Commodity Risk Factors and Intertemporal

April 2014

Adrian Fernadez-Perez Auckland University of Technology, Auckland

Ana-Maria Fuertes Cass Business School, City University, London

Joëlle Miffre EDHEC Business School, Nice Abstract This paper proposes a commodity-based specification of the Intertemporal CAPM (ICAPM) that uses state variables grounded on the theories of storage and hedging pressure. Accordingly, factor-mimicking portfolios are formed by taking long positions in backwardated contracts and positions in contangoed contracts according to either term structure, hedging pressure or momentum signals. The resulting portfolio returns are able to predict changes in the investment opportunity set of agents over long horizons in a way that is consistent with rational pricing by risk-adverse investors. Relative to several traditional implementations of the ICAPM, the proposed specification prices reasonably well a large cross-section of assets that include , fixed income securities and commodity futures contracts.

Keywords: ICAPM; Theory of Storage; Hedging Pressure Hypothesis. JEL classification: G13, G14. This version: April 2014.

EDHEC is one of the top five business schools in France. Its reputation is built on the high quality of its faculty and the privileged relationship with professionals that the school has cultivated since its establishment in 1906. EDHEC Business School has decided to draw on its extensive knowledge of the professional environment and has therefore focused its research on themes that satisfy the needs of professionals.

EDHEC pursues an active research policy in the field of . EDHEC-Risk Institute carries out numerous research programmes in the areas of asset allocation and risk management in both the 2 traditional and alternative investment universes. Copyright © 2014 EDHEC 1. Introduction The literature on commodity futures pricing centres around the theory of storage (Kaldor, 1939; Working, 1949; Brennan, 1958) and the hedging pressure hypothesis (Cootner, 1960; Hirshleifer, 1988). The theory of storage links the slope of the term structure of commodity futures prices (hereafter, TS) to the incentive of agents to own the physical commodity. With high inventories, the term structure is upward-sloping and markets are contangoed; inventory holders can buy the commodity spot at a cheap price and sell it forward at a profit that more than compensates them for storage and financing costs. Conversely, when inventories are low, the futures curve slopes downward and markets are backwardated; then the benefits of owning the physical asset (known as convenience yield) exceed storage and financing costs. In line with this theory, Fama and French (1987) show that the difference between contemporaneous futures and spot prices (the basis) depends on interest rates and seasonals in convenience yields. Erb and Harvey (2006), Gorton and Rouwenhorst (2006), and Gorton et al. (2012) also support the theory of storage by showing that the risk premium of commodity futures can be modeled as a function of either the basis or the level of inventories.1

The hedging pressure hypothesis relates the evolution of commodity futures prices to the net positions of hedgers and speculators. Futures prices are expected to rise when hedgers are net short and speculators are net long; this market state is known as backwardation. Conversely, futures prices are expected to fall when hedgers are net long and speculators net short; this is known as . Hedging pressure (hereafter, HP) has been shown to play a critical role as determinant of commodity futures risk premia (Carter et al., 1983; Bessembinder, 1992; de Roon et al., 2000; Basu and Miffre, 2013). 2

Although cognitive biases that generate overreaction and mean-reversion have been shown to explain momentum profits in commodity futures markets (hereafter, Mom), there is also evidence that winners exhibit backwardated features such as low inventory, while losers present characteristics associated with contango such as upward-sloping term structures (Erb and Harvey, 2006; Miffre and Rallis, 2007; Gorton et al., 2012; Dewally et al., 2013). The momentum effect relates therefore, like term structure and hedging pressure, to the notions of backwardation and contango. Not surprisingly, the returns of long-short mimicking portfolios based on either TS, HP and Mom signals can explain cross-sectional commodity futures returns (Basu and Miffre, 2013; Bakshi et al., 2013; Szymanowska et al., 2014).

This article proposes a novel empirical implementation of Merton (1973)’s intertemporal capital asset pricing model (ICAPM) that uses the returns of TS, HP and Mom portfolios as state variables. We demonstrate that the TS, HP and Mom portfolio returns are found to satisfy the restrictions that the ICAPM places on both the time-series and cross-section behaviour of state variables. On the one hand, an increase (decrease) in the TS, HP and (Mom) portfolio returns predicts deteriorating investment opportunities. On the other hand, assets that correlate positively with the TS, HP and (Mom) state variables command negative (positive) risk premia. In line with ICAPM theory, these results indicate that rational investors are willing to pay a higher price on assets that hedge intertemporal risk and a lower price on assets that underperform when market conditions are poor. The proposed ICAPM can price a large cross-section of assets, including stocks, fixed income securities and commodity futures, rather competitively vis-à-vis extant ICAPM implementations. Our study closely relates to a burgeoning literature that shows that states of backwardation and contango not only price traditional assets but also forecast economic activity (Baker and Routledge, 2012; Bakshi et al., 2013; Koijen et al., 2013b; Yang, 2013).3

1 - Alternatively, the behaviour of commodity spot and futures prices has been studied in a risk-neutral world using competitive rational expectations models of storage (Deaton and Laroque, 1992; Routledge et al., 2000). In these models, the non-negativity constraint on inventory is crucial to understanding the dynamics of the spot price and the shape of the forward curve. Extensions of storage models that allow for a risk premium can be found in Casassus et al. (2009) and Baker (2012). 2 - The sharp increase in commodity assets under management post-2004 revived the debate on the function of speculators as both liquidity and risk-bearing providers and on their potential influence on futures prices, and cross-market linkages (Stoll and Whaley, 2010; Brunetti et al., 2011; Büyüksahin and Robe, 2014). Theoretical perspectives are provided in Baker (2012) and Baker and Routledge (2012) who show that endogenous demand shocks can explain the recent trend in prices. 3 - Baker and Routledge (2012) present evidence that excess returns are higher when the crude oil futures curve slopes downward. Bakshi et al. (2013) document that the commodity TS and Mom 3 state variables forecast economic growth and traditional asset returns. Koijen et al. (2013b) argue that TS relates to global recessions and liquidity crises, while Yang (2013) relates it to a factor that mimics producers’ investment shocks. In addition, our article relates to a broader literature that treats as risk factors or leading economic indicators commodity-based variables such as commodity futures returns, open interests or the Baltic Dry Index (Boons et al., 2012; Hong and Yogo, 2012; Hou and Szymanowska, 2013; Bakshi et al., 2014).

