A Markowitz Optimization of Commodity Futures Portfolios

A. Basic Statistics

In Table I>, we provide the basic statistics for the 39 futures contracts and the equally weighted naïve portfolio over the period 1994–2010. We find that SIF and commodity futures both have high returns and high standard deviations, whereas currency and interest rate futures have low returns and low standard deviations, which is consistent with previous studies on the distribution of futures returns. The equally weighted naïve portfolio possesses an average return and a low risk compared to most of the individual futures contracts, showing the potential diversification benefit of investing in different types of futures contracts.

B. Correlation Analysis

In Table II, we summarize the within- and between-group average correlations over the 17 annual periods from 1994 to 2010 (more comprehensive correlation results are available upon request). The within-group correlations are calculated by averaging the pairwise correlations among all the futures in each futures group over each of the 17 annual periods (e.g., all agricultural futures); the between-group correlation for any pair of groups is obtained by averaging the correlations between individual futures in the two groups over the individual 17 years (e.g., correlating an agricultural future with a SIF).10 In Table II, we show that all of the within-group correlations are high, except for currency and agricultural futures.11 The SIFs within-group correlation ranks the highest, with a value of 0.74, confirming previous findings that the equity markets are more integrated than in past decades. Compared to the within-group correlations, the correlations between different groups are exceptionally low, ranging from −0.21 to 0.20.12 Overall, the correlation analysis shows that diversifying among different groups of futures helps to reduce risk.

C. Ex-ante Optimal Portfolios

In this section, we examine the diversification benefits of commodity futures by employing the optimal mean-variance Markowitz portfolio technique. Two sets of efficient frontiers are constructed using the Markowitz model: the traditional benchmark portfolio (consisting of SIFs and interest rate contracts) and the complete portfolio (consisting of all individual futures contracts). In order to examine the stability and diversification benefits of the portfolios over time, we rebalance the portfolios each year.13 For evaluation purposes, the Sharpe ratios, Sortino ratios, and the VaR and MVaR values of the tangent portfolios are determined over each of the 17 annual periods.

The efficient frontiers of the traditional and complete portfolios are constructed for each year, as well as the risk and return values of the naive portfolio and each individual futures contract. Figure 1A–D illustrates the 1994, 1998, 2002, and 2006 efficient frontiers (the others are omitted due to space considerations). Figure 1A–D shows that the interest rate and currency futures contracts typically have low risk and low return, SIFs possess medium risk and return, agricultural futures generate medium to high risk and return, and energy futures are associated with high risk and high return. The naïve equally weighted portfolio for year t (i.e., the year of the ex-ante portfolio) has a lower risk value than most individual futures contracts, but the naïve portfolio only dominates about half of the individual futures contracts on a risk-return basis. For each year in our study the traditional benchmark efficient frontier (logically) dominates the individual SIF and interest rate futures, showing its diversification benefits relative to individual SIF and interest rate futures contracts on an ex-ante basis. Similarly, the efficient frontier of the ex-ante complete portfolio dominates all of the individual futures contracts for each year. In addition, the complete efficient set dominates the naïve portfolio and the traditional benchmark frontier for all of the 17 years, except 1998.14 Moreover, the shape and position of the complete portfolio efficient frontier is very similar from one year to the next. However, the traditional benchmark efficient frontier changes from year to year. Thus, in an ex-ante sense, adding commodity futures to a traditional stock and bond portfolio provides superior diversification and risk-return portfolio benefits over only using a stock and bond portfolio; moreover, the stability of the location of the efficient set improves when commodities are added to the stock and bond only portfolio.15

The tangent portfolio is often considered to be the most important optimal portfolio, because it possesses the highest Sharpe ratio among all of the optimal portfolio choices. In Figure 2A, we provide the Sharpe ratios for the tangent portfolios for each year based on the ex-ante complete and traditional benchmark efficient frontiers, as well as the complete ex-post and the naively weighted portfolios. The Sharpe ratios for the ex-ante complete tangent portfolio are considerably larger than those for the ex-ante traditional tangent portfolios. On average, the Sharpe ratio for the complete tangent portfolio is 8.05 times the traditional tangent portfolio, with this difference being statistically significant.16 Thus, these results show that a significant benefit can be achieved by adding commodities to the traditional portfolio on an ex-ante basis. However, the Sharpe ratio is more variable over time for the complete tangent portfolio than for the traditional tangent portfolio. This is mainly due to commodity futures being more volatile than interest rate futures. Figure 2B and C, in which we provide the Sortino ratios and the potential VaR losses (at three standard deviations for a $100 investment), also support a similar conclusion of the superiority of the ex-ante complete portfolio.17 The MVaR results are very similar to the VaR results (for all different portfolios); therefore, they are not reported here.18

