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## Abstract

Term premiums, defined as the excess return of long-dated contracts over short-dated contracts, in commodity futures are strongly predictable, both in the time series and in the cross section, by roll yield spreads. Strategies that exploit this predictability show sizable Sharpe ratios and are uncorrelated with strategies that exploit predictability in risk premiums using the basis in futures prices, that is, use contango and backwardation conditions in futures market to develop their strategies.

The term structure of futures prices has a strong impact on the returns earned by investors in those markets. For example, the literature shows that buying futures in backwardation and selling futures in contango is, on average, a profitable strategy. In other words, the risk premium of commodity futures is time-varying and can be forecasted by the slope of the term structure (basis or roll yield).^{1} In upward-sloping term structures (contango), contract prices have a tendency to fall as time passes and contracts approach expiration. In downward-sloping term structures (backwardation), the exact opposite argument is true: contract prices have a tendency to go up as contracts approach expiration.

This article shows that the term structure of futures prices contains significantly more information than just its slope. As in the fixed-income literature, individual segments of the term structure can be used to extract the extra information. For instance, when the first segment of the term structure has a steeper slope than the second segment, this suggests that the price of the first contract has a stronger tendency to fall than the price of the second contract. A simple strategy to benefit from this configuration would be to hold a long position in the second contract and a short position in the first one. The expected return differential between the long-dated contract and the short-dated one is defined as the term premium—one of the main variables of interest in this article.

From a practical point of view, we use commodity futures to study whether risk premiums and term premiums are predictable in the time series or in the cross section. Based on these results, we then proceed by constructing trading strategies that explore the predictability in risk and term premiums of different commodities, taking long positions in those with high expected values and short positions in those with low expected values. The Sharpe ratios of the trading strategies based on term premiums are significantly higher than the Sharpe ratios of the risk premium strategies usually studied in the literature (Erb and Harvey [2006], Gorton and Rouwenhorst [2006]). Moreover, these two types of trading strategies are uncorrelated; portfolios combining them enjoy significant diversification benefits and earn even higher Sharpe ratios.

From a theoretical point of view, our predictability tests are motivated by a solid foundation developed in the fixed-income literature and extended here to investigate the term structure of futures contracts. This approach has two important benefits. First, it guides the selection of variables and significantly reduces the criticisms that our predictability tests and trading strategies are the result of data mining. Second, as we show later, even a lack of predictability in risk and term premiums would be interesting theoretically, because it indicates strong predictability in other important variables, such as changes in spot prices and changes in the basis.

**THEORY**

Fama and French [1987] divided the theories of commodity futures prices into two camps. The first camp is related to the theory of storage of Kaldor [1939], Working [1948], Brennan [1958], and Telser [1958], which explains differences between spot prices and futures prices using three different costs (or benefits) accrued to holders of futures: interest earned on principal, warehousing costs avoided by not having to store the commodity, and a convenience yield forgone by not having inventories readily available. The second camp, exemplified by Cootner [1960], Dusak [1973], Breeden [1980], and Hazuka [1984], shows that futures prices contain information about expected risk premiums and a forecast of future spot prices. In this article, we focus on theories in the second camp because of their straightforward practical use: information about expected risk premiums can be directly translated into trading strategies, as we show in the following.

According to the risk premium theory, the current log price of a futures contract can be written as its expected log price next period minus an expected return premium (or price discount), p:

1In particular, Fama and French [1987] show that the price of a short-term contract,
, is
using the fact that futures contracts converge to the spot price, *s _{t}
*, at expiration. Therefore, the price of the short-term contract reflects the expected spot price next period minus its expected return premium (price discount). Because spot prices are imprecise or simply unavailable in the case of commodities, we follow Fama and French [1987] and use the price of the nearby contract, denoted by
, as a proxy for the spot price.

Notice that the definition in Equation (1) makes expected risk premiums identical to one-period expected excess returns: . In other words, predictability in risk premiums translates directly into predictability in excess returns.

