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Using Durable Consumption Risk to ExplainCommodities Returns
Deepa Dhume∗
Harvard University
November 11, 2010
Job Market Paper
Abstract
Commodities futures contracts have low correlations with market returns but highaverage returns. This is puzzling because the capital asset pricing model predicts thatassets that are uncorrelated with market returns will have low average returns. Applyingthe consumption-based capital asset pricing model does little to help, as commoditiesare also uncorrelated with nondurable consumption growth. In this paper, I show thatadding durable consumption growth to a multi-factor consumption-based asset pricingmodel can explain high commodities returns. Using a more comprehensive data setthan has been used in previous studies of commodities and asset pricing models, I showthat commodities returns covary strongly with durable consumption growth, which hasbeen shown to be an important driver of returns to other assets. Furthermore, sys-tematic durables risk can explain the cross section of returns to commodity portfoliossorted on the futures-spot spread, returns momentum, and spot price volatility. Finally,I demonstrate that commodities’ high durables risk may be explained by the businesscycle properties of commodities and durable consumption growth.
JEL Classifications: G12, G13, E21, E32, E44Keywords: asset pricing, commodity, futures, durable consumption
∗Department of Economics, Harvard University. Email: dhume@fas.harvard.edu. I am grateful toKenneth Rogoff and John Campbell for their invaluable guidance. I also thank Craig Burnside, MartinFeldstein, Jeffrey Frankel, Stefano Giglio, David Mericle, Gloria Sheu, Luis Viceira, and Motohiro Yogofor helpful comments and discussions.
1 Introduction
“Hogs, orange juice, sugar, and coffee.... It sounds like a breakfast menu as
investment plan, but it could be the best money you ever spend, particularly
if U.S. stocks continue to flounder. After all, breakfast is the most important
meal of the day.” – Jim Rogers, The Breakfast of Champions, July 3, 2002
Commodities have generated a lot of interest of late because of their high returns
and increasing importance as an asset class. Yet, returns to commodities are not
well understood or measured, and until recently, even the existence of positive returns
was debated. Given the deepening markets for commodities futures contracts and
the recent commodity price bubble, it is clear that understanding this asset class will
be helpful to both hedgers, who invest in commodities futures markets to balance
exposure to commodity price risk, and speculators, who invest in commodities futures
markets to make a financial gain. Whether or not Jim Rogers is correct to claim
that commodities are the investors’ breakfast of champions, understanding the risks of
commodities investments is as important as understanding the returns.
This paper makes three key contributions to the literature on explaining commodi-
ties returns using asset pricing models. First, I demonstrate that inclusion of durable
consumption growth as a factor improves the fit of multi-factor asset pricing models and
allows us to explain the time series and cross sectional characteristics of commodities
returns within a theoretically grounded model of systematic financial risk. Second, I
show that returns to portfolios of commodities sorted on known correlates of returns
are consistent with high durables risk. In the cross section of returns, commodities
with a low basis, high returns momentum, or high spot price volatility tend to have
higher covariance with durable consumption growth and higher returns. Finally, I show
that the high risk exposure of commodities returns to durable consumption growth may
be explained by the similar business cycle characteristics of commodities returns and
durable consumption growth.
This paper examines the return to buying a futures contract for 35 individual com-
modities over the period 1959 to 2008. On average, the index of all commodities has
earned 5.7% returns over the entire sample period. However, the return varies sub-
stantially across commodities, ranging from 22.9% per annum for propane futures to
negative 2.1% for zinc futures. By grouping commodities by type, we can see that the
portfolio of energy, animal, and metals commodities have each earned positive returns
1
of 6.1% or more, while agriculture and soft commodities have earned lower positive
returns.
To resolve the apparent contradiction between commodities’ high returns and low
correlation with equity markets, I demonstrate that the consumption-based asset pric-
ing model by Yogo (2006) is able to explain the high returns to commodities. The
model begins with an intertemporal household optimization problem with choices over
durable and nondurable consumption. When combined with portfolio choice theory,
the model predicts that the factors affecting marginal utility include nondurable and
durable consumption growth and returns to market wealth. Building upon this model,
previous empirical work has found that the inclusion of durable consumption growth
as a factor is able to explain the well-documented value premium in equity markets,
as well as the sizable returns to portfolios of high interest rate currencies (Yogo, 2006;
Lustig and Verdelhan, 2007).
The inclusion of durable consumption growth as a factor is the key to explaining
the high returns to commodities. According to the model, assets whose returns are
highly correlated with the factors affecting marginal utility should have higher returns
on average because they do not help investors hedge against downturns in consumption
growth and wealth. While commodities returns have low correlations with equity re-
turns and nondurable consumption growth, they exhibit strong correlation with durable
consumption growth. Therefore, the high returns to commodities we observe are in fact
consistent with the a multi-factor risk-based model, because they compensate the in-
vestor for exposure to durable consumption growth risk.
To test the prediction that assets with higher factor risk have higher returns, I first
estimate the model on portfolios of commodities grouped by type. To these 5 portfolios,
I add a broad cross section of potential assets in which an individual may choose to
invest, including 11 equity portfolios, 6 bond portfolios, 3 international indexes, and 6
currency portfolios. The results indicate that the model is able to predict the returns
to commodities. The generally high R2 (above 0.90) and low prediction error (less than
0.90 percentage points) indicate that the returns to commodities can be explained by
their levels of factor risk. Furthermore, the implied preference parameters of the model
are consistent with previous findings.
Given the success of the model in predicting commodities returns for portfolios
based on type, I next explore which characteristics of commodities may be driving
these returns and risk amounts. There is a long history in finance of grouping equities
into portfolios based on characteristics that successfully predict returns, including size,
2
book-to-market value, and industry type. Sorting assets into portfolios allows us to
study the relationship between the underlying sorting characteristic and asset returns
by eliminating the other idiosyncratic drivers of returns that can be diversified within
each portfolio. Motivated by recent work by Gorton et al. (2007), I focus on the
basis, the spot price volatility, and the return momentum as sorting characteristics for
commodities, to see whether portfolios formed on these characteristics have returns
that are predictable using standard asset pricing models. The results indicate that the
known high returns to low basis, high momentum, and high volatility commodities are
consistent with the high durables risk these portfolios have, both in the time series and
cross section of commodities returns.
While the model is able to measure factor risk and use it to predict commodities re-
turns, it does not provide an explanation for why commodities returns have a relatively
high covariance with durables consumption growth. To answer this further question,
I examine the business cycle properties of commodities returns. Gorton and Rouwen-
horst (2006) note that commodities returns behave differently than stock and bond
returns over the business cycle. Following their analysis of the full index of commodi-
ties and stock market returns, I add an examination of the business cycle properties of
the five commodities type portfolios and the consumption-based risk factors of durable
and nondurable consumption growth.
The results indicate that commodities follow a business cycle pattern similar to
that of durables consumption growth. While nondurable consumption growth and
market returns peak in early expansion, durable consumption growth and commodity
portfolios returns peak in late expansion. In addition, while nondurables and market
returns trough in early recession, durables and commodity portfolios trough in late
recession. According to the model, investors demand high average returns from assets
that pay poorly when marginal utility is high. Commodities have high returns on
average because they deliver low returns when durable consumption growth is low and
marginal utility is high.
There is little agreement in the literature about how to explain these high returns
to commodities. One issue is that the magnitude of average returns differs depending
on the sample period and the set of individual commodities included. However, recent
work has found that commodities have earned positive average returns over an extended
sample period including recent years. For example, both Gorton and Rouwenhorst
(2006) and Hong and Yogo (2010) find that an index of commodities earns high returns
3
with low variance when the sample period is extended to 2004 or later. However, there
is little agreement in the literature about how to explain these returns.
One strand in the commodities literature argues that commodities returns compen-
sate investors for systematic risk exposure within an integrated market. In support of
this view, Bessembinder and Chan (1992) show that many of the same variables that
have forecast power for equity and bond markets also have forecast power for agricul-
ture and metals futures markets. More recently, Bollinger and Kind (2010) have found
that the convenience yield, which is the benefit that accrues to the owner of a physical
inventory, can be explained by risk exposure to standard asset pricing factors such as
the returns to the S&P 500 index and an index of world government bonds.
While a number of papers have successfully linked commodities returns to systematic
financial risk, they have been unable to do so within the context of the Capital Asset
Pricing Model. According to the CAPM, assets which are highly correlated with market
returns should have high returns on average. In contrast, assets which have a low
correlation with market returns tend to have low returns, because they provide a hedge
for investors and reduce portfolio volatility. This model would predict that commodities,
which have high returns on average, have a high correlation with market returns. Yet,
we know that “commodities tend to zig when the equity markets zag” (Rogers, 2002).
Indeed, while the return to my index of 35 commodities futures is 5.7% per annum over
the sample period of 1959 to 2008, the empirical correlation coefficient between this
commodities index and a broad market index of equity returns is 0.07. Gorton and
Rouwenhorst (2006) similarly find that while their index of commodities has returns
comparable to those of equities, it also has a zero or negative correlation with equity
and bond returns over time.
Given these empirical facts, it is unsurprising that previous literature has found that
the CAPM is unable to explain the high returns to commodities (Bodie and Rosansky,
1980; Dusak, 1973). Using data covering fifteen years of corn, soybean, and wheat
futures, Dusak (1973) argues that commodities’ low returns and low correlation with
market wealth are consistent with the CAPM. However, this finding has not held up
over time as researchers have found that when studying more commodities and a longer
sample period, commodities continue to have a low correlation with market wealth but
earn high returns. In one such paper, using quarterly observations on 23 individual
commodities futures contracts between 1950 and 1976, Bodie and Rosansky (1980) find
a negative relationship between average returns and the market correlation for the cross
section of commodities. Finally, using data on prices for corn, soybeans, and wheat,
4
Jagannathan (1985) demonstrates that the consumption-based intertemporal CAPM is
rejected for commodities as well. However, in his conclusion he notes that one weakness
in his work is that his consumption data is limited to nondurable consumption due
to the difficulty of measuring the service flow from durable goods. He notes that if
durable goods are an important component of consumption, this data limitation could
be significant.
Because these efforts to explain commodities returns using systematic risk-based fac-
tors have met with only limited success, an older view of commodities as a segmented
market has sustained continued interest. Keynes’ Theory of Normal Backwardation
casts futures markets as a tool used by commodities hedgers to transfer risk to spec-
ulators. Since most hedgers are risk averse commodities sellers, they offer speculators
a futures price lower than the expected spot price in return for increased certainty.
Excess hedging pressure from commodity sellers can result in a futures price below the
expected spot price in equilibrium, which implies positive returns on average. However,
if speculators are risk neutral, they may bid up the futures price until it equals the
expected spot price, reducing returns to zero. The empirical finding of high returns to
commodities indicates that speculators require a positive return, or risk premium, for
assuming the exposure to spot price risk. Under the segmented markets theory, this
risk premium is impacted by commodity-specific variables such as the basis, inventory
levels, open interest, and hedging pressure.
A number of papers have tried to measure the relative importance of the two sets of
variables: the commodities-specific variables and the systematic financial risk variables.
Evidence for a segmented view of commodities markets can be found in Bessembinder
(1992), who shows that the returns to agricultural futures contracts are affected by
hedging pressure even after controlling for variables that measure systematic risk. Ad-
ditionally, Hong and Yogo (2010) show that open interest is a stronger predictor of
commodities returns than systematic risk variables such as the yield spread and the
short rate. They use this evidence to argue that market segmentation is an important
driver of commodities returns. In contrast, Etula (2009) highlights the broker-dealer
characteristic of risk-appetite to show that systematic risk factors can affect the willing-
ness of speculators to take on spot price risk, and so can affect the balance of hedging
pressure and the risk premium demanded by speculators in commodities markets.
This paper fills a gap in the literature on systematic financial risk in commodities
markets by demonstrating that a consumption-based asset pricing model can explain
commodities returns, as long as durable consumption growth is included as a factor.
5
By using a theoretically grounded model of household investment and portfolio alloca-
tion, it supports earlier findings that systematic financial risk is an important driver
of commodities returns. However, it also confirms earlier findings that the CAPM and
CCAPM are insufficient to explain commodities returns because it demonstrates that
including market returns, nondurable consumption, or even both of these as factors is
not enough to explain commodities returns without the inclusion of durable consump-
tion growth as a separate factor. Lastly, this paper also demonstrates that the finding
of systematic risk in commodities returns is consistent with the findings in the market
segmentation literature. In particular, it shows that commodities which have the prop-
erties that are well-known correlates of commodities returns, including low basis, high
returns momentum, and high volatility, also have high durables risk exposure.
The rest of the paper is organized as follows: In Section 2, I detail how commodities
returns are constructed from spot and futures prices and provide descriptive statistics
on the conditional and unconditional returns to investing in individual commodities
and portfolios. Section 3 outlines the Yogo (2006) model of household investment and
portfolio choice that results in the three factor consumption-based asset pricing model.
In Sections 4 and 5, I describe the time series and cross sectional results of the model
using commodities type portfolios and sorted portfolios. Section 7 demonstrates robust-
ness of the model and Section 8 explores the business cycle properties that may drive
the strong correlation between commodities and durable goods. Section 9 concludes.
2 Commodities Data
Despite the claim by Jim Rogers (2002) that investing in commodities could be “the best
money you ever spend,” returns to commodities are not well understood or measured.
While older literature finds mixed evidence for the existence of a risk premium, more
recent work has found that commodities have earned positive average returns over an
extended sample period including recent years. For example, Gorton and Rouwenhorst
(2006) find that an index of commodities earns high returns with low variance over the
period 1959 to 2004. Hong and Yogo (2010) find a similarly high mean-variance ratio
but use cash market spot prices, which can be unreliable. Before examining the ability
of asset pricing models to explain commodities futures returns, it is helpful to look at
the data sources and construction of commodities returns. In addition to descriptive
6
statistics, this section contains preliminary analysis of the size and predictability of
commodities futures returns.
2.1 Data Sources and Construction
A futures contract specifies on date t the quantity, quality, delivery location, and price
of a commodity to be traded on date t + h. While some commodities have contracts
expiring every month, others have contract months that are regularly spaced (every 3
months, for example) or irregularly spaced (7 months out of the year). A number of
recent papers have turned to the Commodities Research Bureau (CRB) for data on
these contracts (Gorton and Rouwenhorst, 2006; Gorton et al., 2007; Hong and Yogo,
2010). For each of these contracts, the CRB maintains a database that includes daily
prices on each contract from its opening to its expiration. This paper examines futures
contracts on 35 individual commodities that span a broad range of commodity types,
including grains, softs, metals, animal, and energy. While futures contracts became
available as early as 1959 for some grains, many futures markets did not begin until
much later. Crude oil contracts, for example, became available only in 1983, and rice
contracts began trading in 1986. Appendix A provides further details on commodities
data availability and Table A.1 lists individual futures contract coverage, including the
exchange the contract is traded on, the start and end date for contract availability,
available contract months, and number of monthly observations.
Using daily price data on these individual futures contracts, I create monthly series
by extracting prices on the last trading day of each month. The monthly data for each
contract are then used to generate two main series for each commodity: a spot price
series and a 3-month futures series. While data on cash transactions are available for
most commodities, I follow Gorton et al. (2007) and use near-month futures prices as
spot prices instead. The spot price for a given month is therefore defined as the price on
the last trading day of the month for the near-month futures contract, i.e. the futures
contract which is the next to expire. Using the near-month futures price as a measure of
the spot price allows me to focus on the change in the futures price over time, unclouded
by other differences between futures and cash prices such as differences in transaction
or transportation costs, liquidity, delivery cost, or delivery location. Furthermore, since
the delivery window for a contract often overlaps with the final month of trading, the
near-month futures price should often be nearly equal to the spot price in the cash
market. One weakness in using near-month prices is that due to the irregular spacing
7
of commodities futures contracts, for a given commodity the near-month price may
reflect a contract that expires after a few months, rather than within a few days.
To extract data series for the 3-month futures price, I take the price on the last
trading day of the month for the contract which is next to expire 3 months in the
future. For example, the January 2005 price for a 3-month futures contract will be the
price on the last trading day in January 2005 of the contract that is next to expire on
the last trading day in April 2005. This contract may actually expire on the last trading
day of April 2005 or as late as May or even June, depending on contract availability
for that particular commodity.
Using these two variables, I construct three important series for the basis, the spot
return, and the excess (futures) return. The basis is defined as the percentage difference
between the contemporaneous futures and spot prices:
Basist ≡Ft,t+3 − St
St, (1)
where Ft,t+3 is the price on date t of the futures contract that is next to expire on date
t+ 3, and St is the spot (near-month) price on date t. The spot price return is defined
as the percentage difference between the current and lagged spot price:
SpotPriceReturnt ≡St − St−3St−3
. (2)
The spot price return is not a financial return that can be earned by an investor, because
an investor who holds a commodity over the period t−3 to t will pay a storage cost and
earn a convenience yield, which is the benefit earned by someone who holds physical
inventory of a commodity. Instead, the financial return that can be earned is the excess
return, which is defined as the percentage difference between the current spot price and
the lagged futures price:
ExcessReturnt ≡St − Ft−3,tFt−3,t
. (3)
We call this value an excess return because it measures the profit that can be made by
taking a long position in a futures contract at date t− 3 and holding the contract until
8
maturity at time t.1 While technically no money must change hands for an investor to
enter into such a contract, in practice, investors are often required to post margin on
these investments. Even so, this margin is generally in the form of Treasury bills, so
that the return as measured is still in excess of the risk free rate.
