A Stochasstic Model for Hedge Fund Relative Returns

Abstract

A stochastic difference equation of the form Xn = AnXn-1 + Bn is proposed to model the annual returns Xn of a hedge fund relative to other funds in the same strategy group in year n, and is fit to data from the TASS database over the period 2000 to 2004. In the proposed model, fAng and fBng are independent sequences of independent and identically distributed random variables,allowing general distributions, with An and Bn independent of Xn¡1, E[Bn] = 0 and E[An] = r, 0 < r < 1. The key model parameters are the year-to-year persistence factor ° and the noise variance vb2 = V ar(Bn). The model was chosen primarily to capture the observed persistence, which ranges from 0.11 to 0.49 across eleven different hedge-fund strategies, according to regression analysis, and the observed stationary variance v2 = V ar(Xn). The constant-persistence normal-noise special case with An = r and Bn (and thus Xn) normal provides a good ¯t for some strategies, but not for others, largely because in those other cases the observed relative-return distribution has a heavy tail. The heavy-tail case is successfully modelled within the same general framework in two ways: first, by a constant-persistence stable-noise model, in which Bn (and thus Xn) has a non-normal stable law (having infinite variance) and, second, by stochastic-persistence non-normal-noise models. The model is evaluated by comparing model predictions with observed values of (i) the relative-return distribution, (ii) the lag-1 auto-correlation and (iii) the hitting probabilities of high and low thresholds within a five-year period. These models are appealing because they can involve relatively few parameters, they are tractable, and they fit the limited and somewhat unreliable data reasonably well.