The building blocks to extract the information contained in the term structure of futures contracts are roll yields, , calculated as the inverse slope between the prices of two consecutive contracts:

2Using the inverse slope simply aligns the sign of the roll yield with the sign of the expected return in subsequent tests: the roll yield has a positive sign when the curve is downward sloping (backwardation) and a negative sign when the curve is upward sloping (contango). Contango hurts the performance of long positions in futures, because futures prices tend to fall over time (negative roll yield and return). Backwardation is beneficial to investors with long positions in futures, because futures prices have a tendency to move up over time (positive roll yield and return). An alternative definition of roll yields is the slope between each contract and the front contract—also known as the basis—but our choice of defining roll yields between neighboring contracts simplifies the notation. In the fixed-income literature, as defined in Equation (2) is usually referred to as the forward rate.

**Predictability in Risk Premiums**

How can we use the information in roll yields to test whether risk premiums are predictable? Fama and French [1987] show that subtracting *s _{t}
* from both sides of
results in

where ?*s*
_{t+1} represents changes in *s _{t}
*. This expression is an ex post identity and not a behavioral model of how investors set their expectations. It holds in the time series as well as in the cross section. Most importantly, it shows that the short-term roll yield contains information about the expected excess return on the short-term contract (contract 1), about the expected change in the spot price of the commodity, or about a combination of both. The negative sign in front of

*E*[?

_{t}*s*

_{t}

_{+1}] comes from the fact that predictability in spot returns reduces predictability in excess returns. For instance, if the short-term roll yield is 100 bps but spot prices are expected to go

*down*by 50 bps, then the expected return is only 50 bps.

Fama and French [1987] test the information contained in the term structure by regressing excess returns and changes in spot prices on past short-term roll yields:

4How can the results from this regression be useful to investors? If we find that ß is significantly different from zero, this information implies that the expected risk premium, , is time-varying and, most importantly, is a function of the short-term roll yield, . Investors could use this information to choose whether to go long or short a certain futures contract depending on its roll yield.

In their empirical tests, Fama and French [1987] studied the information content of futures prices for individual commodities separately, but concluded that “the large variances of realized premiums mean that average premiums that often seem economically large are usually insufficient to infer that expected premiums are nonzero, especially in the data for individual commodities.” We improve on the lack of statistical power in tests of individual commodities in multiple ways. First, we pool commodities together and run joint tests to increase significance. Second, studying a pool of commodities allows us to investigate both time-series and cross-sectional predictability. As we show in the empirical part of the article, on many occasions the term structure contains significantly more information about the cross section, resulting in more powerful tests.

Our most important contribution, however, comes from studying *term premiums*, which remain relatively unexplored in the literature. As we show next, term premiums can be isolated by taking opposite long and short positions in different contracts of the same commodity, thereby removing exposure to variation in spot prices and reducing volatility significantly.

**Predictability in Term Premiums**

The motivation for our second set of tests comes from the fact that investing in a single contract exposes the investor to changes in spot prices, which are very volatile. Therefore, finding an investment strategy that removes this exposure to spot prices likely reduces risk and improves performance. Of course, hedging out this exposure directly by shorting the physical commodity is impossible in most cases. In this section, we derive a simple alternative that consists of holding a long position in one contract and a short position in a different contract.

The proof that such long–short strategies remove exposure to spot prices is straightforward. Start with the definition of in Equation (1) and use some algebra to obtain

5where represents changes in . Now, take the difference between Equation (5) and Equation (3) to obtain

6In fixed-income theory, the expression is usually referred to as the term premium, because it represents the additional risk premium earned by buying a long-term instrument relative to buying a short-term instrument. For this reason, we use the same term here. The term premium doesn’t always have to be positive; in fact, a strategy that exploits predictability in the term premium would buy the short-term futures contract and sell the long-term futures contract when the term premium is negative. Notice also that the term premium completely removes any exposure to spot prices. However, the trade-off is that it introduces a new source of risk: changes in the short-term roll yield, .

