# Statistiske metoder i finansiel risikostyring: Anvendelse af ekstremværditeori og copulaer i estimation af Value-at-Risk

Jonas Hal

Student thesis: Master thesis

## Abstract

Value at Risk (VaR) is one of the most used risk measures today. The validity of a VaR measure is closely connected with how well the statistical model underlying VaR captures the structure in the empirical data. In practical application, it is very common to base VaR on the assumption that all the risk factors changes in a given portfolio are Gaussian and that their joint distribution function is Gaussian too. The Gaussian assumption makes the calculation easy and it is often possible to nd an analytical solution which makes VaR calcu- lation straightforward. But empirical ndings suggest that this is not a good approximation. The reason for this is that nancial data are characterized as being leptokurtic, skewed and heteroskedastic. And often one observes more extreme observations in nancial data than can be explained by the Gaussian distribution and VaR estimates are thereby underesti- mated. That is the capital reserve based on VaR will be too small. Another point is that the Gaussian distribution is strictly symmetrical thus making it unsuitable for skewed empirical distribution. Chapter one is an introduction to VaR and how it is calculated. In order to calculate VaR you need to have a joint probability function for the risk factors of your portfolio as well as a speci cation of the loss operator. A loss operator is a function that maps changes in risk factors to loss of the portfolio. While the selection and estimation of the statistical model underlying VaR is the focus of chapter 3 and 4, chapter 2 looks at how to transform the joint probability distribution for the risk factors to a probability function for portfolio losses. I show that in certain special cases it is possible to get an analytical expression for the loss distribution but most often given the special characteristic of nancial data one has to use Monte Carlo simulation in order to get a VaR estimate. This chapter also discuss VaR as a relevant risk measure. The statistical model underlying VaR will be based on a copula approach. A copula is a function which couples given marginal distributions to form a joint distribution function. Chapter 4 illustrate the dependence structure of 3 di¤erent copula functions, namely the Gaussian, t and Gumbel copula. Besides introducing a whole new array of dependence struc- tures, the interesting aspect of a copula is that it makes it possible to separate the marginal distribution from the dependence structure, thus splitting the model speci cation in two. From a mathematical viewpoint the copula approach introduces nothing new. From a sta- tistical viewpoint, however the selecting and estimation process are eased, since we can pick the marginal distribution function independently from each other and from the dependence structure. I show how the dependence structure can be estimated even without knowledge of the parametric form for the marginal distributions functions. Of course we still have to select a copula function and in practical application here lies the challenge. I also show - via a small monte carlo simulation study - that one could choose the copula with the lowest AIC from a list of copulas. The use of copula theory allows us to look at the marginal distributions individually.With respect to VaR calculation, we are mainly interested in the tail of the marginal distributions. But since most observation lies in the centre (around the mean), the estimation is of course most accurate here. This inherent problem can be overcome by looking at extreme value theory. More speci cly a theory known as Peaks over Threshold (POT) allows us to estimate the tail distribution solely. In chapter 3 I show that POT is based on classical extreme value theory and under some weak assumptions the theory tells us the tail approximate, a distribution known as Generalized Pareto Distribution (GPD). The approximation gets better the further out in the tail one is looking. Of course the futher out in the tail the number of observations will be scarce. One must address this dilemma in choosing the starting point for the tail distribution and this issue will also be dealt with in chapter 3. The main contribution from chapter 3 is that it is possible under some very general assumptions to use GPD as an approximation for the tail distribution. The copula/POT approach is not just another assumption on the parametric form of the underlying model but a new more exible framework which can incorporate the special char- acteristics of nancial assets. To illustrate this approach, I use empirical data to estimate a daily VaR based on a copula approach. The marginal distribution is modelled by a mix distri- bution where the tail is modelled by a GPD and for the rest, I used a Gaussian distribution. I compared VaR based on this model by VaR based on the traditional variance-covariance approach. The ndings suggest that the copula/POT approach gives a more accurate VaR even though the variance-covariance method wasn t that far o¤ in the back testing period. Other ndings suggest that the speci cation of the copula is important especially for the accuracy of the model.