

24

Table 7 Portfolio performance comparison Multiple commodities 1991-2012

Sample estimates Shrinkage estimatesBenchmark Expanded Benchmark Expanded

Panel A Portfolio performanceMean 213 127 224 190SD 678 633 776 750Sharpe 031 020 029 025p-value 050 054

Panel B Optimal weights US equities 019 011 027 024US bonds 055 047 034 028Global equities 005 003 017 015Global bonds 021 017 023 025Commodities (total) 021 009

Cocoa 004 002Coffee 001 001Cotton 000 000Sugar 002 001Feeder cattle 005 002Hogs 000 001Live cattle 005 001Corn 000 000Soybeans 002 000Soy oil 001 001Wheat CBOT 000 000Wheat KBOT 001 000

Note Benchmark portfolio consists of US equities US bonds global equities and global bonds andis expanded by including all twelve individual commodities Mean and SD are based on the annualizedportfolio excess returns Sharpe ratio (Sharpe) is defined as the ratio between Mean and SD Thep-values are computed based on Ledoit and Wolf (2008) with the null hypothesis that the Sharpe ratioobtained from benchmark portfolio is equal to that derived from expanded portfolios The out-of-sampleoptimization is based on a 60-month estimation window (K = 60) and monthly rebalancing Optimalweights are averaged across periods The shrinkage factor δ is assumed to be 02 when shrinkage estimatesare used

25

Table 8 Sharpe ratio performance Subsamples

Benchmarkportfolio



1991-2003 054 053 054 110 056 0602004-2007 020 028 105 047 033 0392008-2012 001 -013 -019 007 008 016


1991-2003 030 030 031 061 031 0332004-2007 088 086 088 093 086 0912008-2012 012 012 011 005 011 013

Note Benchmark portfolio consists of US equities US bonds global equities and global bonds and isexpanded by one commodity at a timeThe out-of-sample optimization is based on a 60-month estimationwindow (K = 60) and monthly rebalancing The shrinkage factor δ is assumed to be 02 when shrinkageestimates are used

26


Lei Yan

and

Philip Garcia lowast

Paper presented at the NCR-134 Conference on Applied Commodity PriceAnalysis Forecasting and Market Risk Management

St Louis Missouri April 21-22 2014

Copyright 2014 by Lei Yan and Philip Garcia All rights reserved Readers may makeverbatim copies of this document for non-commercial purposes by any means

provided that this copyright notice appears on all such copies

lowastLei Yan is a PhD student and Philip Garcia is the TA Hieronymus Distinguished Chair in FuturesMarkets both in the Department of Agricultural and Consumer Economics at the University of Illinois atUrbana-Champaign Correspondence may be sent to leiyan2illinoisedu We thank Anton BekkermanJeffrey Dorfman Randall Fortenbery Scott Irwin Berna Karali Paul Peterson Dwight Sanders and par-ticipants at the NCCC-134 conference for helpful discussions and comments Any remaining errors are ourresponsibility alone




Introduction




1





2

Related Literature




3




4


Data






5



Rt = ln(ptptminus1

)times 100 (1)







6






7


Methods





maxw


(2)



wlowast =1






8







σ2m

(5)






9



σ2m

ι = [1 1 1]prime (6)



414

14



Implementation




10



Results





11







12



Robust analysis





13





Shrinkage factor


Subsamples



14












15

Concluding remarks




16





20Q1202013Europepdf

17

References
















18

















19











20






USbonds

Globalequities

Globalbonds


US equities 1991-2012 002 078 009 020 023 -010 0141991-2003 003 069 -001 -002 000 -014 0062004-2007 -014 070 -009 -024 -011 -024 0082008-2012 006 091 030 057 055 -003 029

US bonds 1991-2012 006 068 -001 -002 -008 0041991-2003 002 066 004 -004 -002 0022004-2007 002 068 002 012 012 0022008-2012 015 074 -007 001 -028 010

Global equities 1991-2012 032 037 040 003 0161991-2003 023 018 017 001 0042004-2007 027 009 022 012 0072008-2012 048 063 062 001 033

Global bonds 1991-2012 014 016 005 0071991-2003 012 007 004 0002004-2007 020 030 023 0042008-2012 016 026 -002 020


21


Benchmarkportfolio



Mean 345 345 357 377 344 361SD 354 354 351 327 341 347Sharpe 098 098 102 115 101 104


Mean 344 344 344 362 344 345SD 670 671 671 612 671 671Sharpe 051 051 051 059 051 051


22



Panel A In-sample



US equities 0189 0195 0188 0138 0177 0266 0268 0267 0258 0263(0276) (0273) (0239) (0184) (0257) (0070) (0068) (0068) (0045) (0069)

US bonds 0546 0542 0601 0513 0564 0338 0340 0339 0341 0332(0332) (0322) (0342) (0297) (0360) (0128) (0127) (0127) (0124) (0129)

Global equities 0050 0038 0009 0032 0047 0167 0164 0163 0126 0166(0128) (0098) (0028) (0087) (0120) (0071) (0067) (0068) (0069) (0070)

Global bonds 0215 0194 0075 0090 0178 0230 0226 0227 0184 0231(0199) (0192) (0131) (0136) (0204) (0080) (0079) (0079) (0078) (0081)

Commodities 0031 0128 0227 0035 0003 0004 0091 0009(0041) (0108) (0114) (0044) (0009) (0010) (0057) (0013)


23


Benchmarkportfolio











24







25


Benchmarkportfolio



1991-2003 054 053 054 110 056 0602004-2007 020 028 105 047 033 0392008-2012 001 -013 -019 007 008 016


1991-2003 030 030 031 061 031 0332004-2007 088 086 088 093 086 0912008-2012 012 012 011 005 011 013


26




Introduction




1





2

Related Literature




3




4


Data






5



Rt = ln(ptptminus1

)times 100 (1)







6






7


Methods





maxw


(2)



wlowast =1






8







σ2m

(5)






9



σ2m

ι = [1 1 1]prime (6)



414

14



Implementation




10



Results





11







12



Robust analysis





13





Shrinkage factor


Subsamples



14












15

Concluding remarks




16





20Q1202013Europepdf

17

References
















18

















19











20






USbonds

Globalequities

Globalbonds


US equities 1991-2012 002 078 009 020 023 -010 0141991-2003 003 069 -001 -002 000 -014 0062004-2007 -014 070 -009 -024 -011 -024 0082008-2012 006 091 030 057 055 -003 029

US bonds 1991-2012 006 068 -001 -002 -008 0041991-2003 002 066 004 -004 -002 0022004-2007 002 068 002 012 012 0022008-2012 015 074 -007 001 -028 010

Global equities 1991-2012 032 037 040 003 0161991-2003 023 018 017 001 0042004-2007 027 009 022 012 0072008-2012 048 063 062 001 033

Global bonds 1991-2012 014 016 005 0071991-2003 012 007 004 0002004-2007 020 030 023 0042008-2012 016 026 -002 020


21


Benchmarkportfolio



Mean 345 345 357 377 344 361SD 354 354 351 327 341 347Sharpe 098 098 102 115 101 104


Mean 344 344 344 362 344 345SD 670 671 671 612 671 671Sharpe 051 051 051 059 051 051


22



Panel A In-sample



US equities 0189 0195 0188 0138 0177 0266 0268 0267 0258 0263(0276) (0273) (0239) (0184) (0257) (0070) (0068) (0068) (0045) (0069)

US bonds 0546 0542 0601 0513 0564 0338 0340 0339 0341 0332(0332) (0322) (0342) (0297) (0360) (0128) (0127) (0127) (0124) (0129)

Global equities 0050 0038 0009 0032 0047 0167 0164 0163 0126 0166(0128) (0098) (0028) (0087) (0120) (0071) (0067) (0068) (0069) (0070)

