A Stochastic Model for Hedge Fund Relative Returns

50 Pages Posted: 14 May 2008 Last revised: 22 Jun 2016

See all articles by Emanuel Derman

Emanuel Derman

Columbia University

Kun Soo Park

Columbia University

Ward Whitt

Columbia University

Date Written: April 24, 2008

Abstract

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 < gamma < 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.

Keywords: hedge funds, stochastic processes

JEL Classification: G12

Suggested Citation

Derman, Emanuel and Park, Kun Soo and Whitt, Ward, A Stochastic Model for Hedge Fund Relative Returns (April 24, 2008). Available at SSRN: https://ssrn.com/abstract=1127050 or http://dx.doi.org/10.2139/ssrn.1127050

Emanuel Derman (Contact Author)

Columbia University ( email )

3022 Broadway
New York, NY 10027
United States
212 854 9883 (Phone)

HOME PAGE: http://www.emanuelderman.com

Kun Soo Park

Columbia University ( email )

500 West 120th Street
New York, NY 10027
United States

HOME PAGE: http://www.columbia.edu/~kp2143

Ward Whitt

Columbia University ( email )

500 West 120th Street
New York, NY 10027
United States

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