We test whether term premiums are predictable using a regression of return differentials on yield spreads:

7The results of the above regression have multiple practical applications to investors. Evidence that *b* is significantly different from zero implies that the expected term premium is time-varying and a function of the yield spread. For instance, with that information in mind, investors can more carefully execute contract selection, i.e., decide whether to buy a long-term or a short-term contract. Alternatively, they can also create long–short strategies that buy the long-term contract and sell the short-term one, or vice-versa, depending on the yield spread. As we show next, these long–short strategies based on yield spreads have Sharpe ratios that are larger than those obtained from traditional strategies based on short-term roll yields (i.e., contango and backwardation). Moreover, because these two types of strategies are uncorrelated, combining them into portfolios results in higher Sharpe ratios.

Most commodities have a large number of contracts trading on any given day, but we focus on the first *two* contracts for multiple reasons. First, using only two contracts reduces the complexity in the notation and the number of empirical tests, making it considerably easier to understand our arguments and results. Second, contract liquidity, in terms of either volume or open interest, tends to decrease with tenure. Thus, making use of highly liquid contracts alleviates the concern that our results are caused by a liquidity premium. Finally, increasing the number of contracts in this study would likely result in stronger effects. Therefore, we see our results as lower bounds on what one could obtain by using a larger contract universe.

Finally, we use the regressions in Equation (4) and Equation (7) to study commodity futures, but note that they can also be used with futures of different underlying assets. For instance, when buying a futures contract on a foreign currency, investors receive a yield represented by the interest rate differential between the two countries, known in advance, plus the expected future appreciation or depreciation of the foreign currency relative to the U.S. dollar.

**EMPIRICAL TESTS**

**Data**

Futures prices for the five metals traded on the London Metals Exchange (LME) come from Bloomberg. Prices for all other futures come from the Commodity Research Bureau (CRB).

Futures contracts vary significantly on many dimensions: schedule of available contracts, liquidity, maturity, etc. For this reason, we follow the contract selection schedule of the S&P GSCI, one of the most widely followed commodity indexes.^{2} This gives us confidence that we are not only selecting the most liquid contracts but also selecting them at the right point in time. The only minor difference in implementation comes from the fact that the S&P GSCI rolls out of the nearby contract and into the next contract over a five-day period starting on the fifth business day of the month. We follow the approach commonly used in the literature of rolling the entire position in the last day of the previous month. In practice, this means that we start and finish the contract roll process five days before the S&P GSCI starts.

Although the formulas derived in the previous section used *log* returns to simplify the notation, in the empirical tests we use simple monthly excess returns,

to avoid any criticisms that the performance of our strategies is driven by second-order effects.

Monthly roll yields are calculated as in Equation (2). However, for those commodities that don’t have a full monthly schedule of futures contracts, we adjust their roll yields by the number of months between consecutive contracts. For instance, with the log prices of the nearby and short-term contracts denoted by and , the roll yield for the short-term contract becomes

9Further, when the time distance between consecutive contracts is longer than one month, i.e., *N*
_{1} - *N*
_{0} = 2, we keep the roll yield constant for that entire time period. An alternative choice would be to wait until
expires and then recalculate the roll yield as the slope between
and the next contract,
. This second method produces no material difference in our results. We prefer the first method because it always calculates the roll yield between a particular contract and the *previous* contract, which better reflects the definition derived from the risk premium theory (see Equation [2]).

**Summary Statistics**

Exhibit 1 presents the summary statistics for the main variables studied in the article. We divide the 27 commodities into six sectors: energy, grains, industrial metals, livestock, precious metals, and softs. The first column shows the number of monthly observations available for each commodity. The sample period goes from January 1966, the first year to contain at least 10 commodities, to December 2014. For the 10 commodities available since 1966, we have almost 49 years of data with 588 monthly observations. Energy futures became available in the late 1980s or early 1990s, explaining the slightly shorter samples for those periods. Data availability for the five metals traded on the London Metals Exchange starts in 1997, reducing their monthly observations to about 200.^{3}

The next two sets of columns show annualized means and standard deviations of excess returns, , and short-term roll yields, . Most commodities (24 out of 27) had positive mean excess returns, but these came with relatively high volatilities. Natural gas was both the worst-performing commodity (-6.8%) and the most volatile (49.3%). The best-performing commodity was nickel (17.0%), followed by gasoline (16.1%), and the least-volatile one was feeder cattle (16.1%).