Often, we express these variables using the approximation that (X − Y )/Y ≈log(X)− log(Y ). Using lowercase letters to denote logs, we have:
ExcessReturnt ≈ (st − ft−3,t)
SpotReturnt ≈ (st − st−3)
Basist ≈ (ft,t+3 − st)
These approximations allow us to express the excess return as approximately equal to
the spot return minus the lagged basis:
(st − ft−3,t) = (st − st−3)− (ft−3,t − st−3) (4)
⇒
ExcessReturnt ≈ SpotReturnt −Basist−3 (5)
While this approximation helps us understand the relationship among these three vari-
ables, using log approximations for returns would require the additional step of adjusting
excess returns and portfolio aggregation by one half the variance of the returns. For
this reason, and because average simple returns are the relevant returns for investors, I
use average simple returns as expressed in equation 3 for all further estimations in this
paper.
2.2 Conditional and Unconditional Returns
To investigate the returns to investing in commodities, I examine excess returns for both
individual commodities and portfolios of commodities. Because patterns of returns can
differ widely across commodities, I aggregate commodities into portfolios based on
commodity type to better uncover common patterns. As a benchmark, Figures 1 and
2(a) depict the price index and returns for an equal weighted portfolio of commodities
1Given the structure of the futures contracts, the excess return will always be measured as thechange in price for the same contract over a horizon of 3 months. While these 3 months are intendedto be the last 3 months of the life of the contract, when there is a missing contract, the excess returnwill be measured as the 3-month return that occurs earlier in the life of the next contract to expire.
9
futures plotted against the same measures for a broad equity market portfolio.2 As the
figures show, the index of commodities futures prices has grown unevenly over time.
The commodities index has similar mean and volatility as the market portfolio, though
the two are not highly correlated.
Drawing heavily from analogous investigations in the exchange rate literature, here
I present results from two tests of forecast efficiency. The tests examine whether the
futures price Ft,t+3 is a good predictor of the future spot price St+3. Following Gilmore
and Hayashi (2008), I first perform an unconditional test, which examines whether
excess returns are statistically different from zero on average. Formally, I test:
H0 : Et
(St+3 − Ft,t+3
Ft,t+3
)= 0.
Because the overlapping forecast windows introduce serial correlation in the observa-
tions of the excess returns, the t-statistic is constructed using Newey-West heteroskedas-
ticity and autocorrelation consistent (HAC) standard errors with a maximum of 4 lags.
The results in Table 1 indicate that there are unconditional returns to investing
in commodities futures for only 8 of 35 commodities tested. While 28 commodities
have positive returns on average, the high variance of these returns prevents many
of these positive returns from being statistically significant.3 In contrast, once these
individual commodities are grouped into portfolios, we can see a stronger pattern of
positive unconditional returns. In Table 1, we can see that the index of all commodities
has positive significant returns of 5.7% per annum. In addition, the energy and animal
portfolios have significant unconditional returns of 14.3% and 6.2%, respectively, while
the metal portfolio has returns of 6.3% that are significant at the 0.10 level.
Secondly, I perform the conditional test of forecast efficiency, which examines whether
the lagged basis can be used to predict excess returns. Since excess returns are approxi-
mately equal to the difference between spot returns and the lagged basis, the hypothesis
that the excess return equals zero is equivalent to the hypothesis that the spot return
equals the lagged basis. For the conditional test, I estimate the Fama (1984) regression
2The broad equity market index is further described in Appendix C. Details on portfolios con-struction can be found in Appendix A and summary statistics for individual commodities and typeportfolios are provided in Tables A.2 and A.3.
3Unless otherwise stated, significance refers to statistical significance at the α = 0.05 level.
10
and perform the associated hypothesis tests:
SpotReturnt+3 = α + β(Basist) + εt+3 (6)
H0 : α = 0 & H0 : β = 1
Since the spot return equals the percentage change in the spot rate, we can think of this
as the gross return (before storage costs) to buying a commodity at time t, holding it,
and selling it at time t+ 3. This return should be equal to the gross return on buying
a commodity at time t and signing a futures contract at time t to sell the commodity
at time t+ 3.
The estimated parameter α is sometimes said to measure the non-time varying
risk premium, because under the assumption that β = 1, α measures the average
unconditional excess return. The hypothesis test H0 : β = 1 is called the conditional
test of excess returns because the trading strategy implied by the result is conditional
on the spot and futures prices at time t. For (α = 0, β < 1), investors should sell a
futures contract (take a short position) when Ft,t+3 − St > 0 to earn excess returns
(Ft,t+3− St+3)/Ft,t+3 at time t+ 3. Conversely, investors should buy a futures contract
(take a long position) when Ft,t+3 − St < 0 to earn excess return (St+3 − Ft,t+3)/Ft,t+3
at time t+ 3.4
The conditional test for excess returns focuses on the spot return and the basis as
the key variables. Here, I present results from the conditional test for the 35 individual
commodities in my sample, using daily price data and the maximum sample period
for each commodity. Results in Table 2 indicate that we reject H0 : α = 0 for 6
commodities, all of which have point estimates of α > 0. These results are a strong
indication of a positive excess return, or a non-time varying risk premium, for these
commodities on average. An additional 6 commodities reject H0 : α = 0 at the 0.10
level of significance, and all but 2 of the commodities have positive point estimates for
α.
While much of the analogous exchange rate literature tries to explain why estimates
for β are usually negative, this pattern does not hold for commodities. Rather, β is
between 0 and 1 for most commodities. The only negative point estimates for β are for
4 of the 10 metals (aluminum, platinum, palladium, and tin), which along with the β’s
4When α 6= 0, the trading strategy is almost the same, but the cutoff is no longer Ft,t+3 − St = 0.Instead, an investor should take a short position in commodities futures when Basist is large and takea long position in futures when Basist is small. The precise cutoff point for “large” and “small” isBasist = SpotReturnt+3, which is when Basist = α/(1− β).
11
for gold, lead, and copper are the seven lowest values for β. At the upper end, there
are 7 point estimates for which β is greater than 1, including 4 of 6 energy commodities
(crude oil, coal, propane, and gasoline). Overall, I reject H0 : β = 1 for 11 of the 35
commodities, including soybean oil, oats, rice, cocoa, lumber, coffee, copper, aluminum,
nickel, tin, and live cattle. The results provide statistical evidence of a time-varying
risk premium for these 11 commodities, and suggest that a time-varying risk premium
might exist for 8 additional commodities which have point estimates of β < 0.75 but
are not statistically significant.
To summarize, these results demonstrate statistically significant unconditional re-
turns to a number of commodities, and unconditional returns that are economically
significant (greater than 5%) though not statistically significant for an additional few
commodities. Furthermore, the conditional test of returns provides evidence that a
number of individual commodities have time-varying risk premia. Table A.2 indicates
that the correlation of returns across commodities can be low even within commodities
of the same type. Because shocks in one commodity market may not affect returns in
another, the potential to earn high returns with low volatility by investing in portfo-
lios of commodities is especially high. Indeed, the returns to portfolios of commodities
grouped by commodity type supports this hypothesis, as the grains, metal, and energy
portfolios earn positive significant returns, as does the index of all commodities. The
remainder of this paper investigates the extent to which these returns to commodity
portfolios may be associated with systematic factor risk.
3 Model of Household Consumption and Portfolio
Choice
In this section, I describe the consumption-based asset pricing model as developed by
Yogo (2006). In the model, an intertemporal household optimization problem with
choices over durable and nondurable consumption is combined with portfolio choice
theory. The resulting Euler equation is used to develop a linear factor model that is
estimated using two-step GMM.
12
3.1 Household’s Optimization Problem
The consumption-based asset pricing model derived by Yogo (2006) begins with a house-
hold optimization problem with a durable consumption good. In each period t, the
household purchases Ct units of a nondurable consumption good and Et units of a
durable consumption good. Pt is the price of the durable good in units of the non-
durable good. The nondurable good is entirely consumed in the period of purchase,
whereas the durable good provides service flows for more than one period. The service
flow from the stock of durables, Dt, is related to durables expenditure by the law of
motion:
Dt = (1− φ)Dt−1 + Et, (7)
where φ ∈ (0, 1) is the depreciation rate. Each period, the representative household
maximizes a utility function of the CES form:
u(C,D) = [(1− α)C1−1/ρ + αD1−1/ρ]1/(1−1/ρ), (8)
where α ∈ (0, 1) is the weight on durables and ρ ≥ 0 is the elasticity of substitution
between nondurable and durable consumption.
There are N+1 tradeable assets in the economy, indexed by i = 0, . . . , N . In period
t, the household invests Bit units of wealth Wt in asset i, which earns Ri,t+1 in period
t+1. The household’s intraperiod budget constraint requires that total saving in assets
equals household wealth minus that period’s expenditures:
N∑i=0
Bit = Wt − Ct − PtEt. (9)
The household also optimizes across time by maximizing intertemporal utility given
by the recursive function:
Ut = {(1− δ)u(Ct, Dt)1−1/σ + δ[Et(U
1−γt+1 )]1/κ}1/(1−1/σ), (10)
where κ ≡ (1− γ)/(1− 1σ). The parameter δ ∈ (0, 1) is the subjective discount factor,
σ > 0 is the elasticity of intertemporal substitution (EIS), and γ > 0 determines
the relative risk aversion. Finally, the intertemporal budget constraint describes the
13
evolution of household wealth:
Wt+1 =N∑i=0
BitRi,t+1. (11)
In sum, given wealth Wt and the stock of durable goods Dt−1, the household chooses
consumption and saving {Ct, Et, B0t, . . . , BNt} to maximize its utility (equation (10))
subject to the three constraints (equations (7), (9), and (11)).
After recasting this household optimization problem as one of portfolio choice, Yogo
derives the Euler equation:
Et[Mt+1(Rei,t+1)] = 0, (12)
where Rei,t+1 is the return to asset i in excess of the risk free rate5, and the stochastic
discount factor Mt+1 is the intertemporal marginal rate of substitution:
Mt+1 =
[δ
(Ct+1
Ct
)−1/σ(v(Dt+1/Ct+1)
v(Dt/Ct)
)1/ρ−1/σ
R1−1/κW,t+1
]κ(13)
where
v
(D
C
)=
[1− α + α
(D
C
)1−1/ρ]1/(1−1/ρ)
,
and RW,t+1 is the return on the market portfolio. According to the model, assets whose
returns are highly correlated with consumption growth and returns to market wealth
should have higher returns on average, because they have high returns when marginal
utility is low. Conversely, assets whose returns have low correlations with these factors
should have low returns on average, because they provide a good hedge during periods
of high marginal utility.
3.2 Estimating the Linear Factor Model
Yogo (2006) demonstrates that the above model can be approximated as a linear factor
model in nondurable consumption growth (∆ct), durable consumption growth (∆dt),
and market returns (rWt):
E[Reit] = b1Cov(∆ct, R
eit) + b2Cov(∆dt, R
eit) + b3Cov(rWt, R
eit). (14)
5In the case of commodities futures returns, Rei,t+1 is equal to the excess return on commodities asdefined in equation (3).
14
The risk prices b are a function of the preference parameters in the model:
b =
b1b2b3
=
κ[1/σ + α(1/ρ− 1/σ)]
κα(1/σ − 1/ρ)
1− κ
. (15)
Furthermore, this model can be estimated using two-step GMM. Define the vector of
factors, ft = [∆ct,∆dt, rWt]′, and its expectation µf = E[ft]. We can estimate the
model using the moment function:
e(zt, θ) =
[Ret −Re
t (ft − µf )′b(ft − µf )
](16)
and weighting matrices:
W1 =
[kIN 0
0 Σ̂−1ff
]and W2 = S−1,
where k > 0 is a constant and Σ̂ff is a consistent estimator of Σff , the variance-
covariance matrix of ft.6
The two-step GMM is solved using constrained linear minimization. Following Yogo
(2006), I impose two constraints on the risk prices that arise from their relationships
with the preference parameters of the model. Using Equation 15, we have that σ =
(1−b3)/(b1+b2), γ = b1+b2+b3, and α = b2/[b1+b2+(b3−1)/ρ].7 The first constraint,
that b1 + b2 + b3 ≥ 0, ensures that the risk aversion (γ) is nonnegative. The second
constraint, that b3 ≤ 0, ensures that the elasticity of intertemporal substitution (σ) is
nonnegative when b2 and b3 are nonnegative.8
The commodities returns in this and all further estimations of the linear factor model
are the excess returns as defined and constructed in Section 2 and as summarized in
Table A.2. The factors used in the estimation, which are described in detail in Appendix
6The initial weighting matrix W1 puts equal weight on the first N moments. The constant k =[det(Σ̂ii)]
1/N is inversely related to the variance-covariance matrix of the initial values for the firstN moments. See Yogo (2006), Appendix C and Cochrane (2005), Chapter 13 for discussions of themoment function and choices for weighting matrices.
7Because α and ρ cannot be separately identified, I follow Yogo (2006) and use ρ = 0.79.8While the results indicate that the parameters are rarely constrained in the final solutions, the
restrictions do sometimes impact the initial values for the parameters, which are calculated as theconstrained OLS estimates of the coefficients of the linear factor model in equation 14.
15
C and summarized in Table A.4, are constructed following Yogo (2006) with only slight
modification. The monthly observations are converted to overlapping quarters to match
commodities returns, so that the return associated with a given month reflects the
market return or durable or nondurable consumption growth for the three months
leading up to and including that month. The mean return on market wealth is 4.9%,
with an annualized volatility of 15.9%. In contrast, the mean returns on nondurable
and durable consumption are 2.1% and 3.9%, and the volatilities are much lower at
1.1% each. Unsurprisingly, the correlation coefficient between nondurable consumption
and durable consumption is largest among the factors, at ρ = 0.28. The correlation
between nondurable consumption and wealth is somewhat lower (ρ = 0.19), and the
correlation for durable consumption and wealth is close to 0 (ρ = 0.03).
4 Testing the Model using Commodity Type Port-
folios
To estimate the linear factor model for commodities, I use portfolios of commodities in
order to diversify away some of the commodity-specific idiosyncratic risk. There are a
number of potential sorting characteristics for commodities, just as there are for equities.
While the next section will explore characteristics of commodity portfolios sorted on
the basis, returns momentum, and volatility, in this section I focus on commodity type
portfolios. Tables 3 and A.3 list summary statistics for commodity portfolios aggregated
by commodity type. The returns to these portfolios are examined in detail in Section 2.
Of the 5 type portfolios, energy commodities unsurprisingly have the highest average
excess return, at 14.3%, while grains have the lowest average excess returns, at 2.9%.
The full index of commodities has significantly positive returns of 5.7%. Given the high
volatility of individual energy commodities and high correlation across these contracts,
it is unsurprising that the annualized volatility of the energy portfolio is the highest, at
33.2%. In contrast, the low correlation across the full set of commodities reduces the
volatility of the index of full commodities to 13.6%. In Table A.4, we can see that over
the sample period 1959 to 2008, commodities have a higher return and lower volatility
than the return to the broad market index of equity returns.
16
4.1 Estimating Risk Exposure using Time Series Regressions
Before turning to the results of the linear factor model, I investigate the time series
properties of commodities returns by examining the multiple betas for each portfolio.
These multiple betas are estimated using a separate time series regression for each
commodity portfolio.9 The regression coefficients measure the amount of risk each
commodity has relative to each factor within the three factor model:
Reit = βi0 + βic∆ct + βid∆dt + βiwrWt + εit (17)
The magnitude and significance of the multiple betas for each commodity can indicate
which factors are most important in the asset pricing of commodities futures. The
number of observations for each regression depends upon the data availability for that
particular type portfolio, and ranges from 307 for energy to 591 for grains. As before,
I adjust for serial correlation using Newey-West standard errors with 4 lags.
In Panel A of Table 3, we can see the first indication that durable consumption
growth is an important factor for commodities returns. For the index of all commodities,
we have a negative βc, and positive values for βd and βw. Of these, only βd is statistically
significant. The results for the five type portfolios also point to the importance of
durable consumption growth, as the estimate of βd is significant for the softs, metal,
and animal portfolios. Point estimates for βd are generally the largest in magnitude
as well, ranging from 2.09 for softs to 4.80 for metals. The only other statistically
significant factor beta is the market wealth beta for the metal portfolio (βw = 0.30).
However, there is no discernible pattern in market wealth risk, as the βw estimate is
negative for energy and grains portfolios and positive for softs, animals, and metals.
Finally, the results for the five type portfolios indicate that nondurable consumption
risk also differs quite a bit across commodity type, as the multiple beta for nondurable
consumption ranges from -5.86 for energy to 0.14 for animals.
9Though these multiple betas can also be estimated using the GMM estimation of the linear factormodel, the GMM estimates are affected by the choice of test assets used in the cross section and theprecision with which the GMM is able to estimate the population means of the factor returns. In orderto examine factor multiple betas which are free of these complicating factors, in this section I presentthe results from time series regressions instead.
17
4.2 Evidence and Evaluation for the Broader Cross Section
While examining the individual assets’ multiple betas helps us understand the amount
of risk exposure each asset has relative to the factors included, the linear factor model
allows us to determine how well these risk amounts and the estimated risk prices can
predict the average excess returns. In other words, the linear factor model allows us to
test the prediction that assets with higher betas and higher amounts of risk will have
higher excess returns.
In order to test whether this model can predict the returns of commodity portfolios,
I estimate the linear factor model using a broad cross section of potential assets in which
an individual may choose to invest. In addition to the 6 commodity portfolios, each
cross section may also include 6 Fama-French equity portfolios sorted on size and book-
to-market value, 5 equity portfolios sorted on industry characteristics, 6 bond portfolios,
3 indexes of international equities, and 6 foreign currency portfolios. A description of
portfolio construction and summary statistics for these assets can be found in Appendix
D.
