Maximum Loss for Risk Measurement of Portfolios

Effective risk management requires adequate risk measurement. A basic problem herein is the quantification of market risks: what is the overall effect on a portfolio if market rates change? First, a mathematical problem statement is given and the concept of “Maximum Loss” (ML) is introduced as a method for identifying the worst case in a given scenario space, called “Trust Region”. Next, a technique for calculating efficiently 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 finding.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 Profit” (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 profit and loss functions by 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 the true profit and loss distribution and allow statements about the spread of ML and MP.