Aggregation of Dependent Risk with Specific Marginals
by the Family of Koehler-Symanowski Distributions
Paola Palmitesta Corrado Provasi
Department of Quantitative Methods Department of Statistical Sciences
University of Sienna, Italy University of Padua, Italy
June, 2004
Preliminary Version
Abstract
This paper studies the family of Koehler and Symanovski multivariate distributions
with specific marginals, as skewed Student t, generalized secant hyperbolic and gener-
alized exponential power distributions, in order to model financial returns and measure
dependent risks. This family of distributions can be specified using the cumulative dis-
tribution function adding interaction terms to the case of independence. Moreover, it
can be also derived using a particular transformation of independent gamma functions.
The advantage of using this distributions respect to others lies in the opportunity to
model complex dependence structures among subsets of marginals, as shown with a
Monte Carlo study, and to aggregate dependent risks of some market indices.
Keywords: Asset Returns, Risk Management, Skew Marginals, Monte Carlo Simula-
tion, IFM method.
1 Introduction
Many problems in Finance, including risk management, optimal asset allocation and
derivative pricing, require an understanding of the volatility and correlations of asset
returns. In these cases it can be necessary to represent empricial data with a parametric
distribution. In literature many distributions can be found able to model univariate
data, but they cannot be easily extended to represent multivariate populations. In this
context, the most used multivariate distribution in the aggregation of dependent risks
is the normal distribution, which nevertheless has the drawback to be not very flexible
and in many cases not appropriate to model returns.
An important tool to account of individual risks, the copula function, has been
introduced in finance by Embrechts, McNeil and Straumann (1999, 2002), who have
explained some essential concepts of dependence which have affected the construction
of methods for the risk management industry (Embrechts, Lindskog, McNeil, 2003;
Rosenberg, Schuermann, 2004). According to these specifications, this paper studies
a method to obtain an analytical form of the joint distribution of asset returns in a
portfolio based on the family of distributions introduced by Koehler and Symanowski
(KS) (1995). This family of distributions is defined by the cdf adding interaction terms
to the case of independence and it permits to specify arbitrary marginals. Moreover, it
can be also derived using a particular transformation of independent exponential and
gamma random variables. The advantage of using this distribution respect to others lies
in the opportunity to model complex dependence structures among subsets of marginals,
as shown with a Monte Carlo study, and to aggregate dependent risks.