The numbers for roll yields, on the other hand, were very different. Only 10 out of 27 commodities had positive mean roll yields, implying that the other 17 commodities were in contango during the majority of their sample periods. The commodity with the strongest average backwardation was gasoline (7.7%), whereas natural gas had the strongest average contango (-21.3%). Roll yields were also significantly less volatile than excess returns, with standard deviations around 5% in most cases. Natural gas had the highest volatility (21%) and both gold and silver had the lowest (1.1%).^{4} These rankings should not be surprising, since the theory of storage relates convenience yields to the level and cost of inventories. Because natural gas is extremely expensive to transport and store, the magnitude of its roll yield is large and swings from positive to negative following extreme changes in inventories. Precious metals, on the other hand, are relatively inexpensive to store, causing their roll yields to be very stable and driven mostly by interest-rate variation.^{5} Silver, and especially gold, are not perishable like other commodities and are used mostly as storage of value; therefore, their inventory levels are relatively constant over time.

Equation (3) shows that variation in excess returns must come either from roll yields or from spot returns. Since roll yields have significantly lower volatilities than excess returns, we conclude that the majority of the volatility of excess returns must come from spot returns. This is expected, given that physical commodities are known to experience extreme and rapid price fluctuations. What is perhaps more surprising is the reduced role played by average spot returns in explaining average excess returns *across* different commodities.^{6} For instance, natural gas had both the lowest average excess return and the lowest average roll yield, while gasoline had both the second highest average excess return and the highest average roll yield.

To quantify the relative importance of average roll yields and average spot returns in the cross section, Exhibit 2 shows a scatterplot of average
versus average
over a sample period that begins in 1998, the first year with data available for all commodities. We choose a common starting year because performance measures are highly dependent on starting and final dates, but also because it provides a robustness check for the results in Exhibit 1. A similar plot using the entire history available, i.e., using the numbers from Exhibit 1, does not change any of the conclusions that follow. Almost all variation in average excess returns across commodities was driven by variation in average roll yields, with a correlation of 90%. Of course, it is impossible to know in advance which commodities are going to have the best average roll yields over the next few years, but strategies based on *current* roll yields are common in the literature, and we study them in the next section.

A possible explanation for the reduced role played by spot returns is that although spot prices are very volatile over short horizons, in the long run they tend to behave similarly, reflecting general price trends—inflation—in commodities and in the economy as a whole. Since excess returns are equal to spot returns in the absence of roll yields, the intercept in the line of best fit in Exhibit 2 provides an indication of the average spot return common to all commodities over the sample period 1998–2014: 10.9%. The fact that this number is higher than most average excess returns in Exhibit 1 underscores the (usually negative) influence of roll yields on the long-term performance of commodity futures.

The last two sets of columns in Exhibit 1 show annualized means and standard deviations for return differentials,
, and yield spreads,
. Fifteen out of the 27 commodities had positive average return differentials, which might come as a surprise to investors who believe that the unconditional term premium should *always* be positive. In practical terms, for 12 out of the 27 commodities, a strategy of always rolling into short-term contracts outperformed a strategy of always rolling into long-term contracts. Nonetheless, as we show later in the article, it is possible to forecast with some precision *when* a long-term contract will outperform a short-term contract, i.e., we show that the conditional term premium is time-varying.

Natural gas had the highest unconditional return differential (4.7 percent), while gasoline had the lowest (-2.2 percent). Return differentials had significantly smaller standard deviations than excess returns, which can be explained by the fact that was significantly less volatile than . (Equation [3] and Equation [6] show that excess returns are exposed to changes in the spot price, whereas return differentials are exposed to changes in the roll yield).

We also investigate the relative importance of average yield spreads and average roll yield changes in explaining cross-sectional variation in average return differentials. Using the shorter sample that starts in 1998 for all commodities, Exhibit 3 shows that the correlation between and was 75%, even in the presence of sugar (SB) and lean hogs (LH), which seem to be outliers in the chart. The intercept of the line of best fit, an estimate of across commodities (Equation [6]), was small and positive at 0.6%, indicating that the average roll yield of the different commodities has become slightly more positive—less contango—since 1998.