Global bonds 0215 0194 0075 0090 0178 0230 0226 0227 0184 0231(0199) (0192) (0131) (0136) (0204) (0080) (0079) (0079) (0078) (0081)

Commodities 0031 0128 0227 0035 0003 0004 0091 0009(0041) (0108) (0114) (0044) (0009) (0010) (0057) (0013)


23


Benchmarkportfolio











24







25


Benchmarkportfolio



1991-2003 054 053 054 110 056 0602004-2007 020 028 105 047 033 0392008-2012 001 -013 -019 007 008 016


1991-2003 030 030 031 061 031 0332004-2007 088 086 088 093 086 0912008-2012 012 012 011 005 011 013


26





2

Related Literature




3




4


Data






5



Rt = ln(ptptminus1

)times 100 (1)







6






7


Methods





maxw


(2)



wlowast =1






8







σ2m

(5)






9



σ2m

ι = [1 1 1]prime (6)



414

14



Implementation




10



Results





11







12



Robust analysis





13





Shrinkage factor


Subsamples



14












15

Concluding remarks




16





20Q1202013Europepdf

17

References
















18

















19











20






USbonds

Globalequities

Globalbonds


US equities 1991-2012 002 078 009 020 023 -010 0141991-2003 003 069 -001 -002 000 -014 0062004-2007 -014 070 -009 -024 -011 -024 0082008-2012 006 091 030 057 055 -003 029

US bonds 1991-2012 006 068 -001 -002 -008 0041991-2003 002 066 004 -004 -002 0022004-2007 002 068 002 012 012 0022008-2012 015 074 -007 001 -028 010

Global equities 1991-2012 032 037 040 003 0161991-2003 023 018 017 001 0042004-2007 027 009 022 012 0072008-2012 048 063 062 001 033

Global bonds 1991-2012 014 016 005 0071991-2003 012 007 004 0002004-2007 020 030 023 0042008-2012 016 026 -002 020


21


Benchmarkportfolio



Mean 345 345 357 377 344 361SD 354 354 351 327 341 347Sharpe 098 098 102 115 101 104


Mean 344 344 344 362 344 345SD 670 671 671 612 671 671Sharpe 051 051 051 059 051 051


22



Panel A In-sample



US equities 0189 0195 0188 0138 0177 0266 0268 0267 0258 0263(0276) (0273) (0239) (0184) (0257) (0070) (0068) (0068) (0045) (0069)

US bonds 0546 0542 0601 0513 0564 0338 0340 0339 0341 0332(0332) (0322) (0342) (0297) (0360) (0128) (0127) (0127) (0124) (0129)

Global equities 0050 0038 0009 0032 0047 0167 0164 0163 0126 0166(0128) (0098) (0028) (0087) (0120) (0071) (0067) (0068) (0069) (0070)

Global bonds 0215 0194 0075 0090 0178 0230 0226 0227 0184 0231(0199) (0192) (0131) (0136) (0204) (0080) (0079) (0079) (0078) (0081)

Commodities 0031 0128 0227 0035 0003 0004 0091 0009(0041) (0108) (0114) (0044) (0009) (0010) (0057) (0013)


23


Benchmarkportfolio











24







25


Benchmarkportfolio



1991-2003 054 053 054 110 056 0602004-2007 020 028 105 047 033 0392008-2012 001 -013 -019 007 008 016


1991-2003 030 030 031 061 031 0332004-2007 088 086 088 093 086 0912008-2012 012 012 011 005 011 013


26

Related Literature




3




4


Data






5



Rt = ln(ptptminus1

)times 100 (1)







6






7


Methods





maxw


(2)



wlowast =1






8







σ2m

(5)






9



σ2m

ι = [1 1 1]prime (6)



414

14



Implementation




10



Results





11







12



Robust analysis





13





Shrinkage factor


Subsamples



14












15

Concluding remarks




16





20Q1202013Europepdf

17

References
















18

















19











20






USbonds

Globalequities

Globalbonds


US equities 1991-2012 002 078 009 020 023 -010 0141991-2003 003 069 -001 -002 000 -014 0062004-2007 -014 070 -009 -024 -011 -024 0082008-2012 006 091 030 057 055 -003 029

US bonds 1991-2012 006 068 -001 -002 -008 0041991-2003 002 066 004 -004 -002 0022004-2007 002 068 002 012 012 0022008-2012 015 074 -007 001 -028 010

Global equities 1991-2012 032 037 040 003 0161991-2003 023 018 017 001 0042004-2007 027 009 022 012 0072008-2012 048 063 062 001 033

Global bonds 1991-2012 014 016 005 0071991-2003 012 007 004 0002004-2007 020 030 023 0042008-2012 016 026 -002 020


21


Benchmarkportfolio



Mean 345 345 357 377 344 361SD 354 354 351 327 341 347Sharpe 098 098 102 115 101 104


Mean 344 344 344 362 344 345SD 670 671 671 612 671 671Sharpe 051 051 051 059 051 051


22



Panel A In-sample



US equities 0189 0195 0188 0138 0177 0266 0268 0267 0258 0263(0276) (0273) (0239) (0184) (0257) (0070) (0068) (0068) (0045) (0069)

US bonds 0546 0542 0601 0513 0564 0338 0340 0339 0341 0332(0332) (0322) (0342) (0297) (0360) (0128) (0127) (0127) (0124) (0129)

Global equities 0050 0038 0009 0032 0047 0167 0164 0163 0126 0166(0128) (0098) (0028) (0087) (0120) (0071) (0067) (0068) (0069) (0070)

Global bonds 0215 0194 0075 0090 0178 0230 0226 0227 0184 0231(0199) (0192) (0131) (0136) (0204) (0080) (0079) (0079) (0078) (0081)

Commodities 0031 0128 0227 0035 0003 0004 0091 0009(0041) (0108) (0114) (0044) (0009) (0010) (0057) (0013)


23


Benchmarkportfolio











24







25


Benchmarkportfolio



1991-2003 054 053 054 110 056 0602004-2007 020 028 105 047 033 0392008-2012 001 -013 -019 007 008 016


1991-2003 030 030 031 061 031 0332004-2007 088 086 088 093 086 0912008-2012 012 012 011 005 011 013


26




4


Data






5



Rt = ln(ptptminus1

)times 100 (1)







6






7


Methods





maxw


(2)



wlowast =1






8







σ2m

(5)






9



σ2m

ι = [1 1 1]prime (6)



414

14



Implementation




10



Results





11







12



Robust analysis





13





Shrinkage factor


Subsamples



14












15

Concluding remarks




16





20Q1202013Europepdf

17

References
















18

















19











20






USbonds

Globalequities

Globalbonds


US equities 1991-2012 002 078 009 020 023 -010 0141991-2003 003 069 -001 -002 000 -014 0062004-2007 -014 070 -009 -024 -011 -024 0082008-2012 006 091 030 057 055 -003 029

US bonds 1991-2012 006 068 -001 -002 -008 0041991-2003 002 066 004 -004 -002 0022004-2007 002 068 002 012 012 0022008-2012 015 074 -007 001 -028 010

Global equities 1991-2012 032 037 040 003 0161991-2003 023 018 017 001 0042004-2007 027 009 022 012 0072008-2012 048 063 062 001 033

Global bonds 1991-2012 014 016 005 0071991-2003 012 007 004 0002004-2007 020 030 023 0042008-2012 016 026 -002 020


21


Benchmarkportfolio



Mean 345 345 357 377 344 361SD 354 354 351 327 341 347Sharpe 098 098 102 115 101 104


Mean 344 344 344 362 344 345SD 670 671 671 612 671 671Sharpe 051 051 051 059 051 051


22



Panel A In-sample



US equities 0189 0195 0188 0138 0177 0266 0268 0267 0258 0263(0276) (0273) (0239) (0184) (0257) (0070) (0068) (0068) (0045) (0069)

