Monte Carlo methods for mean-risk optimization and portfolio selection

• Published in: Computational management science 2012-02, Vol.9 (1), p.3-29
• Main Authors: Xu, Huifu, Zhang, Dali
• Format: Article
• Language: English
• Subjects: Approximation; Business and Management; Computer simulation; Decision making; Estimators; Expected values; Mathematical analysis; Mathematical models; Methods; Monte Carlo simulation; Numerical analysis; Operations Research/Decision Theory; Optimization; Original Paper; Portfolio management; Programming; Random variables; Risk management; Sample size; Samples; Statistical analysis; Statistical methods; Studies; Taxation
• ISSN: 1619-697X; 1619-6988
• DOI: 10.1007/s10287-010-0123-6

Stochastic programming is a well-known instrument to model many risk management problems in finance. In this paper we consider a stochastic programming model where the objective function is the variance of a random function and the constraint function is the expected value of the random function. Instead of using popular scenario tree methods, we apply the well-known sample average approximation (SAA) method to solve it. An advantage of SAA is that it can be implemented without knowing the distribution of the random data. We investigate the asymptotic properties of statistical estimators obtained from the SAA problem including examining the rate of convergence of optimal solutions of the SAA problem as sample size increases. By using the classical penalty function technique and recent results on uniform exponential convergence of sample average random functions, we show that under some mild conditions the statistical estimator of the optimal solution converges to its true counterpart at an exponential rate. We apply the proposed model and the numerical method to a portfolio management problem and present some numerical results.