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

  • * Corresponding author: Liu Yang

    * Corresponding author: Liu Yang 
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  • 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.


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  • 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 000713 600115 600111 600000
    expectation 0.0051 0.0040 0.0030 0.0100 0.0049
    variance 0.0031 0.0042 0.0050 0.0060 0.0036
    code of stock 600252 600011 600362 000401 600267
    expectation 0.0072 0.0028 0.0073 0.0052 0.0050
    variance 0.0061 0.0027 0.0065 0.0046 0.0030
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV
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