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

Haixiang Yao; Zhongfei Li; Xun Li; Yan Zeng

doi:10.3934/jimo.2016072

Article Contents

2017, Volume 13, Issue 3: 1273-1290. Doi: 10.3934/jimo.2016072

This issue Previous Article Equilibrium analysis of an opportunistic spectrum access mechanism with imperfect sensing results Next Article Double well potential function and its optimization in the

$N$

-dimensional real space-part Ⅰ

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

Received: March 2015

Revised: April 2016

Published: September 2017

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.

Abstract / Introduction Full Text(HTML) Figure(0) / Table(1) Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 90C26; Secondary: 91B28, 49N15.

Citation:

Full Text(HTML)

Table 1. Computational results for the out-of-sample empirical analysis

$i$	$\widehat{Shpf}_{opt} $	$\widehat{Shp}_{opt}$	$\widehat{Shp}_{1/N} $	$I_{Shpf}$	$\widehat{PDT}$
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

| Show Table

DownLoad: CSV

Related Papers

Cited by

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.
[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.
[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.
[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.