On Polyhedral Coherent Risk Measures and Portfolio Optimization Problems

Abstract

Introduction. The problem of decision-making under risk and uncertainty lies in the use of adequate criteria for assessing their optimality, in particular, in an adequate risk assessment. Various functions are known that are used as risk measures. For technical systems, the probability of an accident (failure) is used, in insurance – the probability of bankruptcy, in finance – Value-at-Risk, etc. At present, the concept of a coherent risk measure (CRM), in which its basic properties are postulated, is widely recognized. The paper considers CRMs and their subset, the polyhedral CRMs (PCMRs), which have attractive properties and contain a number of important risk measures. Such risk measures are well defined on complete information about the stochastic distributions of random variables. However, applications usually contain only partial such information from observational data. This only allows one to describe the stochastic distribution by an ambiguity set (AS). For such a case, robust PCMR constructions intended for risk assessment at AS are considered in the paper. The computation of such PCRM constructions in the form of linear programming problems (LP) is described. To demonstrate the use of the PCRM apparatus, the problems of portfolio optimization on reward-risk ratio are considered, where reward and risk are estimated by the average return and some PCRM respectively for known stochastic distributions, and by their robust constructions under uncertainty with AS. It is described how in both these cases the portfolio optimization problems are reduced to appropriate LP problems. The purpose of the paper is to describe the PCRM apparatus for assessing risks under uncertainty with AS and demonstrating the effectiveness of its application to linear problems on the example of portfolio optimization problems. Results. The use of the PCRM apparatus for the case of uncertainty with AS in the form of appropriate robust constructions and their application to portfolio optimization problems on reward-risk ratio is described. The conditions under which these portfolio problems are reduced to the corresponding LP tasks are formulated. Conclusions. The PCRM apparatus can be effectively applied to linear optimization problems under uncertainty with AS, which is demonstrated by the example of portfolio optimization problems. The reduction of portfolio problems to LP problems allows one to effectively solve them using standard methods. Keywords: coherent risk measure, polyhedral coherent risk measure, CVaR, ambiguity set, portfolio optimization, linear programming problem.