GMM estimation requires that all included test assets have data availability over
the entire sample period chosen. Therefore, a cross section that includes the energy
portfolio must be limited to the sample period June 1983 to December 2008, which
leaves over two decades of data on most other commodities unused. In order to make
maximum use of the data available, I therefore estimate the model on a long sample
that covers March 1965 (when the animal portfolio begins) to December 2008 and a
short sample that covers January 1984 to December 2007, when data on the energy
portfolio, currency portfolios, and international equities are available. For the long
sample, the set of test assets includes all test assets for which the data is available for
the entire sample period: the grains, softs, animal, and metals portfolios, the 6 Fama-
French portfolios, the 5 industry-sorted equity portfolios, and the 6 bond portfolios.
For the short sample, I add to these the energy portfolio, the 3 international indexes
and the 6 currency portfolios.
The cross sectional results of the model, which can be found in Table 5, demonstrate
that the model describes the data well. The first estimation with the long sample gives
positive significant risk prices for durable and nondurable consumption growth, but a
negative risk price for market wealth. The high R2 (0.88) and low mean absolute error
(0.87 percentage points) are based on first-stage estimates for the 21 test assets, and
are visible in Figure 3(a), which graphs the actual returns to each test asset against
18
the predicted return. Furthermore, the J-test of overidentifying restrictions does not
reject the model (p = 0.98). For the short sample, the estimated risk prices for durable
consumption growth and market wealth are positive and significant, while the risk
price for nondurable consumption growth is negative and significant. The J-test does
not reject this model either (p = 1.00), and the R2 is an even higher 0.93. Figure 3(b)
graphs the actual and predicted returns for the short sample estimation.
According to the theory, these estimated risk prices can be used to calculate the
values of the preference parameters of the model. For the long sample, the risk prices
imply an elasticity of intertemporal substitution (σ) of 0.01, a coefficient of risk aversion
(γ) of 167.68, and a weight on durables (α) of 0.35. In the short sample, the estimate
for the elasticity of intertemporal substitution is similarly low (0.01), while the estimate
for the weight on durables is higher than for the long sample (1.52). Additionally, in
the short sample, the estimate for the coefficient of risk aversion is lower than for the
long sample, but still high (48.96). Yogo (2006) explains that the high coefficient of
risk aversion is an expression of the equity premium puzzle, which is driven by the low
volatility of nondurable and durable consumption relative to asset returns.
These results indicate that the consumption-based asset pricing model including
durable and nondurable consumption growth can predict returns to commodity type
portfolios and a broad cross section of assets. This result is at odds with earlier pa-
pers on commodities, which found that asset pricing models were not able to explain
commodities returns. However, it is consistent with conventional wisdom in the fi-
nance literature that while factor models might perform poorly when trying to explain
differences among individual assets, they are generally more successful at explaining
differences in returns across broad classes of assets. By grouping commodities into
portfolios based on type and then including these assets with many assets of diverse
origin, I am able to estimate a consumption-based multi-factor asset pricing model that
successfully predicts commodities returns.
Furthermore, these results indicate that durable consumption growth is an impor-
tant factor in the pricing of commodity portfolios. In time series regressions, coefficient
on the durable goods factor has the largest magnitude and most significance for com-
modity portfolios, indicating a strong correlation between commodities returns and
durable consumption growth. In the cross section, while theory would predict all risk
prices to be positive, the risk price on durable goods is the only consistently positive
significant one. This positive risk price is consistent with the theory that durables con-
sumption growth is an important factor in explaining the cross section of asset returns,
19
because it indicates that assets with higher durables risk have higher returns. While
the high returns to commodities and low correlation with nondurable consumption and
equity returns was previously puzzling, these results demonstrate that the high returns
to commodities can be explained by their high durables risk.
5 Characteristics driving commodities returns
5.1 Theory, Motivation, Construction
Given the success of the model in predicting commodities returns for portfolios based on
type, I further investigate which characteristics of commodities may be driving these
returns and risk amounts. There is a long history in finance of sorting equities into
portfolios according to characteristics that can predict returns, including size, book-
to-market value, and industry type, as in the Fama-French test assets used above.
Sorting assets into portfolios allows us to study the relationship between the underlying
sorting characteristic and asset returns by eliminating the other idiosyncratic drivers of
returns that can be diversified within each portfolio. For example, equity portfolios that
are sorted on size should contain equities with similar distributions of book-to-market
ratios. Comparing these size portfolios, and especially the spread between the extreme
portfolios, allows us to focus on the risk-return tradeoff of the size characteristic. These
ideas have also been applied by Lustig and Verdelhan (2007), who explain the returns
to investing in currency futures by sorting currencies into portfolios based on interest
rate differentials.
To identify potential sorting characteristics for commodities, I draw on a broad set
of theories on commodities returns, backwardation, and the storage model. In partic-
ular, this work is motivated by Gorton et al. (2007), who examine the empirical link
between inventories and the risk premium. They demonstrate patterns of returns to
portfolios of commodities sorted on inventories and priced-based signals of inventories,
such as the futures basis, spot return momentum, and excess return momentum. How-
ever, they stop short of testing the relationship between these predictors of returns and
conventional asset pricing models. In this paper, I focus on spot price volatility, the
basis, and excess return momentum as sorting characteristics, to see whether commod-
ity portfolios formed on these characteristics have returns that are predictable using
standard asset pricing models.
20
The first characteristic I use for sorting is the spot price volatility. If we view the
return to buying a futures contract as a reward for taking on the risk associated with
future spot price volatility, then we should expect returns to increase in periods of high
volatility and high risk. Furthermore, if past volatility is a predictor of future volatility,
when we sort on past volatility, we should see an increase in excess returns for the high
volatility portfolios. If spot price volatility also covaries predictably with the factors in
the consumption-based model, then we would find predictable patterns in the returns
to volatility-sorted portfolios. To measure recent volatility, I calculate for the end of
each month the three month coefficient of variation, which equals the variance of the
daily spot prices over the previous three months divided by the mean of these daily spot
prices. Finally, I sort commodities each month into five portfolios using the demeaned
value of recent spot price volatility.10
Given the focus on the basis as a predictor of spot returns in the Fama (1984)
regression (equation (6)), the basis is a natural choice for the second sorting character-
istic. More recently, Hong and Yogo (2010) and Gorton et al. (2007) have studied the
basis as one of the determinants of commodity returns and note that excess returns are
generally higher when the basis is low. They argue that a low basis is often a signal
of low inventories, which cause spot prices to rise more than futures prices. Since low
inventories can result in higher spot price risk, they also require higher excess returns.
By sorting commodities into portfolios using the basis, I measure the relationship be-
tween the basis and futures returns in a cross sectional, rather than time series, setting.
I construct the set of basis-sorted commodity portfolios by sorting commodities each
month using the demeaned monthly basis for each commodity.
The final sorting characteristic is returns momentum. In addition to being an im-
portant predictor of returns for equities, Gorton et al. (2007) point out that past returns
are likely indicative of unanticipated shocks to supply or demand. Since these shocks
may have lasting effects on prices and inventories, it is reasonable to think that past
returns, to the extent that they carry information about the state of inventories, will
be a good predictor of future returns. Indeed, Hong and Yogo (2010) and Gorton et al.
(2007) find high momentum commodities have higher returns. To capture the informa-
tion reflected in the history of excess returns, I construct a set of five portfolios sorted
on the demeaned value of the average of the previous twelve months of excess returns.
10Further details on portfolio construction can be found in Appendix B.
21
In addition to the five portfolios in each of the above sets, I add a sixth portfolio to
each set to measure the returns to a trading strategy that takes advantage of the spread
in returns across the extreme portfolios. For the basis sorted portfolios, this means going
long in the low basis portfolio and short in the high basis portfolio. For the momentum
and volatility sorted portfolios, it means going long in the high momentum/volatility
portfolio and short in the low momentum/volatility portfolio.
5.2 Summary Statistics and Time Series Properties of Com-
modity Portfolios
We can learn about the returns to these sorting characteristics by examining both the
average excess returns and the pattern of betas across the five portfolios. The trends
described in this section are visually represented by Figures 4(a)-4(c), which graph
the returns and time series betas for each portfolio. As shown in Figure 4(a), the
volatility-sorted portfolios indicate the patterns expected, with returns that increase
monotonically from the low volatility portfolio to the high volatility portfolio. The
high volatility portfolio has statistically significant returns of 24.7%, while the high-
minus-low portfolio generates a positive return of 26.8% and a Sharpe ratio of 0.80.
Looking at the pattern of betas across the 5 sorted portfolios in Panel B of Table 3
and Figure 4(a), it is clear that higher returns are consistent with higher durables risk,
as the positive, statistically significant values for βd increase with excess returns and
momentum. In other words, the returns to the high momentum portfolio covary more
strongly with durable consumption growth than the returns to the low momentum
portfolio. In contrast, the βw generally decrease with momentum, indicating that the
low momentum portfolio returns covary more strongly with returns to market wealth
than the high momentum portfolio returns. While none of the βw on the individual
portfolios is significant, the return to the high-minus-low portfolio, which measures the
spread in the returns to the momentum-sorted portfolios, is negative. Finally, the table
and figure show that βc does not vary consistently with the basis or returns.11
11Since nondurable and durable consumption are positively correlated, the multiple betas presentedhere could be quite different from simple betas calculated from univariate time series regressions.Simple betas are provided for in Table A.8 for comparison. While market and durables betas retainsimilar magnitudes and qualitative patterns, many nondurables betas are in fact positive in a univariatesetting where the durables factor is not included in the regression. However, since the nondurablesbetas still do not demonstrate a pattern of increasing with average returns across sorted portfolios,they remain unable to explain the cross section of commodities returns.
22
For the portfolios sorted on the basis, we can see in Panel C of Table 3 and Figure
4(b) that returns increase monotonically as the basis decreases, and the low-minus-high
portfolio has a significant positive return of 10.0%. Again, the pattern of exposure
to durable consumption risk is consistent with the pattern of returns, as βd generally
rises as the basis falls and returns increase. In addition, βd is the only positive beta
coefficient for the spread portfolio. In contrast, βc and βw are declining or U-shaped as
the basis falls, and are negative for the basis spread portfolio.
Finally, in looking at Panel D of Table 3 and Figure 4(c), which show the returns
and time series properties for the momentum-sorted portfolios, we have that returns
are monotonically increasing with momentum and are highest for the high momentum
portfolio. We also have that the values for βd are generally increasing with momentum,
while the values for βc and βw are lower for the high momentum portfolios. Finally,
we can see that for the spread portfolio, only βd is positive. All these results indicate
that for the momentum-sorted portfolios, high returns are again most closely associated
with high durables risk.
5.3 Cross sectional results for portfolios
I add these sorted commodity portfolios to the set of test assets and estimate the model
using the same long and short sample periods. The cross section of test assets used
now includes the full set of assets from the previous estimations as well as the 15 sorted
commodity portfolios (three sets of five sorted portfolios). The results in Table 5 show
that for both estimations, the risk price of durables remains positive and significant.
However, as in the estimations with only the type portfolios, the risk price on market
wealth is negative for the long sample, and the risk price on nondurable consumption
growth is negative for the short sample. As before, the models have high values for
the R2 (0.85 and 0.96) and low values for the mean absolute error (0.82 and 0.39). In
addition, neither of the models is rejected by the J-test of overidentifying restrictions.
Though these models fit the data well, the implied preference parameters are again
sometimes outside of the expected range. The elasticity of intertemporal substitution
is still close to zero (0.06 or lower), but within the expected range of σ ∈ (0, 1). While
the estimate for the weight on durables of 7.7 for the short sample is well outside the
expected range of α ∈ (0, 1), the estimate of 0.81 is within range for the long sample.
Finally, the coefficient of risk aversion γ is lower than in the estimations without the
sorted portfolios, but still very high (69.38 and 9.03).
23
A summary of these results is best represented by Figures 3(c) and 3(d), which plot
the realized average excess returns against the predicted average excess returns. While
the momentum and basis-sorted portfolios have returns that range from 1% to 12%
per annum, the realized returns for the volatility-sorted portfolios have a much larger
range, from -2% to 25%. Figure 3(c) illustrates that the model is able to accurately
predict this range of returns. Finally, Figure 3(d) demonstrates that the model also fits
well when using a short sample period.
6 Nested Special Cases and other Benchmark Mod-
els
The three factor model presented in Section 3 nests a number of well-known special
cases. In this section I present results for these linear factor models, each of which uses
only a subset of the three factors included in the main model and imposes corresponding
restrictions on the preference parameters of the model. I also present results for the
Fama-French three factor model as an additional benchmark. The results demonstrate
that durable consumption growth is an important factor in explaining the cross section
of asset returns.
Secondly, I present two robustness checks for the GMM estimations above. First,
in order to examine whether the risk prices change over time, I provide GMM esti-
mation using the same cross section of test assets over different sample periods. The
results indicate that while estimates of risk prices are not stable over time, the risk
price for durable consumption is the only one that is positive and significant in every
sample period. Finally, because consumption data may not be well-measured at the
monthly level, I present results using only nonoverlapping quarters. While data using
nonoverlapping quarters has the benefit of a lower autocorrelation across consecutive
observations, it also has the drawback of having a smaller number of observations in
the time series, which restricts further the number of test assets which may be included
in a sample of a given length. Even so, the results are consistent with those estimated
using overlapping quarterly data.
24
6.1 Nested Special Cases
The full three factor model presented in Section 3 nests a number of well-known asset
pricing models as special cases. Using the short cross section with sorted portfolios
(column 4 of Table 4), I present results for linear factor models which use one or two of
the three factors of the full model. The columns of Table 6 correspond to the following
special cases:
1. CAPM: The standard CAPM uses only market wealth as a factor. Preference
parameters are restricted such that b3 = γ, b1 = b2 = 0, σ = ρ, and σ →∞.
2. CCAPM: The consumption CAPM uses only nondurable consumption as a factor,
and restricts preference parameters to b1 = γ, b2 = b3 = 0, σ = 1/γ = ρ and α = 0.
3. DCAPM: The durable CAPM uses only durable consumption as a factor. Pref-
erence parameter restrictions are the same as those in the CCAPM, as this
model simply substitutes durable consumption growth for nondurable consump-
tion growth: b2 = γ, b1 = b3 = 0, σ = 1/γ = ρ and α = 1.
4. EZ-CCAPM: The Epstein-Zin CAPM uses nondurable consumption and market
wealth as factors. Once again defining κ ≡ (1 − γ)/(1 − 1σ), we have preference
parameter restrictions: b1 = κ/σ, σ = ρ, b2 = 0, and b3 = 1− κ.
5. CD-CAPM: The nonseparable expected utility model uses nondurable and durable
consumption as factors. Preference parameters are restricted such that σ = 1/γ,
b1 = γ + α(1/ρ− γ), b2 = α(γ − 1/ρ), b3 = 0, and ρ = 0.79 by assumption.
6. EZ-DCAPM: Results for the full three factor model are reproduced for ease of
reference. As before, b1 = κ[1/σ+α(1/ρ−1/σ)], b2 = κα(1/σ−1/ρ), b3 = 1−κσ,
and ρ = 0.79 by assumption.
Comparing the first three models which have one factor each, it is clear that the
durable CAPM has the most explanatory power. The durable CAPM is the only model
for which the R2 is higher than 0.9 and the mean absolute error is below 1%. The
relatively strong fit of the model can be seen in Figure 5(c) as compared to Figures
5(a) and 5(b). In fact, the durable CAPM with only one factor has a higher R2 and
lower mean absolute error than the EZ-CCAPM, which has nondurable consumption
and market wealth as factors.
25
Comparing the results of the DCAPM to the results in columns 5 and 6 of Table
6, we can see that the R2 falls from 0.985 to 0.977 and then rises to 0.991. Similarly,
the mean absolute error rises when the nondurable consumption factor is added, going
from 0.457 to 0.584, but then falls to 0.391 when the market returns factor is added.
When using a linear factor model to explain returns to a cross section of international
assets and commodity portfolios, these results indicate that once durable consumption
is included as a factor, the addition of nondurable consumption does not add much ex-
planatory power to the model. Additionally, these results indicate that adding durable
consumption to a model with only nondurable consumption and market returns as
factors does improve explanatory power.
One final important benchmark model is the three factor Fama-French model, which
uses market returns as well as the size and value spread portfolios as factors. The SMB
factor measures the spread in returns between equities with small and large market
capitalizations, while the HML factor measures the spread in returns between equities
with high and low book-to-market values. The results in column 7 of Table 6 show that
the three factor Fama-French model performs better than many of the nested mod-
els discussed above. However, it is outperformed by all models which include durable
consumption growth as a factor, including the one-factor DCAPM, the two-factor CD-
CAPM, and the full three-factor model with nondurable consumption, durable con-
sumption, and market returns.
6.2 Risk Prices over Time
The cross sectional results in Table 5 show large differences in risk prices across the
four specifications. These differences could be due to either the inclusion of different
assets in each cross section or the different sample periods used for each estimation. In
order to further investigate whether the risk prices have changed over time, I estimate
the model using an early and late sample period for the same cross section.
Table 7 provides results using the set of assets that are available in the early sam-
ple period, which includes the domestic equities and bonds, and all the commodity
portfolios except for energy (column 3 of Table 4). In addition to the early and late
samples, the results for the full sample period are reproduced for ease of reference. The
estimated risk price for each factor differs across sample periods. While the changes in
parameter estimates may reflect changes in the actual risk price over time, they could
also reflect the difficulty in estimating the parameters using limited sample periods. In
26
fact, the full sample risk prices are not always an intermediate value between the two
short sample estimates. However, despite the inconsistency across sample periods, it
is important to note that the durables risk price is the only risk price that remains
significant and positive for all sample periods.
