Conditional value-at-risk: optimization approach. (English) Zbl 0989.91052

Uryasev, Stanislav (ed.) et al., Stochastic optimization: Algorithms and applications. Conference, Univ. of Florida, Tallahassee, FL, USA, February 20-22, 2000. Dordrecht: Kluwer Academic Publishers. Appl. Optim. 54, 411-435 (2001).
Summary: 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 coherent 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.
For the entire collection see [Zbl 0964.00055].


91G70 Statistical methods; risk measures
91G10 Portfolio theory
90C15 Stochastic programming