US bonds 0546 0542 0601 0513 0564 0338 0340 0339 0341 0332(0332) (0322) (0342) (0297) (0360) (0128) (0127) (0127) (0124) (0129)

Global equities 0050 0038 0009 0032 0047 0167 0164 0163 0126 0166(0128) (0098) (0028) (0087) (0120) (0071) (0067) (0068) (0069) (0070)

Global bonds 0215 0194 0075 0090 0178 0230 0226 0227 0184 0231(0199) (0192) (0131) (0136) (0204) (0080) (0079) (0079) (0078) (0081)

Commodities 0031 0128 0227 0035 0003 0004 0091 0009(0041) (0108) (0114) (0044) (0009) (0010) (0057) (0013)


23


Benchmarkportfolio











24







25


Benchmarkportfolio



1991-2003 054 053 054 110 056 0602004-2007 020 028 105 047 033 0392008-2012 001 -013 -019 007 008 016


1991-2003 030 030 031 061 031 0332004-2007 088 086 088 093 086 0912008-2012 012 012 011 005 011 013


26


Data






5



Rt = ln(ptptminus1

)times 100 (1)







6






7


Methods





maxw


(2)



wlowast =1






8







σ2m

(5)






9



σ2m

ι = [1 1 1]prime (6)



414

14



Implementation




10



Results





11







12



Robust analysis





13





Shrinkage factor


Subsamples



14












15

Concluding remarks




16





20Q1202013Europepdf

17

References
















18

















19











20






USbonds

Globalequities

Globalbonds


US equities 1991-2012 002 078 009 020 023 -010 0141991-2003 003 069 -001 -002 000 -014 0062004-2007 -014 070 -009 -024 -011 -024 0082008-2012 006 091 030 057 055 -003 029

US bonds 1991-2012 006 068 -001 -002 -008 0041991-2003 002 066 004 -004 -002 0022004-2007 002 068 002 012 012 0022008-2012 015 074 -007 001 -028 010

Global equities 1991-2012 032 037 040 003 0161991-2003 023 018 017 001 0042004-2007 027 009 022 012 0072008-2012 048 063 062 001 033

Global bonds 1991-2012 014 016 005 0071991-2003 012 007 004 0002004-2007 020 030 023 0042008-2012 016 026 -002 020


21


Benchmarkportfolio



Mean 345 345 357 377 344 361SD 354 354 351 327 341 347Sharpe 098 098 102 115 101 104


Mean 344 344 344 362 344 345SD 670 671 671 612 671 671Sharpe 051 051 051 059 051 051


22



Panel A In-sample



US equities 0189 0195 0188 0138 0177 0266 0268 0267 0258 0263(0276) (0273) (0239) (0184) (0257) (0070) (0068) (0068) (0045) (0069)

US bonds 0546 0542 0601 0513 0564 0338 0340 0339 0341 0332(0332) (0322) (0342) (0297) (0360) (0128) (0127) (0127) (0124) (0129)

Global equities 0050 0038 0009 0032 0047 0167 0164 0163 0126 0166(0128) (0098) (0028) (0087) (0120) (0071) (0067) (0068) (0069) (0070)

Global bonds 0215 0194 0075 0090 0178 0230 0226 0227 0184 0231(0199) (0192) (0131) (0136) (0204) (0080) (0079) (0079) (0078) (0081)

Commodities 0031 0128 0227 0035 0003 0004 0091 0009(0041) (0108) (0114) (0044) (0009) (0010) (0057) (0013)


23


Benchmarkportfolio











24







25


Benchmarkportfolio



1991-2003 054 053 054 110 056 0602004-2007 020 028 105 047 033 0392008-2012 001 -013 -019 007 008 016


1991-2003 030 030 031 061 031 0332004-2007 088 086 088 093 086 0912008-2012 012 012 011 005 011 013


26



Rt = ln(ptptminus1

)times 100 (1)







6






7


Methods





maxw


(2)



wlowast =1






8







σ2m

(5)






9



σ2m

ι = [1 1 1]prime (6)



414

14



Implementation




10



Results





11







12



Robust analysis





13





Shrinkage factor


Subsamples



14












15

Concluding remarks




16





20Q1202013Europepdf

17

References
















18

















19











20






USbonds

Globalequities

Globalbonds


US equities 1991-2012 002 078 009 020 023 -010 0141991-2003 003 069 -001 -002 000 -014 0062004-2007 -014 070 -009 -024 -011 -024 0082008-2012 006 091 030 057 055 -003 029

US bonds 1991-2012 006 068 -001 -002 -008 0041991-2003 002 066 004 -004 -002 0022004-2007 002 068 002 012 012 0022008-2012 015 074 -007 001 -028 010

Global equities 1991-2012 032 037 040 003 0161991-2003 023 018 017 001 0042004-2007 027 009 022 012 0072008-2012 048 063 062 001 033

Global bonds 1991-2012 014 016 005 0071991-2003 012 007 004 0002004-2007 020 030 023 0042008-2012 016 026 -002 020


21


Benchmarkportfolio



Mean 345 345 357 377 344 361SD 354 354 351 327 341 347Sharpe 098 098 102 115 101 104


Mean 344 344 344 362 344 345SD 670 671 671 612 671 671Sharpe 051 051 051 059 051 051


22



Panel A In-sample



US equities 0189 0195 0188 0138 0177 0266 0268 0267 0258 0263(0276) (0273) (0239) (0184) (0257) (0070) (0068) (0068) (0045) (0069)

US bonds 0546 0542 0601 0513 0564 0338 0340 0339 0341 0332(0332) (0322) (0342) (0297) (0360) (0128) (0127) (0127) (0124) (0129)

Global equities 0050 0038 0009 0032 0047 0167 0164 0163 0126 0166(0128) (0098) (0028) (0087) (0120) (0071) (0067) (0068) (0069) (0070)

Global bonds 0215 0194 0075 0090 0178 0230 0226 0227 0184 0231(0199) (0192) (0131) (0136) (0204) (0080) (0079) (0079) (0078) (0081)

Commodities 0031 0128 0227 0035 0003 0004 0091 0009(0041) (0108) (0114) (0044) (0009) (0010) (0057) (0013)


23


Benchmarkportfolio











24







25


Benchmarkportfolio



1991-2003 054 053 054 110 056 0602004-2007 020 028 105 047 033 0392008-2012 001 -013 -019 007 008 016


1991-2003 030 030 031 061 031 0332004-2007 088 086 088 093 086 0912008-2012 012 012 011 005 011 013


26






7


Methods





maxw


(2)



wlowast =1






8







σ2m

(5)






9



σ2m

ι = [1 1 1]prime (6)



414

14



Implementation




10



Results





11







12



Robust analysis





13





Shrinkage factor


Subsamples



14












15

Concluding remarks




16





20Q1202013Europepdf

17

References
















18

















19











20






USbonds

Globalequities

Globalbonds


US equities 1991-2012 002 078 009 020 023 -010 0141991-2003 003 069 -001 -002 000 -014 0062004-2007 -014 070 -009 -024 -011 -024 0082008-2012 006 091 030 057 055 -003 029

US bonds 1991-2012 006 068 -001 -002 -008 0041991-2003 002 066 004 -004 -002 0022004-2007 002 068 002 012 012 0022008-2012 015 074 -007 001 -028 010

Global equities 1991-2012 032 037 040 003 0161991-2003 023 018 017 001 0042004-2007 027 009 022 012 0072008-2012 048 063 062 001 033

Global bonds 1991-2012 014 016 005 0071991-2003 012 007 004 0002004-2007 020 030 023 0042008-2012 016 026 -002 020


21


Benchmarkportfolio



Mean 345 345 357 377 344 361SD 354 354 351 327 341 347Sharpe 098 098 102 115 101 104


Mean 344 344 344 362 344 345SD 670 671 671 612 671 671Sharpe 051 051 051 059 051 051