6.3 Nonoverlapping Quarters
In Table 8, I present results using nonoverlapping quarterly data. These cross sectional
estimates use the same long and short sample periods as in the estimates using over-
lapping quarterly data. However, while the cross section of test assets in the two long
sample estimations remain the same as before, the short sample estimations must be
adjusted. As noted by Burnside (2007), GMM estimation requires the number of test
assets to be relatively small compared to the sample length. When using nonoverlap-
ping quarters, the sample length for the short sample period is small (T=96), so the
cross section must be limited.
One way to limit the number of test assets is to use only the index of all commodities
rather than the 5 type portfolios. Additionally, I replace the 5 sorted portfolios for
each of the sorting variables with the spread portfolios (High-Minus-Low Momentum
and Volatility, and Low-Minus-High Basis). To demonstrate that the fit of the model is
robust to various choices in the cross section of test assets, I estimate the model on two
very different cross sections. The first is a set of primarily domestic test assets, which
adds to the 4 commodity portfolios the 6 Fama-French equity portfolios, 5 industry-
sorted equity portfolios, 6 bond portfolios, one index of international equities, and
one currency portfolio that measures the spread between high and low interest rate
currencies. The second cross section is an internationally focused one, and includes the
4 commodity portfolios as well as the Small-Minus-Big equity portfolio, the High-Minus-
Low equity portfolio, the 6 bond portfolios, 3 international indexes, and 7 currency
portfolios.
For these 4 estimations, the results are little changed from before. While the risk
price on nondurables and market wealth change signs for the long samples as compared
to the previous set of estimations, the risk price on durables remains positive and
significant for all 4 estimations. In addition, the R2 remains high (0.92 or higher),
and the mean absolute error remains roughly the same as before. The parameter
values remain of a similar magnitude: relatively small for the elasticity of intertemporal
substitution (hitting the lower bound of 0 for the short domestic sample), the coefficient
27
of risk aversion remains high, and the weight on durables is higher than the expected
range of α ∈ (0, 1). Finally, while the p-value for the J-test of overidentifying restrictions
falls to below 0.05 for the long sample with sorted commodity portfolios, the J-test fails
to reject the model for the other three estimations.
7 OLS Estimation and Out of Sample Tests
In this section, I present OLS estimates of the linear factor model. GMM estimation
has a number of advantages over OLS estimation. For example, using the moment
conditions in equation 16 allows the population means of the factors to be different
from their sample averages, and using the weighting matrix 3.2 and allows the moment
conditions to have different weights. However, the GMM estimates allow for the risk
amounts, or betas, to change at every period, inducting an extremely good fit for the
model. Additionally, the risk prices can be sensitive to small changes in estimation pro-
cedure, including choice of initial parameters or variance-covariance matrix estimator.
In contrast, the OLS estimation can provide results from a simpler procedure that is
also more comparable to a number of previous studies.
Finally, I present two types of out of sample predictions. Using the OLS estimation
procedure, I estimate either risk amounts or risk prices from an earlier sample period.
These early sample parameters are used to predict the returns to a broad cross section
of assets in a later period. Secondly, in order to provide out of sample results that
are more comparable with the main results of the model, I estimate the linear factor
model using GMM on three estimation samples. The parameters from these benchmark
estimations are then used to predict the average returns in a later sample period for
commodities and other portfolios not in the original sample.
7.1 Estimation using Ordinary Least Squares
While the estimation up to this point has followed the procedures developed by Yogo
(2006), in this section I present results for an estimation using simple ordinary least
squares regression. Using the short cross section with sorted portfolios, (column 4 of
Table 3), I first run a time series regression of each test asset on the three factors:
Reit = βi0 + βic∆ct + βid∆dt + βiwrWt + εit (18)
28
Next, I estimate the risk prices using a single cross sectional regression of the average
return to each test asset on the estimated time series betas:
Reit = λcβ̂
ic + λdβ̂
id + λwβ̂
iw + εi. (19)
Column 1 of Table 9 presents the results of this cross sectional regression.12 While
the coefficient on durables risk is positive, it is not statistically distinguishable from
zero. Meanwhile, the risk price of market returns is positive and significant, and the
risk price on nondurable growth is negative and significant. The R-squared for this
regression is 0.22, and mean absolute error indicates that the cross sectional regression
predicts the average returns to commodity portfolios about as well as the returns to
all the test assets, on average. Figure 6(a) graphs the actual returns to each test asset
against the predicted return for the OLS estimation.
Given that the risk prices change depending on the cross section of test assets, I
also estimate the cross sectional regression using only the 20 commodity portfolios. In
the results reported in column 2, we can see that the risk price on durables is the only
positive one, though it is still not significant. Additionally, the R-squared is even higher,
at 0.51. These results provide additional evidence that while a number of assets have
returns that increase with market risk, the average returns to commodity portfolios
instead increase with durable consumption risk.
Finally, I provide two more OLS estimations for comparison. Column 3 provides
results for the cross section of 25 Fama-French equity portfolios sorted on size and
book-to-market value. For this estimation, the risk price of market returns is the only
positive one. However, the R-squared is negative. These results, along with Figure 6(c),
provide evidence that OLS estimation of the linear factor model does not perform well
even for the benchmark test assets. In comparison with these results, the OLS results
for commodities look even stronger. Column 4 and Figure 6(d) provide results for the
EZ-CCAPM, which is the nested model with only nondurable consumption and market
returns as factors. The poor fit of this model provide further evidence that even in an
OLS estimation, the durables factor helps explain the returns to commodity portfolios.
12Since the regressors here are betas, rather than covariances, the risk prices (λ) are not necessarilyequal to the risk prices (b) in the previous cross sectional analyses. Consequently, they also do nothave the same relationship with the preference parameters of the model in Section 3.
29
7.2 Out of Sample Tests using OLS
Another way to check the robustness of the results is to examine how the model per-
forms out of sample. In this section, I perform three of out of sample tests using OLS
regressions. For each test, either risk amounts or risk prices are estimated using an early
sample period. These early sample parameters are used to predict the cross section of
average returns in a later period. These out of sample estimations are performed using
the short cross section with sorted portfolios (column 4 of Table 4).
In the first out of sample test, I use the OLS cross sectional regression to test
whether betas from an earlier period are able to predict average returns in a later
period. I first estimate the time series betas from equation 18 for each portfolio in the
cross section by using the first half of the available sample period (January 1984 to
December 1995). Next, I regress the average return to each portfolio over the second
half of the sample period (January 1996 to December 2007) on these early sample betas
in the cross sectional regression (equation 19).
For both the full cross section (column 3) and the subset of commodity portfolios
(column 4), the results in Table 10 indicate that the risk price for durable consumption
is significant and positive. As compared to the in-sample OLS estimation (reproduced
in column 1), the full cross section has slightly higher mean absolute error and a lower
R-squared. In contrast, the subset of just the commodity portfolios has a slightly
lower mean absolute error and a higher R-squared than in the in-sample OLS estima-
tion (column 2). Figures 7(a) and 7(b) demonstrate that the fit of the model for the
out of sample estimation are similar to the corresponding in-sample OLS estimation,
represented by Figures 6(a) and 6(b).
For the second out of sample test, I estimate the risk prices using the first half of
the available sample period (January 1984 to December 1995). This requires using the
same time series betas from the early sample period as in the previous estimation. I
estimate the early sample risk prices by regressing the average return to each portfolio
over the first half of the period on these early sample betas, using the cross sectional
regression (equation 19). These risk prices are reported in columns 5 and 6 of Table
10. Finally, I estimate the time series betas using the second half of the sample period
(January 1996 to December 2007), and use the early sample risk prices and the late
sample betas to predict the average returns to each portfolio. The measures of fit in
columns 5 and 6 of Table 10 are calculated using these predicted average returns and
the actual average returns over the late sample period.
30
The results indicate that the risk prices calculated in the early sample do not help
predict the average returns in the late sample period. The mean absolute error and
R-squared values in columns 5 and 6 of Table 10 are quite poor compared to previous
in-sample and out of sample estimations. These results are consistent with previous
evidence that risk prices seem to change over time.
7.3 Out of Sample Tests using GMM
While the OLS out of sample tests allow evaluation of the model using either risk
prices or risk amounts estimated from an early sample, the results are comparable only
to the in-sample OLS estimates and not to the in-sample GMM estimations. In this
section, I provide out of sample estimations that are more directly comparable to the
main results, which are estimated by GMM. Here, I use various samples to estimate the
linear factor model (equation 14) by GMM, using the moments in equation 16. I use
the estimated risk prices and factor population means to predict the average returns in
a later sample period for commodities and other portfolios not in the original sample.
Finally, I examine how well these predicted returns fit the actual cross section of returns
using various measures of fit.
In the first set of out of sample tests, I use GMM to estimate the risk prices for
market returns, nondurable and durable consumption. Using an early sample period
of January 1959 to December 1983, I estimate the model on the 25 benchmark Fama-
French equity portfolios sorted on size and book-to-market values. The results of this
estimation are reported in column 1 of Table 11. Using the parameter estimates for
risk prices (b) and factor means (µ), I calculate the predicted values and residuals for a
broad cross section of test assets. In particular, I use the short cross section with sorted
portfolios (column 4 of Table 4), except that I omit the 6 Fama-French portfolios since
they are similar to the 25 benchmark portfolios used in the estimation of the parameters.
The results in Panel A of Table 12 indicate that the fit of the model is not as strong
when the parameters are estimated out of sample. While the J-test does not reject
the hypothesis that the pricing errors are jointly zero for the full cross section of out
of sample assets in column 1, this hypothesis is rejected for the subset that includes
only the commodity portfolios (column 2). The R-squared and the mean absolute error
indicate that the out of sample predictions have much worse fit than the in-sample
predictions, as reported in Table 5. Figures 8(a) and 8(b) graph the actual returns
to each test asset against the predicted return from the GMM estimation. They show
31
that this out of sample estimation generally over-predicts the returns to the test assets,
especially for assets with a high actual return.
These results are consistent with previous evidence that risk prices change depending
on the sample period and cross section of test assets included. A second out of sample
test estimates the risk prices using the same set of 25 Fama-French portfolios but
extends the sample period to December 2008. The risk prices, reported in column 2
of Table 11, are somewhat changed. Panel B of Table 12 shows that these risk prices
are better able to predict average returns. While the R-squared is better for the full
sample, it is worse for the set of commodity portfolios. However, the J-test does not
reject the model for either set of portfolios. Figures 8(c) and 8(d) provide graphical
illustrations.
The third out of sample test estimates the risk prices using the set of assets that are
available in the early sample period, which includes the domestic equities and bonds,
and all the commodity portfolios except for energy (column 3 of Table 4). Using these
risk prices, reported in column 3 of Table 11, the out of sample tests have a much higher
R-squared and much lower mean absolute error. These results, graphically represented
by Figures 8(e) and 8(f), are comparable to the in-sample estimates in Table 5. However,
the J-test rejects the hypothesis that the pricing errors are jointly zero for the full set
of test assets and for the subset of commodity portfolios.
8 Business Cycle Properties
The strong fit of the consumption-based multi-factor model indicates that the returns
to commodities and other assets can be explained by their covariance with the three
factors that affect investors’ marginal utility: nondurable and durable consumption
growth, and returns to market wealth. In particular, the results indicate that durable
consumption growth is a key factor in explaining the returns to commodities. While the
Yogo (2006) model of household optimization explains why assets with durables risk
require high returns, it leaves open the question of why commodities have high durables
risk. In this section, I examine the business cycle properties of commodities returns
in order to further explore the link between commodities and durable consumption
growth.
First, returning to the model presented in Section 3, we can see why assets with
high durables risk require high returns. When utility is not additively separable in
32
nondurable and durable consumption, marginal utility has an extra multiplicative term,
v(D/C)κ(1/ρ−1/σ). When ρ > σ and κ > 0 (σ < 1 and γ > 1), marginal utility of
nondurable consumption rises when durable consumption falls. The results support
this theory, as the parameters fall well within these restrictions. The high durables risk
of commodities means that commodities have low returns when durable consumption
growth is low. This is precisely when marginal utility of consumption is high. Therefore,
returns to these risky commodities must be high on average for households to invest in
them.
Next, in order to explain the high durables risk of commodities, I turn to examining
their business cycle properties. Gorton and Rouwenhorst (2006) note that commodities
returns behave differently than stock and bond returns over the business cycle. Using
the dates determined by the NBER Business Cycle Dating Committee, they classify
each month as being part of one of the four phases of the business cycle, splitting each
expansion and recession into “early” and “late” periods by simply dividing the full
period in half. For each series (stocks, bonds, and an index of commodities returns),
they take the average return over all months in each phase to determine the peaks and
troughs. They find that while stocks peak in early expansion, commodities returns peak
in late expansion. In addition, stock returns trough in early recession while commodities
returns trough in late recession. These results show that while stocks and commodities
both have higher returns in expansions than in recessions, splitting the business cycle
into four phases can better illuminate the low correlation between commodities and
equities.
Following their work, I perform a similar analysis on the five commodities type port-
folios and the consumption-based risk factors of durable and nondurable consumption
growth. The results in Table 13 indicate that returns to each commodity type portfo-
lio and the returns to the index of all commodities follow a pattern similar to that of
durable consumption growth. In addition, nondurable consumption growth follows the
same pattern as market returns. While nondurables and market returns peak in early
expansion, durables and commodity portfolios returns peak in late expansion. While
nondurables and market returns trough in early recession, durables and commodity
portfolios trough in late recession.
Though these patterns are not evidence of a causal link between durable consump-
tion growth and commodities returns, they do indicate a timing pattern that may
explain the strong correlation between durables and commodities. Commodity portfo-
lios deliver high returns when durable consumption growth is high, which means they
33
have weak hedging properties. According to the model, commodities therefore must
have high returns on average. Furthermore, this analysis provides corroborating ev-
idence that commodities returns have low risk exposure to nondurable consumption
growth and market returns, which may further explain why previous attempts at using
the CAPM or consumption-based CAPM were unsuccessful.
This similar timing of durable consumption growth and commodities returns may
be caused by a basic link between the use of commodities in the production of durable
goods. Another possibility is that these two series have similar behavior over the busi-
ness cycle because each has a cycle that slightly lags that of nondurable consumption,
but for different reasons. For instance, the returns to commodities may be driven by
the level of inventories of commodities. If the timing of this inventory is like that of the
general inventory cycle, which lags the usual business cycle indicators, then commodi-
ties returns will also lag the usual business cycle. If durable consumption growth also
lags the usual business cycle for unrelated reasons, we will see high covariance between
commodities returns and durable consumption growth. A covariance between com-
modities and durables consumption growth that is driven by similar timing rather than
a causal link is still consistent with the model. According to the model, the investor
cares about getting high returns when marginal utility is high. Commodity portfolios
which deliver high returns when durable consumption growth is high must have high
returns on average, because they have weak hedging properties.
9 Conclusion
Using a consumption-based multi-factor asset pricing model, I have demonstrated that
the high returns to commodities can be explained by their high correlation with durable
goods consumption growth. Previous papers have been unable to explain the high
returns to commodities and have also been unable to reconcile their high returns with
their low correlation with nondurable consumption and market returns. In this paper,
I have shown that adding durable consumption growth as a factor can resolve both
issues. Because commodities returns are highly correlated with durable consumption
growth, in the context of the model, the high average returns to commodities provide
compensation for the high durables risk a commodities investor assumes. By including
durable goods as a factor in the model, I am able to measure the strong correlation
34
between commodities and durables and accurately predict the high average returns to
commodities.
In time series regressions, the correlation between commodities returns and durables
results in significant positive coefficients on the durable consumption growth factor.
Grouping commodities by type allows me to diversify some idiosyncratic elements of the
individual commodities futures markets, and better analyze the underlying patterns.
Furthermore, when commodities are sorted into portfolios using known correlates of
commodities returns, including the basis, momentum, and volatility, the coefficient on
durable goods increases with sorted portfolio returns. These results indicate that the
importance of durable goods as a factor is consistent with existing explanations of
commodities returns.
In the cross section, after including durable goods in the household investors’ utility
function, I use the linear factor model to predict average returns to each asset in the
cross section. The results indicate that the three factor model is very successful at
predicting the range of returns across a variety of test assets. Furthermore, the GMM
estimates for the preference parameters of the model indicate that durable goods play
an important role in the utility of household investors. Additionally, by testing the same
cross section of test assets on the nested special cases of the model, I am also able to
demonstrate that inclusion of durable consumption growth as a factor is important for
explaining the cross section of returns, relative to the nondurable consumption growth
and market returns factors.
Finally, by examining commodities returns over the business cycle, I demonstrate
that returns to each commodity type portfolio and the returns to the index of all
commodities follow a pattern similar to that of durables consumption growth. While
nondurables and market returns peak in early expansion, durables and commodity port-
folios returns peak in late expansion. In addition, while nondurables and market returns
trough in early recession, durables and commodity portfolios trough in late recession.
These results begin to indicate one way in which durables growth and commodities re-
turns may be linked, and may suggest a causal explanation for the importance of durable
consumption growth as a factor for explaining the high returns to commodities.