The unconditional risk and term premiums studied in this section contain valuable information about static (buy-and-hold) investments. In the next sections, however, we show that dynamic strategies are more interesting from both theoretical and practical points of view. We study the information contained in roll yields and yield spreads about time-varying risk and term premiums and how it can be used to build simple dynamic strategies.

**Predictability and Trading Strategies**

From a theoretical point of view, the difference between cross-sectional and time-series predictability is subtle, but in practice it can be significant. In time-series tests, one is interested in how the risk premium and the term premium of a particular commodity vary over time. Of course, that doesn’t imply that commodities can’t be combined into portfolios, but any information about other commodities is irrelevant. In cross-sectional tests, on the other hand, the information contained in other commodities is essential; the main interest lies in whether the risk premium and term premium of a particular commodity are above or below those of other commodities. These differences become clearer when we present the strategies in the next two sections.

**Time-series predictability.** We start our tests with time-series tests and strategies. Exhibit 4 shows coefficients, *t*-statistics and *R*
^{2}s for two sets of forecasting regressions following Equation (4) and Equation (7). In the interest of space, we report results for commodity futures pooled into six different sectors or into a large group containing all of them. Standard errors are corrected for heteroskedasticity and cross-sectional correlation in the residuals. We remove the intercepts from all regressions to avoid the look-ahead bias in the estimation of full-sample means. In other words, forecasting regressions with intercepts measure variation relative to a long-term mean estimated using the entire sample, whereas forecasting regressions without intercepts measure variation relative to a mean of zero. Results including intercepts or fixed-effects for individual commodity futures are immaterially different and available from the author by request.

The first three columns tell us not only whether expected risk premiums are time-varying, but also how the information in roll yields is translated into expected risk premiums. The 0.87 coefficient (*t*-stat = 1.83) of industrial metals, for instance, shows that a roll yield of 100 bps translates, on average, into a *positive* excess return of 87 bps. At first sight, this coefficient might indicate a highly profitable trading strategy, but the *R*
^{2} of less than 1% reveals a high amount of uncertainty involved in the strategy. Pooling all commodities together results in a coefficient of 0.12, barely significant at the 10% level (*t*-stat = 1.67), and with a minuscule *R*
^{2} of 0.07%. The low *R*
^{2}s in the commodity regressions should not be surprising, given that spot returns are significantly more volatile than roll yields over time, as discussed in the previous section. Summarizing, the results in the first three columns of Exhibit 4 indicate that using roll yields to time excess returns does not seem to be a very promising strategy, at least in the time series.

The last three columns in Exhibit 4 show how the information in yield spreads translates into expected term premiums. The regression coefficient of 0.08 (*t*-stat = 3.71) for all commodities, for instance, shows that when the yield spread is 100 bps, the expected term premium is, on average, 8 bps. Although the coefficients of the second set of regressions (*b*) are not particularly larger than those of the first set of regressions (ß), their *t*-statistics and *R*
^{2}s are higher for all commodity sectors. (Grains are the only exception, but the coefficients are statistically *insignificant* in both cases). The practical implication of these results is that trading strategies exploiting time-series variation in term premiums should have lower risk and higher Sharpe ratios than trading strategies exploiting time-series variation in risk premiums.

Exhibit 5 shows that the information contained in roll yields and yield spreads can be used to construct dynamic strategies in a very straightforward way. In the case of roll yields, we buy the short-term futures contract on each commodity if the short-term yield is positive (backwardation), and we sell it if the short-term yield is negative (contango). In the case of yield spreads, we buy the second contract and sell the short-term contract if the yield spread is positive, and we sell the second contract and buy the short-term contract if the yield spread is negative. As benchmarks for these two dynamic strategies, we create two static strategies that simply invest in the short-term contract, or in the spread between the second and the short-term contract, at the beginning of every month. The average returns of these two static strategies can be interpreted as the unconditional risk and term premiums. The average returns of the two dynamic strategies, on the other hand, tell us whether roll yields and yield spreads contain sufficient information to explore time-varying expected risk and term premiums. Notice that all strategies can be constructed for each commodity separately, but in the interest of space and to enjoy the benefits of diversification, we combine different commodities into sector portfolios—or a portfolio containing all of them—by equal-weighting the individual commodities.