In what follows, Section 2 outlines the ICAPM theory and implied testable restrictions. Section 3 describes the data and presents the state variables proposed that relate to term structure, hedging pressure and momentum. Section 4 puts forward formal tests of various ICAPM restrictions. Section 5 provides robustness checks and Section 6 concludes.

2. Asset Pricing Framework 2.1 ICAPM Theory Merton (1973) proposes a theoretical pricing framework for the cross-section of asset returns, known as ICAPM, where investment opportunities are time-varying as dictated by a vector of

stochastic state variables zt. Rational investors demand positive risk premium for exposure to the intertemporal risk associated with unwelcome shocks in these state variables and are willing to accept a negative risk premium on assets that hedge such risk. The ICAPM theory postulates the following equilibrium relationship between risks and (1)

where E(.) denotes expectation, Ri is the return on asset i (i=1,…,N) in excess of the risk-free rate, M is the , ∆z is the innovation in the state variable z that captures shifts in investment opportunities. The right-hand side formalises two distinct quantities of risk

(covariances). The static market risk of the asset is given by σi,M, the covariance between the excess returns of asset i and the changes in current aggregate wealth proxied by the market

excess returns. The second covariance term, σi,∆z, represents intertemporal risk, where the word “intertemporal” serves to emphasize that investors wish to hedge shortfalls in consumption and

adverse changes in investment opportunities over time. The parameters γM and γz are the prices of market risk and intertemporal risk, respectively. Equation (1) becomes the standard CAPM

when γz=0. The parameter γM can be further interpreted as the relative risk aversion (RRA) of a representative rational investor.

2.2 Testable restrictions on IPCAM state variables Merton’s (1973) ICAPM theory does not directly identify the state variables. This leads to the perverse situation where the ICAPM could be misused as a “fishing license” for ad hoc risk factors (Fama, 1991). Yet, as argued by Cochrane (2005) and Maio and Santa-Clara (2012), the non- market factors should proxy for innovations in the state variables which, according to Merton’s theory, must meet three important testable restrictions. The first ICAPM restriction implies that the state variables must anticipate changes in the investment opportunity set. To probe it, as in Maio and Santa-Clara (2012), we estimate the following long-horizon predictive regressions for aggregate market returns and variances (2)

(3)

where t=1,…,T are months and T is the effective sample size (the original number of months minus the horizon q). The regressand in (2) is the market portfolio excess return continuously

compounded from month t+1 to month t+q; formally RMt,t+q = RMt+1 +… + RMt+q. The regressand in (3) is defined as VMt,t+q = VMt+1 +… + VMt+q where VMt represents the realised variance or sum of daily squared returns on month t. The candidate state variables included in the ICAPM

specification of choice are gathered in the K×1 vector zt=(z1,t,…,zK,t)’. The parameters of the 4 predictive regressions are consistently estimated by OLS. Inferences are based on Newey and West (1987) covariance matrix to accommodate the residual autocorrelation induced by overlapping effects in the predictands.

The second ICAPM restriction refers to the direction of long-run predictability which ought to be consistent with cross-sectional asset pricing by risk-averse investors. Suppose that decreases in a state variable zj,t predict a deterioration of economic conditions; that is, lower expected aggregate returns and higher volatility of returns. Then its innovation (the risk factor) should command a positive risk premium in cross-sectional tests. The intuition is that such an asset, which offers low returns when investment opportunities are deemed to worsen, does not hedge reinvestment risk. The third ICAPM restriction is that the implied RRA should be between 1 and 10 to be economically plausible (see Mehra and Prescott, 1985).

In order to assess the first two ICAPM restrictions, we need to construct innovations in the state variables. Following Campbell (1996), Petkova (2006) and Maio (2013) among others, we do so through the following vector autoregressive (VAR) model

(4) for months t=1,…,T where RMt and zt are as defined above; A is a (K+1)(K+1) matrix of unknown parameters which are consistently estimated by OLS. The innovations in each of the state variables are the residual sequences orthogonalised with respect to the market excess returns, and standardised so that all of them have the same standard deviation as .

The multi-factor pricing model proposed as ICAPM is directly estimable in expected return- covariance form using the generalised method of moments (GMM) of Hansen (1982). We set up a GMM system with N+K+1 moment conditions; the first N moment conditions are the sample counterparts of equation (1) for risky assets i=1,…, N. The remaining conditions are intended to account for estimation risk in the factor means. Formally, we have

(5) where Ri,t+1 is the excess return on testing asset i (for i=1,…,N) from month t to month t+1; RMt+1 is the excess return on the market portfolio and ft+1 = ƒ1,t+1,…, ƒK,t+1)’ defines the vector of innovations in the K state variables. This estimation approach is conceptually equivalent to running a regression of average excess returns on covariances, in line with the equilibrium equation (1). Thus the GMM approach allows us to estimate directly the market price of risk or implied RRA parameter γM and the intertemporal or “hedging” risk prices associated with the K state variables. The (K+1)×1 vector contains the unconditional means of the risk factors. Robust GMM t-statistics are based on the Newey and West (1987) heteroskedasticity and autocorrelation consistent (HAC) covariance matrix.4

We gauge the economic value of the pricing model using three criteria. The first criterion is the mean absolute error, , where is the pricing error for asset i defined as the difference between its average excess return and its expected excess return according to the GMM estimates. The second measure is the coefficient of determination (inspired by the OLS method) defined as the degrees-of-freedom adjusted fraction of the cross-sectional variation in average excess returns captured by the ICAPM; formally . The third criterion is a J test for the null hypothesis that the sum of squared pricing errors is zero; the test statistic

4 - The implementation of the ICAPM in expected return-covariance form through GMM has been deployed in various studies with differences in how the innovations are proxied (Maio and Santa-Clara, 2012; Maio, 2013; Hammami and Lindahl, 2014). We follow Maio (2013) in proxying the innovations by the residuals of an OLS-estimated VAR(1) model. Hammami and Lindahl (2014) conduct their 5 analysis using the same two-step residual-based approach and show that the joint estimation of the VAR model and the ICAPM by GMM (which precludes both the orthogonalisation and standardisation of the innovations) does not make a material difference to their main results. is asymptotically chi-squared with N-(K+1) degrees of freedom; is the N×N spectral density (or long-run covariance) matrix of the sample moments based on .