35
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37
Table 1: Test for Unconditional Excess Returns
Commodity Mean Ann. Sharpe Start(% p.a.) Volatility Ratio p-value T Date
Grains 2.93 21.8 0.14 0.31 591 Oct ’59Corn -2.09 24.3 -0.09 0.52 591 Sept ’59Soybeans 5.24 27.8 0.19 0.15 591 Sept ’59Soybean Oil 7.80* 32.3 0.24 0.08 591 Sept ’59Soybean Meal 8.68** 31.1 0.28 0.03 590 Dec ’59Wheat -0.12 25.4 -0.01 0.97 591 Sept ’59Oats -1.22 28.2 -0.04 0.74 590 Dec ’59Rough Rice -0.36 34.6 -0.01 0.96 266 Nov ’86
Softs 3.50 17.9 0.20 0.15 583 June ’60Cocoa 3.91 32.8 0.12 0.38 587 Mar ’60Cotton 1.51 24.6 0.06 0.66 583 Jul ’60Sugar 7.36 48.7 0.15 0.29 572 Jul ’61Orange Juice 4.44 33.1 0.13 0.35 500 May ’67Lumber -1.53 28.4 -0.05 0.72 467 Mar ’70Coffee 7.04 41.4 0.17 0.29 431 Mar ’73
Metal 6.27* 24.2 0.26 0.07 542 Nov ’63Silver 1.70 33.3 0.05 0.72 542 Aug ’63Gold -0.88 19.9 -0.04 0.79 406 Jan ’75Copper 11.52*** 31.3 0.37 0.01 591 Oct ’59Aluminum 4.03 28.6 0.14 0.33 487 Aug ’99Platinum 8.33 38.1 0.22 0.19 377 Jul ’68Palladium 2.21 20.1 0.11 0.72 113 Mar ’77Zinc -2.14 29.7 -0.07 0.74 251 Feb ’88Nickel 12.88 45.4 0.28 0.17 250 Mar ’88Tin 3.00 24.3 0.12 0.60 232 Sept ’89Lead 7.50 31.3 0.24 0.31 210 Jul ’91
Animal 6.16** 18.9 0.33 0.02 526 Mar ’65Pork Bellies 3.12 32.7 0.10 0.47 539 Feb ’64Lean Hogs 7.08** 24.8 0.29 0.04 510 Jul ’66Live Cattle 6.01*** 16.9 0.36 0.01 526 Apr ’65Feeder Cattle 3.72 16.4 0.23 0.14 441 Mar ’72Milk 10.17 32.2 0.32 0.24 149 Apr ’96Butter 9.66 34.4 0.28 0.30 139 Feb ’97
Energy 14.32** 33.2 0.43 0.02 307 June ’83Heating Oil 13.56** 35.7 0.38 0.02 359 Feb ’79Crude Oil 14.12** 35.2 0.40 0.03 307 June ’83Gasoline 20.50*** 35.7 0.57 0.00 262 Feb ’85Propane 22.94*** 49.1 0.47 0.01 254 Dec ’87Natural Gas 8.09 49.0 0.17 0.43 222 June ’90Coal 5.61 35.6 0.16 0.68 87 Sept ’01
All Commodities 5.69*** 13.6 0.42 0.00 591 Oct ’59
Notes: ∗p < 0.10, ∗∗p < 0.05, and ∗∗∗p < 0.01. Excess returns are calculated as(St − Ft−3,t)/Ft−3,t. Volatilities, standard errors, and p-values are calculated usingNewey-West standard errors with 4 lags.
38
Table 2: Test for Conditional Excess Returns
Commodity Mean Coefficients Joint test p-values(% p.a.) T α β H0 : α = 0 & β = 1
GrainsCorn -2.09 590 -0.003 0.891 0.660Soybeans 5.24 590 0.013 0.975 0.308Soybean Oil 7.80* 590 0.018* 0.250*** 0.003***Soybean Meal 8.68** 587 0.021** 0.903 0.107Wheat -0.12 590 0.001 0.864 0.614Oats -1.22 587 0.001 0.710** 0.120Rough Rice -0.36 264 -0.014 1.518*** 0.009***
SoftsCocoa 3.91 584 0.016 0.335** 0.054*Cotton 1.51 580 0.004 0.930 0.677Sugar 7.36 569 0.023 0.645 0.278Orange Juice 4.44 498 0.012 0.663 0.199Lumber -1.53 464 0.002 0.695*** 0.033**Coffee 7.04 428 0.019 0.464** 0.041**
MetalSilver 1.70 542 0.015 0.469 0.341Gold -0.88 406 0.015 0.037 0.579Copper 11.52*** 589 0.020** 0.162*** 0.000***Aluminum 4.03 484 0.018* -0.142** 0.058*Platinum 8.33 376 0.028* -0.018 0.083*Palladium 2.21 111 0.013 -0.569 0.598Zinc -2.14 251 0.002 0.696 0.505Nickel 12.88 250 0.041* 3.201*** 0.000***Tin 3.00 232 0.011 -0.154** 0.078*Lead 7.50 210 0.022 0.101* 0.118
AnimalPork Bellies 3.12 536 0.007 1.118 0.463Lean Hogs 7.08** 507 0.016* 0.914 0.053*Live Cattle 6.01*** 523 0.013** 0.770** 0.000***Feeder Cattle 3.72 438 0.009 0.970 0.331Milk 10.17 147 0.020 0.835 0.213Butter 9.66 136 0.020 0.909 0.706
EnergyHeating Oil 13.56** 357 0.030** 0.788 0.009***Crude Oil 14.12** 306 0.035* 1.036 0.079*Gasoline 20.50*** 261 0.050*** 1.047 0.005***Propane 22.94*** 251 0.064*** 1.260 0.023**Natural Gas 8.09 222 0.024 0.846 0.326Coal 5.61 86 0.014 1.113 0.798
Notes: Table reports results for the Fama Regression: SpotReturnt+3 =α + β(Basist) + εt+3. Stars on the point estimates of α and β indicate significancefor the hypotheses H0 : α = 0 and H0 : β = 1, respectively. The final column reportsthe p-value for the test of the joint hypothesis H0 : α = 0 & β = 1. ∗p < 0.10,∗∗p < 0.05, and ∗∗∗p < 0.01. Standard errors are calculated using Newey-Weststandard errors with 4 lags.
39
Table 3: Portfolio Characteristics
T Excess Return Risk AmountMean Ann. Sharpe Multiple Betas(% p.a.) Volatility Ratio p-value βw βc βd
Panel A: Type PortfoliosGrains 591 2.93 21.75 0.14 0.31 -0.07 -1.07 2.39Softs 583 3.50 17.89 0.20 0.15 0.10 -0.55 2.09**Metal 542 6.27* 24.19 0.26 0.07 0.30*** -0.79 4.80***Animal 526 6.16** 18.91 0.33 0.02 0.06 0.14 3.09**Energy 307 14.32** 33.20 0.43 0.02 -0.12 -5.86* 4.18All Commodities 591 5.69*** 13.58 0.42 0.00 0.07 -1.11 2.96***Panel B: Volatility-sorted PortfoliosLow 573 -2.13 13.88 -0.15 0.19 0.07 -0.73 1.90**2 573 -0.43 12.85 -0.03 0.77 0.07 -0.45 2.85***3 573 0.21 15.30 0.01 0.91 0.05 -0.74 3.62***4 573 7.66*** 19.20 0.40 0.00 0.06 -1.15 3.07**High 573 24.67*** 33.48 0.74 0.00 0.04 -3.60 5.79***Spread (HML) 573 26.80*** 33.40 0.80 0.00 -0.02 -2.86 3.89**Panel C: Basis-sorted PortfoliosLow 569 11.84*** 22.91 0.52 0.00 -0.12 -1.04 3.44**2 569 7.58*** 18.97 0.40 0.00 0.07 -1.23 3.74***3 569 5.03** 18.81 0.27 0.04 0.16* -2.17* 4.69***4 569 2.54 17.69 0.14 0.25 0.12 -1.27 3.37***High 569 1.80 17.73 0.10 0.39 0.02 -0.52 2.09**Spread (LMH) 569 10.04*** 24.71 0.41 0.00 -0.13 -0.52 1.35Panel D: Momentum-sorted PortfoliosLow 559 0.57 20.16 0.03 0.82 0.14 -0.18 1.922 559 4.56** 17.68 0.26 0.05 0.05 -0.85 3.46***3 559 5.78*** 15.85 0.37 0.01 0.03 -1.85** 4.11***4 559 7.30*** 19.91 0.37 0.01 0.07 -2.25* 4.04***High 559 11.88*** 25.84 0.46 0.00 0.02 -1.72 3.43*Spread (HML) 559 11.31*** 28.29 0.40 0.00 -0.11 -1.54 1.51
Notes: ∗p < 0.10, ∗∗p < 0.05, and ∗∗∗p < 0.01. Volatilities, standard errors, and p-values arecalculated using Newey-West standard errors with 4 lags. Reported multiple betas are estimatedusing a separate time series regression for each portfolio: Reit = βi0 +βic∆ct+βid∆dt+βiwrWt+ εit.
40
Tab
le4:
Cro
ssSec
tion
Det
ails
(1)
(2)
(3)
(4)
(5)
(6)
Lon
gShor
tL
ong
Shor
tN
onov
erla
ppin
gT
est
Ass
etSam
ple
Per
iod
Dom
esti
cIn
tern
atio
nal
All
com
modit
ies
Oct
’59-
Dec
’08
XX
Gra
ins
Oct
’59-
Dec
’08
XX
XX
Sof
tsJun
’60-
Dec
’08
XX
XX
Met
alN
ov’6
3-D
ec’0
8X
XX
XA
nim
alM
ar’6
5-D
ec’0
8X
XX
XE
ner
gyJun
’83-
Dec
’08
XX
Bas
is-S
orte
d(5
)A
ug
’61-
Dec
’08
XX
Bas
isSpre
adA
ug
’61-
Dec
’08
XX
Mom
entu
m-S
orte
d(5
)Jun
’62-
Dec
’08
XX
Mom
entu
mSpre
adJun
’62-
Dec
’08
XX
Vol
atilit
y-S
orte
d(5
)A
pr
’61-
Dec
’08
XX
Vol
atilit
ySpre
adA
pr
’61-
Dec
’08
XX
Fam
a-F
rench
Equit
yP
ortf
olio
s(6
)Jan
’59-
Dec
’08
XX
XX
XSM
B,
HM
LSpre
adP
ortf
olio
s(2
)Jan
’59-
Dec
’08
XIn
dust
ry-S
orte
dE
quit
yP
ortf
olio
s(5
)Jan
’59-
Dec
’08
XX
XX
XB
ond
Por
tfol
ios
(6)
Jan
’59-
Dec
’08
XX
XX
XX
Inte
rnat
ional
Equit
yIn
dex
Mar
’75-
Dec
’07
XX
XX
Val
ue-
Sor
ted
Inte
rnat
ional
Index
es(2
)M
ar’7
5-D
ec’0
7X
XX
Curr
ency
Por
tfol
ios
(6)
Jan
’84-
Dec
’08
XX
XC
urr
ency
Spre
adJan
’84-
Dec
’08
XN
um
ber
ofte
stas
sets
2131
3646
2321
Num
ber
ofco
mm
odit
yp
ortf
olio
s4
519
204
4Sta
rtD
ate
Mar
’65
Jan
’84
Mar
’65
Jan
’84
Jan
’84
Jan
’84
End
Dat
eD
ec’0
8D
ec’0
7D
ec’0
8D
ec’0
7D
ec’0
7D
ec’0
7Sam
ple
lengt
h(m
onth
s)52
628
852
628
896
96
41
Table 5: Cross-sectional results using two-step GMM
(1) (2) (3) (4)Type Portfolios Sorted Commodity Portfolios
Long Sample Short Sample Long Sample Short SampleMarket Returns -0.161 0.343*** -0.088 0.518***
(0.505) (0.107) (0.145) (0.129)Nondurable Growth 110.509*** -23.810*** 14.182** -51.962***
(23.149) (7.369) (6.199) (3.061)Durable Growth 57.327*** 72.432*** 55.291*** 60.470***
(17.695) (5.010) (5.756) (2.188)EIS (σ) 0.007** 0.014*** 0.016*** 0.057***
(0.003) (0.002) (0.002) (0.021)Risk Aversion (γ) 167.675*** 48.964*** 69.384*** 9.026***
(21.672) (4.953) (7.080) (2.870)Durables Weight (α) 0.345 1.516 0.812 7.656
(16.377) (9.053) (5.147) (18.970)Portfolios (N) 21 31 36 46Commodity portfolios 4 5 17 20Sample Length (T) 526 288 526 288R-squared 0.876 0.934 0.852 0.957MAE (%) 0.869 0.552 0.819 0.391Commodities MAE (%) 1.221 0.693 1.106 0.423J-test 7.932 3.081 18.446 17.746(p-value) (0.980) (1.000) (0.981) (1.000)
Notes: ∗p < 0.10, ∗∗p < 0.05, and ∗∗∗p < 0.01. Standard errors are calculated using the VARHACprocedure and automatic lag length selection using the AIC. R-squared values and mean absolutepricing errors (MAE) are calculated using first stage estimates for the N test assets. J-test iscalculated using second stage estimates for the N test assets. Because α and ρ cannot be separatelyidentified, I follow Yogo (2006) and use ρ = 0.79 for parameter calculations. Long sample coversMarch 1965 to December 2008. Short sample covers January 1984 to December 2007. See Table 4for list of test assets included in each cross sectional estimation.
42
Tab
le6:
Com
par
ison
wit
hB
ench
mar
kM
odel
s
(1)
(2)
(3)
(4)
(5)
(6)
(7)
Model
:C
AP
MC
CA
PM
DC
AP
ME
Z-C
CA
PM
CD
-CA
PM
EZ
-DC
AP
MF
ama-
Fre
nch
Fac
tors
:W
CD
W,
CC
,D
W,
C,
DW
,SM
B,H
ML
Mar
ket
Ret
urn
s3.
809*
**–
–-0
.533
–0.
518*
**5.
633*
**(0
.288
)–
–(0
.463
)–
(0.1
29)
(0.1
64)
Non
dura
ble
Gro
wth
–18
1.68
9***
–22
1.69
8***
-5.0
04*
-51.
962*
**SM
B:
1.99
5***
–(7
.871
)–
(9.3
90)
(2.8
54)
(3.0
61)
(0.1
65)
Dura
ble
Gro
wth
––
58.0
85**
*–
75.0
11**
*60
.470
***
HM
L:
0.81
5–
–(1
.370
)–
(2.1
70)
(2.1
88)
(0.1
56)
EIS
(σ)
∞0.
006*
**0.
017*
**0.
007*
**0.
014*
**0.
057*
**–
–(0
.000
)(0
.000
)(0
.002
)(0
.000
)(0
.021
)–
Ris
kA
vers
ion
(γ)
3.80
9***
181.
689*
**58
.085
***
221.
166*
**70
.006
***
9.02
6***
–(0
.288
)(7
.871
)(1
.370
)(9
.478
)(2
.349
)(2
.870
)–
Dura
ble
sW
eigh
t(α
)–
01
01.
091
7.65
6–
––
––
(2.8
81)
(18.
970)
–P
ortf
olio
s(N
)46
4646
4646
4646
Com
modit
yp
ortf
olio
s20
2020
2020
2020
Sam
ple
Len
gth
(T)
288
288
288
288
288
288
288
R-s
quar
ed0.
402
0.82
50.
985
0.87
90.
977
0.99
10.
915
MA
E(%
)2.
572
1.32
20.
457
1.28
90.
584
0.39
10.
984
Com
modit
ies
MA
E(%
)3.
511
1.80
10.
424
1.29
80.
523
0.42
30.
684
J-t
est
36.0
0454
.553
5.02
755
.955
5.61
417
.746
31.5
94(p
-val
ue)
(0.8
29)
(0.1
56)
(1.0
00)
(0.1
07)
(1.0
00)
(1.0
00)
(0.9
01)
Not
es:
∗ p<
0.10
,∗∗p<
0.05
,an
d∗∗
∗ p<
0.0
1.
Sam
ple
per
iod
isJanu
ary
1984
toD
ecem
ber
2007.
See
Tab
le4,
colu
mn
6fo
rli
stof
test
asse
tsin
clu
ded
inth
ecr
oss
sect
ion
.E
stim
atio
nan
dpre
fere
nce
para
met
erre
stri
ctio
ns
are
as
des
crib
edin
Sec
tion
6.1
:(1
)C
CA
PM
:b 1
=γ
,b 2
=b 3
=0,σ
=1/γ
=ρ
andα
=0.
(2)
DC
AP
M:b 2
=γ
,b 1
=b 3
=0,σ
=1/γ
=ρ
an
dα
=1.
(3)
CA
PM
:b 3
=γ
,b 1
=b 2
=0,σ
=ρ,
an
dσ→∞
.(4
)E
Z-C
CA
PM
:κ≡
(1−γ
)/(1−
1 σ),b 1
=κ/σ
,σ
=ρ,b 2
=0,
an
db 3
=1−κ
.(5
):C
D-C
AP
M:σ
=1/γ
,b 1
=γ
+α
(1/ρ−γ
),b 2
=α
(γ−
1/ρ),b 3
=0,
andρ
=0.
79.
(6)
EZ
-DC
AP
M:b 1
=κ
[1/σ
+α
(1/ρ−
1/σ
)],b 2
=κα
(1/σ−
1/ρ),b 3
=1−κσ
,an
dρ
=0.
79.
See
also
not
esto
Tab
le5.