Exhibit 5, Panel A shows results for strategies based on short-term yields. Since the average excess returns of most individual commodities were positive (Exhibit 1), the average excess returns of all static sector portfolios are positive as well. The average return of the portfolio containing all commodities, for instance, was 6.2%, with a Sharpe ratio of 0.44.

Analyzing the performance of the dynamic strategies reveals mixed results in the case of commodities, with higher Sharpe ratios in three of the six sectors (energy, grains, and softs). Interestingly, the dynamic strategy with all commodities had exactly the same Sharpe ratio of 0.44 as the static strategy, but with slightly lower average excess return and volatility. The last column, showing the percentage of months with positive roll yields, confirms the evidence in Exhibit 1 that most commodities spent the majority of time in contango: all sectors had less than 50% of positive roll-yield observations. Summarizing the results in Panel A of Exhibit 5, strategies that exploit time-series variation in risk premiums did not add value in the case of commodity futures.

Exhibit 5, Panel B shows results for strategies based on yield spreads. Static strategies—buying the second contract and selling the short-term contract—performed poorly in most cases, as can be seen by the zero average excess return of the portfolio including all commodities. The only two exceptions were livestock, with a Sharpe ratio of 0.30, and industrial metals, with a Sharpe ratio of 1.29. When analyzing the performance of the dynamic strategies—buying or selling the return spread between the first two contracts depending on the sign of the yield spread between them—we saw a substantial improvement in Sharpe ratios across all portfolios. In particular, the portfolio including all commodities had a Sharpe ratio of 0.96. The improved Sharpe ratios in Panel B of Exhibit 5 confirm the predictability in term premiums shown in the regressions in Exhibit 4. The last column shows that the sign of yield spreads was slightly biased toward positive values, but not too far from a 50/50 distribution in most cases.

Notice that the strategies in Panel B of Exhibit 5 experienced significantly lower standard deviations than those in Panel A of Exhibit 5, underscoring our earlier comments that return spreads eliminate exposure to spot price movements, the most volatile component of commodity futures’ excess returns.

Summarizing the results in this section, our regression and portfolio tests revealed relatively strong evidence in favor of time-series predictability in *term premiums* of commodity futures. In the case of time-series predictability in *risk premiums*, however, we found no strong evidence. Predictability in term premiums could be useful to investors in at least two scenarios. First, they can be used to construct long–short trading strategies similar to the ones we presented in Exhibit 5. Second, they can be used to select whether to hold short-term or long-term contracts in long-only strategies.

In the next section, we investigate whether cross-sectional tests reveal more or less predictability in risk and term premiums.

**Cross-sectional predictability.** Following the same approach as the one used in the previous section, we test predictability first by using regressions and then more directly by constructing trading strategies. Similar to the literature on cross-sectional tests in equity markets, we use Fama–MacBeth regressions to study the relationship between short-term roll yield and excess returns and between yield spreads and return differentials. Recall that the coefficients of Fama–MacBeth regressions can be interpreted as the average return of a dollar-neutral long–short portfolio weighted in proportion to the independent variable. Therefore, if there’s a large cross-sectional variation in the independent variable, the portfolios can take a significant amount of leverage and concentration, which will be evident in the magnitudes of the coefficients. For this reason, it is common to focus on the *t*-statistics of the coefficients, which are proportional to the Sharpe ratios of the long–short portfolios and thus unaffected by leverage. For instance, given that our sample has 49 years, converting to annual Sharpe ratios can be easily done by dividing each *t*-statistic by
.