The second ICAPM restriction is met by a given candidate state variable if the sign of the estimated risk price coincides with the sign of the predictive slope in the market return equation (2), and is opposite to the sign of the predictive slope in the market variance equation (3). Finally,

we assess the economic plausibility of the estimated γM as RRA proxy.

3. State Variables and Data As noted above, the choice of state variables introduces a subjective element in any ICAPM implementation since Merton’s (1973) theory does not identify them. Section 3.1 below details our ICAPM implementation and Section 3.2 presents eight ICAPM specifications from the literature. Section 3.3 outlines the test assets and the market portfolio proxy. The dataset spans the period from January 1989 to August 2011 (272 monthly observations).

3.1 Backwardation, contango and intertemporal risk Our main line of argument is that the returns of long-short portfolios that capture backwardation and contango contain relevant information on changes in future investment and consumption opportunities. As such, the resulting factor-mimicking portfolio returns lend themselves as plausible candidates for ICAPM state variables. This proposition is well aligned with a growing literature that links backwardation and contango with economic activity (Baker and Routledge, 2012; Bakshi et al., 2013; Koijen et al., 2013b; Yang, 2013).

End-of-month crude oil futures prices are displayed in Figure 1. Shaded areas signify months with downward-sloping futures curves in Panel A, months with net long speculators in Panel B and months with positive 12-month past average returns in Panel C; as noted earlier, these three signals have been shown to capture backwardation and hence, albeit with a slight abuse of terminology, we refer to the shaded areas as ‘backwardated’ months. The three panels show a good correspondence between backwardation and rising futures prices.

To quantify this evidence, we create four dummies. One is a ‘positive return’ indicator that takes a value of 1 if the monthly change in the price of crude oil futures is positive and 0 otherwise. The other three variables are backwardation indicators: is equal to 1 if the front- end futures curve slopes downward, is equal to 1 if speculators are net long, and is equal to 1 if 12-month past average return is positive (and 0 otherwise). The significantly positive correlation of with at 0.30 (p-value=0.00), with at 0.16 (p-value=0.01) and with at 0.11 (p-value=0.06) further shows that crude oil futures prices tend to rise when the market is in backwardation.

Figure 1. Historical crude oil futures prices and backwardation. The figure plots at monthly frequency the futures prices of crude oil (continuous line) and backwardated months (shaded areas) defined as those months when front-end futures curves slope downward (Panel A), speculators are net long (Panel B), or 12-month average returns are positive (Panel C). The sample period is January 1989 to August 2011. Panel A: Backwardation states identified from Term Structure signal

6 Panel B: Backwardation states identified from Hedging Pressure signal

Panel C: Backwardation states identified from Momentum signal

Our next task is to construct the long-short portfolios. The TS mimicking portfolio buys the backwardated contracts with the highest roll yields and shorts the contangoed contracts with the lowest roll yields; the roll yield is measured as the difference in the time t logarithmic prices of the nearby and second nearest contracts. Following Basu and Miffre (2013), we define hedging pressure for each of two categories of traders (hedgers and speculators) as the percentage of their long positions relative to their total open interest.

Accordingly, the HP mimicking portfolio takes long (short) positions in the commodities for which hedgers’ hedging pressure is the lowest (highest) and speculators’ hedging pressure is the highest (lowest). Finally, our third mimicking portfolio, Mom, buys the commodities with the best past performance and shorts the commodities with the worst past performance.

The following principles are maintained throughout the paper in constructing the factor- mimicking portfolios. At the time of portfolio formation, the long portfolio contains the 20% most backwardated contracts based on a given signal; likewise, the short portfolio includes the 20% most contangoed contracts.5 The ranking period over which the TS, HP or Mom signals are averaged is 12 months; the period over which the long-short portfolio is held is one month. The positions are fully-collateralised, meaning that the long-short return equals half the return of the long portfolio minus half the return of the short portfolio. The constituents are equally- weighted and chosen out of a cross-section of 27 commodity futures; the latter is dictated by the availability of large hedgers and speculators positions in the Aggregated Commitment of Traders report at the Commodity Futures Trading Commission.6

3.2 Traditional IPCAM state variables We compare our commodity-based ICAPM implementation with eight traditional multi-factor models in the ICAPM literature (a non-exhaustive but representative list) that can be grouped as follows. In the first group, we include ICAPM applications in which the non-market factors proxy for the innovations in state variables that emanate from the predictability literature.

5 - The 20% composition of the long and short HP portfolios is attained via a double-sort based on hedgers’ HP with the 50th quantile as breakpoint and then speculators’ HP using the 40th quantile. 6 - The 27 commodities include 12 agricultural products (cocoa, coffee C, corn, cotton n°2, concentrated frozen orange juice, rough rice, oats, soybean meal, soybean oil, soybeans, sugar n°11, wheat), 5 energy contracts (electricity, gasoline, heating oil, light sweet crude oil, natural gas), 4 livestock commodities (feeder cattle, frozen pork bellies, lean hogs, live cattle), 5 metals (copper, gold, palladium, 7 platinum, silver), and lumber. The price series are obtained from Datastream employing front contracts up to one month before maturity. The first model is the ICAPM version of Campbell and Vuolteenaho (2004) in which investment opportunities are characterised by the slope of the Treasury (TERM), the aggregate price-earnings ratio (PE) and the small- value spread (VS). The second model is the ICAPM of Hahn and Lee (2006) which uses TERM and the corporate bond default spread (DEF) as state variables. The third model is the ICAPM of Petkova (2006) which uses TERM, DEF, Treasury bill rate (TBILL) and the dividend yield (DY). The fourth model is the ICAPM of Bali and Engle (2010) which includes TERM, DEF and the Federal Reserve fund rate (FED). The fifth model is the ICAPM of Koijen et al. (2013a) with TERM and the bond risk premia factor of Cochrane and Piazzesi (2005; hereafter, CP factor).

In the second group, we include multi-factor models that originate in the empirical equity pricing literature and can been reconciled with ICAPM theory (Fama and French, 1996; Petkova, 2006; Maio and Santa Clara, 2012). The first model is the Fama and French (1993) three-factor model that treats the returns of size (SMB) and value (HML) sorted portfolios as state variables. The second model is the Carhart (1997) version that considers an additional state variable related to equity momentum (Eq-Mom). The third model is that proposed by Pastor and Stambaugh (2003) that includes SMB, HML and a proxy for liquidity (L). Appendix A provides definitions and data sources on all of the above state variables.

