\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

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

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

    1 Corresponding author. Tel: +86 20 84111989. Fax: +86 20 84114823 
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 Full Text(HTML) Figure(0) / Table(1) Related Papers Cited by
  • 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.

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

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • 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
     | Show Table
    DownLoad: CSV
  • [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. BieleckiH. Q. JinS. 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. ChenH. 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. ChenJ. 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. CuiJ. J. GaoX. 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. CvitanicA. 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. DeMiguelL. 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. LiX. 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. YaoZ. 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. YaoZ. 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. ZengD. 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. ZhuD. 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.
  • 加载中

Tables(1)

SHARE

Article Metrics

HTML views(912) PDF downloads(197) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return