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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

Haixiang Yao 1, , Zhongfei Li 2,, , Xun Li 3, and  Yan Zeng 4,

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.

Keywords: Continuous-time mean-variance model, efficient investment strategy, efficient frontier, sharpe ratio, Hamilton-Jacobi-Bellman equation.
Mathematics Subject Classification: Primary: 90C26; Secondary: 91B28, 49N15.
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
References:
[1]

D. Bayley and M. L. de Prado, The Sharpe Ratio Efficient Frontier, Journal of Risk, 15 (2012), 3-44.  doi: 10.2139/ssrn.1821643.  Google Scholar

[2]

T. R. Bielecki, H. Q. Jin, S. R. Pliska and X. Y. Zhou, Continuous-time mean--variance portfolio selection with bankruptcy prohibition, Mathematical Finance, 15 (2005), 213-244.  doi: 10.1111/j.0960-1627.2005.00218.x.  Google Scholar

[3]

P. Chen, H. L. Yang and G. Yin, Markowitz's mean-variance asset-liability management with regime switching: A continuous-time model, nsurance: Mathematics and Economics, 43 (2008), 456-465.  doi: 10.1016/j.insmatheco.2008.09.001.  Google Scholar

[4]

Z. P. Chen, J. Liu and G. Li, Time consistent policy of multi-period mean-variance problem in stochastic markets, Journal of Industrial and Management Optimization, 12 (2016), 229-249.  doi: 10.3934/jimo.2016.12.229.  Google Scholar

[5]

C. H. Chiu and X. Y. Zhou, The premium of dynamic trading, Quantitative Finance, 11 (2011), 115-123.  doi: 10.1080/14697681003685589.  Google Scholar

[6]

V. Chow and C. W. Lai, Conditional Sharpe Ratios Finance Research Letters, inpress, 2014, available online, http://dx.doi.org/10.1016/j.frl.2014.11.001. Google Scholar

[7]

X. Y. Cui, J. J. Gao and D. Li, Continuous-time mean-variance portfolio selection with finite transactions, Stochastic analysis and applications to finance, Interdiscip. Math. Sci. , World Sci. Publ. , Hackensack, NJ, 13 (2012), 77–98. doi: 10.1142/9789814383585_0005.  Google Scholar

[8]

X. Y. Cui, J. J. Gao, X. Li and D. Li, Optimal multi-period mean--variance policy under no-shorting constraint, European Journal of Operational Research, 234 (2014), 459-468.  doi: 10.1016/j.ejor.2013.02.040.  Google Scholar

[9]

J. Cvitanic, A. Lazrak and T. Wang, Implications of the sharpe ratio as a performance measure in multi-period settings, Journal of Economic Dynamics and Control, 32 (2008), 1622-1649.  doi: 10.1016/j.jedc.2007.06.009.  Google Scholar

[10]

D. M. Danga and P. A. Forsyth, Better than pre-commitment mean-variance portfolio allocation strategies: A semi-self-financing Hamilton-Jacobi-Bellman equation approach, European Journal of Operational Research, 250 (2016), 827-841.  doi: 10.1016/j.ejor.2015.10.015.  Google Scholar

[11]

V. DeMiguel, L. Garlappi and R. Uppal, Optimal versus Naive Diversification: How ineficient is the 1/N portfolio strategy?, Review of Financial Studies, 22 (2009), 1915-1953.  doi: 10.1093/acprof:oso/9780199744282.003.0034.  Google Scholar

[12]

K. Dowd, Adjusting for risk: An improved Sharpe ratio, International Review of Economics and Finance, 9 (2000), 209-222.  doi: 10.1016/S1059-0560(00)00063-0.  Google Scholar

[13]

W. H. Fleming and H. M. Soner, Controlled Markov processes and viscosity solutions, 2ed. Springer, New York, 2006.  Google Scholar

[14]

D. Li and W. L. Ng, Optimal dynamic portfolio selection: Multiperiod mean-variance formulation, Mathematical Finance, 10 (2000), 387-406.  doi: 10.1111/1467-9965.00100.  Google Scholar

[15]

X. Li, X. Y. Zhou and A. E. B. Lim, Dynamic mean--variance portfolio selection with no-shorting constraints, SIAM Journal on Control and Optimization, 40 (2002), 1540-1555.  doi: 10.1137/S0363012900378504.  Google Scholar

[16]

H. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91.  doi: 10.1111/j.1540-6261.1952.tb01525.x.  Google Scholar

[17]

