Providing Against the Worst: Risk Capital for Worst Case Scenarios

Abstract

Maximum Loss was one of the risk measures proposed as alternatives to Value at Risk following criticism that Value at Risk is not coherent. Although the power of Maximum Loss as a theoretically sound and practically useful risk measure is recognised for non-linear portfolios, there are arguments that for linear portfolios Maximum Loss does not convey more information than Value at Risk. This paper argues for the usefulness of Maximum Loss even for linear portfolios.

In particular, (1) we show that for linear portfolios under elliptic distributions Maximum Loss is proportional to Value at Risk, and to Expected Shortfall, with the proportionality constant not depending on the portfolio composition, (2) we give a new example of Value at Risk violating subadditivity, using a portfolio of simple European options, (3) we give an example that for non-linear portfolios Maximum Loss need not even approximately be explained by the sum of Maximum Loss contributions of the individual risk factors, (4) we propose an intuitive, computationally easy way how to improve average returns of linear portfolios while reducing worst case losses.