43
Table 7: Risk Prices over Time
(1) (2) (3)Early Sample Late Sample Full Sample
Jun.’62-Dec.’83 Jan.’84-Dec.’08 Jun.’62-Dec.’08Market Returns 0.545*** -0.081 -0.088
(0.054) (0.101) (0.145)Nondurable Growth -3.840*** -36.038*** 14.182**
(1.133) (5.971) (6.199)Durable Growth 38.138*** 62.001*** 55.291***
(1.182) (3.255) (5.756)EIS (σ) 0.013*** 0.042*** 0.016***
(0.002) (0.010) (0.002)Risk Aversion (γ) 34.843*** 25.882*** 69.384***
(1.735) (6.613) (7.080)Durables Weight (α) 1.131 2.521 0.812
(1.292) (13.705) (5.147)Portfolios (N) 36 36 36Commodity portfolios 19 19 19Sample Length (T) 226 300 526R-squared 0.973 0.984 0.959MAE (%) 0.930 0.446 0.819Commodities MAE (%) 1.247 0.498 1.106J-test 6.430 19.265 18.446(p-value) (1.000) (0.973) (0.981)
Notes: ∗p < 0.10, ∗∗p < 0.05, and ∗∗∗p < 0.01. See Table 4, Column 3 for list of test assetsincluded in the cross section. See also notes to Table 5.
44
Table 8: Results using nonoverlapping quarters
(1) (2) (3) (4)Type Portfolios Sorted Commodity Portfolios
Long Sample Long Sample Short Domestic Short InternationalMarket Returns 0.342 0.170 1.000*** 0.466
(0.323) (0.152) (0.168) (0.457)Nondurable Growth -58.875*** -51.363*** -44.116*** -84.887***
(11.763) (6.158) (6.204) (16.435)Durable Growth 107.448*** 59.302*** 86.956*** 114.277***
(14.430) (4.108) (4.242) (7.898)EIS (σ) 0.014** 0.105* 0.000 0.018
(0.006) (0.061) (0.004) (0.020)Risk Aversion (γ) 48.915*** 8.109* 43.840*** 29.856
(14.725) (4.459) (6.535) (19.699)Durables Weight (α) 2.251 8.609 2.030 3.980
(24.850) (29.455) (12.106) (71.298)Portfolios (N) 21 36 23 21Commodity portfolios 4 19 4 4Sample Length (T) 176 176 96 96R-squared 0.876 0.934 0.852 0.957MAE (%) 0.869 0.552 0.819 0.391Commodities MAE (%) 1.221 0.693 1.106 0.423J-test 7.932 3.081 18.446 17.746(p-value) (0.980) (1.000) (0.981) (1.000)
Notes: ∗p < 0.10, ∗∗p < 0.05, and ∗∗∗p < 0.01. Long sample covers March 1965 to December 2008.Short sample covers January 1984 to December 2007. See Table 4 for list of test assets included ineach cross sectional estimation. See also notes to Table 5.
45
Table 9: Cross-sectional results using OLS
(1) (2) (3) (4)Model: EZ-DCAPM EZ-DCAPM EZ-DCAPM EZ-CCAPMFactors: W, C, D W, C, D W, C, D W, CMarket Returns 1.995*** -6.311** 1.215*** 2.291***
(0.324) (2.245) (0.369) (0.346)Nondurable Growth -0.581*** -0.216 -0.848*** -0.495***
(0.128) (0.214) (0.298) (0.139)Durable Growth 0.223 0.357 -0.808 –
(0.142) (0.221) (0.329) –Portfolios (N) 46 20 25 46Commodity Portfolios 20 20 0 20Sample Length (T) 288 288 288 288R-squared 0.216 0.510 -0.632 0.013MAE (%) 3.271 2.494 2.891 3.618Commodities MAE (%) 3.284 2.494 – 3.760
Notes: Table reports OLS estimates of the risk prices (multiplied by a factor of 100) from the
regression Reit = λcβ̂ic + λdβ̂
id + λwβ̂
iw + εi.
∗p < 0.10, ∗∗p < 0.05, and ∗∗∗p < 0.01. Sample period isJanuary 1984 to December 2007. See column 4 of Table 4 for list of test assets included in the crosssection for column 1. Of these, only the commodity portfolios are included in column 2. Column3 uses the 25 Fama-French portfolios sorted on size and book-to-market values, and column 4 usesthe same cross section as column 1, but excludes the durable consumption growth factor.
46
Tab
le10
:O
LS
Out
ofSam
ple
Est
imat
ion
(1)
(2)
(3)
(4)
(5)
(6)
Full
Sam
ple
Ear
lySam
ple
Bet
asE
arly
Sam
ple
Ris
kP
rice
sO
LS
Est
imat
ion
Lat
eSam
ple
Ris
kP
rice
sL
ate
Sam
ple
Bet
asA
llp
ortf
olio
sC
omm
odit
ies
All
por
tfol
ios
Com
modit
ies
All
por
tfol
ios
Com
modit
ies
Mar
ket
Ret
urn
s1.
995*
**-6
.311
**1.
921*
**-4
.505
***
2.36
0***
-2.7
44**
(0.3
24)
(2.2
45)
(0.3
75)
(1.0
26)
(0.3
24)
(2.2
45)
Non
dura
ble
Gro
wth
-0.5
81**
*-0
.216
-0.4
60**
*0.
019
-0.2
96**
*-0
.045
(0.1
28)
(0.2
14)
(0.1
02)
(0.0
94)
(0.1
28)
(0.2
14)
Dura
ble
Gro
wth
0.22
30.
357
0.25
0**
1.22
3***
0.27
10.
193
(0.1
42)
(0.2
21)
(0.1
10)
(0.2
11)
(0.1
42)
(0.2
21)
Por
tfol
ios
(N)
4620
4620
4620
Com
modit
yP
ortf
olio
s20
2020
2020
20Sam
ple
Len
gth
(T)
288
288
288
288
288
288
R-s
quar
ed0.
216
0.51
00.
014
0.75
0-1
.800
-2.1
80M
AE
(%)
3.27
12.
494
4.40
22.
348
6.75
59.
307
Com
modit
ies
MA
E(%
)3.
284
2.49
45.
373
2.34
89.
803
9.30
7
Not
es:
Tab
lere
por
tsO
LS
esti
mat
esof
the
risk
pri
ces
(mu
ltip
lied
by
afa
ctor
of
100)
from
the
regre
ssio
nRe it
=λcβ̂i c
+λdβ̂i d
+λwβ̂i w
+ε i
.∗ p<
0.10
,∗∗p<
0.05
,an
d∗∗
∗ p<
0.01
.S
eeco
lum
n4
of
Tab
le4
for
list
of
test
ass
ets
incl
ud
edin
the
cross
sect
ion
for
colu
mn
s1,
3,
an
d5.
Of
thes
e,on
lyth
eco
mm
od
ity
por
tfol
ios
are
incl
ud
edin
colu
mn
s2,
4,
an
d6.
Fu
llsa
mp
lep
erio
dis
Janu
ary
1984
toD
ecem
ber
2007,
earl
ysa
mp
leis
Jan
uar
y19
84to
Dec
emb
er19
95,
and
late
sam
ple
isJanu
ary
1996
toD
ecem
ber
2007.
Colu
mn
s1
an
d2
pro
vid
efu
llsa
mp
lees
tim
ate
s.C
olu
mn
s3
and
4re
gres
sla
tesa
mp
leav
erag
ere
turn
son
earl
ysa
mp
leti
me
seri
esb
etas.
Colu
mn
s5
and
6re
port
risk
pri
ces
esti
mate
du
sin
gea
rly
sam
ple
tim
ese
ries
bet
asan
dea
rly
sam
ple
aver
age
retu
rns.
Th
ese
earl
ysa
mp
leri
skp
rice
sare
use
dw
ith
late
sam
ple
tim
ese
ries
be-
tas
top
red
ict
aver
age
retu
rns.
Mea
sure
sof
fit
com
pare
thes
ep
red
icte
dre
turn
sw
ith
act
ualav
erage
retu
rns.
See
Sec
tion
7.2
for
furt
her
det
ail
s.
47
Table 11: GMM Estimation for Out of Sample Predictions
(1) (2) (3)25 Fama-French Portfolios Broad Cross Section
Jan.’59-Dec.’83 Jan.’59-Dec.’08 Jun.’62-Dec.’83Market Returns 0.543 -0.412 0.545***
(0.494) (0.790) (0.054)Nondurable Growth -12.552 -64.332** -3.840***
(12.430) (32.753) (1.133)Durable Growth 112.094*** 206.517*** 38.138***
(6.821) (18.800) (1.182)EIS (σ) 0.005 0.010* 0.013***
(0.005) (0.006) (0.002)Risk Aversion (γ) 100.084*** 141.773*** 34.843***
(13.064) (31.837) (1.735)Durables Weight (α) 1.133 1.471 1.131
(13.632) (44.292) (1.292)Portfolios (N) 25 25 36Commodity portfolios 0 0 19Sample Length (T) 300 600 226R-squared 0.946 0.933 0.973MAE (%) 0.687 0.571 0.930Commodities MAE (%) – – 1.247J-test 12.508 24.017 6.430(p-value) (0.946) (0.346) (1.000)
Notes: ∗p < 0.10, ∗∗p < 0.05, and ∗∗∗p < 0.01. Columns 1 and 2 use the 25 Fama-French portfoliossorted on size and book-to-market values. Column 3 uses the assets included in column 3 of Table4. See also notes to Table 5.
48
Table 12: Out of Sample Predictions
Panel A: 25 Fama-French Early Sample Risk Prices(1) (2)
All Out of Sample Assets Commodity PortfoliosPortfolios (N) 40 20Commodity Portfolios 20 20Sample Length (T) 288 288R-squared 0.605 0.731MAE (%) 2.330 –Commodities MAE (%) 1.867 1.867J-test 30.383 29.535(p-value) (0.771) (0.030)
Panel B: 25 Fama-French Full Sample Risk Prices(1) (2)
All Out of Sample Assets Commodity PortfoliosPortfolios (N) 40 20Commodity Portfolios 20 20Sample Length (T) 288 288R-squared 0.670 0.571MAE (%) 2.061 –Commodities MAE (%) 2.346 2.346J-test 39.196 21.668(p-value) (0.372) (0.198)
Panel C: Broad Cross Section Early Sample Risk Prices(1) (2)
All Out of Sample Assets Commodity PortfoliosPortfolios (N) 40 20Commodity Portfolios 20 20Sample Length (T) 288 288R-squared 0.983 0.973MAE (%) 0.486 –Commodities MAE (%) 0.542 0.542J-test 64.418 32.012(p-value) (0.003) (0.015)
Table reports measures of fit comparing actual average returns to returns predicted usingestimated parameters reported in Table 11. Assets included in the cross section include allthose listed in column 4 of Table 4, except for the 6 Fama-French equity portfolios whichare excluded due to their similarity with the 25 Fama-French benchmark portfolios. SeeSection 7.3 for further details.
49
Table 13: Business Cycle Characteristics
Expansion Recession Mean Start TEarly Late Early Late (% p.a.) Date
Market Wealth 12.42 5.26 -26.45 -6.07 4.94 Jan ’59 600Nondurables 2.42 2.27 -0.67 2.12 2.08 Jan ’59 600
Durables 3.76 4.93 1.82 1.01 3.94 Jan ’59 600Grains 2.44 6.60 -8.90 1.88 3.24 Oct ’59 591
Softs 4.79 6.47 -3.95 -15.22 3.50 June ’60 583Metal 3.11 15.71 5.50 -45.85 6.09 Nov ’63 542
Animal 5.35 11.01 -14.18 2.41 6.16 Mar ’65 526Energy 10.03 20.02 42.68 -73.81 14.32 June ’83 307
Commodity Index 4.58 10.26 -2.31 -11.55 5.44 Oct ’59 591
Table reports average return over all months in each phase of the business cycle for each factor
or commodity portfolio. Bold text indicates peak or trough of each series.
50
Figure 1: Spot Price and Market Returns Indexes
Figure 2: Commodities Index and Factor Returns
(a) Commodities and Market Returns
(b) Nondurable and Durable Consumption Growth
51
Figure 3: GMM Cross Sectional Results
(a) Type Portfolios, Long Sample (b) Type Portfolios, Short Sample
(c) Sorted Portfolios, Long Sample (d) Sorted Portfolios, Short Sample
Notes: Each figure presents results for the corresponding column of Table 5. Figures graph therealized return to each portfolio against the predicted return according to two-step GMM estimation.See Table 4 for detailed list of test assets included in each cross section.
52
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54
Figure 6: OLS Cross Sectional Results
(a) All Test Assets (b) Commodities Only
(c) 25 Fama-French Portfolios (d) Using only C,W as factors
Notes: Each figure presents results for the corresponding column of Table 9. Figures graph therealized return to each portfolio against the predicted return according to the estimation of thespecified model.
55
Figure 7: OLS Out of Sample Results
(a) Early Sample Betas, Late Sample Risk PricesAll Test Assets
(b) Early Sample Betas, Late Sample Risk PricesCommodities Only
(c) Early Sample Risk Prices,Late Sample Betas, All Test Assets
(d) Early Sample Risk Prices,Late Sample Betas, Commodities Only
Notes: Each figure presents results for the corresponding column of Table 10. Figures graph therealized return to each portfolio against the predicted return according to the estimation of thespecified model.
56
Figure 8: GMM Out of Sample Results
(a) FF25 Early Sample Risk Prices,All Test Assets
(b) FF25 Early Sample Risk Prices,Commodities Only
(c) FF25 Full Sample Risk Prices,All Test Assets
(d) FF25 Full Sample Risk Prices,Commodities Only
(e) Broad Cross SectionEarly Sample Risk Prices, All Test Assets
(f) Broad Cross SectionEarly Sample Risk Prices, Commodities Only
Notes: Each figure presents results for the corresponding column of Table 12. Figures graph therealized return to each portfolio against the predicted return according to the estimation of thespecified model.
57
Appendix A Commodities Data
This appendix provides summary statistics and data description for individual com-
modities and portfolios of commodities aggregated by type. Data comes from the Com-
modities Research Bureau (CRB), which provides daily prices on each contract from its
opening to its expiration. Because the CRB covers commodities futures markets that
are still in existence today, the set of included commodities is subject to survivor bias.
However, as noted by Gorton and Rouwenhorst (2006), commodities markets fail due
to lack of interest, unlike equities which usually fail due to poor performance and low
returns. Interest in a given commodities futures market is not necessarily correlated
with returns, since a negative return on a long futures position implies that the corre-
sponding short investor earns positive returns. In fact, while the futures markets for
corn and wheat have consistently yielded very low or even negative returns, they are
two of the oldest and most liquid markets.
Table A.1 lists individual futures contract coverage, including the exchange the
contract is traded on, the start and end date for contract availability, available contract
months, and number of monthly observations. To be consistent with related research,
this paper includes the set of CRB commodities with generally large production and
trading volumes (Gorton and Rouwenhorst, 2006; Gorton et al., 2007; Hong and Yogo,
2010). Following Gorton and Rouwenhorst (2006), I add to the set of commodities
futures 4 metal contracts from the London Metals Exchange. Since these commodities
are traded as 3-month forward contracts, their monthly prices are not extracted as
described in Section 2. Instead, for zinc, nickel, tin, and lead, the 3-month forward
price is used for the 3-month futures price, and the spot price is taken from the cash
forward price, both provided by CRB.
Table A.2 lists for each commodity the mean percent per annum and annualized
volatility for the excess return, spot return, and basis at the 3-month horizon. Excess
returns vary substantially across commodities, ranging from 22.9% for gasoline futures
to -2.1% for zinc futures. Volatility of the excess return can also be quite different across
commodities, ranging from 16.4% for feeder cattle to 49.1% for propane. The implied
Sharpe ratio for these assets, which is the ratio of mean return to volatility often used
to measure the return/risk tradeoff, ranges from -0.09 for corn to 0.57 for gasoline.
The difference between the contemporaneous futures and spot price (as measured by
the basis) also differs quite a bit across commodities, ranging from -9.3% for gasoline to
15.1% for natural gas. The volatility of the basis has a slightly smaller range than the
58
volatility of the excess returns, with a minimum of 1.8% for aluminum and a maximum
of 31.2% for natural gas. This is consistent with the observation by Choe (1990) that
“[f]utures prices tend to move more closely with current spot prices than with the
expected futures price.” Indeed, we can see in Table A.2 that the volatility of the excess
returns is greater than the volatility of the basis for every commodity. Furthermore, the
volatility of the spot return is similar to that of the excess return, since both measures
describe the behavior of prices over the three month horizon. Feeder cattle again has
the minimum spot return volatility at 17.7%, while natural gas has the maximum of
56.8%. Finally, the lowest spot return stands at 3.8% for aluminum, while the highest
is 22.4% for natural gas.
Table A.1 indicates the type and start date for individual commodities, including
grains, softs, metals, animal, and energy. Using these classifications, I construct six
equally-weighted type portfolios: one for each of the five types and one index for all
commodities. Because individual futures markets start at different times, commodities
enter and exit the portfolios based on data availability, and all portfolios start when
returns for at least two component commodities are available.
Table A.3 lists summary statistics for these commodity type portfolios. Of the
5 type portfolios, energy commodities unsurprisingly have the highest average excess
return, at 14.3%, while grains have the lowest average excess returns, at 3.2%. The
full index of commodities has returns of 5.4%. Given the high volatility of individual
energy commodities and high correlation across these contracts, it is unsurprising that
the annualized volatility of the energy portfolio is the highest, at 33.2%. In contrast,
Table A.3 shows the pairwise correlations among the type portfolios, the highest of
which is 0.35. The relatively low correlation across the full set of commodities reduces
the volatility of the index of all commodities to 13.5%.
