Article Contents
Article Contents

# A smoothing SAA algorithm for a portfolio choice model based on second-order stochastic dominance measures

• * Corresponding author: Liu Yang
• In this paper, we provide a smoothing sample average approximation (SAA) method to solve a portfolio choice model based on second-order stochastic dominance (SSD) measure. Introducing a second-order stochastic dominance constraint in portfolio choice is theoretically attractive since all risk-averse investors would prefer a dominating portfolio. However, how to get the best choice among SSD efficient portfolios which is based on a stochastic optimization model is a challenge. We use the sample average to approximate the expected return rate function in the model and get a linear/nonlinear programming when the benchmark has discrete distribution. Then we propose a smoothing penalty algorithm to solve this problem. Meanwhile, we investigate the convergence of the optimal value of the transformed model and show that the optimal value converges to its counterpart with probability approaching to one at exponential rate as the sample size increases. By comparing the numerical results of the smoothing SAA algorithm with the common linear programming (LP) algorithm, we find that the smoothing algorithm has better performance than the LP algorithm in three aspects: (ⅰ)the smoothing SAA method can avoid the infinite constraints in the transformed models and the size of the smoothing algorithm model will not increase as the sample grows; (ⅱ)the smoothing SAA algorithm can deal with the nonlinear portfolio models with nonlinear transaction cost function; (ⅲ) the smoothing algorithm can get the global optimal solution because the smoothing function maintains the original convexity.

Mathematics Subject Classification: Primary: 90C15, 90C05, 90C30.

 Citation:

• Figure 1.  The expected returns for different smoothing parameters

Figure 2.  Comparison of the CPU time for LP and SMOOTH

Figure 3.  The expected returns for different transaction cost ratio

Figure 4.  The expected returns for different penalty parameter and sample size

Table 1.  Expectation and variance of return rates

 code of stock 600690 713 600115 600111 600000 expectation 0.0051 0.004 0.003 0.01 0.0049 variance 0.0031 0.0042 0.005 0.006 0.0036 code of stock 600252 600011 600362 401 600267 expectation 0.0072 0.0028 0.0073 0.0052 0.005 variance 0.0061 0.0027 0.0065 0.0046 0.003

Table 2.  Comparison of the numerical results for LP and SMOOTH

 Method Problem $x$ E$[\cdot]$ LP No-cost (0, 0, 0, 0.7744, 0, 0.2165, 0, 0.0091, 0, 0) 0.0095 SMOOTH No-cost (0, 0, 0, 0.9592, 0, 0.0348, 0, 0.006, 0, 0) 0.0089 LP cost (0, 0, 0, 0.6620, 0, 0.2554, 0, 0.0642, 0, 0) 0.0075 SMOOTH cost (0, 0.0586, 0.0809, 0.1435, 0, 0, 0.4344, 0, 0, 0.2608) 0.0067
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