'Maximal' Convenience Yield Model Implied by Commodity Futures

We develop a three-factor Gaussian model of commodity spot prices, convenience yields and interest rates, which extends previous research (e.g., Brennan (1991), Gibson and Schwartz (1990), Schwartz (1997), Ross (1997), Schwartz and Smith (2000)) in two ways. First, the model is maximal, and thus nests all previously proposed specifications. Second, we allow for time-varying risk-premia. We show that previous models have implicitly imposed unnecessary restrictions on the unconditional correlation structure of commodity prices, convenience yields and interest rates. Using data on copper, crude oil, silver and gold commodity futures, we empirically estimate the model using maximum likelihood. We find both features of the model to be economically and empirically significant. In particular, we find strong evidence for spot-price level dependence in convenience yields, which implies mean-reversion in spot prices under the risk-neutral measure, and is consistent with the "theory of storage." We also find evidence for time-varying risk-premia, which implies mean-reversion of commodity prices under the physical measure albeit with different strength and long-term mean. The model thus disentangles the different sources of mean-reversion in spot commodity prices. The results suggest that the relative contribution of both effects (level dependent convenience yield vs. time-varying risk-premia) to mean reversion depends on the nature of the commodity. We find that for metals like gold and silver, negative correlation between risk-premia and spot prices explains most of the mean reversion, whereas for oil almost all of the mean-reversion in spot prices is attributable to convenience yields.