22



Panel A In-sample



US equities 0189 0195 0188 0138 0177 0266 0268 0267 0258 0263(0276) (0273) (0239) (0184) (0257) (0070) (0068) (0068) (0045) (0069)

US bonds 0546 0542 0601 0513 0564 0338 0340 0339 0341 0332(0332) (0322) (0342) (0297) (0360) (0128) (0127) (0127) (0124) (0129)

Global equities 0050 0038 0009 0032 0047 0167 0164 0163 0126 0166(0128) (0098) (0028) (0087) (0120) (0071) (0067) (0068) (0069) (0070)

Global bonds 0215 0194 0075 0090 0178 0230 0226 0227 0184 0231(0199) (0192) (0131) (0136) (0204) (0080) (0079) (0079) (0078) (0081)

Commodities 0031 0128 0227 0035 0003 0004 0091 0009(0041) (0108) (0114) (0044) (0009) (0010) (0057) (0013)


23


Benchmarkportfolio











24







25


Benchmarkportfolio



1991-2003 054 053 054 110 056 0602004-2007 020 028 105 047 033 0392008-2012 001 -013 -019 007 008 016


1991-2003 030 030 031 061 031 0332004-2007 088 086 088 093 086 0912008-2012 012 012 011 005 011 013


26


Methods





maxw


(2)



wlowast =1






8







σ2m

(5)






9



σ2m

ι = [1 1 1]prime (6)



414

14



Implementation




10



Results





11







12



Robust analysis





13





Shrinkage factor


Subsamples



14












15

Concluding remarks




16





20Q1202013Europepdf

17

References
















18

















19











20






USbonds

Globalequities

Globalbonds


US equities 1991-2012 002 078 009 020 023 -010 0141991-2003 003 069 -001 -002 000 -014 0062004-2007 -014 070 -009 -024 -011 -024 0082008-2012 006 091 030 057 055 -003 029

US bonds 1991-2012 006 068 -001 -002 -008 0041991-2003 002 066 004 -004 -002 0022004-2007 002 068 002 012 012 0022008-2012 015 074 -007 001 -028 010

Global equities 1991-2012 032 037 040 003 0161991-2003 023 018 017 001 0042004-2007 027 009 022 012 0072008-2012 048 063 062 001 033

Global bonds 1991-2012 014 016 005 0071991-2003 012 007 004 0002004-2007 020 030 023 0042008-2012 016 026 -002 020


21


Benchmarkportfolio



Mean 345 345 357 377 344 361SD 354 354 351 327 341 347Sharpe 098 098 102 115 101 104


Mean 344 344 344 362 344 345SD 670 671 671 612 671 671Sharpe 051 051 051 059 051 051


22



Panel A In-sample



US equities 0189 0195 0188 0138 0177 0266 0268 0267 0258 0263(0276) (0273) (0239) (0184) (0257) (0070) (0068) (0068) (0045) (0069)

US bonds 0546 0542 0601 0513 0564 0338 0340 0339 0341 0332(0332) (0322) (0342) (0297) (0360) (0128) (0127) (0127) (0124) (0129)

Global equities 0050 0038 0009 0032 0047 0167 0164 0163 0126 0166(0128) (0098) (0028) (0087) (0120) (0071) (0067) (0068) (0069) (0070)

Global bonds 0215 0194 0075 0090 0178 0230 0226 0227 0184 0231(0199) (0192) (0131) (0136) (0204) (0080) (0079) (0079) (0078) (0081)

Commodities 0031 0128 0227 0035 0003 0004 0091 0009(0041) (0108) (0114) (0044) (0009) (0010) (0057) (0013)


23


Benchmarkportfolio











24







25


Benchmarkportfolio



1991-2003 054 053 054 110 056 0602004-2007 020 028 105 047 033 0392008-2012 001 -013 -019 007 008 016


1991-2003 030 030 031 061 031 0332004-2007 088 086 088 093 086 0912008-2012 012 012 011 005 011 013


26







σ2m

(5)






9



σ2m

ι = [1 1 1]prime (6)



414

14



Implementation




10



Results





11







12



Robust analysis





13





Shrinkage factor


Subsamples



14












15

Concluding remarks




16





20Q1202013Europepdf

17

References
















18

















19











20






USbonds

Globalequities

Globalbonds


US equities 1991-2012 002 078 009 020 023 -010 0141991-2003 003 069 -001 -002 000 -014 0062004-2007 -014 070 -009 -024 -011 -024 0082008-2012 006 091 030 057 055 -003 029

US bonds 1991-2012 006 068 -001 -002 -008 0041991-2003 002 066 004 -004 -002 0022004-2007 002 068 002 012 012 0022008-2012 015 074 -007 001 -028 010

Global equities 1991-2012 032 037 040 003 0161991-2003 023 018 017 001 0042004-2007 027 009 022 012 0072008-2012 048 063 062 001 033

Global bonds 1991-2012 014 016 005 0071991-2003 012 007 004 0002004-2007 020 030 023 0042008-2012 016 026 -002 020


21


Benchmarkportfolio



Mean 345 345 357 377 344 361SD 354 354 351 327 341 347Sharpe 098 098 102 115 101 104


Mean 344 344 344 362 344 345SD 670 671 671 612 671 671Sharpe 051 051 051 059 051 051


22



Panel A In-sample



US equities 0189 0195 0188 0138 0177 0266 0268 0267 0258 0263(0276) (0273) (0239) (0184) (0257) (0070) (0068) (0068) (0045) (0069)

US bonds 0546 0542 0601 0513 0564 0338 0340 0339 0341 0332(0332) (0322) (0342) (0297) (0360) (0128) (0127) (0127) (0124) (0129)

Global equities 0050 0038 0009 0032 0047 0167 0164 0163 0126 0166(0128) (0098) (0028) (0087) (0120) (0071) (0067) (0068) (0069) (0070)

Global bonds 0215 0194 0075 0090 0178 0230 0226 0227 0184 0231(0199) (0192) (0131) (0136) (0204) (0080) (0079) (0079) (0078) (0081)

Commodities 0031 0128 0227 0035 0003 0004 0091 0009(0041) (0108) (0114) (0044) (0009) (0010) (0057) (0013)


23


Benchmarkportfolio











24







25


Benchmarkportfolio



1991-2003 054 053 054 110 056 0602004-2007 020 028 105 047 033 0392008-2012 001 -013 -019 007 008 016


1991-2003 030 030 031 061 031 0332004-2007 088 086 088 093 086 0912008-2012 012 012 011 005 011 013


26



σ2m

ι = [1 1 1]prime (6)



414

14



Implementation




10



Results





11







12



Robust analysis





13





Shrinkage factor


Subsamples



14












15

Concluding remarks




16





20Q1202013Europepdf

17

References
















18

















19











20






USbonds

Globalequities

Globalbonds


US equities 1991-2012 002 078 009 020 023 -010 0141991-2003 003 069 -001 -002 000 -014 0062004-2007 -014 070 -009 -024 -011 -024 0082008-2012 006 091 030 057 055 -003 029

US bonds 1991-2012 006 068 -001 -002 -008 0041991-2003 002 066 004 -004 -002 0022004-2007 002 068 002 012 012 0022008-2012 015 074 -007 001 -028 010

Global equities 1991-2012 032 037 040 003 0161991-2003 023 018 017 001 0042004-2007 027 009 022 012 0072008-2012 048 063 062 001 033

Global bonds 1991-2012 014 016 005 0071991-2003 012 007 004 0002004-2007 020 030 023 0042008-2012 016 026 -002 020


21


Benchmarkportfolio



Mean 345 345 357 377 344 361SD 354 354 351 327 341 347Sharpe 098 098 102 115 101 104


Mean 344 344 344 362 344 345SD 670 671 671 612 671 671Sharpe 051 051 051 059 051 051