3.3 Market portfolio and testing assets The market portfolio contains every individual asset in the economy and is inherently unobservable. We proxy it with a combination of stock, bond and commodity indices in proportions that are broadly in line with the weight of each asset class as a proportion of total wealth; 50% loading is given to the U.S. value-weighted index, available from Kenneth French’s library, 40% loading is given to Barclays bond index, and 10% loading to the Standard & Poor’s Goldman Sachs Commodity Index (S&P-GSCI). Monthly data on the bond and commodity indices are obtained from Bloomberg.

Table I presents summary statistics for the market portfolio (Panel A) and the state variables (Panels B to D). The annualised market excess returns are 5.78% on average (t-ratio of 3.17). The excess returns of the TS, HP and Mom portfolios in Panel B have significantly positive means of 5.21%, 5.81% and 8.02% per annum (t-ratios of 2.37, 3.09 and 3.01, respectively). The pairwise return correlations are positive but low at 0.05 for TS and HP, 0.38 for TS and Mom, and 0.30 for HP and Mom, which motivates the joint inclusion of the three associated factors in the ICAPM. The statistics in Panels C and D for the traditional state variables are similar to those reported elsewhere (e.g., Maio and Santa-Clara, 2012).

We use 25 equity portfolios, 6 bond portfolios and the S&P-GSCI as testing assets (N=32). Returns are expressed in excess of the one-month Treasury bill. The equity portfolios are based on all the CRSP NYSE/AMEX/NASDAQ stocks sorted by size and book-to-market (25 SBM), as available from Kenneth French’s library. The bond portfolios are U.S. Treasury bond indices with maturities of 1, 2, 5, 10, 20 and 30 years, downloaded from the CRSP database. In robustness checks, we consider other choices of market proxy and testing assets.

8 Table I. Summary statistics for the market portfolio and the state variables. Panels A and B report summary statistics for the annualised excess returns of the market portfolio and commodity-based state variables based on term structure (TS), hedging pressure (HP) and momentum (Mom). Panel C summarizes the excess returns of equity-based state variables based on size (SMB), value (HML), and momentum (Eq-Mom), all in annualised form, and for the non-traded liquidity factor of Pastor and Stambaugh (2003; L). Panel D pertains to state variables from the predictability literature. The market portfolio is proxied by a combination of the Fama-French equity index (50% weight), the Barclays bond index (40%) and the S&P-GSCI (10%). The risk-free rate is proxied by the one-month Treasury-bill rate. Appendix A provides detailed definitions for the state variable reported in Panels C and D. Mean StDev. Min. Max.

Panel A: Market 0.0578 0.0869 -0.1606 0.0609

Panel B: State variables from the commodity pricing literature TS 0.0521 0.1044 -0.0921 0.0908 HP 0.0581 0.0896 -0.0577 0.0962 Mom 0.0802 0.1270 -0.1044 0.1256

Panel C: State variables from the equity pricing literature SMB 0.0198 0.1177 -0.1639 0.2202 HML 0.0264 0.1121 -0.1268 0.1387 Eq-Mom 0.0799 0.1764 -0.3472 0.1839 L 0.0022 0.0627 -0.2704 0.2874

Panel D: State variables from the predictability literature TERM 0.0182 0.0121 -0.0053 0.0376 PE 25.2339 7.3296 13.3200 44.2000 VS 1.4945 0.6060 -1.9280 3.3411 DEF 0.0095 0.0042 0.0055 0.0338 TBILL 0.0369 0.0224 0.0002 0.0882 DY 0.0217 0.0071 0.0108 0.0403 FED 0.0399 0.0245 0.0007 0.0985 CP 0.0122 0.0162 -0.0458 0.0583

4. Empirical Results 4.1 Are the predictive regressions consistent with IPCAM theory? Can the candidate state variables forecast future changes in investment opportunities over long horizons? To address this question, the empirical commodity and equity state variables (TS, HP, Mom, SMB, HML, Eq-Mom, L) are transformed to cumulative sums over p (current and past) months; for instance, we use as predictor in equations (2) and (3) the TS state variable where p={12, 24, 36, 60} and zTS,j is the time j excess return of the TS mimicking portfolio. Two reasons motivate this transformation. First, the cumulated variables mimic rather well the slow changes in investment opportunities, akin to traditional state variables from the predictability literature. Figure 2 illustrates this argument. Panel A of the figure shows the dynamics in future ex-post investment opportunities as proxied by market returns and variances continuously compounded from months t+1 to t+60; these variables are the regressands in equations (2) and (3). Panel B plots one commodity state variable, TS, and one equity state variable, SMB, in levels and transformed (60-month cumulated), alongside a traditional state variable from the predictability literature (TERM).

A second related motivation is that, in order for the long-horizon regressions to be statistically consistent, predictands and predictors ought to have comparable persistence; namely, they must behave similarly regarding how long they memorise the effect of a shock. To gauge persistence, we compute the first-order autocorrelation coefficient and deploy two unit root tests. One test is based on the augmented Dickey and Fuller (1979; ADF) test statistic that is designed to assess the null hypothesis of unit root or non-stationary I(1) behavior versus the alternative of mean- reversion or I(0) behaviour. Another is the Kwiatkowski et al. (1992; KPSS) test where the above

9 null and alternative hypotheses are reversed. Table II shows the results for predictands (Panel A) and predictors (Panels B to D).

Figure 2. Evolution of market returns, market variances and predictors. Panel A plots the continuously compounded market returns (RM60) and variances of market returns (VM60) cumulated over the next 60 months. Panel B plots one of the state variables from the empirical commodity pricing literature (TS), one of the state variables from the empirical equity pricing literature (SMB), and one of the state variables from the predictability literature (TERM). The commodity and equity state variables are plotted both untransformed (TS and SMB) and cumulated over 60 current and previous months (TS60 and SMB60). The frequency of the observations is monthly over the period from January 1989 to August 2011 which amounts to 272 data points less the 60 observations lost due to the cumulation.