It is interesting to note that the average returns to each type portfolio is decreasing
in the number of observations available for that type portfolio. If commodities returns
have changed over time, this pattern could be caused by low returns in the early years of
the grains and softs commodities markets, when other markets had not yet formed. To
investigate this possibility, I provide summary statistics for commodities type portfolios
for an early and late sample period in Panels A2 and A3 of Table A.5. Panel A3 indicates
that the ordering of the portfolios by average returns remains the same even when the
sample is limited to the later period. Additionally, Panel A2 indicates that the returns
to a given type portfolio are probably not constant over time, as every portfolio has
much higher returns in the early period. However, these results indicate that the high
59
returns to the full index of commodities over the entire sample period is neither driven
solely by the later period nor driven solely by the extremely high returns to the energy
portfolio.
Appendix B Sorted Commodity Portfolios
This section will describe the construction of the sorted commodity portfolios. I con-
struct three sets of sorted portfolios, using the basis, the returns momentum, and the
spot price volatility as the sorting characteristics. Each set of portfolios includes 5
sorted portfolios, and a sixth portfolio that measures the return to the spread of the
sorted portfolios by taking the return to the high return portfolio minus the low return
portfolio. In addition, each set of portfolios begins with data on the sorting variable is
available for at least 10 commodities.
The first set of portfolios is sorted on the basis. The sample period for this set of
portfolios begins in August 1961. As in Section 2, I calculate the basis as Basist ≡Ft,t+h−St
St. Before sorting the commodities, I demean the monthly basis for each com-
modity using the mean basis from the previous 60 months.13 Therefore, a commodity
in the “high basis” portfolio has a high basis relative to the average basis value for
that commodity, not relative to other commodities. For a given month, I take all com-
modities for which the basis is available and order them from smallest to largest. The
commodities with basis values in the lowest quintile are put into the first portfolio, the
second quintile go into the second portfolio and so on. I calculate the returns to these
portfolios over the next three months as the mean percentage return to all commodities
in a given portfolio.
The second set of portfolios is sorted on the spot price volatility and begins in
April 1961. To measure recent volatility, I calculate for the end of each month the
three month coefficient of variation, which equals the variance of the daily spot prices
over the previous three months divided by the mean of these daily spot prices. Again,
I subtract the mean coefficient of variation for each commodity from its previous 60
monthly values. Next, I sort the commodities and measure returns exactly as done
with the basis portfolios. The final set of portfolios is sorted on the returns momentum
and begins in June 1962. Since the measure of momentum is the mean of the previous
12 months of returns, the sorting data is available for these portfolios beginning a bit
13When fewer than 60 previous observations are available, I subtract the mean of the first 60 obser-vations instead.
60
later than for the other sets of portfolios. Again, I subtract the moving average of the
momentum measure for each commodity from the monthly observations before sorting
the commodities into portfolios and measuring returns over the next three months.
Because the commodities are being re-sorted monthly but are measured in overlap-
ping quarters, the actual trading strategy that would reflect these returns would require
three sets of 5 portfolios each for each of the sorting variables. The first set would be
resorted in January, April, July, and October, and the second and third sets would be
offset by one and two months. The returns to my portfolios equal the average return
to these three sets of portfolios.
To learn about the returns to these sorted portfolios, we can examine the average
excess returns as well as the pattern of betas across the five portfolios. However,
in order for these properties to reflect the returns to the sorting characteristic (low
basis, high momentum, etc), we require variation in the composition of each portfolio
so that the results are not being driven by persistent cross sectional differences in
the individual commodities in the sample. One way to check the amount of portfolio
switching is to examine whether commodities move across portfolios over time. In
my sample, virtually all the commodities are included in every one of the 5 basis
portfolios, 5 momentum portfolios, and 5 volatility portfolios for at least one month
during the sample (the only exceptions being aluminum and gold, which never reach
the highest basis portfolio). While this is an extreme level of portfolio traveling, it
occurs almost by construction. First, the measures of volatility, momentum, and basis
are either measured as percentages or are otherwise independent of the price levels of
the commodities. In addition, because I subtract the mean values from the monthly
observations, each commodity is likely to have a basis (or momentum or volatility) that
is high or low relative to its recent history at some point in the sample.
Another way to measure the amount of variation across portfolios is to measure the
average frequency of portfolio switches per month (Lustig et al., 2008). The monthly
switch frequency is measured as the number of commodities leaving their assigned
portfolio at the end of the month for another portfolio, divided by the number of com-
modities in all portfolios that month. I average over these observations to get the
average frequency of portfolio switches for each set of portfolios. The sample period for
the portfolios covers the months when there are at least 10 commodities available to be
sorted in the 5 portfolios. Commodities are added to the sample as they become avail-
able, and the sample reaches a maximum of 35 commodities. On average, the volatility
sorted portfolios each have 13.7 portfolio switches out of 24.6 commodities each month,
61
and an average frequency of 0.55, which implies that just over half the commodities
move portfolios each month. The corner portfolios are a bit more persistent; the average
switching frequency for each portfolio from low to high is 0.40, 0.60, 0.66, 0.64, 0.44.
The numbers for the basis sorted portfolios are very similar, as there are 12.8 port-
folio switches out of 24.6 commodities each month, and an average frequency of 0.52.
Again, there is a bit more persistence in the corners, as the average switching frequency
across portfolios is: 0.39, 0.59, 0.62, 0.58, 0.41. Finally, the momentum portfolios are
unsurprisingly a bit more persistent but still have 6.0 portfolios switches out of 24.3
commodities each month, and an average frequency of 0.25. The average frequencies
for the low to high portfolios are: 0.14, 0.29, 0.34, 0.33, 0.14.
Given that the motivation for using these three sorting variables draws upon a
closely related set of theories, it is important to know how correlated the three sets of
portfolios are. To check the joint distribution of portfolios across the various pairs of
sorting variables, Table A.7 reports the percent of commodities that have a given sorted
portfolio combination. For example, the top left cell of Panel A reports the percent of
all monthly observations of commodities that fall in the high basis portfolio and the
low volatility portfolio. If the two sorting distributions were completely independent,
we would expect to see an equal distribution of commodities across all cells in a given
panel, with a frequency of about 4%.
Unsurprisingly, the chi-squared test of the hypothesis that the rows and columns
are independent rejects the hypothesis for each panel. We can see that the commodi-
ties in the high basis portfolio are slightly more likely to have a low volatility, while
commodities in the low basis portfolio are likely to have a high volatility. However,
we can see that in Panel A, the frequencies are still generally close to 4%. The most
obvious deviation from independent sorting is in the extreme portfolios of Panel C,
which show that high basis commodities are very likely to have low momentum, and
vice versa. These results indicate that the basis and momentum sorting reflect very
similar properties of commodities. However, the volatility sort appears to contain some
amount of new information relative to the basis and momentum sorting.
Appendix C Factor Data
The market portfolio factor is calculated by subtracting the risk-free rate (the one-
month Treasury bill rate from Ibbotson Associates) from the return on a value-weighted
62
portfolio of all NYSE, AMEX, and NASDAQ stocks obtained from CRSP via the French
Data Library (French, 2010). Calculation of nondurable and durable consumption
growth uses data from the National Income and Product Accounts Tables available
from the U.S. Bureau of Economic Analysis, and closely follows the procedure detailed
by Yogo (2006) with modifications made only for the calculation of the monthly rather
than quarterly series. Nondurable consumption includes food, clothing, housing, util-
ities, transportation, and medical care. Durable consumption includes motor vehicles,
furniture, appliances, electronics, and jewelry. For nondurable consumption growth,
because monthly data for real personal consumption expenditures by major type of
product are available in chained dollar estimates starting only in 1995 (NIPA Table
2.8.6), I calculate this series by scaling the data series expressed in quantity indexes
found in NIPA Table 2.8.3 by the 2000 current dollar values for nondurable goods and
services. These values are then divided by monthly population estimates from the U.S.
Census Bureau.14
The series for durable consumption captures not just the monthly expenditure on
durable goods (Et), but the monthly flow of services from new and existing durable
goods:
Dt = (1− φ)Dt−1 + Et, (20)
To calculate this series, I first calculate a baseline chained 2000-dollar estimate of the
net stock of fixed asset and consumer durable goods in the year before the data series
begins (1958) by taking the quantity index number for this data point from NIPA
Table 1.2 and multiplying by the net stock of consumer durable goods at current cost
in 2000, from NIPA Table 1.1. Next, I calculate the chained dollar estimates of real
personal consumption expenditure on durable goods as above, by scaling the data series
expressed in quantity indexes found in NIPA Table 2.8.3 by the 2000 current dollar
values for durable goods and services found in NIPA Table 2.3.5. Finally, I calculate
the monthly flow of services from durable goods according to Equation 20. In order to
adjust for the monthly calculation of this series, I apply a depreciation rate of φ = 2%
per month in lieu of Yogo’s 6% per quarter. As with nondurables, these values are
divided by the monthly population estimates to get a measure of durable goods per
capita.
14Outside this paragraph, “nondurable consumption” refers to personal consumption expenditureon nondurable goods and services.
63
Table A.9 presents the levels and shares of expenditure and consumption of durable
and nondurable goods. While the flow of services from nondurable goods and services
is equivalent to the amount purchased in a given period (ct), the flow of services from
durable goods (dt) depends on present and past durable goods expenditures. In 2008,
the share of nominal personal consumption expenditure (PCE) on nondurable goods and
services (ct) was about 89% ($29,700 per capita), while only 11% ($3,600 per capita) was
spent on durable goods (et). Using nominal expenditure data, these expenditure shares
have changed little since the beginning of the sample period. In contrast, the shares of
the flow of services from durable and nondurable goods have changed over time, with
an increasing share of consumption coming from durable goods. In 2008, the share of
durable goods consumption was 39% ($14,600 per capita in chained (2000) dollars),
while the share of nondurable goods and services consumption was 61% ($23,300 per
capita). In 1959, these shares were 20% for durables ($2,100 per capita) and 80% for
nondurables ($8,300 per capita).
The monthly growth in per capita nondurable consumption expenditure is calculated
as the percentage change, or log difference in this series, using the “beginning of period”
timing convention as in Yogo (2006).As with nondurables, these values are divided by
the monthly population estimates before the monthly growth in durable consumption is
calculated as the percentage change in this series. Finally, in order to make the returns
on each of these factors comparable to the returns on holding commodities for the 3-
month horizon, for each of the three factors (market returns, nondurable consumption
growth, and durable consumption growth), monthly returns were compounded so that
each monthly observation reflects the returns over the 3-month period ending that
month. As with commodity returns, using monthly observations of quarterly returns
results in overlapping returns and high serial correlation.
Summary statistics for the three factors are presented in Table A.4. While all
factors have means between 2 and 5%, the volatility of the market return is 15.9%,
which is much higher than the volatility of 1.1% for each of the consumption variables.
Largely stemming from the overlapping windows, all factors have relatively high first-
order autocorrelation, ranging from 0.61 to 0.98. In addition, because the durable
goods series is constructed as a continuous flow of services that depends heavily on
past expenditure, the autocorrelation of durable consumption is extremely high even
at the quarterly level, at 0.91.
64
Appendix D Test Assets
The cross sectional regression includes a wide set of test assets in addition to the
commodity portfolios of various types, so that the risk prices estimated can be based
on a broad set of asset choices. All equity-based test assets can be found in the French
Data Library (French, 2010). Data for currency portfolios were obtained from (Lustig et
al., 2010). Data and descriptions of bond portfolios were obtained from CRSP. Because
all of these data are reported as monthly returns, like the factor data, they must be
converted to overlapping quarters to match commodities returns. Monthly observations
are compounded over 3 month periods, such that the observation for a given month
reflects the compounded return over the three months leading up to and including that
month.
The first set of test assets is the standard set of six Fama-French value-weighted
equity portfolios, which sort all NYSE, AMEX, and NASDAQ stocks by size and book-
to-market value. Stocks are sorted into the Small or Big portfolios depending on whether
their market equity is below or above the median market equity of the NYSE stocks.
Using breakpoints that are the 30th and 70th percentile of book-to-market values for
NYSE stocks, the stocks are also sorted into Growth, Neutral, or Value portfolios de-
pending on whether their book-to-market value (ratio of book equity to market equity)
is low, medium, or high. The six Fama-French portfolios are formed as the intersection
of these two sorting criteria.
These six Fama-French portfolios are used to generate the standard Fama-French
factor portfolios that measure the returns to the size and value premium. The SMB
portfolio measures the average return on the three small portfolios minus the average
return on the three big portfolios, while the HML portfolio measures the average return
on the two value portfolios minus the average return on the two growth portfolios.
The second set of test assets is a set of 5 industry-sorted portfolios. These value-
weighted portfolios are constructed by sorting NYSE, AMEX and NASDAQ equities
their four-digit SIC codes. Returns to the first and second set of test assets are available
for the entire sample, which begins in 1959. The third set of test assets is the 6 bond
portfolios, sorted by maturity date. The short term portfolio contain bonds that mature
in less than one year, and the longest-term portfolio contains bonds that mature in 5 to
10 years. The portfolio returns are calculated as an equal-weighted average of returns
of the bonds in each portfolio.
65
The fourth set of test assets includes three international equity portfolios. The
first is a weighted average of 20 country equity portfolios. For this index, each of the
component country portfolios is constructed as a value-weighted index of the dollar
returns on equities from all firms in that country for which book-to-market data is
available. Countries are added to the portfolio once their data becomes available,
and averaging across countries is done using EAFE weights, which are developed by
Morgan Stanley Capital International to measure the market capitalization share of each
country. The next two test assets are the high and low book-to-market international
portfolios. The high book-to-market portfolio is formed by first forming the high book-
to-market portfolio for each country, and then taking a weighted average of available
country portfolios using EAFE weights as for the full international index. The low book-
to-market portfolio is constructed analogously. All three international equity portfolios
begin in 1975 with 14 countries: Australia, Belgium, France, Germany, Hong Kong,
Italy, Japan, Netherlands, Norway, Singapore, Spain, Sweden, Switzerland, and the
United Kingdom. Later additions include Austria (1987), Finland (1988), New Zealand
(1988), Denmark (1989), and Ireland (1991). The last addition is Malaysia in 1994.
Malaysia is also the only country to leave the sample early (in October 2001), before
the end of the full sample in December 2007.
The fifth set of test assets is a set of 7 currency portfolios constructed for and de-
scribed in Lustig et al. (2008). These portfolios measure the return to buying a foreign
currency in the forward market and selling it one month later. The currencies are
sorted into 6 portfolios using the forward discount (f − s), and the seventh currency
portfolio is a spread portfolio that measures the return to the high forward discount
portfolio minus the return to the low forward discount portfolio. The full sample period
is November 1983 to December 2008. The particular currencies included in a given year
depends on data availability, but the full sample includes Australia, Austria, Belgium,
Canada, Hong Kong, Czech Republic, Denmark, Euro area, Finland, France, Germany,
Greece, Hungary, India, Indonesia, Ireland, Italy, Japan, Kuwait, Malaysia, Mexico,
Netherlands, New Zealand, Norway, Philippines, Poland, Portugal, Saudi Arabia, Sin-
gapore, South Africa, South Korea, Spain, Sweden, Switzerland, Taiwan, Thailand, and
the United Kingdom.
The average of the annualized monthly returns for the 6 Fama-French portfolios
and the 5 industry-sorted portfolios range from 10.8% to 18.3%. Annualized volatility
ranges from 13.9% to 25.8%, and all portfolios have significantly positive returns. The
bond portfolios have much lower returns that are increasing with maturity and range
66
from 6.0% to 8.4%. The volatility also increases with maturity, ranging from 1.7% to
9.9%, and all bond portfolio returns are statistically significant. Annualized returns
for the 19 developed country portfolios are also all positive and statistically significant,
ranging from 8.8% to 19.1%, with volatility ranging from 15.9% to 32.8%. However,
the Malaysia equity index is an outlier, with mean returns of 0.3% and high volatility
of 47.1%. The international indices for all equities, and high and low book-to-market
values have returns of 13.6%, 17.7%, and 11.5%, respectively, and volatility around 17
to 18%. Finally, for the currency portfolios sorted on the forward discount, the returns
range from negative 0.8% returns for the low portfolio up to 5.1% for the high portfolio,
which gives us a 5.9% return to the spread portfolio. The volatilities for these portfolios
are generally increasing with the forward discount, ranging from 7.5% for the second
portfolio up to 10.0% for the high portfolio.