R. C. Merton, An analytic derivation of the efficient portfolio frontier, Journal of Financial and Quantitative Analysis, 7 (1972), 1851-1872.  doi: 10.2307/2329621.  Google Scholar

[18]

M. Schuster and B. R. Auer, A note on empirical Sharpe ratio dynamics, Economics Letters, 116 (2012), 124-128.  doi: 10.1016/j.econlet.2012.02.005.  Google Scholar

[19]

W. F. Sharpe, Capital asset prices: A theory of market equilibrium under conditions of risk, Journal of Finance, 19 (1964), 425-442.   Google Scholar

[20]

W. F. Sharpe, Mutual fund performance, Journal of Business, 39 (1966), 119-138.  doi: 10.1086/294846.  Google Scholar

[21]

W. F. Sharpe, The Sharpe ratio, The Journal of Portfolio Management, 21 (1994), 49-58.  doi: 10.3905/jpm.1994.409501.  Google Scholar

[22]

A. D. Roy, Safety first and the holding of assets, Econometrica, 20 (1952), 431-449.  doi: 10.2307/1907413.  Google Scholar

[23]

Z. Wang and S. Y. Liu, Multi-period mean-variance portfolio selection with fixed and proportional transaction costs Journal of Industrial and Management Optimization, 9 (2013), 643-657. doi: 10.3934/jimo.2013.9.643.  Google Scholar

[24]

H. X. Yao, Z. F. Li and S. M. Chen, Continuous-time mean-variance portfolio selection with only risky assets, Economic Modelling, 36 (2014), 244-251.  doi: 10.1016/j.econmod.2013.09.041.  Google Scholar

[25]

H. X. Yao, Z. F. Li and Y. Z. Lai, Dynamic mean-variance asset allocation with stochastic interest rate and inflation rate, Journal of Industrial and Management Optimization, 12 (2016), 187-209.  doi: 10.3934/jimo.2016.12.187.  Google Scholar

[26]

V. Zakamouline and S. Koekebakker, Portfolio performance evaluation with generalized Sharpe ratios: Beyond the mean and variance, Journal of Banking and Finance, 33 (2009), 1242-1254.   Google Scholar

[27]

Y. Zeng, D. P. Li and A. L. Gu, Robust equilibrium reinsurance-investment strategy for a mean-variance insurer in a model with jumps, Insurance: Mathematics and Economics, 66 (2016), 138-152.  doi: 10.1016/j.insmatheco.2015.10.012.  Google Scholar

[28]

X. Y. Zhou and D. Li, Continuous-time mean-variance portfolio selection: A stochastic LQ framework, Applied Mathematics and Optimization, 42 (2000), 19-33.  doi: 10.1007/s002450010003.  Google Scholar

[29]

S. S. Zhu, D. Li and S. Y. Wang, Risk control over bankruptcy in dynamic portfolio selection: A generalized mean-variance formulation, IEEE Transactions on Automatic Control, 49 (2004), 447-457.  doi: 10.1109/TAC.2004.824474.  Google Scholar

show all references

References:
[1]

D. Bayley and M. L. de Prado, The Sharpe Ratio Efficient Frontier, Journal of Risk, 15 (2012), 3-44.  doi: 10.2139/ssrn.1821643.  Google Scholar

[2]

T. R. Bielecki, H. Q. Jin, S. R. Pliska and X. Y. Zhou, Continuous-time mean--variance portfolio selection with bankruptcy prohibition, Mathematical Finance, 15 (2005), 213-244.  doi: 10.1111/j.0960-1627.2005.00218.x.  Google Scholar

[3]

P. Chen, H. L. Yang and G. Yin, Markowitz's mean-variance asset-liability management with regime switching: A continuous-time model, nsurance: Mathematics and Economics, 43 (2008), 456-465.  doi: 10.1016/j.insmatheco.2008.09.001.  Google Scholar

[4]

Z. P. Chen, J. Liu and G. Li, Time consistent policy of multi-period mean-variance problem in stochastic markets, Journal of Industrial and Management Optimization, 12 (2016), 229-249.  doi: 10.3934/jimo.2016.12.229.  Google Scholar

[5]

C. H. Chiu and X. Y. Zhou, The premium of dynamic trading, Quantitative Finance, 11 (2011), 115-123.  doi: 10.1080/14697681003685589.  Google Scholar

[6]

V. Chow and C. W. Lai, Conditional Sharpe Ratios Finance Research Letters, inpress, 2014, available online, http://dx.doi.org/10.1016/j.frl.2014.11.001. Google Scholar

[7]

