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Equilibrium analysis of an opportunistic spectrum access mechanism with imperfect sensing results
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 |
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.
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. |
[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. |
[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. |
[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. |
[5] |
C. H. Chiu and X. Y. Zhou,
The premium of dynamic trading, Quantitative Finance, 11 (2011), 115-123.
doi: 10.1080/14697681003685589. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[13] |
W. H. Fleming and H. M. Soner, Controlled Markov processes and viscosity solutions, 2ed. Springer, New York, 2006. |
[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. |
[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. |
[16] |
H. Markowitz,
Portfolio selection, Journal of Finance, 7 (1952), 77-91.
doi: 10.1111/j.1540-6261.1952.tb01525.x. |
[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. |
[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. |
[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. |
[21] |
W. F. Sharpe,
The Sharpe ratio, The Journal of Portfolio Management, 21 (1994), 49-58.
doi: 10.3905/jpm.1994.409501. |
[22] |
A. D. Roy,
Safety first and the holding of assets, Econometrica, 20 (1952), 431-449.
doi: 10.2307/1907413. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[5] |
C. H. Chiu and X. Y. Zhou,
The premium of dynamic trading, Quantitative Finance, 11 (2011), 115-123.
doi: 10.1080/14697681003685589. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[13] |
W. H. Fleming and H. M. Soner, Controlled Markov processes and viscosity solutions, 2ed. Springer, New York, 2006. |
[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. |
[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. |
[16] |
H. Markowitz,
Portfolio selection, Journal of Finance, 7 (1952), 77-91.
doi: 10.1111/j.1540-6261.1952.tb01525.x. |
[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. |
[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. |
[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. |
[21] |
W. F. Sharpe,
The Sharpe ratio, The Journal of Portfolio Management, 21 (1994), 49-58.
doi: 10.3905/jpm.1994.409501. |
[22] |
A. D. Roy,
Safety first and the holding of assets, Econometrica, 20 (1952), 431-449.
doi: 10.2307/1907413. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
| | | | | |
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 |
| | | | | |
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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