By: Harry Kat & Helder Palaro, Cass Business School, City University London
Published: November 23, 2005
Kat & Palaro’s hedge fund replication technique was first introduced in this 2005 paper. In it, they chronicle the performance-fueled genesis of today’s hedge fund industry and then juxtapose this against the recent trend toward hedge funds as a diversifier, not simply a return enhancer.
This reliance on hedge funds’ diversification properties, however, exposed the industry’s delicate underbelly – the ability to approximate these uncorrelated returns by using various, highly ubiquitous, passive investments or even by just dynamic trading strategies. And if hedge funds could be replicated using highly transparent and liquid passive tools or trading strategies, then investors wouldn’t have to worry about expensive due diligence, lock-ups, style-drift, transparency and capacity limitations – not to mention management & performance fees.
Like many papers of its ilk, this one references Sharpe’s seminal 1992 style analysis research. But Kat & Palaro find that a simple factor model has significant shortcomings:
“The problem when applying the above approach [factor analysis] in a hedge fund context is that in practice we often have little idea how a hedge fund’s returns are actually generated, i.e. which risk factors to use. As a result, factor models typically explain only 25-30% of the variation in hedge fund returns, which compares quite unfavourably with the 90-95% that is typical for mutual funds. Although the procedure works better for portfolios of hedge funds, funds of funds and hedge fund indices, where much of the idiosyncratic risk is diversified away, factor models do not appear to offer a particularly fruitful alternative when looking to replicate hedge fund returns accurately.”
While factor replication techniques focus on picking the right factors, Kat & Palaro’s approach focuses on trading a basic “reserve asset” in the right way. (In their subsequent tool, fundcreator.com, the “reserve asset” is simply an equal weighting of various passive investments, but could be any number of highly liquid assets). Kat is the former head of equity derivatives for Bank of America’s European group. So he is at ease with the creation of exotic options using sophisticated trading rules.
Since investors pick hedge funds not only because of their return characteristics, but also because of their lack of correlation to traditional investments, Kat & Palaro’s methodology yields “clone” funds with very particular correlation characteristics, as well as unique skewedness and kurtosis. But the problem is, these correlations tend to be asymmetric and difficult to model using traditional measures. So to account for these hard-to-model aspects of hedge funds, the authors use “copulas” to describe the unique attributes of the hedge funds they aim to replicate.
Kat & Palaro’s copulas are basically cubic matrices of data that integrate the probability distributions of two variables (basic 2D example at right).
The probability distributions of the independent variable (e.g. two stocks) are represented along two axes while the third axis represents the probability of an outcome given the various possible outcomes of the two aforementioned return distributions. By explicitly modeling the various outcomes for each possible input distribution, we are able to account for correlations that are not constant.
This technique is particularly useful in analyzing the correlations of seemingly uncorrelated investments during times of distress. Obviously, a sole correlation value could not adequately characterize the relationship between two usually-uncorrelated securities if they both fell in tandem only when the market was down over 4% in a month. Copulas allow for the explicit modeling of such asymmetrical behavior.
The rest of the paper is fairly complex (at least to us). Thankfully, Kat & Palaro have written a slightly more readable guide called “Replication and Evaluation of Fund of Hedge Funds Returns”.
As an illustrative example in that paper, the authors construct a “replicated pay-off” that mimics the performance of a hypothetical hedge fund. This return distribution is represented as the combination of an “investor’s portfolio” (defined in subsequent illustrations as 50% S&P500, 50% long-dated US Treasury Bonds) and a “reserve asset” (the core, traded asset). The resulting pay-off matrix can be represented by what is essentially a three-dimensional scatter-plot.
In this diagram, Kat & Palaro (later simply referred to as “KP” – like “TomKat”, or “Brangelina”) show how a “reserve asset” is dynamically traded in such a way as to produce a “replicated pay-off” when mixed with the investor’s portfolio.
Mathematical complexity aside, the bottom line is that to truly replicate hedge funds, KP argues that one needs to replicate not only their performance characteristics, but their correlation to an investor’s existing portfolio. And since you’ll never find the funds (or factors) that will provide the exact correlation characteristics you require, don’t bother looking. Instead, just construct a trading strategy that will deliver the correlation, skewedness and kurtosis you’re looking for.