X. Y. Cui, J. J. Gao and D. Li, Continuous-time mean-variance portfolio selection with finite transactions, Stochastic analysis and applications to finance, Interdiscip. Math. Sci. , World Sci. Publ. , Hackensack, NJ, 13 (2012), 77–98. doi: 10.1142/9789814383585_0005.  Google Scholar

[8]

X. Y. Cui, J. J. Gao, X. Li and D. Li, Optimal multi-period mean--variance policy under no-shorting constraint, European Journal of Operational Research, 234 (2014), 459-468.  doi: 10.1016/j.ejor.2013.02.040.  Google Scholar

[9]

J. Cvitanic, A. Lazrak and T. Wang, Implications of the sharpe ratio as a performance measure in multi-period settings, Journal of Economic Dynamics and Control, 32 (2008), 1622-1649.  doi: 10.1016/j.jedc.2007.06.009.  Google Scholar

[10]

D. M. Danga and P. A. Forsyth, Better than pre-commitment mean-variance portfolio allocation strategies: A semi-self-financing Hamilton-Jacobi-Bellman equation approach, European Journal of Operational Research, 250 (2016), 827-841.  doi: 10.1016/j.ejor.2015.10.015.  Google Scholar

[11]

V. DeMiguel, L. Garlappi and R. Uppal, Optimal versus Naive Diversification: How ineficient is the 1/N portfolio strategy?, Review of Financial Studies, 22 (2009), 1915-1953.  doi: 10.1093/acprof:oso/9780199744282.003.0034.  Google Scholar

[12]

K. Dowd, Adjusting for risk: An improved Sharpe ratio, International Review of Economics and Finance, 9 (2000), 209-222.  doi: 10.1016/S1059-0560(00)00063-0.  Google Scholar

[13]

W. H. Fleming and H. M. Soner, Controlled Markov processes and viscosity solutions, 2ed. Springer, New York, 2006.  Google Scholar

[14]

D. Li and W. L. Ng, Optimal dynamic portfolio selection: Multiperiod mean-variance formulation, Mathematical Finance, 10 (2000), 387-406.  doi: 10.1111/1467-9965.00100.  Google Scholar

[15]

X. Li, X. Y. Zhou and A. E. B. Lim, Dynamic mean--variance portfolio selection with no-shorting constraints, SIAM Journal on Control and Optimization, 40 (2002), 1540-1555.  doi: 10.1137/S0363012900378504.  Google Scholar

[16]

H. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91.  doi: 10.1111/j.1540-6261.1952.tb01525.x.  Google Scholar

[17]

R. C. Merton, An analytic derivation of the efficient portfolio frontier, Journal of Financial and Quantitative Analysis, 7 (1972), 1851-1872.  doi: 10.2307/2329621.  Google Scholar

[18]

M. Schuster and B. R. Auer, A note on empirical Sharpe ratio dynamics, Economics Letters, 116 (2012), 124-128.  doi: 10.1016/j.econlet.2012.02.005.  Google Scholar

[19]

W. F. Sharpe, Capital asset prices: A theory of market equilibrium under conditions of risk, Journal of Finance, 19 (1964), 425-442.   Google Scholar

[20]

W. F. Sharpe, Mutual fund performance, Journal of Business, 39 (1966), 119-138.  doi: 10.1086/294846.  Google Scholar

[21]

W. F. Sharpe, The Sharpe ratio, The Journal of Portfolio Management, 21 (1994), 49-58.  doi: 10.3905/jpm.1994.409501.  Google Scholar

[22]

A. D. Roy, Safety first and the holding of assets, Econometrica, 20 (1952), 431-449.  doi: 10.2307/1907413.  Google Scholar

[23]

Z. Wang and S. Y. Liu, Multi-period mean-variance portfolio selection with fixed and proportional transaction costs Journal of Industrial and Management Optimization, 9 (2013), 643-657. doi: 10.3934/jimo.2013.9.643.  Google Scholar

[24]

H. X. Yao, Z. F. Li and S. M. Chen, Continuous-time mean-variance portfolio selection with only risky assets, Economic Modelling, 36 (2014), 244-251.  doi: 10.1016/j.econmod.2013.09.041.  Google Scholar

[25]

H. X. Yao, Z. F. Li and Y. Z. Lai, Dynamic mean-variance asset allocation with stochastic interest rate and inflation rate, Journal of Industrial and Management Optimization, 12 (2016), 187-209.  doi: 10.3934/jimo.2016.12.187.  Google Scholar

[26]

V. Zakamouline and S. Koekebakker, Portfolio performance evaluation with generalized Sharpe ratios: Beyond the mean and variance, Journal of Banking and Finance, 33 (2009), 1242-1254.   Google Scholar

