by Daniel Hanson, with contributions by Steve Su (author of the GLDEX package). Part 1 of a series.
As most readers are well aware, market return data tends to have heavier tails than that which can be captured by a normal distribution; furthermore, skewness will not be captured either. For this reason, a four parameter distribution such as the Generalized Lambda Distribution (GLD) can give us a more realistic representation of the behavior of market returns, including a more accurate measure of expected loss in risk management applications as compared to the normal distribution.
This is not to say that the normal distribution should be thrown in the dustbin, as the underlying stochastic calculus, based on Brownian Motion, remains a very convenient tool in modeling derivatives pricing and risk exposures (see earlier blog article here), but like all modeling methods, it has its strengths and weaknesses.
As noted in the book Financial Risk Modelling and Portfolio Optimization with R (Pfaff, Ch 6: Suitable distributions for returns) (publisher information provided here), the GLD is one of the recommended distributions to consider in order “to model not just the tail behavior of the losses, but the entire return distribution. This need arises when, for example, returns have to be sampled for Monte Carlo type applications.” The author provides descriptions and examples of several R packages freely available on the CRAN website, namely Davies, fBasics, gld, and lmomco. Another package, also freely available on CRAN, is the GLDEX package, which is the package we will use in the current article. It contains a rich offering of functions and is well documented. In addition, the author of the GLDEX package, Dr Steve Su, has kindly provided assistance in the writing of this article. He has also published a very useful and related article in the Journal of Statistical Software (JSS) (2007), to which we will refer in the discussion below.
Brief Background on the GLD
The four parameters of the GLD are, not surprisingly, λ1, λ2, λ3, and λ4. Without going into theoretical details, suffice it to say that λ1 and λ2 are measures of location and scale respectively, while the skewness and kurtosis of the distribution are determined by λ3 and λ4.
Furthermore, there are two forms of the GLD that are implemented in GLDEX, namely those of Ramberg and Schmeiser (1974), and Freimer, Mudholkar, Kollia, and Lin (1988). These are commonly abbreviated as RS and FMKL. As the FMKL form is the more modern of the two, we will focus on it in the discussion that follows. An additional reference frequently cited in the literature related to the GLD in finance is the paper by Chalabi, Scott, and Wurtz, freely available here on the rmetrics website.
Fitting a Time Series of Financial Returns to the GLD
As Steve Su points out in his 2007 JSS article on the GLDEX package (see link above), there are three basic steps that are useful in determining the quality of the GLD fit. The first two, as we shall see, can be competing objectives in determining the fit. The GLDEX package provides functionality for each.