Panel A: Evolution of market returns and market variances

Panel B: Evolution of predictors in non-cumulated and cumulated formats

10 Table II. Persistence analysis of variables involved in long-horizon predictability regressions. This table reports the first-order autocorrelation φ(1) and unit root/stationarity tests for predictands and predictors in the subsequent long-horizon predictability regressions. The predictands are the aggregate market return (RM) and market variance (VM) continuously compounding over q future months (Panel A); the market portfolio is proxied by a combination of the Fama-French equity portfolio (50% weight), the Barclays bond index (40%) and the S&P-GSCI (10%). The predictors are state variables from i) the empirical commodity pricing literature (Panel B), ii) the empirical equity pricing literature (Panel C) and iii) the predictability literature (Panel D). p is the number of months over which the state variables are cumulated. Appendix A describes the state variables reported in Panels C and D. ADF is the augmented Dickey and Fuller (1979) test for the null hypothesis of unit root behavior, denoted I(1), versus the alternative of mean-reversion, denoted I(0). KPSS is the Kwiatkowski et al. (1992) test for the converse hypotheses. The deterministic specification in the ADF test regression comprises a constant and linear trend, and the lag order is selected using the SBC. The KPSS test is based on the Bartlett kernel with Newey- West optimal bandwidth selection. The original sample period is 1989:01-2011:08. *, ** and *** denote rejection of the null at the 10%, 5% and 1% levels.

φ(1) ADF test KPSS test φ(1) ADF test KPSS test Panel A: Market returns and variances Panel C: State variables from the equity pricing literature RM SMB q =12 0.927 -2.6583 I(1) 0.087 I(0) p =1 -0.034 -11.520 *** I(0) 0.052 I(0) q =24 0.962 -2.4827 I(1) 0.244 I(0) p =12 0.897 -3.076 I(1) 0.133 * I(1) q =36 0.970 -2.0306 I(1) 0.585 ** I(1) p =24 0.932 -2.895 I(1) 0.254 *** I(1) q =60 0.973 -2.0306 I(1) 0.766 *** I(1) p =36 0.958 -2.168 I(1) 0.315 *** I(1) p =60 0.976 -1.726 I(1) 0.437 *** I(1) VM q =12 0.985 -1.1844 I(1) 0.547 ** I(1) HML q =24 0.993 -2.5737 I(1) 0.727 ** I(1) p =1 0.154 -10.475 *** I(0) 0.076 I(0) q =36 0.984 -1.2124 I(1) 0.943 *** I(1) p =12 0.943 -3.065 I(1) 0.145 * I(1) q =60 0.983 -1.4645 I(1) 1.031 *** I(1) p =24 0.965 -2.576 I(1) 0.198 ** I(1) p =36 0.964 -2.064 I(1) 0.329 *** I(1) p =60 0.969 -1.298 I(1) 0.413 *** I(1) Panel B: State variables from the commodity pricing literature TS Eq-Mom p =1 -0.005 -16.391 *** I(0) 0.065 I(0) p =1 0.084 -12.074 *** I(0) 0.041 I(0) p =12 0.900 -3.176 * I(0) 0.102 I(0) p =12 0.937 -3.823 ** I(0) 0.115 I(0) p =24 0.926 -2.726 I(1) 0.241 I(0) p =24 0.956 -3.147 * I(0) 0.369 *** I(1) p =36 0.933 -3.158 I(1) 0.403 * I(1) p =36 0.955 -2.166 I(1) 0.531 *** I(1) p =60 0.957 -2.119 I(1) 0.384 * I(1) p =60 0.971 -2.161 I(1) 0.688 *** I(1) HP L p =1 0.041 -15.705 *** I(0) 0.081 I(0) p =1 -0.111 -12.585 *** I(0) 0.054 I(0) p =12 0.913 -3.279 * I(0) 0.158 ** I(1) p =12 0.920 -3.420 * I(0) 0.098 I(0) p =24 0.958 -2.938 I(1) 0.236 *** I(1) p =24 0.955 -2.321 I(1) 0.163 ** I(1) p =36 0.968 -2.204 I(1) 0.367 *** I(1) p =36 0.971 -2.251 I(1) 0.282 *** I(1) p =60 0.973 -2.142 I(1) 0.383 *** I(1) p =60 0.977 -1.836 I(1) 0.465 *** I(1)

Mom Panel D: State variables from the predictability literature p =1 0.081 -15.766 *** I(0) 0.138 I(0) TERM 0.978 -2.369 I(1) 0.127 I(0) p =12 0.937 -3.557 ** I(0) 0.200 I(0) PE 0.987 -1.332 I(1) 0.530 ** I(1) p =24 0.969 -2.510 I(1) 0.249 I(0) VS 0.782 -4.570 *** I(0) 0.032 I(0) p =36 0.969 -2.108 I(1) 0.352 * I(1) DEF 0.964 -4.080 *** I(0) 0.586 ** I(1) p =60 0.970 -1.887 I(1) 0.640 ** I(1) TBILL 0.983 -2.719 I(1) 1.235 *** I(1) DY 0.983 -1.529 I(1) 1.209 *** I(1) FED 0.984 -2.916 I(1) 1.153 *** I(1) CP 0.965 -3.510 I(1) 0.157 ** I(1)

The autocorrelation coefficients suggest that the predictands, RMt,t+q and VMt,t+q, with horizon q of 12, 24, 36 and 60 months (Panel A) have comparable degrees of persistence to traditional state variables from the predictability literature (Panel D). However, the empirical commodity and equity state variables (Panels B and C) require cumulation over at least p=12 months to achieve similar persistence as predictands. The unit root tests further confirm the cumulation of empirical commodity and equity state variables as suitable so that predictands and predictors in the long- horizon regressions (2) and (3) have similar orders of integration.

Table III summarises the long-horizon predictability analysis for the commodity state variables using all combinations of horizon q={12, 24, 36, 60} and cumulative length p={12, 24, 36, 60}. With some exceptions at the lowest horizons, the signs of the predictive slopes are negative for both TS and HP but positive for Mom in the market return equation; positive for TS and HP but negative for Mom in the market variance equation. The contrasting direction of predictability

11 for the Mom state variable mirrors the wisdom that commodity momentum not only relates to backwardation/contango risk but also reflects various behavioural phenomena such as overreaction (see e.g., Barberis et al., 1998). These results are well aligned with the earlier evidence in Bakshi et al. (2013) on the opposite direction of predictability of TS and Mom factors over the business cycle and traditional assets returns.