Exhibit 6 reports intercepts, coefficients, *t*-statistics (in parentheses) and average *R*
^{2}s for Fama–MacBeth regressions at a monthly frequency. Each row represents a separate regression, with the dependent variable specified in the first column. For instance, the first row shows results for a Fama–MacBeth regression of
on
. How can we interpret the 0.18 coefficient in this regression? First, it shows that the trading strategy implied by the Fama–MacBeth regression—buying commodity futures with high roll yields and selling commodity futures with low roll yields—had an average monthly return of 18%. Of course, this impressive performance comes from highly leveraged and concentrated positions resulting from large cross-sectional differences in yields. In the portfolio tests presented in the following, we form trading strategies with no leverage, obtaining average returns more in line with expectations. The *t*-statistic of 2.85 reveals that the long–short trading strategy had an annual Sharpe ratio of 2.85 ÷ 7 = 0.41, which is comparable to previous results in this article and in the literature. A second interpretation of the coefficient shows that if the futures contract on Commodity A has a roll yield 100 bps higher than the futures contract on Commodity B, on average, we can expect the contract on Commodity A to outperform the contract on Commodity B by 18 bps: we earn the 100 bps owing to the difference in roll yields, but spot prices move in the opposite direction, reducing the return by 82 bps, i.e., the spot return on Commodity A is 82 bps lower than the spot return on Commodity B.

The second row shows results for a Fama–MacBeth regression of
on
. The idea behind this regression is to go long the return differential—buy the second contract and sell the first contract—if the yield differential of a particular commodity is above the average yield differential across all commodities, and sell the return differential—buy the first contract and sell the second contract—otherwise. The regression coefficient shows a monthly average return of 10% with a *t*-statistic of 5.44. The annualized Sharpe ratio of 5.44 ÷ 7 = 0.78 is impressive, but slightly below the 0.96 in Panel B of Exhibit 5 (“All Commodities”).

The third and fourth rows investigate the importance of using the yield *differential* as the variable to forecast term premiums. For instance, Szymanowska et al. [2014] construct similar strategies based on return differentials to isolate term premiums, but then sort commodities on the magnitude of the short-term yield,
, alone. As the third regression shows, the short-term yield contains some information about expected term premiums, but the *t*-statistic of -2.88 is approximately only half of the 5.44 obtained in the second regression, which uses the yield differential as the independent variable. The last row confirms the result by including both
and
in the regression. The *t*-statistic on
now jumps to -5.37—which is very close to 5.44—and the similar magnitudes of the coefficients—at 0.09 and -0.10—show that using the yield differential is very close to the optimal combination between the two roll yield variables.

Because Fama–MacBeth regressions might take significant amounts of leverage and highly concentrated weights when the independent variables assume extreme values, we also investigate the predictability of risk premiums and term premiums in the cross section using long–short strategies, also known as factor portfolios. At the end of each month, we sort all commodities into three portfolios (low, medium, and high) based on either or , and then equal-weight or over the following month. Exhibit 7 presents annualized averages, annualized standard deviations, and Sharpe ratios for each of the three portfolios and for the long–short factor portfolio calculated as the difference between the high and low portfolios. To avoid any confusion, we refer to the high-minus-low portfolio based on as the risk premium portfolio and the high-minus-low portfolio based on as the term premium portfolio. Panel A of Exhibit 7 shows that the risk premium portfolio has a Sharpe ratio of 0.67, which is significantly higher than the 0.44 obtained in Panel A of Exhibit 5 (“All Commodities”), indicating more predictability of risk premiums in the cross section than in the time series. Panel B of Exhibit 7 shows a Sharpe ratio of 0.92 for the term premium portfolio, which is very similar to the 0.96 obtained in Panel B of Exhibit 5 (“All Commodities”), indicating a similar amount of predictability of term premiums in the cross section as in the time series.

Exhibit 8 shows the annualized rolling three-year returns of each of the two factors. Although the risk premium factor strongly outperformed for long periods of time, it also suffered occasional drawdowns of up to 10%–15%. The term premium factor, on the other hand, was remarkably stable over time, ranging between 0%–10% most of the time; the only periods of drawdown happened between 1974 and 1977, when fewer commodities were available, and briefly in 1986.