22



Panel A In-sample



US equities 0189 0195 0188 0138 0177 0266 0268 0267 0258 0263(0276) (0273) (0239) (0184) (0257) (0070) (0068) (0068) (0045) (0069)

US bonds 0546 0542 0601 0513 0564 0338 0340 0339 0341 0332(0332) (0322) (0342) (0297) (0360) (0128) (0127) (0127) (0124) (0129)

Global equities 0050 0038 0009 0032 0047 0167 0164 0163 0126 0166(0128) (0098) (0028) (0087) (0120) (0071) (0067) (0068) (0069) (0070)

Global bonds 0215 0194 0075 0090 0178 0230 0226 0227 0184 0231(0199) (0192) (0131) (0136) (0204) (0080) (0079) (0079) (0078) (0081)

Commodities 0031 0128 0227 0035 0003 0004 0091 0009(0041) (0108) (0114) (0044) (0009) (0010) (0057) (0013)


23


Benchmarkportfolio











24







25


Benchmarkportfolio



1991-2003 054 053 054 110 056 0602004-2007 020 028 105 047 033 0392008-2012 001 -013 -019 007 008 016


1991-2003 030 030 031 061 031 0332004-2007 088 086 088 093 086 0912008-2012 012 012 011 005 011 013


26



Results





11







12



Robust analysis





13





Shrinkage factor


Subsamples



14












15

Concluding remarks




16





20Q1202013Europepdf

17

References
















18

















19











20






USbonds

Globalequities

Globalbonds


US equities 1991-2012 002 078 009 020 023 -010 0141991-2003 003 069 -001 -002 000 -014 0062004-2007 -014 070 -009 -024 -011 -024 0082008-2012 006 091 030 057 055 -003 029

US bonds 1991-2012 006 068 -001 -002 -008 0041991-2003 002 066 004 -004 -002 0022004-2007 002 068 002 012 012 0022008-2012 015 074 -007 001 -028 010

Global equities 1991-2012 032 037 040 003 0161991-2003 023 018 017 001 0042004-2007 027 009 022 012 0072008-2012 048 063 062 001 033

Global bonds 1991-2012 014 016 005 0071991-2003 012 007 004 0002004-2007 020 030 023 0042008-2012 016 026 -002 020


21


Benchmarkportfolio



Mean 345 345 357 377 344 361SD 354 354 351 327 341 347Sharpe 098 098 102 115 101 104


Mean 344 344 344 362 344 345SD 670 671 671 612 671 671Sharpe 051 051 051 059 051 051


22



Panel A In-sample



US equities 0189 0195 0188 0138 0177 0266 0268 0267 0258 0263(0276) (0273) (0239) (0184) (0257) (0070) (0068) (0068) (0045) (0069)

US bonds 0546 0542 0601 0513 0564 0338 0340 0339 0341 0332(0332) (0322) (0342) (0297) (0360) (0128) (0127) (0127) (0124) (0129)

Global equities 0050 0038 0009 0032 0047 0167 0164 0163 0126 0166(0128) (0098) (0028) (0087) (0120) (0071) (0067) (0068) (0069) (0070)

Global bonds 0215 0194 0075 0090 0178 0230 0226 0227 0184 0231(0199) (0192) (0131) (0136) (0204) (0080) (0079) (0079) (0078) (0081)

Commodities 0031 0128 0227 0035 0003 0004 0091 0009(0041) (0108) (0114) (0044) (0009) (0010) (0057) (0013)


23


Benchmarkportfolio











24







25


Benchmarkportfolio



1991-2003 054 053 054 110 056 0602004-2007 020 028 105 047 033 0392008-2012 001 -013 -019 007 008 016


1991-2003 030 030 031 061 031 0332004-2007 088 086 088 093 086 0912008-2012 012 012 011 005 011 013


26







12



Robust analysis





13





Shrinkage factor


Subsamples



14












15

Concluding remarks




16





20Q1202013Europepdf

17

References
















18

















19











20






USbonds

Globalequities

Globalbonds


US equities 1991-2012 002 078 009 020 023 -010 0141991-2003 003 069 -001 -002 000 -014 0062004-2007 -014 070 -009 -024 -011 -024 0082008-2012 006 091 030 057 055 -003 029

US bonds 1991-2012 006 068 -001 -002 -008 0041991-2003 002 066 004 -004 -002 0022004-2007 002 068 002 012 012 0022008-2012 015 074 -007 001 -028 010

Global equities 1991-2012 032 037 040 003 0161991-2003 023 018 017 001 0042004-2007 027 009 022 012 0072008-2012 048 063 062 001 033

Global bonds 1991-2012 014 016 005 0071991-2003 012 007 004 0002004-2007 020 030 023 0042008-2012 016 026 -002 020


21


Benchmarkportfolio



Mean 345 345 357 377 344 361SD 354 354 351 327 341 347Sharpe 098 098 102 115 101 104


Mean 344 344 344 362 344 345SD 670 671 671 612 671 671Sharpe 051 051 051 059 051 051


22



Panel A In-sample



US equities 0189 0195 0188 0138 0177 0266 0268 0267 0258 0263(0276) (0273) (0239) (0184) (0257) (0070) (0068) (0068) (0045) (0069)

US bonds 0546 0542 0601 0513 0564 0338 0340 0339 0341 0332(0332) (0322) (0342) (0297) (0360) (0128) (0127) (0127) (0124) (0129)

Global equities 0050 0038 0009 0032 0047 0167 0164 0163 0126 0166(0128) (0098) (0028) (0087) (0120) (0071) (0067) (0068) (0069) (0070)

Global bonds 0215 0194 0075 0090 0178 0230 0226 0227 0184 0231(0199) (0192) (0131) (0136) (0204) (0080) (0079) (0079) (0078) (0081)

Commodities 0031 0128 0227 0035 0003 0004 0091 0009(0041) (0108) (0114) (0044) (0009) (0010) (0057) (0013)


23


Benchmarkportfolio











24







25


Benchmarkportfolio



1991-2003 054 053 054 110 056 0602004-2007 020 028 105 047 033 0392008-2012 001 -013 -019 007 008 016


1991-2003 030 030 031 061 031 0332004-2007 088 086 088 093 086 0912008-2012 012 012 011 005 011 013


26



Robust analysis





13





Shrinkage factor


Subsamples



14












15

Concluding remarks




16





20Q1202013Europepdf

17

References
















18

















19











20






USbonds

Globalequities

Globalbonds


US equities 1991-2012 002 078 009 020 023 -010 0141991-2003 003 069 -001 -002 000 -014 0062004-2007 -014 070 -009 -024 -011 -024 0082008-2012 006 091 030 057 055 -003 029

US bonds 1991-2012 006 068 -001 -002 -008 0041991-2003 002 066 004 -004 -002 0022004-2007 002 068 002 012 012 0022008-2012 015 074 -007 001 -028 010

Global equities 1991-2012 032 037 040 003 0161991-2003 023 018 017 001 0042004-2007 027 009 022 012 0072008-2012 048 063 062 001 033

Global bonds 1991-2012 014 016 005 0071991-2003 012 007 004 0002004-2007 020 030 023 0042008-2012 016 026 -002 020


21


Benchmarkportfolio



Mean 345 345 357 377 344 361SD 354 354 351 327 341 347Sharpe 098 098 102 115 101 104


Mean 344 344 344 362 344 345SD 670 671 671 612 671 671Sharpe 051 051 051 059 051 051


22



Panel A In-sample



US equities 0189 0195 0188 0138 0177 0266 0268 0267 0258 0263(0276) (0273) (0239) (0184) (0257) (0070) (0068) (0068) (0045) (0069)

US bonds 0546 0542 0601 0513 0564 0338 0340 0339 0341 0332(0332) (0322) (0342) (0297) (0360) (0128) (0127) (0127) (0124) (0129)