If a portfolio's composition and individual component weights are stable from year to year, then an investor benefits from the reduced risk of investing in the complete portfolio. Here we examine the weights of the components of the Markowitz ex-ante portfolios from year to year as one measure of stability. In Tables III-V, we provide the asset weights of the portfolio's components for the high-, medium-, and low-risk levels for each of the 17 years of the study. The risk levels are chosen based on the standard deviations of the optimal portfolios.19 The results show that the portfolios’ component assets, as well as theirs weights, generally change drastically from year t to year t + 1, especially for the high-risk level portfolio. For the low-risk portfolios approximately half of the component assets for year t remain in the optimal portfolio for year t + 1, although the weights can vary considerably.20 Thus, if the stability of the component assets (and the resultant risk-return performance over time) is the primary goal, then the low-risk portfolio is preferable.

D. Ex-post Results

Relative to the stability of the weights of the ex-ante portfolio, investors are even more concerned about the stability of the realized return and risk measures between the ex-ante and ex-post portfolios. This section examines these ex-post stability results.

Investors should prefer a Markowitz diversification with commodity futures only if the ex-post Markowitz results provide return and risk diversification benefits that are superior to a naïve diversification approach. In Figure 1A–D, we provide the ex-post as well as the ex-ante efficient frontier results for years 1994, 1998, 2002, and 2006. Figure 1A–D shows that the ex-post optimal portfolio results significantly underperform relative to the ex-ante portfolios (as expected, except for the high-risk portfolio for 1998), and even underperform relative to the original year's performance of some of the individual futures contracts.

More interesting is the comparison of the ex-post results to the traditional benchmark ex-ante portfolio results and to the naïve portfolio. In order to examine the resultant ex-post “optimal” portfolios at time t + 1, we apply the optimal weights at year t to the data in year t + 1 to obtain the ex-post “optimal” year t results (e.g., the component weights of the optimal portfolios from 1994 are applied to data from 1995 to obtain what we call the 1994 ex-post results).21 An important conclusion for the portfolio comparison is that the ex-post optimal portfolio resultant frontiers lie above the traditional stock and bond ex-ante efficient frontiers for the majority of optimal portfolios on the frontiers, suggesting that adding commodity futures to a portfolio can provide important ex-post risk-return benefits. Moreover, comparing the naïve portfolio's return and risk for year t + 1 to the ex-post efficient frontier (which typically occurs at the low-risk end of the efficient frontier) we find the ex-post t + 1 resultant frontier possesses as good as, if not better than, risk-return combinations for 10 of the 16 years studied. Finally, the higher risk portfolios on the ex-post efficient frontier portfolios typically do not possess substantially higher ex-post returns (again, 1998 is the exception). Overall, benefits from using the Markowitz portfolios in an attempt to achieve superior risk-return results do appear to be limited to the lower risk portfolios.

Another approach to evaluate the ex-post results is to compare the Sharpe ratios of portfolios. Theoretically, investors desire the portfolio providing the highest Sharpe ratio. Therefore, we obtain the return and risk of the ex-post tangent portfolio by applying the weights of the tangent portfolio for year t to the data for year t + 1.22 The ex-post tangent portfolio for year t + 1 is compared to the naive portfolio results for year t + 1. The ex-post tangent portfolios’ Sharpe ratios in Figure 2A are larger than the naïve portfolios’ Sharpe ratios for eight of the 16 years of this study, although the differences are generally small.23 However, the Sharpe ratio has several problems. First, when negative returns exist, ranking portfolios by their negative ratios provides incorrect information, because highly volatile markets will possess “better” (smaller negative) ratios. Second, Brooks and Kat (2002) point out that the Sharpe ratio substantially overestimates the true performance of portfolios when significant skewness and excess kurtosis exist. In order to overcome these issues we complement the Sharpe ratio with the Sortino ratio, as well as address the VaR and MVaR (four-moment) measures.

Figure 2B presents the Sortino ratio. The results are very similar to those of the Sharpe ratio. Figure 2C shows the VaR values for three standard deviations for a $100 investment, illustrating that the ex-post tangent portfolio has less potential loss than the naive portfolio for every year in our sample except 1995. The MVaR results are not given here because they are very similar to the VaR results. Thus, although the ex-post tangent portfolio provides a similar reward to standard deviation performance to the naïve portfolio, it has less downside risk. Consequently, we conclude that ex-post optimal commodity futures portfolios can provide risk-return benefits relative to the naïve portfolio, at least based on the VaR and MVaR criteria.

