Mathematics of Risk Measures. And the measures of the Basel Committee.

Abstract

Risk measurement within financial institutions remains of the utmost importance in practice. In the last few years it has become evident that a mathematical model should consider two steps of the risk measurement procedure; the estimation of the loss distribution and the construction of a risk measure that summarizes the risk of a position. In 1997 Artzner et all. gave rise to a whole new theory concerning risk measures with their axiomatic approach to coherent risk measures. In my thesis I extend this axiomatic approach to include not only mathematical properties of risk measures but also their statical properties. I will treat two important classes of risk measures and argue that a risk measure should belong to both these classes in order to deal with all aspects of the risk measurement procedure. I will discuss Value-at-Risk, Expected Shortfall en Expectile Value-at-Risk and compare these to the risk measures introduced by the Basel Committee.