Table III. Multiple long-horizon predictability regressions for commodity state variables. The table presents q-horizon regression of market returns and variances (predictands) on the commodity TS, HP and Mom state variables (predictors). The horizon for the predictands is q={12, 24, 36, 60} months. The predictors are defined as cumulative sums over the current and p-1 past months, p={12, 24, 36, 60}. The sample period is 1989:01-2011:08 which amounts to 272 months minus the observations lost due to horizon and cumulation lengths. The market portfolio as proxy for the investment opportunity set is a combination of the Fama-French equity index (50% weight), Barclays bond index (40% weight) and S&P-GSCI (10% weight). All regressions include an (unreported) intercept. The table reports the OLS predictive slopes and the autocorrelation robust Newey-West (1987) t-statistics in parentheses. The bottom part of each panel reports as diagnostics the adjusted time-series coefficient of determination (%) and the F test for the null hypothesis that the predictors jointly have zero predictive power. *, ** and *** denote test rejection at the 10%, 5% and 1% levels, respectively. Significant predictive slopes at the 10%, 5% or 1% levels are denoted in bold.

The long-horizon predictability analysis for traditional ICAPM state variables is summarised in Table IV. For space constrains, we focus on horizons q={36, 60} for predictands and cumulative length p=60 for the equity state variables; the results for the remaining q and p combinations are qualitatively similar. This evidence confirms that traditional state variables are able to anticipate future changes in investment opportunities. The market return regressions (Panel A) reveal significant predictive power for Eq-Mom, TERM, DY, FED and CP with positive slopes, and for PE with negative slope. Five out of these six state variables are significant predictors of future market variance (Panel B) with reversed signs, as one would expect; the exception is DY which is not significant.7

12 7 - Univariate long-run predictability regressions essentially confirm these findings, albeit with weaker statistical significance. Moreover, the Engle-Granger (1987) cointegration test for the null hypothesis of I(1) errors provides reasonable evidence that the multiple regressions reported in Tables III and IV are not spurious. Detailed results are available upon request. Table IV. Multiple long-horizon predictability regressions for traditional ICAPM state variables. The table presents q-horizon regression of market returns and variances (predictands) on various candidate state variables (predictors) according to traditional multifactor models that have been interpreted as ICAPMs; details on the models and variable definitions are provided in Appendix A. The state variables from the empirical literature on equity pricing are defined as cumulative sums over p=60 months. The sample period is 1989:01-2011:08 which amounts to 272 months minus the observations lost due to horizon and cumulation lengths. The market portfolio used as proxy for the investment opportunity set is a combination of the Fama-French equity index (50% weight), Barclays bond index (40% weight) and S&P-GSCI (10% weight). All regressions include an (unreported) intercept. The table reports the OLS predictive slopes with HAC type Newey-West (1987) t-statistics in parentheses. The bottom part of each panel reports as diagnostics the adjusted coefficient of determination (%) and the F test for the null hypothesis that the predictors jointly have zero predictive power. *, ** and *** are used to denote test rejection at the 10%, 5% and 1% levels, respectively. Bold denotes significant predictive slopes at conventional levels.

The model diagnostics obtained for the long-horizon predictability regressions reveal various noteworthy aspects. First, it is easier to predict the second moment than the first moment of asset returns, as borne out by the higher for the market variance predictability equation than for the market return equation. Second, the predictive power suggested by the is stronger for the longest horizon (q=60 months) than for the shorter horizon (q=12 months). Third, for a given horizon q the predictive power of empirical commodity (and equity) state variables improves with the cumulation length p. Finally, standard F tests confirm the joint significance of the state variables as predictors of market returns and variances. Hence, all nine pricing kernels meet the first long-horizon predictability criterion for consistency with the Merton (1973) ICAPM theory.8

8 - Our predictability analysis is conducted in-sample as in Maio and Santa-Clara (2012) since the main aim is to compare the signs of the predictive slopes with the signs of the covariance risk prices asso- 13 ciated with the state variables (second ICAPM restriction, discussed next). An in-sample analysis allows us to employ the same span of data to estimate both sets of parameters. 4.2 Are the factor risk prices consistent with IPCAM theory ? Having confirmed that the commodity state variables have long-run predictive power over market returns and variances, we turn to the next two ICAPM restrictions. The second ICAPM restriction refers to the compatibility of the signs of the predictive slopes and the signs of the intertemporal risk prices, and the third ICAPM restriction to the economic plausibility of the implied RRA. Table V shows the covariance risk prices obtained by one-step GMM. We begin by discussing the first column of the table that pertains to the novel ICAPM implementation that uses innovations to commodity state variables as intertemporal risk factors; the remaining eight columns pertain to traditional ICAPM implementations.

Table V. Prices of covariance risk from nine candidate ICAPMs. The table reports the one-step GMM estimation results for the postulated commodity-based ICAPM and for eight traditional multi-factor models that have been interpreted as ICAPMs. Appendix A provides details on the traditional models and variable definitions. The testing assets are 25 equity portfolios sorted by size and book-to-market (25 SBM), 6 U.S. Treasury-bond indices with maturities of 1, 2, 5, 10, 20 and 30 years, and the S&P-GSCI. γM is the price of covariance risk associated with the market (or implied RRA level). The proposed ICAPM implementation reported in Panel A treats as risk factors the innovations in the TS, HP and Mom state variables with associated prices of covariance risk denoted γTS, γHP and γMom. The remaining γ are the prices of covariance risk associated with the risk factors originating from the equity pricing and predictability literatures. Bold γ denote significant covariance risk prices at the 10% level or better. Robust GMM t-statistics (in parentheses) are reported based on the Bartlett kernel with Newey-West optimal bandwidth selection. The bottom part of the table reports as model diagnostics the mean absolute pricing error (MAE), the degrees-of-freedom adjusted fraction of the cross-sectional variance in average excess returns explained by the model ( ), and the chi-squared J test statistic for the null hypothesis that the sum of squared pricing errors is zero. The sample covers the period 1989:01-2011:08. *, ** and *** denote test rejection at the 10%, 5% and 1% levels, respectively.

The GMM estimates of the covariance risk prices γTS and γHP are negative, in line with the negative (positive) predictive slopes for market returns (variances). This indicates that rational agents accept

lower expected returns on assets that are able to hedge intertemporal risk. The estimate for γMom is positive which is consistent with the positive (negative) link between the Mom state variable and future market returns (variances) documented in the long-run predictability analysis. Hence, rational investors require a positive premium for holding assets that are poor hedges against intertemporal risk.

