# American Institute of Mathematical Sciences

November  2020, 16(6): 2581-2602. doi: 10.3934/jimo.2019071

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

 1 School of Economics and Management, Nanjing University of Science and Technology, Nanjing, 210094, China 2 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong 3 Jiangsu Provincial Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China

* Corresponding author: Hailin Sun

Received  December 2017 Revised  March 2019 Published  July 2019

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

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.

Citation: Meng Xue, Yun Shi, Hailin Sun. Portfolio optimization with relaxation of stochastic second order dominance constraints via conditional value at risk. Journal of Industrial & Management Optimization, 2020, 16 (6) : 2581-2602. doi: 10.3934/jimo.2019071
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##### References:
In-sample back testing with NDX data with NDX index as benchmark
NDX: ex-post compounded daily returns (01/06/2016 - 30/9/2016), index returns as benchmark
NDX: ex-post compounded daily returns (01/06/2016 - 30/9/2016), MV returns as benchmark
S&P 500: ex-post compounded daily returns (01/06/2016 - 30/9/2016), index returns as benchmark
S&P 500: ex-post compounded daily returns (01/06/2016 - 30/9/2016), MV returns as benchmark
FTSE 100: ex-post compounded daily returns (01/06/2016 - 30/9/2016), index returns as benchmark
FTSE 100: ex-post compounded daily returns (01/06/2016 - 30/9/2016), MV returns as benchmark
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
 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
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
 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
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
 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
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
 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
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