[27]

Y. Zeng, D. P. Li and A. L. Gu, Robust equilibrium reinsurance-investment strategy for a mean-variance insurer in a model with jumps, Insurance: Mathematics and Economics, 66 (2016), 138-152.  doi: 10.1016/j.insmatheco.2015.10.012.  Google Scholar

[28]

X. Y. Zhou and D. Li, Continuous-time mean-variance portfolio selection: A stochastic LQ framework, Applied Mathematics and Optimization, 42 (2000), 19-33.  doi: 10.1007/s002450010003.  Google Scholar

[29]

S. S. Zhu, D. Li and S. Y. Wang, Risk control over bankruptcy in dynamic portfolio selection: A generalized mean-variance formulation, IEEE Transactions on Automatic Control, 49 (2004), 447-457.  doi: 10.1109/TAC.2004.824474.  Google Scholar

Table 1.  Computational results for the out-of-sample empirical analysis
$i$ $\widehat{Shpf}_{opt} $ $\widehat{Shp}_{opt}$ $\widehat{Shp}_{1/N} $ $I_{Shpf}$ $\widehat{PDT}$
11.26231.2397-0.336410.0226
21.13061.1307-0.17630-0.0001
31.35551.3595-0.21050-0.0040
41.15081.1485-0.003010.0023
51.44551.44160.094410.0040
60.88800.88820.06590-0.0001
70.92470.92430.121410.0004
80.90420.9036-0.006510.0006
90.97481.0066-0.06230-0.0318
101.06511.0712-0.11530-0.0060
111.12881.1171-0.206110.0118
121.11661.1051-0.205510.0115
131.05951.0398-0.182010.0196
140.82220.8001-0.079210.0222
150.83770.8350-0.085810.0027
160.87810.8348-0.190010.0432
170.81070.7647-0.082110.0460
180.99100.9260-0.026410.0650
190.88870.8830-0.056510.0057
200.79120.7805-0.009010.0107
210.85060.85020.081110.0004
221.03961.04230.05820-0.0026
231.20971.2066-0.002610.0032
241.14861.1012-0.061910.0474
250.87860.84990.061610.0287
260.76110.7211-0.088910.0400
270.75470.7319-0.126810.0228
280.82160.7850-0.074210.0366
290.88150.8742-0.002310.0073
300.98910.9116-0.014410.0775
310.93420.8812-0.180710.0530
320.90170.90050.009510.0011
330.97550.97780.05520-0.0023
341.15431.15630.10360-0.002
350.96070.94810.336710.0126
361.09401.09340.430810.0006
371.30441.29590.481310.0085
381.21161.20090.595110.0107
390.85970.85480.430810.0049
400.82780.81410.129210.0137
Table Options

Download as excel

$i$ $\widehat{Shpf}_{opt} $ $\widehat{Shp}_{opt}$ $\widehat{Shp}_{1/N} $ $I_{Shpf}$ $\widehat{PDT}$
11.26231.2397-0.336410.0226
21.13061.1307-0.17630-0.0001
31.35551.3595-0.21050-0.0040
41.15081.1485-0.003010.0023
51.44551.44160.094410.0040
60.88800.88820.06590-0.0001
70.92470.92430.121410.0004
80.90420.9036-0.006510.0006
90.97481.0066-0.06230-0.0318
101.06511.0712-0.11530-0.0060
111.12881.1171-0.206110.0118
121.11661.1051-0.205510.0115
131.05951.0398-0.182010.0196
140.82220.8001-0.079210.0222
150.83770.8350-0.085810.0027
160.87810.8348-0.190010.0432
170.81070.7647-0.082110.0460
180.99100.9260-0.026410.0650
190.88870.8830-0.056510.0057
200.79120.7805-0.009010.0107
210.85060.85020.081110.0004
221.03961.04230.05820-0.0026
231.20971.2066-0.002610.0032
241.14861.1012-0.061910.0474
250.87860.84990.061610.0287
260.76110.7211-0.088910.0400
270.75470.7319-0.126810.0228
280.82160.7850-0.074210.0366
290.88150.8742-0.002310.0073
300.98910.9116-0.014410.0775
310.93420.8812-0.180710.0530
320.90170.90050.009510.0011
330.97550.97780.05520-0.0023
341.15431.15630.10360-0.002
350.96070.94810.336710.0126
361.09401.09340.430810.0006
371.30441.29590.481310.0085
381.21161.20090.595110.0107
390.85970.85480.430810.0049
400.82780.81410.129210.0137
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