14 Our results essentially confirm the evidence in Maio and Santa-Clara (2012) that traditional models that employ state variables from the predictability literature generally fail to satisfy the second ICAPM restriction. To illustrate, let us select one traditional state variable from the predictability literature, TERM, and another from the equity pricing literature, HML. The long- horizon predictability analysis suggested that a current fall in either TERM or HML anticipates a decline in future investment opportunities; thus, assets that covary positively with innovations to these state variables do not hedge reinvestment risk. Both HML and TERM should command a positive risk premium. Yet, this ICAPM restriction is met for the HML risk factor (Panel B), but not for the TERM risk factor (Panel C). Likewise, the DY and FED risk factors are implausibly priced according to the signs of the predictive slopes. With regards to the third ICAPM restriction, all ICAPMs considered produce economically plausible RRA estimates that range from 4.07 to 7.27.

We also compare the pricing kernels regarding their ability to capture the cross-sectional variation in the average excess returns of the 32 base assets. The relevant diagnostics for this purpose are reported in the last three rows of Table V. Firstly, the novel ICAPM implementation that uses commodity state variables achieves the lowest pricing error (MAE) and the largest explanatory power ( ) among all the pricing kernels under study. Secondly, the J test statistic does not reject the null hypothesis that the sum of squared pricing errors is zero for the commodity-based ICAPM, but it is significant for various traditional models. Finally, as a graphical summary of pricing ability, the graphs in Figure 3 plot the realised average excess return of each of the 32 testing assets (vertical axis) against the corresponding prediction from each model (horizontal axis). The points are relatively close to the 45° line for the postulated ICAPM specification which further suggests that it compares well with extant ICAPM specifications in terms of pricing ability.

Figure 3. Individual pricing errors of nine candidate ICAPMs. This figure plots the average excess returns in percentage per annum (p.a.) of 32 test assets against the corresponding excess returns predicted from nine ICAPM specifications. The test assets are the 25 SBM portfolios, 6 U.S. Treasury-bond indices with maturities of 1, 2, 5, 10, 20 and 30 years, and the S&P-GSCI. The sample period is January 1989 to August 2011. The commodity-based ICAPM considers as risk factors the market portfolio and the innovations in the commodity TS, HP and Mom state variables. The remaining models are described in Appendix A.

Summing up, the empirical evidence reported in the paper suggests that the commodity-based ICAPM specification proposed produces both reasonable RRA estimates and intertemporal risk prices that are consistent with the predictive slopes from time-series regressions. Hence, the model is at least as consistent with Merton’s (1973) theory as a number of traditional multi-factor models justified as ICAPMs in the literature. 15 5. Robustness Checks Additional analyses are now conducted to add robustness to our findings regarding the consistency of the commodity-based pricing kernel with ICAPM theory. We pay attention to the choices of methodology, testing assets and benchmark. Table VI reports the results.

5.1. Choice of methodology We estimate the proposed ICAPM specification by one-step GMM, as before, but now including an

intercept γ0 in the first N moment conditions. The constrained estimation (γ0=0) may provide more efficient estimates but the unconstrained moment conditions are more robust to misspecification. If the model is correctly specified, the intercept ought to be indistinguishable from zero. Table VI shows in column one (Panel A) that the intercept is significantly positive, reflecting some misspecification. However, the evidence that the intertemporal risk prices are consistent with the slopes from the time-series regressions is not challenged, even though the HP factor risk price becomes insignificant.

Table VI. Robustness tests for prices of covariance risk. The table reports additional results regarding the choices of methodology (Panel A), testing assets (Panel B) and benchmark (Panel C) for the commodity-based ICAPM that treats as risk factors the market excess returns and the innovations in the term structure (TS), hedging pressure (HP) and momentum (Mom) state variables. γ0 is the model intercept. γM is the price of covariance risk associated with the market portfolio or the implied RRA. γTS, γHP and γMom are the prices of covariance risk associated with the risk factors. Bold font is used to denote significant coefficients at the 10%, 5% or 1% levels. Robust GMM t-statistics (in parentheses) are reported based on the Bartlett kernel with Newey- West optimal bandwidth selection. In the columns pertaining to the Fama-MacBeth (1973) approach, the estimated coefficients are risk prices, the t-statistics in italics are based on the Newey-West (1987) covariance, and the underlined t-statistics are based on the Shanken (1992) covariance. One-step, two-step and iterative are the type of GMM estimation, N is the number of base assets, K is the number of state variables. The testing assets consist of 25 equity portfolios sorted by size and book-to-market (25 SBM), 25 equity portfolios sorted by size and momentum (25 SMom), 6 U.S. Treasury-bond indices with maturities ranging from 1 to 30 years, and the S&P-GSCI. The benchmarks include equity, bond and commodity indices in the specified proportions. The bottom part of the table reports as model diagnostics the mean absolute pricing error (MAE), the degrees-of-freedom adjusted fraction of the cross-sectional variance in average excess returns explained by the model ( ), the weighted least squares (WLS) coefficient of determination ( ), and the chi-squared J test statistic for the null hypothesis that the (weighted) sum of squared pricing errors is zero. The sample covers the period 1989:01-2011:08. *, ** and *** denote test rejection at the 10%, 5% and 1% levels, respectively.

We exclude the K factor means from the GMM system so that now the system is reduced to N+1 equations. This robustness check can be motivated by the fact that the innovations in each of the state variables (that proxy intertemporal risk) are constructed as residuals from a VAR model and hence, their means should be asymptotically zero. The new estimation results, shown in column two (Panel A) of Table VI, are similar to those reported formerly in Table V. If anything, the inclusion of the K means helps in model identification as borne out by the lower MAE of the original N+K+1 system; likewise, the is notably lower (at 38.7%) in the N+1 system than in the N+K+1 system (at 66.5%).

16 We deploy two other more sophisticated GMM estimation approaches: two-step efficient GMM that weighs less the noisier moment conditions, and iterated GMM that has the same asymptotic properties (as two-step GMM) but can have better finite-sample performance. At the first stage of these approaches, the weighting matrix is the identity matrix (as in the baseline or one-step GMM). At the second stage, the weighting matrix for the moment conditions associated with the N assets is the first N×N block of the inverse of the spectral density matrix ; thus base assets with more uncertain moment conditions are given less importance.9 The appropriate measure of explanatory power in these estimation approaches is the weighted least squares (WLS) coefficient of determination, , where and are the N×1 vectors of demeaned pricing errors and demeaned excess returns, respectively, and contains the first N diagonal elements of pertaining to the N testing assets. The J test statistic computes the weighted sum of squared pricing errors.

