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July  2017, 13(3): 1273-1290. doi: 10.3934/jimo.2016072

## Optimal Sharpe ratio in continuous-time markets with and without a risk-free asset

 1 School of Finance, Guangdong University of Foreign Studies, Guangzhou 510006, China 2 Sun Yat-sen Business School, Sun Yat-sen University, Guangzhou 510275, China 3 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China 4 Lingnan (University) College, Sun Yat-sen University, Guangzhou 510275, China

1 Corresponding author. Tel: +86 20 84111989. Fax: +86 20 84114823

Received  March 2015 Revised  April 2016 Published  October 2016

Fund Project: This research is partially supported by the National Natural Science Foundation of China (Nos. 71231008,71471045,71201173 and 71571195), the China Postdoctoral Science Foundation (Nos. 2014M560658,2015T80896), Guangdong Natural Science Funds for Research Teams (2014A030312003), Guangdong Natural Science Funds for Distinguished Young Scholars (2015A030306040), the Characteristic and Innovation Foundation of Guangdong Colleges and Universities (Humanity and Social Science Type), Fok Ying Tung Education Foundation for Young Teachers in the Higher Education Institutions of China (No. 151081), Science and Technology Planning Project of Guangdong Province (No. 2016A070705024), and the Hong Kong RGC grants 15209614 and 15224215.

In this paper, we investigate a continuous-time mean-variance portfolio selection model with only risky assets and its optimal Sharpe ratio in a new way. We obtain closed-form expressions for the efficient investment strategy, the efficient frontier and the optimal Sharpe ratio. Using these results, we further prove that (ⅰ) the efficient frontier with only risky assets is significantly different from the one with inclusion of a risk-free asset and (ⅱ) inclusion of a risk-free asset strictly enhances the optimal Sharpe ratio. Also, we offer an explicit expression for the enhancement of the optimal Sharpe ratio. Finally, we test our theory results using an empirical analysis based on real data of Chinese equity market. Out-of-sample analyses shed light on advantages of our theoretical results established.

Citation: Haixiang Yao, Zhongfei Li, Xun Li, Yan Zeng. Optimal Sharpe ratio in continuous-time markets with and without a risk-free asset. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1273-1290. doi: 10.3934/jimo.2016072
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##### References:
Computational results for the out-of-sample empirical analysis
 $i$ $\widehat{Shpf}_{opt}$ $\widehat{Shp}_{opt}$ $\widehat{Shp}_{1/N}$ $I_{Shpf}$ $\widehat{PDT}$ 1 1.2623 1.2397 -0.3364 1 0.0226 2 1.1306 1.1307 -0.1763 0 -0.0001 3 1.3555 1.3595 -0.2105 0 -0.0040 4 1.1508 1.1485 -0.0030 1 0.0023 5 1.4455 1.4416 0.0944 1 0.0040 6 0.8880 0.8882 0.0659 0 -0.0001 7 0.9247 0.9243 0.1214 1 0.0004 8 0.9042 0.9036 -0.0065 1 0.0006 9 0.9748 1.0066 -0.0623 0 -0.0318 10 1.0651 1.0712 -0.1153 0 -0.0060 11 1.1288 1.1171 -0.2061 1 0.0118 12 1.1166 1.1051 -0.2055 1 0.0115 13 1.0595 1.0398 -0.1820 1 0.0196 14 0.8222 0.8001 -0.0792 1 0.0222 15 0.8377 0.8350 -0.0858 1 0.0027 16 0.8781 0.8348 -0.1900 1 0.0432 17 0.8107 0.7647 -0.0821 1 0.0460 18 0.9910 0.9260 -0.0264 1 0.0650 19 0.8887 0.8830 -0.0565 1 0.0057 20 0.7912 0.7805 -0.0090 1 0.0107 21 0.8506 0.8502 0.0811 1 0.0004 22 1.0396 1.0423 0.0582 0 -0.0026 23 1.2097 1.2066 -0.0026 1 0.0032 24 1.1486 1.1012 -0.0619 1 0.0474 25 0.8786 0.8499 0.0616 1 0.0287 26 0.7611 0.7211 -0.0889 1 0.0400 27 0.7547 0.7319 -0.1268 1 0.0228 28 0.8216 0.7850 -0.0742 1 0.0366 29 0.8815 0.8742 -0.0023 1 0.0073 30 0.9891 0.9116 -0.0144 1 0.0775 31 0.9342 0.8812 -0.1807 1 0.0530 32 0.9017 0.9005 0.0095 1 0.0011 33 0.9755 0.9778 0.0552 0 -0.0023 34 1.1543 1.1563 0.1036 0 -0.002 35 0.9607 0.9481 0.3367 1 0.0126 36 1.0940 1.0934 0.4308 1 0.0006 37 1.3044 1.2959 0.4813 1 0.0085 38 1.2116 1.2009 0.5951 1 0.0107 39 0.8597 0.8548 0.4308 1 0.0049 40 0.8278 0.8141 0.1292 1 0.0137
 $i$ $\widehat{Shpf}_{opt}$ $\widehat{Shp}_{opt}$ $\widehat{Shp}_{1/N}$ $I_{Shpf}$ $\widehat{PDT}$ 1 1.2623 1.2397 -0.3364 1 0.0226 2 1.1306 1.1307 -0.1763 0 -0.0001 3 1.3555 1.3595 -0.2105 0 -0.0040 4 1.1508 1.1485 -0.0030 1 0.0023 5 1.4455 1.4416 0.0944 1 0.0040 6 0.8880 0.8882 0.0659 0 -0.0001 7 0.9247 0.9243 0.1214 1 0.0004 8 0.9042 0.9036 -0.0065 1 0.0006 9 0.9748 1.0066 -0.0623 0 -0.0318 10 1.0651 1.0712 -0.1153 0 -0.0060 11 1.1288 1.1171 -0.2061 1 0.0118 12 1.1166 1.1051 -0.2055 1 0.0115 13 1.0595 1.0398 -0.1820 1 0.0196 14 0.8222 0.8001 -0.0792 1 0.0222 15 0.8377 0.8350 -0.0858 1 0.0027 16 0.8781 0.8348 -0.1900 1 0.0432 17 0.8107 0.7647 -0.0821 1 0.0460 18 0.9910 0.9260 -0.0264 1 0.0650 19 0.8887 0.8830 -0.0565 1 0.0057 20 0.7912 0.7805 -0.0090 1 0.0107 21 0.8506 0.8502 0.0811 1 0.0004 22 1.0396 1.0423 0.0582 0 -0.0026 23 1.2097 1.2066 -0.0026 1 0.0032 24 1.1486 1.1012 -0.0619 1 0.0474 25 0.8786 0.8499 0.0616 1 0.0287 26 0.7611 0.7211 -0.0889 1 0.0400 27 0.7547 0.7319 -0.1268 1 0.0228 28 0.8216 0.7850 -0.0742 1 0.0366 29 0.8815 0.8742 -0.0023 1 0.0073 30 0.9891 0.9116 -0.0144 1 0.0775 31 0.9342 0.8812 -0.1807 1 0.0530 32 0.9017 0.9005 0.0095 1 0.0011 33 0.9755 0.9778 0.0552 0 -0.0023 34 1.1543 1.1563 0.1036 0 -0.002 35 0.9607 0.9481 0.3367 1 0.0126 36 1.0940 1.0934 0.4308 1 0.0006 37 1.3044 1.2959 0.4813 1 0.0085 38 1.2116 1.2009 0.5951 1 0.0107 39 0.8597 0.8548 0.4308 1 0.0049 40 0.8278 0.8141 0.1292 1 0.0137
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