Exhibit 9 investigates the relationship between the two different risk premiums studied in this article and whether one is capable of “explaining” the other in asset pricing tests. Panel A of Exhibit 9 presents the results for a regression of the term premium portfolio on the risk premium portfolio. Both the regression coefficient and *R*
^{2} are very close to zero and the annualized intercept of 3.9% is highly significant (*t*-stat = 6.43). The inverse is also true: Panel B of Exhibit 9 shows that a regression of the risk premium portfolio on the term premium portfolio also results in a highly significant intercept of 13.2% (*t*-stat = 4.72). These two factor regressions confirm that there is almost no relationship between the two types of premiums present in commodity futures markets. In other words, investing in one type of risk premium does not give you exposure to the second one and vice-versa.

Finally, given the conclusion that the two types of risk premium are different, in Exhibit 10 we test a simple first step toward finding a good way to combine them and exploit the low correlation between the two factor portfolios. The challenge in combining these two factor portfolios comes from the fact that their volatilities have very different magnitudes. Panel A of Exhibit 10 shows that the risk premium portfolio has an annualized standard deviation of 18.9%, while that of the term premium portfolio is only 4.2%. As a consequence, a portfolio with weights of similar magnitudes is not diversified enough and is largely dominated by the most volatile component. For instance, the “50/50” column shows that an equal-weight portfolio has a Sharpe ratio of 0.83, which is impressive in itself but lower than the 0.91 of the term premium portfolio. One way to achieve better diversification would be to calculate the portfolio with the ex post maximum Sharpe ratio. However, this approach is not implementable in practice. A viable and still simple alternative is to follow the risk parity approach recently proposed in the literature.^{7} It consists of weighting each component by the inverse of its standard deviation and, if so desired, to target a specific volatility level. To implement this approach, we simply calculate the annualized standard deviation of each factor portfolio using the previous 12 months of data, s_{t}, and target a volatility level of 15% by calculating the return over the next month as
.

Panel B of Exhibit 10 shows the results of this approach separately for each factor portfolio as well as for the 50/50 equal-weight portfolio between them, which is identical to a risk parity portfolio that targets a volatility of 15%. Notice that scaling each return by the inverse of its volatility can also be seen as a form of risk control mechanism, because the strategy reduces exposure to a component in times of higher volatility while increasing exposure to it in times of lower volatility. The standard deviations of both factor portfolios—17.5% and 18.7%—are slightly above the target of 15%, indicating that past 12-month volatility tends to underestimate future volatility. This approach reduces the Sharpe ratio of the risk premium portfolio to 0.53 from 0.65, but increases that of the term premium portfolio to 1.02 from 0.91. More importantly, the simple average between the two portfolios achieves a final Sharpe ratio of 1.11, which is an improvement over both individual portfolios and the simple average in Panel A of Exhibit 10. This last result confirms the importance of having a balanced and well-diversified portfolio that is not dominated by any single component.

**CONCLUSION**

The term structure of commodity futures contains a significant amount of information. Using the roll yield spread between the first and second contracts, we are able to extract that information to forecast term premiums. A factor portfolio motivated by cross-sectional predictability in term premiums shows impressive performance and is also uncorrelated with a factor portfolio constructed based on the slope of the term structure alone (contango and backwardation), implying that portfolios combining those two factors might have even higher Sharpe ratios than would the individual factors alone.

## ENDNOTES

↵

^{1}The literature has used different names for the slope of the term structure of futures prices, such as basis or roll yield. We prefer the second and use it throughout the article, given its analogy to fixed-income theory.↵

^{2}The S&P GSCI methodology is currently available at http://us.spindices.com/indices/commodities/sp-gsci.↵

^{3}It is possible to obtain data for the LME metals starting earlier than 1997, but those come in the form of cash and 3-month futures. Thus, calculating returns on futures requires some form of interpolation, likely biasing the results.↵

^{4}Gold has a slightly lower standard deviation than silver: 1.11% versus 1.14%.↵

^{5}Recall that the holder of a futures contract earns interest on the principal, and therefore the futures price accounts for that with a negative roll yield inversely related to interest rates.↵

^{6}This result was also shown by Erb and Harvey [2006]. We extend and update their numbers here.↵

^{7}For a deeper discussion on risk parity portfolios, see Maillard, Roncalli, and Teïletche [2010] and Chaves et al. [2011].

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