Risk measurement with maximum loss

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 set of scenarios, called ‘Trust Region’. Next, a technique for calculating efficiently the Maximum Loss 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 growing trust regions leads to a sequence of worst case scenarios, which form a complete path; similarly, the path of ‘Maximum Profit’ can be determined. Finally, all these concepts are applied to nonquadratic portfolios: so-called ‘Dynamic Approximations’ are used to replace arbitrary profit and loss functions by a sequence of quadratic functions, which can be handled with efficient solution procedures. A description of the overall algorithm rounds off the discussion of nonlinear portfolios.