Improving Value at Risk Calculations by Using Copulas and Non-Gaussian Margins
Financial risk management
Value at risk (VaR) is of central importance in modern financial risk management. Of the various methods that exist to compute the VaR, the most popular are historical simulation, the variance-covariance method and Monte Carlo (MC) simulation. While historical simulation is not based on particular assumptions as to the behavior of the risk factors, the two other methods assume some kind of multinomial distribution of the risk factors. Therefore the dependence structure between different risk factors is described by the covariance or correlation between these factors.
It is shown in [1, 2] that the concept of correlation entails several pitfalls. As a consequence, copulas are proposed to describe the dependence between n variables with arbitrary marginal distributions. A copula is a function C : [0, 1] n -> [0, 1] with certain special properties [3], so that the joint distribution can be written as IP(R1 ≤ r1, . . . ,Rn ≤ rn) = C (F1(r1), . . . , Fn(rn)). F1, . . . , Fn denote the cumulative probability functions of the n variables. In general, a copula C depends on one or more parameters p1, . . . , pk that determine the dependence between the variables r1, . . . , rn. In this sense, these parameters assume the role of correlations.
The second pitfall that arises from the multinomial distribution ansatz is the fat tail problem of the margins [4, 5]. One way of taking extreme values better into account is to assume that the risk factors obey Student distribution patterns instead of Gaussian patterns.
In this thesis we investigate two risk factors only. We model each risk factor independently using a Student distribution and describe their dependence by both the Frank copula [6] and the Gumbel-Hougaard copula. We present algorithms to estimate the parameters of the margins and the copulas and to generate pseudo random numbers due to copula dependence.
Making use of historical data spanning a period of nineteen years, we compute the VaR using a copula-modi?ed MC algorithm. To see the advantage of this method, we compare our results with VaR results obtained from the three standard methods mentioned at the beginning. Based on backtesting results, we Find that the copula method is more reliable than both traditional MC simulation and the variance-covariance method and about as good as historical simulation.