The results of two-step GMM and iterated GMM, shown in columns 3 and 4 (Panel A) of Table VI reveal limited efficiency gains and make no meaningful difference to the findings; except that the HP factor is not significant. But two-step/iterated GMM estimation raises problems of interpretation in asset pricing. As noted by Cochrane (2005), the merely signifies the explanatory power of models for repackaged portfolios which represent extreme linear combinations of the original portfolios and hence, are of little interest to investors.

Finally, we estimate the ICAPM in expected return-beta form using the two-stage approach of Fama and MacBeth (1973) as deployed by Cochrane (2005) and Maio and Santa-Clara (2012). At stage one, we run individual OLS time-series regressions of portfolio returns on the risk factors, to obtain risk factor loadings (betas). At stage two, we run an OLS cross-sectional regression of average portfolio returns on the betas to obtain the common risk prices. The last two columns of Panel A in Table VI present the (beta) risk prices alongside two t-statistics based, respectively, on the HAC robust Newey and West (1987) covariance and the error-in-variables robust Shanken (1992) covariance.10 The signs of the (beta) risk prices are consistent with the signs of the long- horizon predictive slopes. The ratio of the price of market beta risk to the variance of the market excess returns gives 5.63 and 3.69 in the regressions without and with intercept, respectively, and represents plausible RRA levels.

5.2. Alternative testing assets and market portfolio proxies We consider as alternative sets of testing assets the 25 portfolios sorted by size and momentum (25 SMom) available in Kenneth French’s library alongside the same 6 U.S. Treasury-bond indices and the S&P-GSCI; the 25 portfolios sorted by size and book-to-market (25 SBM); and the 25 size and momentum sorted portfolios (25 SMom). The results reported in Panel B of Table VI do not materially challenge the main findings.

Finally, we consider alternative market portfolio proxies such as the Fama-French equity index, the Fama-French equity index (60% weight) together with the Barclays bond index (40% weight), and the Fama-French equity index (90% weight) together with the S&P-GSCI (10% weight). Panel C of Table VI reports one-step GMM estimates for the risk prices obtained using those market portfolios, which do not meaningfully challenge the findings. Furthermore, the results on the long-horizon predictive regressions based on these market portfolio proxies, gathered in Appendix B, are qualitatively similar to those reported earlier.

6. Conclusions Are commodity-based empirical factors good candidates to model intertemporal risk? We propose a novel empirical implementation of the ICAPM theory of Merton (1973) that employs

9 - Iterated GMM proceeds similarly in a recursive manner until convergence is achieved. For a more detailed exposition of these estimation methods, see Cochrane (2005). 10 - The assumptions underlying the Shanken (1992) covariance estimator are not valid for heteroskedastic asset pricing models. GMM enables correct inferences on the risk prices under weaker conditions than those required by Shanken’s correction. 17 commodity-based state variables motivated by the theories of storage and hedging pressure. Our main contention is that portfolios that are long backwardated commodities and short contangoed commodities qualify as ICAPM state variables because they contain useful information on future economic activity. The empirical evidence reported supports this proposition by confirming crucial testable restrictions that the theory places on both the time-series and cross-section behaviour of the state variables. The commodity-based ICAPM implementation compares favourably to traditional implementations in the literature concerning not only those testable restrictions, but also in its ability to price a cross-section of equity, fixed income and commodity portfolios. Our findings are robust to the estimation method for the pricing kernel and to the choice of testing assets and market portfolio proxy.

This paper adds to a promising strand of the literature that ascribes a role to commodity-related variables (such as the basis and open interests) as leading indicators of economic activity and as sources of priced risk. The evidence reported may inspire further research on the interactions between commodity futures, traditional assets classes and the business cycle.

18 APPENDIX A. Description of ICAPM specifications and candidate state variables. Panel I indicates eight popular multifactor models from the literature that have been justified or can be interpreted as ICAPM applications. Panel II provides variable definitions and sources. The sampling frequency for our empirical analysis is monthly.

APPENDIX B. Robustness tests for multiple long-horizon predictability regressions (market portfolio proxies). The table presents counterpart results to those reported in Table III for alternative proxies of the market portfolio. Exhibit 1 reports results for the Fama-French equity index (100% weight), Exhibit 2 for a combination of the Fama-French equity index (60%) and the Barclays bond index (40%), Exhibit 3 for a combination of the Fama-French equity index (90%) and the S&P-GSCI (10%). All regressions have been estimated including an (unreported) intercept. The table reports the OLS predictive slopes and the autocorrelation robust Newey-West (1987) t-statistics in parentheses. The bottom part of each panel reports as diagnostics the adjusted coefficient of determination (%) and the F test for the hypothesis that the predictors jointly have zero predictive power. The original sample period is 1989:01-2011:08. *, ** and *** denote test rejection at the 10%, 5% and 1% levels, respectively. Significant predictive slopes at conventional levels are in bold.

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22 Founded in 1906, EDHEC Business School offers risk management. EDHEC-Risk Institute also management education at undergraduate, has highly significant executive education graduate, post-graduate and executive activities for professionals. It has an original levels. Holding the AACSB, AMBA and EQUIS PhD in Finance programme which has an accreditations and regularly ranked among executive track for high level professionals. Europe’s leading institutions, EDHEC Business Complementing the core faculty, this unique School delivers degree courses to over 6,000 PhD in Finance programme has highly students from the world over and trains prestigious affiliate faculty from universities 5,500 professionals yearly through executive such as Princeton, Wharton, Oxford, Chicago courses and research events. The School’s and CalTech. ‘Research for Business’ policy focuses on issues that correspond to genuine industry and In 2012, EDHEC-Risk Institute signed two community expectations. strategic partnership agreements with the Operations Research and Financial Engineering Established in 2001, EDHEC-Risk Institute department of Princeton University to set up has become the premier academic centre a joint research programme in the area of for industry-relevant financial research. In risk and investment management, and with partnership with large financial institutions, Yale School of Management to set up joint its team of ninety permanent professors, certified executive training courses in North engineers, and support staff, and forty-eight America and Europe in the area of investment research associates and affiliate professors, management. implements six research programmes and sixteen research chairs and strategic research projects focusing on asset allocation and Copyright © 2014 EDHEC-Risk Institute

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