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Portfolio optimization with relaxation of stochastic second order dominance constraints via conditional value at risk

  • * Corresponding author: Hailin Sun

    * Corresponding author: Hailin Sun

The work is supported by National Natural Science Foundation of China grant 11871276, 11571178 and 11571056

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  • A portfolio optimization model with relaxed second order stochastic dominance (SSD) constraints is presented. The proposed model uses Conditional Value at Risk (CVaR) constraints at probability level $ \beta\in(0,1) $ to relax SSD constraints. The relaxation is justified by theoretical convergence results based on sample average approximation (SAA) method when sample size $ N\to\infty $ and CVaR probability level $ \beta $ tends to 1. SAA method is used to reduce infinite number of inequalities of SSD constraints to finite ones and also to calculate the expectation value. The proposed relaxation on the SSD constraints in portfolio optimization problem is achieved when the probability level $ \beta $ of CVaR takes value less than but close to 1, and the model can then be solved by cutting plane method. The performance and characteristics of the portfolios constructed by solving the proposed model are tested empirically on three sets of market data, and the experimental results are analyzed and discussed. Furthermore, it is shown that with appropriate choices of CVaR probability level $ \beta $, the constructed portfolios are sparse and outperform the portfolios constructed by solving portfolio optimization problems with SSD constraints, with either index portfolios or mean-variance (MV) portfolios as benchmarks.

    Mathematics Subject Classification: 91G10; 90C15.

    Citation:

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  • Figure 1.  In-sample back testing with NDX data with NDX index as benchmark

    Figure 2.  NDX: ex-post compounded daily returns (01/06/2016 - 30/9/2016), index returns as benchmark

    Figure 3.  NDX: ex-post compounded daily returns (01/06/2016 - 30/9/2016), MV returns as benchmark

    Figure 4.  S&P 500: ex-post compounded daily returns (01/06/2016 - 30/9/2016), index returns as benchmark

    Figure 5.  S&P 500: ex-post compounded daily returns (01/06/2016 - 30/9/2016), MV returns as benchmark

    Figure 6.  FTSE 100: ex-post compounded daily returns (01/06/2016 - 30/9/2016), index returns as benchmark

    Figure 7.  FTSE 100: ex-post compounded daily returns (01/06/2016 - 30/9/2016), MV returns as benchmark

    Table 1.  NDX: average daily return, standard deviation, Sharpe Ratio, Sortino Ratio

    mean std Sharpe Ratio Sortino Ratio
    Benchmark: index 0.0009 0.0088 0.1030 0.1389
    SSD 0.0032 0.0118 0.2717 0.4501
    $ CVaR_ {\beta=0.9} $ 0.0033 0.0118 0.2784 0.4670
    $ CVaR_{\beta=0.8} $ 0.0032 0.0117 0.2724 0.4486
    $ CVaR_{\beta=0.7} $ 0.0033 0.0119 0.2810 0.4652
    Benchmark: MV 0.0005 0.0078 0.0658 0.0841
    SSD 0.0026 0.0102 0.2545 0.4129
    $ CVaR_ {\beta=0.9} $ 0.0027 0.0118 0.2696 0.4442
    $ CVaR_{\beta=0.8} $ 0.0030 0.0117 0.2931 0.4916
    $ CVaR_{\beta=0.7} $ 0.0030 0.0102 0.2943 0.5004
     | Show Table
    DownLoad: CSV

    Table 2.  S&P 500: average daily return, standard deviation, Sharpe Ratio, Sortino Ratio

    mean std Sharpe Ratio Sortino Ratio
    Benchmark: index 0.0004 0.0079 0.0534 0.0705
    SSD 0.0017 0.0112 0.1490 0.2264
    $ CVaR_ {\beta=0.9} $ 0.0018 0.0111 0.1606 0.2444
    $ CVaR_{\beta=0.8} $ 0.0016 0.0114 0.1442 0.2171
    $ CVaR_{\beta=0.7} $ 0.0017 0.0115 0.1477 0.2241
    Benchmark: MV 0.0003 0.0060 0.0421 0.0573
    SSD 0.0009 0.0110 0.0802 0.1103
    $ CVaR_ {\beta=0.9} $ 0.0012 0.0110 0.1086 0.1517
    $ CVaR_{\beta=0.8} $ 0.0013 0.0107 0.1196 0.1702
    $ CVaR_{\beta=0.7} $ 0.0015 0.0110 0.1383 0.2001
     | Show Table
    DownLoad: CSV

    Table 3.  FTSE 100: average daily return, standard deviation, Sharpe Ratio, Sortino Ratio

    mean std Sharpe Ratio Sortino Ratio
    Benchmark: index 0.0012 0.0112 0.1080 0.1685
    SSD 0.0017 0.0158 0.1094 0.1848
    $ CVaR_ {\beta=0.9} $ 0.0017 0.0155 0.1099 0.1836
    $ CVaR_{\beta=0.8} $ 0.0020 0.0157 0.1254 0.2131
    $ CVaR_{\beta=0.7} $ 0.0021 0.0164 0.1269 0.2185
    Benchmark: MV 0.0018 0.0095 0.1901 0.3418
    SSD 0.0023 0.0141 0.1606 0.2925
    $ CVaR_ {\beta=0.9} $ 0.0021 0.0134 0.1568 0.2747
    $ CVaR_{\beta=0.8} $ 0.0021 0.0136 0.1578 0.2755
    $ CVaR_{\beta=0.7} $ 0.0024 0.0141 0.1703 0.3058
     | Show Table
    DownLoad: CSV

    Table 4.  Average, minimum and maximum of daily traded basket sizes of different models with both benchmarks in three data sets

    NDX (100) Index MV
    avg. min. max. avg. min. max.
    SSD 4.60 3 9 5.55 3 9
    $ CVaR_{\beta = 0.9} $ 4.65 3 10 5.53 2 9
    $ CVaR_{\beta = 0.8} $ 4.65 3 10 5.51 2 8
    $ CVaR_{\beta = 0.7} $ 4.64 3 9 5.50 3 8
    FTSE (100) Index MV
    avg. min. max. avg. min. max.
    SSD 4.05 2 9 5.16 2 9
    $ CVaR_{\beta = 0.9} $ 3.98 2 9 5.11 2 9
    $ CVaR_{\beta = 0.8} $ 3.90 2 8 5.01 2 8
    $ CVaR_{\beta = 0.7} $ 3.91 3 9 5.17 2 9
    S&P (500) Index MV
    avg. min. max. avg. min. max.
    SSD 5.87 3 10 6.62 4 11
    $ CVaR_{\beta = 0.9} $ 6.07 3 11 6.49 3 12
    $ CVaR_{\beta = 0.8} $ 6.07 3 12 6.70 4 12
    $ CVaR_{\beta = 0.7} $ 6.30 3 11 6.74 4 12
     | Show Table
    DownLoad: CSV
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