Distributional Replication

A function which transforms a continuous random variable such that it has a specified distribution is called a replicating function. We suppose that functions may be assigned a price, and study an optimization problem in which the cheapest approximation to a replicating function is sought. Under suitable regularity conditions, including a bound on the entropy of the set of candidate approximations, we show that the optimal approximation comes close to achieving distributional replication, and close to achieving the minimum cost among replicating functions. We discuss the relevance of our results to the financial literature on hedge fund replication; in this case, the optimal approximation corresponds to the cheapest portfolio of market index options which delivers the hedge fund return distribution.

Factors in Time: Fine-Tuning Hedge Fund Replication

Hedge fund replication has become a cottage industry in investing. Among the most popular hedge fund replication frameworks are factor models based on ordinary least squares (OLS) regression, a development that is no doubt due to its simplicity and familiarity among investment practitioners. Despite their widespread use, the OLS regression-based factor models that form the basis for many hedge fund replication programs are often overfitted to a single sample, severely undercutting their predictive effectiveness. As a remedy to the latter shortcoming, in this article the authors apply the regularization method known as “ridge regression” to the replication of hedge fund strategies. Ridge regression works by formally imbuing a regression with additional bias in exchange for a reduction in the variance between training and test samples. Using a simple yet robust methodology, the authors show how to dynamically calibrate the predictively optimal level of bias without significantly reducing the backward-looking explanatory power of a given model. In doing so, the authors demonstrate that ridge regression can help produce generalizable models that are useful in both the ex post risk analysis and ex ante replication of hedge fund strategies.

Hedge Fund Replication

In theory, analogous to equity indices, hedge fund indices can provide broad exposure to hedge funds in a cost-effective manner. In practice, however, hedge fund indices are difficult to implement because direct investment in hedge funds is impractical. Unlike equities, hedge funds are not traded on liquid secondary markets and are often closed to new investment. A solution is hedge fund replication, which, rather than require direct investment in hedge funds, synthetically recreates hedge fund index returns by investing in portfolios that are exposed to the same underlying economic factors that drive hedge fund returns. This approach provides broad, cost-effective, hedge fund exposure and avoids the practical problems associated with direct hedge fund investment. As a consequence, such hedge fund clones exhibit lower tracking error and substantially higher raw and risk-adjusted returns than both investible and noninvestible hedge fund indices.