Fitting probability distributions to market risk and insurance risk

Abstract

Determining the parametric VaR approach is very important in establishing the probability distribution of a risk factor. We assume that a normal distribution is symmetric; however, it has some limitations. This distribution is used for modelling asymmetric data or data that have only positive values, such as insurance claims. The aim of the paper is to find the best probability distribution for stock exchange index returns and for insurance claims. The paper is structured as follows. Firstly, we describe the typical probability distributions used in finance, namely normal, Student, logistic, gamma, exponential and lognormal distribution, and the methods of verification. Subsequently, parameters of the distribution types are estimated via the maximum likelihood method, and after that we calculate the value at risk. The VaR is calculated even though the time series do not correspond to the stated types of proba-bility distribution; nevertheless, we calculate the value at risk for all the stated types of probability distribution because it is apparent that large mistake can arise if an incorrect type of probability distribution is used.