A Comparison of VaR and CVaR Constraints on Portfolio Selection with the Mean-Variance Model

Authors:
Gordon J. Alexander;Alexandre M. Baptista
Affiliations:
Department of Finance, Carlson School of Management, University of Minnesota, Minneapolis, Minnesota 55455;Department of Finance, School of Business, The George Washington University, Washington, D.C. 20052
Venue:
Management Science
Year:
2004

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Abstract

In this paper, we analyze the portfolio selection implications arising from imposing a value-at-risk (VaR) constraint on the mean-variance model, and compare them with those arising from the imposition of a conditional value-at-risk (CVaR) constraint. We show that for a given confidence level, a CVaR constraint is tighter than a VaR constraint if the CVaR and VaR bounds coincide. Consequently, a CVaR constraint is more effective than a VaR constraint as a tool to control slightly risk-averse agents, but in the absence of a risk-free security, has a perverse effect in that it is more likely to force highly risk-averse agents to select portfolios with larger standard deviations. However, when the CVaR bound is appropriately larger than the VaR bound or when a risk-free security is present, a CVaR constraint "dominates" a VaR constraint as a risk management tool.