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

Meng Xue; Yun Shi; Hailin Sun

doi:10.3934/jimo.2019071

Article Contents

2020, Volume 16, Issue 6: 2581-2602. Doi: 10.3934/jimo.2019071

This issue Previous Article Next Article Pricing dynamic fund protection under a Regime-switching Jump-diffusion model with stochastic protection level

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
^* Corresponding author: Hailin Sun

Received: November 30, 2017

Revised: February 28, 2019

Early access: July 2019

Published: October 2020

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

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Abstract

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.

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Mathematics Subject Classification: 91G10; 90C15.

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

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

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

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

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

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

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

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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

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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

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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

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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
NDX (100)	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
FTSE (100)	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
S&P (500)	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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