Providing against the worst: Risk capital for worst case scenarios

Abstract

Purpose

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 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 for both linear and non‐linear portfolios.

Design/methodology/approach

This is a synthesis of existing theorems. Results are established by means of counterexamples. The worst case based risk‐return management strategy is presented as a case study.

Findings

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. The paper gives a new example of Value at Risk violating subadditivity, using a portfolio of simple European options. For non‐linear portfolios, Maximum Loss need not even approximately be explained by the sum of Maximum Loss contributions of the individual risk factors. Finally, is proposed a strategy of risk‐return management with Maximum Loss.

Research limitations/implications

The paper is restricted to elliptically distributed risk factors. Although Maximum Loss can be defined for more general continuous and even discrete distributions of risk factor changes, the paper does not address this matter.

Practical implications

The paper proposes an intuitive, computationally easy way how to improve average returns of linear portfolios while reducing worst case losses.

Originality/value

One is a synthesis of existing theorems. The counterexample establishing result is the first example of a portfolio of plain vanilla options violating Value at Risk subadditivity, an effect hitherto only known for portfolios of exotic options. Furthermore, the strategy of risk‐return management with Maximum Loss is original.

Citation

Breuer, T. (2006), "Providing against the worst: Risk capital for worst case scenarios", Managerial Finance, Vol. 32 No. 9, pp. 716-730. https://doi.org/10.1108/03074350610681934