Global equities 0050 0038 0009 0032 0047 0167 0164 0163 0126 0166(0128) (0098) (0028) (0087) (0120) (0071) (0067) (0068) (0069) (0070)

Global bonds 0215 0194 0075 0090 0178 0230 0226 0227 0184 0231(0199) (0192) (0131) (0136) (0204) (0080) (0079) (0079) (0078) (0081)

Commodities 0031 0128 0227 0035 0003 0004 0091 0009(0041) (0108) (0114) (0044) (0009) (0010) (0057) (0013)


23


Benchmarkportfolio











24







25


Benchmarkportfolio



1991-2003 054 053 054 110 056 0602004-2007 020 028 105 047 033 0392008-2012 001 -013 -019 007 008 016


1991-2003 030 030 031 061 031 0332004-2007 088 086 088 093 086 0912008-2012 012 012 011 005 011 013


26





Shrinkage factor


Subsamples



14












15

Concluding remarks




16





20Q1202013Europepdf

17

References
















18

















19











20






USbonds

Globalequities

Globalbonds


US equities 1991-2012 002 078 009 020 023 -010 0141991-2003 003 069 -001 -002 000 -014 0062004-2007 -014 070 -009 -024 -011 -024 0082008-2012 006 091 030 057 055 -003 029

US bonds 1991-2012 006 068 -001 -002 -008 0041991-2003 002 066 004 -004 -002 0022004-2007 002 068 002 012 012 0022008-2012 015 074 -007 001 -028 010

Global equities 1991-2012 032 037 040 003 0161991-2003 023 018 017 001 0042004-2007 027 009 022 012 0072008-2012 048 063 062 001 033

Global bonds 1991-2012 014 016 005 0071991-2003 012 007 004 0002004-2007 020 030 023 0042008-2012 016 026 -002 020


21


Benchmarkportfolio



Mean 345 345 357 377 344 361SD 354 354 351 327 341 347Sharpe 098 098 102 115 101 104


Mean 344 344 344 362 344 345SD 670 671 671 612 671 671Sharpe 051 051 051 059 051 051


22



Panel A In-sample



US equities 0189 0195 0188 0138 0177 0266 0268 0267 0258 0263(0276) (0273) (0239) (0184) (0257) (0070) (0068) (0068) (0045) (0069)

US bonds 0546 0542 0601 0513 0564 0338 0340 0339 0341 0332(0332) (0322) (0342) (0297) (0360) (0128) (0127) (0127) (0124) (0129)

Global equities 0050 0038 0009 0032 0047 0167 0164 0163 0126 0166(0128) (0098) (0028) (0087) (0120) (0071) (0067) (0068) (0069) (0070)

Global bonds 0215 0194 0075 0090 0178 0230 0226 0227 0184 0231(0199) (0192) (0131) (0136) (0204) (0080) (0079) (0079) (0078) (0081)

Commodities 0031 0128 0227 0035 0003 0004 0091 0009(0041) (0108) (0114) (0044) (0009) (0010) (0057) (0013)


23


Benchmarkportfolio











24







25


Benchmarkportfolio



1991-2003 054 053 054 110 056 0602004-2007 020 028 105 047 033 0392008-2012 001 -013 -019 007 008 016


1991-2003 030 030 031 061 031 0332004-2007 088 086 088 093 086 0912008-2012 012 012 011 005 011 013


26












15

Concluding remarks




16





20Q1202013Europepdf

17

References
















18

















19











20






USbonds

Globalequities

Globalbonds


US equities 1991-2012 002 078 009 020 023 -010 0141991-2003 003 069 -001 -002 000 -014 0062004-2007 -014 070 -009 -024 -011 -024 0082008-2012 006 091 030 057 055 -003 029

US bonds 1991-2012 006 068 -001 -002 -008 0041991-2003 002 066 004 -004 -002 0022004-2007 002 068 002 012 012 0022008-2012 015 074 -007 001 -028 010

Global equities 1991-2012 032 037 040 003 0161991-2003 023 018 017 001 0042004-2007 027 009 022 012 0072008-2012 048 063 062 001 033

Global bonds 1991-2012 014 016 005 0071991-2003 012 007 004 0002004-2007 020 030 023 0042008-2012 016 026 -002 020


21


Benchmarkportfolio



Mean 345 345 357 377 344 361SD 354 354 351 327 341 347Sharpe 098 098 102 115 101 104


Mean 344 344 344 362 344 345SD 670 671 671 612 671 671Sharpe 051 051 051 059 051 051


22



Panel A In-sample



US equities 0189 0195 0188 0138 0177 0266 0268 0267 0258 0263(0276) (0273) (0239) (0184) (0257) (0070) (0068) (0068) (0045) (0069)

US bonds 0546 0542 0601 0513 0564 0338 0340 0339 0341 0332(0332) (0322) (0342) (0297) (0360) (0128) (0127) (0127) (0124) (0129)

Global equities 0050 0038 0009 0032 0047 0167 0164 0163 0126 0166(0128) (0098) (0028) (0087) (0120) (0071) (0067) (0068) (0069) (0070)

Global bonds 0215 0194 0075 0090 0178 0230 0226 0227 0184 0231(0199) (0192) (0131) (0136) (0204) (0080) (0079) (0079) (0078) (0081)

Commodities 0031 0128 0227 0035 0003 0004 0091 0009(0041) (0108) (0114) (0044) (0009) (0010) (0057) (0013)


23


Benchmarkportfolio











24







25


Benchmarkportfolio



1991-2003 054 053 054 110 056 0602004-2007 020 028 105 047 033 0392008-2012 001 -013 -019 007 008 016


1991-2003 030 030 031 061 031 0332004-2007 088 086 088 093 086 0912008-2012 012 012 011 005 011 013


26

Concluding remarks




16





20Q1202013Europepdf

17

References
















18

















19











20






USbonds

Globalequities

Globalbonds


US equities 1991-2012 002 078 009 020 023 -010 0141991-2003 003 069 -001 -002 000 -014 0062004-2007 -014 070 -009 -024 -011 -024 0082008-2012 006 091 030 057 055 -003 029

US bonds 1991-2012 006 068 -001 -002 -008 0041991-2003 002 066 004 -004 -002 0022004-2007 002 068 002 012 012 0022008-2012 015 074 -007 001 -028 010

Global equities 1991-2012 032 037 040 003 0161991-2003 023 018 017 001 0042004-2007 027 009 022 012 0072008-2012 048 063 062 001 033

Global bonds 1991-2012 014 016 005 0071991-2003 012 007 004 0002004-2007 020 030 023 0042008-2012 016 026 -002 020


21


Benchmarkportfolio



Mean 345 345 357 377 344 361SD 354 354 351 327 341 347Sharpe 098 098 102 115 101 104


Mean 344 344 344 362 344 345SD 670 671 671 612 671 671Sharpe 051 051 051 059 051 051


22



Panel A In-sample



US equities 0189 0195 0188 0138 0177 0266 0268 0267 0258 0263(0276) (0273) (0239) (0184) (0257) (0070) (0068) (0068) (0045) (0069)

US bonds 0546 0542 0601 0513 0564 0338 0340 0339 0341 0332(0332) (0322) (0342) (0297) (0360) (0128) (0127) (0127) (0124) (0129)

Global equities 0050 0038 0009 0032 0047 0167 0164 0163 0126 0166(0128) (0098) (0028) (0087) (0120) (0071) (0067) (0068) (0069) (0070)

Global bonds 0215 0194 0075 0090 0178 0230 0226 0227 0184 0231(0199) (0192) (0131) (0136) (0204) (0080) (0079) (0079) (0078) (0081)

Commodities 0031 0128 0227 0035 0003 0004 0091 0009(0041) (0108) (0114) (0044) (0009) (0010) (0057) (0013)


