A Stochastic Model for Hedge Fund Relative Returns

A stochastic difference equation of the form X_n = A_n X_{n-1}+B_n is proposed to model the annual returns X_n 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, {A_n} and {B_n} are independent sequences of independent and identically distributed random variables, allowing general distributions, with A_n and B_n independent of X_{n-1}, E[B_n] = 0 and E[A_n] = gamma, 0 lt; gamma lt; 1. The key model parameters are the year-to-year persistence factor gamma and the noise variance sigma_b^2 = Var(B_n). 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 sigma^2 = Var(X_n). The constant-persistence normal-noise special case with A_n = gamma and B_n (and thus X_n) normal provides a good fit 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 B_n (and thus X_n) 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