Predictability and the Dynamics of Long Forward Rates

Abstract

Distantly maturing forward rates represent the markets long term (risk neutral) expectations about interest rates. As such, they are the fundamental ingredient of the pricing kernel. In most equilibrium models, interest rates mean revert, and so long forward rates are asymptotically constant. However, from US Treasury STRIPs data, forward rates slope increasingly downwards, and do not attenuate in volatility, as maturity increases beyond about 13 years. This has recently been documented formally by Brown and Schaefer (Working Paper, London Business School, 1999), though it also seems to well known to finance practitioners. Our aim is to build and test an equilibrium model, which incorporates this behavior. The key to our analysis, is first to show that most of the volatility in long forward rates comes from a predictable factor which is mean reverting on a time scale of a few weeks. Our techniques for this are first to apply the Variance Ratio test of Lo and MaKinlay (Review of Financial Studies, Vol 1, Number 1, pp 41 - 66, 1988), and then to implement the filtering procedure of Lo and Wang (Journal of Finance, Vol 50, pp 87 - 129, 1995), which extracts a second, non-predictable, factor from the forward rate time series, to which the forward rates mean revert. To reinforce our results, we also apply these procedures to the returns of US Treasury Bond futures, and extract essentially the same predictable component. Then, we show that the mean reverting aspect of the forward rates, is absent when we transform to risk neutral probabilities. This is natural, otherwise the market would obviate the predictability. We also establish this directly, by analyzing our T Bond futures data, and noting that for futures, the risk neutral dynamic can be extracted directly, since its drift is just minus the initial slope of the futures term structure, and we see that it does not have this predictability. Armed with these insights, we formulate our equilibrium model, which is based on the classical Vasicek Model, and has a single predictable factor. This model applies to forward rates with maturity beyond 13 years, and any interest rates with earlier maturity are irrelevant to the model. In our model, the forward rates mean revert with respect to the objectively realized probabilities, but not with respect to the risk neutral probabilities, and so the consequences of mean reversion are absent in the cross sectional shape, and the volatility term structure, of these forward rates.