67
Table A.1: Contract Specifications
Commodity Exchange Start End Contract Months TGrains
Corn CBOT Sept ’59 Dec ’08 1 3 5 7 9 11 12 591Soybeans CBOT Sept ’59 Dec ’08 1 3 5 7 8 9 11 591Soybean Oil CBOT Sept ’59 Dec ’08 1 3 5 7 8 9 10 12 591Soybean Meal CBOT Dec ’59 Dec ’08 1 3 5 7 8 9 10 12 590Wheat CBOT Sept ’59 Dec ’08 3 5 7 9 12 591Oats CBOT Dec ’59 Dec ’08 3 5 7 9 12 590Rough Rice CBOT Nov ’86 Dec ’08 1 3 5 7 9 11 266
SoftsCocoa ICE Mar ’60 Dec ’08 3 5 7 9 12 587Cotton ICE Jul ’60 Dec ’08 3 5 7 10 12 583Sugar ICE Jul ’61 Dec ’08 3 5 7 10 572Orange Juice ICE May ’67 Dec ’08 1 3 5 7 9 11 500Lumber CME Mar ’70 Dec ’08 1 3 5 7 9 11 467Coffee ICE Mar ’73 Dec ’08 3 5 7 9 12 431
MetalsSilver NYMEX Aug ’63 Dec ’08 1 2 3 4 5 7 9 12 542Gold NYMEX Jan ’75 Dec ’08 2 3 4 6 8 10 12 406Copper NYMEX Oct ’59 Dec ’08 3 5 7 9 12 591Aluminum NYMEX Aug ’99 Dec ’08 3 5 7 9 12 113Platinum NYMEX Jul ’68 Dec ’08 3 6 9 12 487Palladium NYMEX Mar ’77 Dec ’08 3 6 9 12 377Zinc LME Feb ’88 Dec ’08 all (3 mo. forward) 251Nickel LME Mar ’88 Dec ’08 all (3 mo. forward) 250Tin LME Sept ’89 Dec ’08 all (3 mo. forward) 232Lead LME Jul ’91 Dec ’08 all (3 mo. forward) 210
AnimalPork Bellies CME Feb ’64 Dec ’08 2 3 5 7 8 539Lean Hogs CME Jul ’66 Dec ’08 2 4 6 7 8 10 12 510Live Cattle CME Apr ’65 Dec ’08 2 4 6 8 10 12 526Feeder Cattle CME Mar ’72 Dec ’08 1 3 4 5 8 9 10 11 441Milk CME Apr ’96 Dec ’08 all 149Butter CME Feb ’97 Dec ’08 2 4 6 7 9 11 139
EnergyHeating Oil NYMEX Feb ’79 Dec ’08 all 359Crude Oil NYMEX June ’83 Dec ’08 all 307Gasoline NYMEX Feb ’85 Dec ’06 all 262Propane NYMEX Dec ’87 Dec ’08 all 254Natural Gas NYMEX June ’90 Dec ’08 all 222Coal NYMEX Sept ’01 Dec ’08 all 87
68
Table A.2: Individual Commodity Summary Statistics
Excess Return Spot Return Basis CorrelationCommodity T Mean Ann. Mean Ann. Mean Ann. All Within
(% p.a.) Vol. (% p.a.) Vol. (% p.a.) Vol. TypeGrains
Corn 591 -2.09 24.33 5.69 25.64 7.87 8.21 0.21 0.57Soybeans 591 5.24 27.85 6.92 28.70 1.64 8.61 0.23 0.57Soybean Oil 591 7.80 32.30 7.19 30.82 0.00 8.91 0.21 0.49Soybean Meal 590 8.68 31.11 8.00 32.05 -0.44 9.73 0.20 0.49Wheat 591 -0.12 25.40 5.56 26.97 5.80 9.49 0.16 0.43Oats 590 -1.22 28.23 6.16 29.61 7.91 11.21 0.14 0.41Rough Rice 266 -0.36 34.60 12.27 39.26 11.88 10.89 0.10 0.23
SoftsCocoa 587 3.91 32.80 7.72 32.29 4.07 8.31 0.11 0.11Cotton 583 1.51 24.64 4.36 26.90 3.00 11.63 0.15 0.05Sugar 572 7.36 48.71 13.80 49.49 7.14 13.94 0.11 0.08Orange Juice 500 4.44 33.07 6.82 33.68 3.08 10.28 0.02 0.00Lumber 467 -1.53 28.37 6.65 30.38 8.68 14.72 0.10 0.04Coffee 431 7.04 41.37 8.73 41.44 2.46 10.81 0.06 0.03
MetalSilver 542 1.70 33.31 9.86 34.14 8.20 4.52 0.21 0.45Gold 406 -0.88 19.93 6.32 20.39 7.27 2.28 0.17 0.37Copper 591 11.52 31.26 7.64 29.81 -3.09 8.49 0.21 0.43Aluminum 487 4.03 28.60 6.70 28.44 3.38 4.09 0.24 0.48Platinum 377 8.33 38.09 11.01 38.23 2.68 3.95 0.20 0.37Palladium 113 2.21 20.06 3.79 20.09 2.27 1.77 0.35 0.58Zinc 251 -2.14 29.70 9.77 36.66 13.17 26.05 0.14 0.38Nickel 250 12.88 45.41 15.11 65.31 -0.41 15.11 0.14 0.32Tin 232 3.00 24.28 4.21 24.02 1.32 3.20 0.22 0.39Lead 210 7.50 31.27 8.80 30.95 1.49 4.89 0.15 0.33
AnimalPork Bellies 539 6.16 32.69 10.70 41.83 7.24 22.20 0.11 0.33Lean Hogs 510 7.08 24.78 6.17 31.10 -0.36 20.60 0.15 0.41Live Cattle 526 6.01 16.94 4.49 18.54 -1.07 9.96 0.15 0.40Feeder Cattle 441 3.72 16.36 4.20 17.66 0.60 6.57 0.12 0.35Milk 149 10.17 32.19 6.58 37.27 -1.93 22.11 0.09 0.22Butter 139 9.66 34.36 11.96 35.60 4.36 12.83 -0.03 0.14
EnergyHeating Oil 359 13.56 35.72 10.32 37.26 -1.94 14.38 0.16 0.72Crude Oil 307 14.12 35.19 8.81 36.38 -5.07 8.32 0.18 0.69Gasoline 262 20.50 35.71 10.40 38.19 -9.35 16.33 0.10 0.63Propane 254 22.94 49.09 16.85 52.47 -6.97 14.36 0.14 0.63Natural Gas 222 8.09 48.95 22.38 56.83 15.07 31.20 0.12 0.46Coal 87 5.61 35.57 15.90 37.14 9.07 9.78 0.30 0.48
Notes: ∗p < 0.10, ∗∗p < 0.05, and ∗∗∗p < 0.01. Excess returns are calculated as(St − Ft−3,t)/Ft−3,t. Volatilities, standard errors, and p-values are calculated using Newey-West standard errors with 4 lags. Reported multiple betas are estimated using a separate timeseries regression for each portfolio: Reit = βi0 + βic∆ct + βid∆dt + βiwrWt + εit.
69
Tab
leA
.3:
Typ
eP
ortf
olio
Sum
mar
ySta
tist
ics
Mea
nP
ortf
olio
Cor
rela
tion
Avg.
Cor
r.A
uto
corr
elat
ion
Por
tfol
io(%
p.a
.)G
rain
sSof
tsM
etal
Anim
alE
ner
gy(w
ithin
typ
e)1
mon
th3
mon
thT
Gra
ins
2.93
..
..
.0.
450.
670.
0559
1Sof
ts3.
500.
35.
..
.0.
050.
690.
1058
3M
etal
6.27
*0.
220.
34.
..
0.41
0.70
0.13
542
Anim
al6.
16**
0.26
0.04
0.14
..
0.31
0.67
-0.0
252
6E
ner
gy14
.32*
*0.
020.
090.
230.
08.
0.60
0.65
0.00
307
All
Com
modit
ies
5.69
***
0.72
0.63
0.65
0.45
0.58
0.15
0.70
0.10
591
Not
es:
∗ p<
0.1
0,∗∗p<
0.05
,an
d∗∗
∗ p<
0.01.
Sta
nd
ard
erro
rsare
calc
ula
ted
usi
ng
New
ey-W
est
stan
dard
erro
rsw
ith
4la
gs.
Ave
rage
corr
elat
ion
wit
hin
typ
em
easu
res
the
aver
age
pair
wis
eco
rrel
ati
on
am
on
gall
pair
sof
com
mod
itie
sw
ith
ina
giv
enty
pe.
Tab
leA
.4:
Fac
tor
Sum
mar
ySta
tist
ics
Mea
nA
nn.
Shar
pe
Auto
corr
elat
ion
Fac
tor
Cor
rela
tion
(%p.a
.)V
ol.
Rat
io1
mon
th3
mon
th(rWt)
(∆c t
)(∆dt)
Mar
ket
Ret
urn
s(rWt)
4.94
**15
.88
0.31
0.70
0.05
..
.N
ondura
ble
Con
sum
pti
on(∆c t
)2.
08**
*1.
061.
960.
610.
170.
18.
.D
ura
ble
Con
sum
pti
on(∆dt)
3.94
***
1.14
3.45
0.98
0.91
0.03
0.28
.A
llC
omm
odit
ies
5.69
***
13.5
80.
420.
700.
100.
070.
000.
22
Not
es:
∗ p<
0.10
,∗∗p<
0.0
5,an
d∗∗
∗ p<
0.01.
Sta
nd
ard
erro
rsan
dvola
tili
ties
are
calc
ula
ted
usi
ng
New
ey-W
est
stan
dard
erro
rsw
ith
4la
gs.
70
Table A.5: Portfolio Characteristics Over Time
T Excess Return Risk AmountMean Ann. Sharpe Multiple Betas(% p.a.) Volatility Ratio p-value βw βc βd
Panel A1: Full Sample (Oct ’59-Dec ’08)Grains 591 2.93 21.75 0.14 0.31 -0.07 -1.07 2.39Softs 583 3.50 17.89 0.20 0.15 0.10 -0.55 2.09**Metal 542 6.27* 24.19 0.26 0.07 0.30*** -0.79 4.80***Animal 526 6.16** 18.91 0.33 0.02 0.06 0.14 3.09**Energy 307 14.32** 33.20 0.43 0.02 -0.12 -5.86* 4.18All Commodities 591 5.69*** 13.58 0.42 0.00 0.07 -1.11 2.96***Panel A2: Early Sample (Oct ’59-Dec ’83)Grains 291 6.63 23.72 0.28 0.14 -0.18 -2.95* 4.92**Softs 283 8.22* 21.06 0.39 0.05 0.15 -2.69* 4.95***Metal 242 9.78 30.42 0.32 0.13 0.40** -2.68 7.02***Animal 226 9.16** 22.33 0.41 0.05 -0.02 1.17 4.96**Energy 7 – – – – – – –All Commodities 291 7.87*** 15.34 0.51 0.01 0.06 -2.11* 4.61***Panel A3: Late Sample (Jan ’84 - Dec ’08)Grains 300 -0.66 19.52 -0.03 0.86 0.06 1.15 -0.09Softs 300 -0.95 13.96 -0.07 0.70 0.08 1.51 -0.19Metal 300 3.44 17.57 0.20 0.32 0.25* 0.42 3.65**Animal 300 3.90 15.80 0.25 0.18 0.14** -2.28** 2.33Energy 300 14.64** 33.52 0.44 0.02 -0.13 -6.32* 4.57All Commodities 300 3.58 11.57 0.31 0.12 0.09 -0.59 2.00
Notes: ∗p < 0.10, ∗∗p < 0.05, and ∗∗∗p < 0.01. Standard errors are calculated using Newey-Weststandard errors with 4 lags. Reported multiple betas are estimated using a separate time seriesregression for each commodity: Reit = βi0 + βic∆ct + βid∆dt + βiwrWt + εit.
71
Table A.6: Individual Commodity Results
Commodity T Avg. Ex. Risk Amount (Multiple Betas)Return βw βc βd
GrainsCorn 591 -2.09 0.01 -1.60 1.33Soybeans 591 5.24 -0.12 -1.08 3.38Soybean Oil 591 7.80* -0.18 -1.34 1.90Soybean Meal 590 8.68** -0.14 -1.56 4.62*Wheat 591 -0.12 0.04 -0.95 1.77Oats 590 -1.22 -0.02 -0.54 2.60*Rough Rice 266 -0.36 -0.01 3.86 -3.08
SoftsCocoa 587 3.91 -0.20* -1.75 4.94***Cotton 583 1.51 0.24** -1.35 2.09Sugar 572 7.36 -0.06 -3.07 -1.54Orange Juice 500 4.44 -0.20 5.99*** 1.38Lumber 467 -1.53 0.52*** -1.07 4.56***Coffee 431 7.04 0.07 1.78 0.71
MetalSilver 542 1.70 0.38*** -2.27 2.63Gold 406 -0.88 0.10 -2.66 2.17Copper 591 11.52*** 0.17 -0.44 4.84***Aluminum 487 4.03 0.31** -2.05 6.96***Platinum 377 8.33 0.61*** 1.79 3.44Palladium 113 2.21 0.56*** 0.46 2.92Zinc 251 -2.14 0.38** 2.65 1.92Nickel 250 12.88 0.53** 2.08 8.40***Tin 232 3.00 0.45** -4.43* 4.93**Lead 210 7.50 0.45** 3.67 2.85
AnimalPork Bellies 539 6.16** -0.07 1.27 4.40*Lean Hogs 510 7.08** 0.06 -0.88 2.94Live Cattle 526 6.01*** 0.11* 0.24 1.68Feeder Cattle 441 3.72 0.15** 0.48 2.19*Milk 149 10.17 0.40** -4.91 1.51Butter 139 9.66 0.11 -6.90 2.71
EnergyHeating Oil 359 13.56** -0.12 -4.28 5.03**Crude Oil 307 14.12** -0.10 -7.77* 6.13Gasoline 262 20.50*** -0.39* -7.96 2.65Propane 254 22.94*** -0.05 -9.63 5.65Natural Gas 222 8.09 -0.09 1.91 3.77Coal 87 5.61 0.38 -8.79 -2.01
Notes: ∗p < 0.10, ∗∗p < 0.05, and ∗∗∗p < 0.01. Standard errors are calculatedusing Newey-West standard errors with 4 lags. Reported multiple betasare estimated using a separate time series regression for each commodity:Reit = βi0 + βic∆ct + βid∆dt + βiwrWt + εit.
72
Table A.7: Sorted Portfolio Correlations
Panel A: Volatility PortfolioBasis Portfolio Low 2 3 4 High
High 4.4 4.0 3.3 3.7 3.5 18.94 4.3 5.1 4.3 4.0 3.1 20.83 3.5 4.7 5.0 4.2 3.3 20.72 3.2 4.1 4.8 4.9 3.7 20.7
Low 3.3 2.9 3.3 3.9 5.4 18.918.8 20.8 20.8 20.7 18.9 N=13976
χ2 = 350.3, p = 0.0
Panel B: Volatility PortfolioMomentum Portfolio Low 2 3 4 High
Low 5.2 3.8 3.2 3.3 3.3 18.82 3.9 4.8 4.7 4.3 3.1 20.83 3.3 5.0 5.0 4.4 3.1 20.84 3.1 4.4 4.8 4.8 3.7 20.8
High 3.0 2.8 3.3 4.1 5.8 18.918.5 20.8 20.9 20.9 18.9 N=13563
χ2 = 547.6, p = 0.0
Panel C: Momentum PortfolioBasis Portfolio Low 2 3 4 High
High 7.6 4.4 3.1 2.1 1.5 18.64 4.7 5.7 4.7 3.6 2.1 20.93 3.0 4.8 5.4 4.8 2.8 20.82 2.0 3.5 4.6 5.5 5.2 20.9
Low 1.5 2.3 2.9 4.8 7.3 18.818.7 20.8 20.8 20.8 18.9 N=13549
χ2 = 2415.0, p = 0.0
Note: The χ2 statistic and p-value reported for each panel are for the test of thehypothesis that the rows and columns are independent.
73
Table A.8: Simple Betas
Excess Return Risk AmountT (% p.a.) βw βc βd
Panel A: Type PortfoliosEnergy 591 2.93 -0.08 -0.53 2.39Agriculture 583 3.50 0.09 -0.55 2.09**Softs 542 6.27* 0.31*** -0.79 4.80***Animal 526 6.16** 0.07 0.14 3.09**Metal 307 14.32** -0.16 -5.86* 4.18All Commodities 591 5.69*** 0.06 -1.11 2.96***Panel B: Volatility-sorted PortfoliosLow 573 -2.13 0.14 0.78 2.392 573 -0.43 0.05 -0.55 2.09**3 573 0.21 0.02 -0.79 4.80***4 573 7.66*** 0.06 0.14 3.09**High 573 24.67*** 0.02 -5.86* 4.18Spread (HML) 573 26.80*** -0.12 -1.11 2.96***Panel C: Basis-sorted PortfoliosLow 569 11.84*** -0.11 -0.33 2.392 569 7.58*** 0.07 -0.55 2.09**3 569 5.03** 0.15 -0.79 4.80***4 569 2.54 0.12 0.14 3.09**High 569 1.80 0.02 -5.86* 4.18Spread (LMH) 569 10.04*** -0.13 -1.11 2.96***Panel D: Momentum-sorted PortfoliosLow 559 0.57 0.07 0.03 2.392 559 4.56** 0.07 -0.55 2.09**3 559 5.78*** 0.05 -0.79 4.80***4 559 7.30*** 0.06 0.14 3.09**High 559 11.88*** 0.03 -5.86* 4.18Spread (HML) 559 11.31*** -0.04 -1.11 2.96***
Notes: ∗p < 0.10, ∗∗p < 0.05, and ∗∗∗p < 0.01. Reported simple betas areestimated using a separate time series regression for each commodity andfor each factor: Reit = βi0c + βic∆ct + εict, Reit = βi0d + βid∆dt + εidt, andReit = βi0w + βiwrWt + εiwt.
74
Table A.9: Consumption Expenditures and Flow of Services
1959 2008Nominal Expenditures ($/capita)
Nondurables (Ct) 1,535 (86%) 29,721 (89%)Durables (Et) 253 (14%) 3,603 (11%)Total 1,788 33,324
Flow of Services (chained 2000$/capita)Nondurables (Ct) 8,267 (80%) 23,332 (61%)Durables (Dt) 2,056 (20%) 14,610 (39%)Total 10,323 37,942
Note: Levels and shares of expenditure and consumption of durable and nondurablegoods are calculated using data from the U.S. Census Bureau.
75
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