The contribution of the third and fourth moments in explaining the return-generating process in futures markets remains unresolved. This study attempts to resolve this issue by using a four-moment model and by sampling 28 futures contracts and nine market proxies. Such sampling provides wide representation of futures markets and lends a high degree of robustness to the results. Our results show that the second, third and fourth moments are all important in explaining futures returns. Evidence from regression tests show increases in explanatory power as the third and fourth moments are included. The results are robust to the market proxy used.
The nature of the return-generating process in futures markets remains an unresolved issue. Although the distribution of futures returns is well known, empirical tests that examine risk premia in futures markets yield conflicting results. Many studies in the past fail to detect evidence of risk premia in futures markets. For instance, studies that employ a traditional Capital Asset Pricing Model (CAPM) or an Arbitrage Pricing Framework (APT), often fail to detect significant risk premia in futures prices (see, e.g., Dusak, 1973, Baxter et al., 1985 and Ehrardt et al., 1987), and those that use a pricing framework that departs from traditional models provide mixed results. For example, Raynauld and Tessier (1984) and Chang (1985) detect significant risk premia while Junkus (1991) does not. To shed new evidence on the return-generating process in futures markets, we examine 28 futures contracts and use nine different proxies for the market. The wide sampling of futures contracts ensures comprehensive representation of all futures markets, while the use of different market proxies ensures the results are robust.
Early tests of the return-generating process were mainly confined to equity markets. Such tests followed the development of the widely accepted two-moment CAPM, developed by Sharpe (1964), Lintner (1965), Mossin (1966), and later Black (1972). Tests of the CAPM are conducted by Friend and Blume (1970), Black et al. (1972), and Basu (1977), among others. These tests provide much insight into the functioning of financial markets. However, violations of the linear pricing kernel proposed by the CAPM found by some (see, e.g., Chen et al., 1986, Fama and French, 1992 and Fama and French, 1993), prompted the search for alternative pricing models. Suggestions included modifications to the CAPM, such as the addition of higher moments (e.g., Arditti and Levy, 1975).
This study provides a four-moment extension to the two-moment CAPM, much in the spirit of Kraus and Litzenberger's (1976) three-moment extension to the standard CAPM. Fang and Lai (1997) and Dittmar (1999) present four-moment extensions in other contexts. The basic inferences of our model are consistent with these models.
In this study we examine the importance of coskewness and cokurtosis in
explaining futures returns. Studies in futures markets that examine risk premia
using the traditional CAPM framework often fail to detect significant risk premia
in futures prices. Research that relies on models that depart from a traditional
framework find mixed results. Given the current state of the literature, we attempt
to provide insight into the return-generating process in futures markets by using a
four-moment CAPM model.
Although it is almost impossible to know the exact form of the return
generating relationship, the preponderance of the evidence from this study provides
strong support for the inclusion of terms that represent coskewness and
cokurtosis. In the regressions where the excess return on the future is employed as
the dependent variable and regressed against the proxies for the second, third, and
fourth moments, we find that the explanatory power increases as terms for
coskewness and cokurtosis are included. To illustrate, when we employ S&P 500
index futures as the market proxy with the full sample of 28 futures, the average
R2 values increase from 0.07 for the two-moment regression to 0.13 for the
three-moment regression and to 0.22 for the four-moment regression. Dispute over
the appropriate market proxy for futures prompts us to use nine different market
proxies, from weighted to nonweighted futures indexes. Additionally, we also
employ an all-equity index as a market proxy. The results are robust to the market
proxy used. Criticisms of the two-pass methodology are addressed by using
corrective techniques. The corrections applied do not change the results. We also
check the specification of the model and find the model to be well specified. The
overall results confirm that coskewness and cokurtosis are important in explaining
the return-generating process in futures markets. Of practical importance is the
implication for risk computation on futures instruments. These calculations must
account for the pricing of coskewness and cokurtosis.
