Value-at-risk estimation using non-integer degrees of freedom of Student's distribution

ABSTRACT Risk managers often need cumbersome and time-consuming simulation tools to approximate fat-tailed distributions. One such distribution is the Student t-distribution, where the degrees of freedom determine the shape of the distribution. Student’s t-distribution has a nice property in that there is a very simple arithmetic relationship between its kurtosis and its degrees of freedom. The problem is that typical values of kurtosis for financial data correspond to noninteger values of degrees of freedom, which are not available in textbooks. For the purpose of risk management, non-integer values of the degrees of freedom, especially those between four and five, are the most interesting. This paper provides tables of non-integer values of degrees of freedom and demonstrates the importance of this application for risk management.