23


Benchmarkportfolio











24







25


Benchmarkportfolio



1991-2003 054 053 054 110 056 0602004-2007 020 028 105 047 033 0392008-2012 001 -013 -019 007 008 016


1991-2003 030 030 031 061 031 0332004-2007 088 086 088 093 086 0912008-2012 012 012 011 005 011 013


26





20Q1202013Europepdf

17

References
















18

















19











20






USbonds

Globalequities

Globalbonds


US equities 1991-2012 002 078 009 020 023 -010 0141991-2003 003 069 -001 -002 000 -014 0062004-2007 -014 070 -009 -024 -011 -024 0082008-2012 006 091 030 057 055 -003 029

US bonds 1991-2012 006 068 -001 -002 -008 0041991-2003 002 066 004 -004 -002 0022004-2007 002 068 002 012 012 0022008-2012 015 074 -007 001 -028 010

Global equities 1991-2012 032 037 040 003 0161991-2003 023 018 017 001 0042004-2007 027 009 022 012 0072008-2012 048 063 062 001 033

Global bonds 1991-2012 014 016 005 0071991-2003 012 007 004 0002004-2007 020 030 023 0042008-2012 016 026 -002 020


21


Benchmarkportfolio



Mean 345 345 357 377 344 361SD 354 354 351 327 341 347Sharpe 098 098 102 115 101 104


Mean 344 344 344 362 344 345SD 670 671 671 612 671 671Sharpe 051 051 051 059 051 051


22



Panel A In-sample



US equities 0189 0195 0188 0138 0177 0266 0268 0267 0258 0263(0276) (0273) (0239) (0184) (0257) (0070) (0068) (0068) (0045) (0069)

US bonds 0546 0542 0601 0513 0564 0338 0340 0339 0341 0332(0332) (0322) (0342) (0297) (0360) (0128) (0127) (0127) (0124) (0129)

Global equities 0050 0038 0009 0032 0047 0167 0164 0163 0126 0166(0128) (0098) (0028) (0087) (0120) (0071) (0067) (0068) (0069) (0070)

Global bonds 0215 0194 0075 0090 0178 0230 0226 0227 0184 0231(0199) (0192) (0131) (0136) (0204) (0080) (0079) (0079) (0078) (0081)

Commodities 0031 0128 0227 0035 0003 0004 0091 0009(0041) (0108) (0114) (0044) (0009) (0010) (0057) (0013)


23


Benchmarkportfolio











24







25


Benchmarkportfolio



1991-2003 054 053 054 110 056 0602004-2007 020 028 105 047 033 0392008-2012 001 -013 -019 007 008 016


1991-2003 030 030 031 061 031 0332004-2007 088 086 088 093 086 0912008-2012 012 012 011 005 011 013


26

References
















18

















19











20






USbonds

Globalequities

Globalbonds


US equities 1991-2012 002 078 009 020 023 -010 0141991-2003 003 069 -001 -002 000 -014 0062004-2007 -014 070 -009 -024 -011 -024 0082008-2012 006 091 030 057 055 -003 029

US bonds 1991-2012 006 068 -001 -002 -008 0041991-2003 002 066 004 -004 -002 0022004-2007 002 068 002 012 012 0022008-2012 015 074 -007 001 -028 010

Global equities 1991-2012 032 037 040 003 0161991-2003 023 018 017 001 0042004-2007 027 009 022 012 0072008-2012 048 063 062 001 033

Global bonds 1991-2012 014 016 005 0071991-2003 012 007 004 0002004-2007 020 030 023 0042008-2012 016 026 -002 020


21


Benchmarkportfolio



Mean 345 345 357 377 344 361SD 354 354 351 327 341 347Sharpe 098 098 102 115 101 104


Mean 344 344 344 362 344 345SD 670 671 671 612 671 671Sharpe 051 051 051 059 051 051


22



Panel A In-sample



US equities 0189 0195 0188 0138 0177 0266 0268 0267 0258 0263(0276) (0273) (0239) (0184) (0257) (0070) (0068) (0068) (0045) (0069)

US bonds 0546 0542 0601 0513 0564 0338 0340 0339 0341 0332(0332) (0322) (0342) (0297) (0360) (0128) (0127) (0127) (0124) (0129)

Global equities 0050 0038 0009 0032 0047 0167 0164 0163 0126 0166(0128) (0098) (0028) (0087) (0120) (0071) (0067) (0068) (0069) (0070)

Global bonds 0215 0194 0075 0090 0178 0230 0226 0227 0184 0231(0199) (0192) (0131) (0136) (0204) (0080) (0079) (0079) (0078) (0081)

Commodities 0031 0128 0227 0035 0003 0004 0091 0009(0041) (0108) (0114) (0044) (0009) (0010) (0057) (0013)


23


Benchmarkportfolio











24







25


Benchmarkportfolio



1991-2003 054 053 054 110 056 0602004-2007 020 028 105 047 033 0392008-2012 001 -013 -019 007 008 016


1991-2003 030 030 031 061 031 0332004-2007 088 086 088 093 086 0912008-2012 012 012 011 005 011 013


26

















19











20






USbonds

Globalequities

Globalbonds


US equities 1991-2012 002 078 009 020 023 -010 0141991-2003 003 069 -001 -002 000 -014 0062004-2007 -014 070 -009 -024 -011 -024 0082008-2012 006 091 030 057 055 -003 029

US bonds 1991-2012 006 068 -001 -002 -008 0041991-2003 002 066 004 -004 -002 0022004-2007 002 068 002 012 012 0022008-2012 015 074 -007 001 -028 010

Global equities 1991-2012 032 037 040 003 0161991-2003 023 018 017 001 0042004-2007 027 009 022 012 0072008-2012 048 063 062 001 033

Global bonds 1991-2012 014 016 005 0071991-2003 012 007 004 0002004-2007 020 030 023 0042008-2012 016 026 -002 020


21


Benchmarkportfolio



Mean 345 345 357 377 344 361SD 354 354 351 327 341 347Sharpe 098 098 102 115 101 104


Mean 344 344 344 362 344 345SD 670 671 671 612 671 671Sharpe 051 051 051 059 051 051


22



Panel A In-sample



US equities 0189 0195 0188 0138 0177 0266 0268 0267 0258 0263(0276) (0273) (0239) (0184) (0257) (0070) (0068) (0068) (0045) (0069)

US bonds 0546 0542 0601 0513 0564 0338 0340 0339 0341 0332(0332) (0322) (0342) (0297) (0360) (0128) (0127) (0127) (0124) (0129)

Global equities 0050 0038 0009 0032 0047 0167 0164 0163 0126 0166(0128) (0098) (0028) (0087) (0120) (0071) (0067) (0068) (0069) (0070)

Global bonds 0215 0194 0075 0090 0178 0230 0226 0227 0184 0231(0199) (0192) (0131) (0136) (0204) (0080) (0079) (0079) (0078) (0081)

Commodities 0031 0128 0227 0035 0003 0004 0091 0009(0041) (0108) (0114) (0044) (0009) (0010) (0057) (0013)


23


Benchmarkportfolio











24







25


Benchmarkportfolio



1991-2003 054 053 054 110 056 0602004-2007 020 028 105 047 033 0392008-2012 001 -013 -019 007 008 016


1991-2003 030 030 031 061 031 0332004-2007 088 086 088 093 086 0912008-2012 012 012 011 005 011 013


26











20






USbonds

Globalequities

Globalbonds


US equities 1991-2012 002 078 009 020 023 -010 0141991-2003 003 069 -001 -002 000 -014 0062004-2007 -014 070 -009 -024 -011 -024 0082008-2012 006 091 030 057 055 -003 029

US bonds 1991-2012 006 068 -001 -002 -008 0041991-2003 002 066 004 -004 -002 0022004-2007 002 068 002 012 012 0022008-2012 015 074 -007 001 -028 010