Now let us directly compare the stability between the ex-ante and ex-post results via the differences in return and risk from year t to year t + 1. We examine this issue by employing the optimal ex-ante and subsequent ex-post portfolios chosen for the three different risk levels: high, medium, and low. Figure 3A shows the differences in risk and return between the ex-ante and ex-post outcomes. The ex-ante portfolios all have higher returns than the ex-post portfolios, because the ex-ante returns minus the ex-post returns are almost all positive. This shows that the Markowitz optimal portfolio weights contain a substantial upward bias when used to forecast out-of-sample portfolio returns. On the risk side, the risk of the ex-ante portfolio results are somewhat smaller on average than the ex-post portfolio risk, because the majority of the differences between the ex-ante and the ex-post risk values are negative; however, for some periods/portfolios the risk is reduced from year t to year t + 1. The relative stability in the risk values is encouraging for the Markowitz procedure; on the other hand, the reduced returns from the ex-ante portfolio to ex-post results, especially for the high-risk portfolios, suggest a difficulty in using the Markowitz method for optimal forecasting purposes. Our findings in this section are generally in line with a number of studies that only employ equities (Chopra & Ziemba, 1993; Fisher & Statman, 1997; Mailer, Durand, & Lee, 2005). Our results support the conclusion of these studies that the Markowitz model has an upward bias in estimating or forecasting returns and is very sensitive to errors in parameters estimation.

E. The Driving Factor for Instability

The instability of the component assets in the optimal portfolios and the instability of the ex-ante versus ex-post risk and return results for the optimization models prompts us to question what factors cause this instability. To examine this issue, we estimate the following regression to analyze the relative importance of the return, standard deviation, and correlation factors in determining the weights of the individual assets in the optimal portfolio:

(5)

where w_i is the weight of the futures contract, and r_i, , and are the ranks for the return, the risk, and the average correlation of the futures contract relative to all the other futures,24 respectively. is the error term. In order to control for the effect of the risk level of the optimal portfolio, we estimate separate regressions at the high-, medium-, and low-risk portfolios. The high-, medium-, and low-risk Markowitz portfolios are obtained from each year and the regressions are executed using these pooled portfolios. The White and Breusch-Pagan tests show heteroskedastic error terms. After adjusting for heteroskedasticity, the regression results are reported in Table VI.

The regression at each risk level possesses significant F-values at the 1% or better level, with the R² equaling 22%, 23%, and 27% for the high, medium, and low portfolio risk levels. Among the three variables, the return variable is significant at the 5% level or better for all three regressions. The risk variable is significant for the medium- and low-risk portfolios, whereas correlation is not significant at any risk level. We conclude that the return is more important than the risk and correlation factors in determining the asset weights in the optimal portfolios. Consequently, the Markowitz model is more sensitive to returns than to either the risk or correlation factors; therefore, the time-varying returns are associated with the inconsistency between the ex-ante and ex-post results. These results support the graphical results in Figure 3A.

F. Robustness Check

In order to determine whether the Markowitz model results presented here are robust to a less parameter sensitive mean-variance optimization model, we compare the Markowitz results to the Sharpe optimization model results. Figure 1A–D shows both the Sharpe and Markowitz efficient frontier sets. These figures illustrate that the two models generate almost identical ex-ante risk-return efficient frontiers, with the Markowitz results slightly dominating the Sharpe results in most cases. Both efficient frontiers clearly dominate the ex-ante individual futures contracts. In Table VII, we present the weights of the asset components for the Sharpe optimal portfolios across time for the high-risk portfolios. (Due to space limitations, we do not present the weights of the components assets for the low- and medium-risk levels for the Sharpe results.) The portfolios in Table VII> are very similar to the Table III portfolios (the Markowitz results) in terms of the component assets and their weights. In addition, Figure 3B presents the ex-ante versus ex-post Sharpe optimal portfolios for different risk levels. The results in Figure 3B are also similar to the Markowitz results in Figure 3A. Therefore, we conclude that our results are invariant to the use of the Sharpe versus Markowitz models. Because the difference between the Sharpe and Markowitz models is that the former ignores the correlations between individual assets while the later takes them into consideration, this also shows that the correlations between the pairs of individual instruments are not an important factor affecting stability.