Conditional Value-at-Risk: Optimization Approach

  • Stanislav Uryasev
  • R. Tyrrell Rockafellar
Part of the Applied Optimization book series (APOP, volume 54)


A new approach for optimization or hedging of a portfolio of finance instruments to reduce the risks of high losses is suggested and tested with several applications. As a measure of risk, Conditional Value-at-Risk (CVaR) is used. For several important cases, CVaR coincides with the expected shortfall (expected loss exceeding Values-at-Risk). However, generally, CVaR and the expected shortfall are different risk measures. CVaR is a coh erent risk measure both for continuous and discrete distributions. CVaR is a more consistent measure of risk than VaR. Portfolios with low CVaR also have low VaR because CVaR is greater than VaR. The approach is based on a new representation of the performance function, which allows simultaneous calculation of VaR and minimization of CVaR. It can be used in conjunction with analytical or scenario based optimization algorithms If the number of scenarios is fixed, the problem is reduced to a Linear Programming or Nonsmooth Optimization Problem. These techniques allow optimizing portfolios with large numbers of instruments. The approach is tested with two examples: (1) portfolio optimization and comparison with the Minimum Variance approach; (2) hedging of a portfolio of options. The suggested methodology can be used for optimizing of portfolios by investment companies, brokerage firms, mutual funds, and any businesses that evaluate risks. Although the approach is used for portfolio analysis, it is very general and can be applied to any financial or non-financial problems involving optimization of percentiles.


Conditional Value-at-Risk risk management portfolio optimization efficient frontier stochastic programming linear programming options 


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Copyright information

© Springer Science+Business Media Dordrecht 2001

Authors and Affiliations

  • Stanislav Uryasev
    • 1
  • R. Tyrrell Rockafellar
    • 2
  1. 1.Department of Industrial and Systems EngineeringUniversity of FloridaGainesvilleUSA
  2. 2.Department of Applied MathematicsUniversity of WashingtonSeattleUSA

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