Quadratic Maximum Loss for Risk Measurement of Portfolios

E ective risk management requires adequate risk measurement. A basic problem herein is the quanti cation of market risks: what is the overall e ect on a portfolio if market rates change? The rst chapter gives a brief review of the standard risk measure \Value{At{Risk" (VAR) and introduces the concept of \Maximum Loss" (ML) as a method for identifying the worst case in a given scenario space, called \Trust Region". Next, a technique for calculating e ciently ML for quadratic functions is described; the algorithm is based on the Levenberg{Marquardt theorem, which reduces the high{ dimensional optimization problem to a one{dimensional root nding. Following this, the idea of the \Maximum Loss Path" is presented: repetitive calculation of ML for a growing trust region leads to a sequence of worst cases, which form a complete path. Similarly, the paths of \Maximum Pro t" (MP) and \Expected Value" (EV) can be determined; the comparison of them permits judgements on the quality of portfolios. These concepts are also applicable to non{quadratic portfolios by using \Dynamic Approximations", which replace arbitrary pro t and loss functions with a sequence of quadratic functions. Finally, the idea of \Maximum Loss Distribution" is explained. The distributions of ML and MP can be obtained directly from the ML and MP paths. They lead to lower and upper bounds of VAR and allow statements about the spread of ML and MP.