Maximum Loss for Measurement of Market Risk

Effective risk management requires adequate risk measurement. A basic prob¬ lem herein is the quantification of market risk: what is the overall effect on a portfolio's value if the market rates change? To answer this question, two funda¬ mental difficulties have to be managed: first, market rates behave randomly and are correlated. Second, portfolio structures are high-dimensional and typically nonlinear. The established risk measurement techniques can be divided into two categories. The purely stochastic approaches are based on the portfolio's profit and loss (P&L) distribution. The most popular method in this class is Value-at-Risk (VaR), which typically represents the 1 or 5 percent quantile of the P&L distri¬ bution. The Maximum Loss (ML) methodology is a member of the second category, where risk is quantified by the value of some worst case scenario. Many of these worst case based measures examine a finite set of scenarios and do not take account of correlations (e.g. stress testing). The more elaborated methods determine the worst case scenario by solving constrained minimization problems, where the set of feasible scenarios is generally defined by the stochastic characteristics of the market rates. Compared to other worst case techniques, the Maximum Loss methodology uses a very particular choice of feasible domains: the so-called trust regions cover a certain percentage of all future outcomes and their shape reflects the corre¬ lation structure of the market rates. Furthermore, ML uses a polynomial time algorithm to identify the global minimum, whereas other techniques employ rudi¬ mentary optimization procedures. The fact that the Maximum Loss computation is based on a fast and reliable algorithm allows to solve the optimization problem repeatedly, which leads to new insights into the risks of nonlinear portfolios. This thesis is divided into five chapters: the first chapter derives the standard assumptions of risk management from the basics of financial calculus. Some of the established risk measurement techniques are presented in a unifying framework and their basic properties are discussed. The chapter ends with the introduction of the Maximum Loss concept. Chapter 2 treats the modelling of risk factors in the ML approach. We discuss the transformation of market rates into more abstract risk factors, which are used

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