Global equities 1991-2012 032 037 040 003 0161991-2003 023 018 017 001 0042004-2007 027 009 022 012 0072008-2012 048 063 062 001 033

Global bonds 1991-2012 014 016 005 0071991-2003 012 007 004 0002004-2007 020 030 023 0042008-2012 016 026 -002 020


21


Benchmarkportfolio



Mean 345 345 357 377 344 361SD 354 354 351 327 341 347Sharpe 098 098 102 115 101 104


Mean 344 344 344 362 344 345SD 670 671 671 612 671 671Sharpe 051 051 051 059 051 051


22



Panel A In-sample



US equities 0189 0195 0188 0138 0177 0266 0268 0267 0258 0263(0276) (0273) (0239) (0184) (0257) (0070) (0068) (0068) (0045) (0069)

US bonds 0546 0542 0601 0513 0564 0338 0340 0339 0341 0332(0332) (0322) (0342) (0297) (0360) (0128) (0127) (0127) (0124) (0129)

Global equities 0050 0038 0009 0032 0047 0167 0164 0163 0126 0166(0128) (0098) (0028) (0087) (0120) (0071) (0067) (0068) (0069) (0070)

Global bonds 0215 0194 0075 0090 0178 0230 0226 0227 0184 0231(0199) (0192) (0131) (0136) (0204) (0080) (0079) (0079) (0078) (0081)

Commodities 0031 0128 0227 0035 0003 0004 0091 0009(0041) (0108) (0114) (0044) (0009) (0010) (0057) (0013)


23


Benchmarkportfolio











24







25


Benchmarkportfolio



1991-2003 054 053 054 110 056 0602004-2007 020 028 105 047 033 0392008-2012 001 -013 -019 007 008 016


1991-2003 030 030 031 061 031 0332004-2007 088 086 088 093 086 0912008-2012 012 012 011 005 011 013


26






USbonds

Globalequities

Globalbonds


US equities 1991-2012 002 078 009 020 023 -010 0141991-2003 003 069 -001 -002 000 -014 0062004-2007 -014 070 -009 -024 -011 -024 0082008-2012 006 091 030 057 055 -003 029

US bonds 1991-2012 006 068 -001 -002 -008 0041991-2003 002 066 004 -004 -002 0022004-2007 002 068 002 012 012 0022008-2012 015 074 -007 001 -028 010

Global equities 1991-2012 032 037 040 003 0161991-2003 023 018 017 001 0042004-2007 027 009 022 012 0072008-2012 048 063 062 001 033

Global bonds 1991-2012 014 016 005 0071991-2003 012 007 004 0002004-2007 020 030 023 0042008-2012 016 026 -002 020


21


Benchmarkportfolio



Mean 345 345 357 377 344 361SD 354 354 351 327 341 347Sharpe 098 098 102 115 101 104


Mean 344 344 344 362 344 345SD 670 671 671 612 671 671Sharpe 051 051 051 059 051 051


22



Panel A In-sample



US equities 0189 0195 0188 0138 0177 0266 0268 0267 0258 0263(0276) (0273) (0239) (0184) (0257) (0070) (0068) (0068) (0045) (0069)

US bonds 0546 0542 0601 0513 0564 0338 0340 0339 0341 0332(0332) (0322) (0342) (0297) (0360) (0128) (0127) (0127) (0124) (0129)

Global equities 0050 0038 0009 0032 0047 0167 0164 0163 0126 0166(0128) (0098) (0028) (0087) (0120) (0071) (0067) (0068) (0069) (0070)

Global bonds 0215 0194 0075 0090 0178 0230 0226 0227 0184 0231(0199) (0192) (0131) (0136) (0204) (0080) (0079) (0079) (0078) (0081)

Commodities 0031 0128 0227 0035 0003 0004 0091 0009(0041) (0108) (0114) (0044) (0009) (0010) (0057) (0013)


23


Benchmarkportfolio











24







25


Benchmarkportfolio



1991-2003 054 053 054 110 056 0602004-2007 020 028 105 047 033 0392008-2012 001 -013 -019 007 008 016


1991-2003 030 030 031 061 031 0332004-2007 088 086 088 093 086 0912008-2012 012 012 011 005 011 013


26


Benchmarkportfolio



Mean 345 345 357 377 344 361SD 354 354 351 327 341 347Sharpe 098 098 102 115 101 104


Mean 344 344 344 362 344 345SD 670 671 671 612 671 671Sharpe 051 051 051 059 051 051


22



Panel A In-sample



US equities 0189 0195 0188 0138 0177 0266 0268 0267 0258 0263(0276) (0273) (0239) (0184) (0257) (0070) (0068) (0068) (0045) (0069)

US bonds 0546 0542 0601 0513 0564 0338 0340 0339 0341 0332(0332) (0322) (0342) (0297) (0360) (0128) (0127) (0127) (0124) (0129)

Global equities 0050 0038 0009 0032 0047 0167 0164 0163 0126 0166(0128) (0098) (0028) (0087) (0120) (0071) (0067) (0068) (0069) (0070)

Global bonds 0215 0194 0075 0090 0178 0230 0226 0227 0184 0231(0199) (0192) (0131) (0136) (0204) (0080) (0079) (0079) (0078) (0081)

Commodities 0031 0128 0227 0035 0003 0004 0091 0009(0041) (0108) (0114) (0044) (0009) (0010) (0057) (0013)


23


Benchmarkportfolio











24







25


Benchmarkportfolio



1991-2003 054 053 054 110 056 0602004-2007 020 028 105 047 033 0392008-2012 001 -013 -019 007 008 016


1991-2003 030 030 031 061 031 0332004-2007 088 086 088 093 086 0912008-2012 012 012 011 005 011 013


26



Panel A In-sample



US equities 0189 0195 0188 0138 0177 0266 0268 0267 0258 0263(0276) (0273) (0239) (0184) (0257) (0070) (0068) (0068) (0045) (0069)

US bonds 0546 0542 0601 0513 0564 0338 0340 0339 0341 0332(0332) (0322) (0342) (0297) (0360) (0128) (0127) (0127) (0124) (0129)

Global equities 0050 0038 0009 0032 0047 0167 0164 0163 0126 0166(0128) (0098) (0028) (0087) (0120) (0071) (0067) (0068) (0069) (0070)

Global bonds 0215 0194 0075 0090 0178 0230 0226 0227 0184 0231(0199) (0192) (0131) (0136) (0204) (0080) (0079) (0079) (0078) (0081)

Commodities 0031 0128 0227 0035 0003 0004 0091 0009(0041) (0108) (0114) (0044) (0009) (0010) (0057) (0013)


23


Benchmarkportfolio











24







25


Benchmarkportfolio



1991-2003 054 053 054 110 056 0602004-2007 020 028 105 047 033 0392008-2012 001 -013 -019 007 008 016


1991-2003 030 030 031 061 031 0332004-2007 088 086 088 093 086 0912008-2012 012 012 011 005 011 013


26


Benchmarkportfolio











24







25


Benchmarkportfolio



1991-2003 054 053 054 110 056 0602004-2007 020 028 105 047 033 0392008-2012 001 -013 -019 007 008 016


1991-2003 030 030 031 061 031 0332004-2007 088 086 088 093 086 0912008-2012 012 012 011 005 011 013


26







25


Benchmarkportfolio



1991-2003 054 053 054 110 056 0602004-2007 020 028 105 047 033 0392008-2012 001 -013 -019 007 008 016


1991-2003 030 030 031 061 031 0332004-2007 088 086 088 093 086 0912008-2012 012 012 011 005 011 013


26


Benchmarkportfolio



1991-2003 054 053 054 110 056 0602004-2007 020 028 105 047 033 0392008-2012 001 -013 -019 007 008 016


1991-2003 030 030 031 061 031 0332004-2007 088 086 088 093 086 0912008-2012 012 012 011 005 011 013


26