Risk management: an analysis of the low-tail behavior of high frequency data for computing value at risk

Journal of the Academy of Business and Economics, March, 2004 by J. Samuel Baixauli, Susana Alvarez

ABSTRACT

This paper deals with the analysis of the low-tail behaviour of a sample. This issue is very important in the financial field, where the fat-tailedness characterizes many series. In particular, the tail behaviour affects the estimates of expected losses. The aim of this paper is to analyze the performance of the bootstrap methodology to implement goodness-of-fit tests for a certain proportion of a sample based on the modified Cramer-von Mises statistics, assuming the presence of unknown parameters. An empirical application is carried out using the SP500 index for the period from January 1994 to December 2002.

1. INTRODUCTION

One common assumption in the financial literature is the hypothesis of normality. However, a feature which stands out most prominently is that the kurtosis of financial series is much larger than the normal value, especially for the daily series. This reflects the fact that the tails of the distributions of these series are fatter than the tails of the normal distribution. Put differently, large observations occur much more often than one might expect for a normally distributed variable (see, for example, de Vries, 1994, for foreign exchange markets and Campbell et al., 1997, for stock market).

Given the fat-tailedness observed for many financial time-series, the use of the normal distribution is questionable (Praetz, 1972; Kon, 1994; Peiro, 1994, between others). The description of the distribution of the daily stock-returns is a subject of continuous analysis in the financial literature. However, less has been said about the analysis of the behaviour of a certain proportion of the sample. In particular, the analysis of the low-tail behaviour of the sample is very important for risk management. Some sources of risk affecting the performance of financial policies include interest rate risk, exchange rate risk and credit risk. Many different risk measures have been proposed and have been used for investment decisions, supervisory decisions, risk capital allocation, external regulation and efficient bank operations. In the context of asset management, the most popular downside-risk measure is the Value-at-Risk (VaR). The tail behaviour of the distribution of asset returns is relevant for computing VaR due to that the calculation of VaR deals with the estimation of the lower quantiles of a distribution.

The aim of this paper is to analyze the performance of the bootstrap methodology to implement goodness-of-fit tests of a certain proportion of a sample with a Cramer-von Mises type statistic, when the distribution function is known completely except by a vector of unknown parameters, and the test statistic is constructed with the standarized residuals of a linear regression model. The paper is organized as following. Section 2 describes the modified Cramer-von Mises test statistic for censored data. Section 3 presents a simulation experiment to observe the performance of the designed bootstrap procedures to carry out the goodness-of-fit tests with the modified Cramer-von Mises test statistic. In Section 4, empirical evidence is provided using the SP500 index from January 1994 to December 2002. Section 5 concludes.

2. GOODNESS-OF-FIT TESTS FOR CENSORED DATA

It is well-known the fact that, if it is wished to test the null hypothesis [H.sub.0]: F(x)=[F.sub.o](x), where [F.sub.o](x) is a completely specified distribution function, the traditional Cramer-von Mises test statistic (hereafter, [W.sup.2]) can be used. The asymptotic distribution of [W.sup.2sub.n] is distribution free. Pettitt and Stephens (1976) modified the [W.sup.2.sub.n] so that it could be used to test the goodness of fit of a censored data; that is, to test the goodness of fit of a certain proportion of the random sample when the distribution function is known completely, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. Also, they obtained the asymptotic distribution of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] from a straightforward extension of those of Anderson and Darling (1952). However, to assume that the null distribution function is known completely is a very restrictive assumption. For this reason, numerous articles have appeared in the literature related with the asymptotic properties of the Cramer-von Mises type statistics when the null distribution function is F(x,[theta]), a known distribution function but [theta] is a vector of unknown parameters,

[[??]W.sup.2.sub.n]. A comprehensive analysis of the basic theory about the goodness of fit tests when parameters are estimated has been given by Durbin (1973). His main result is that the limit distribution function is no longer free but it depends on the postulated null distribution and, in general on [theta]. It should be also noted that the tabulated asymptotic critical values (Shorack and Wellner, 1986) are inappropriate when parameters of the hypothesized distribution are estimated from the data used for the test. Pettitt (1976) applied Durbin's (1973) results when [theta] is estimated by maximum likelihood procedure from censored data and he derived the modified version of [[??].sup.2.sub.n], denoted [.sub.q][[??].sup.2.sub.n], and its asymptotic properties.