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Abstract
We propose a stochastic difference equation of the form X
n
= A
n
X
n−1 + B
n
to model the annual returns X
n
of a hedge fund relative to other funds in the same strategy group in year n. We fit this model to data from the TASS database over the period 2000 to 2005. We let {A
n
} and {B
n
} be independent sequences of independent and identically distributed random variables, allowing general distributions, with A
n
and B
n
independent of X
n−1, where E[B
n
] = 0. This model is appealing because it can involve relatively few parameters, can be analysed, and can be fitted to the limited and somewhat unreliable data reasonably well. The key model parameters are the year-to-year persistence factor γ ≡ E[A
n
] and the noise variance . 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. The constant-persistence normal-noise special case with A
n
= γ 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. We show that the heavy-tail case can also be successfully modelled within the same general framework. 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 the five-year period.
Acknowledgement
Ward Whitt was partially supported by NSF grant DMI-0457095.