Optimal Sharpe ratio in continuous-time markets with and without a risk-free asset. (English) Zbl 1361.90047

Summary: 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.


91G10 Portfolio theory
93E20 Optimal stochastic control
Full Text: DOI


[1] D. Bayley, The Sharpe Ratio Efficient Frontier,, Journal of Risk, 15, 3, (2012)
[2] T. R. Bielecki, Continuous-time mean-variance portfolio selection with bankruptcy prohibition,, Mathematical Finance, 15, 213, (2005) · Zbl 1153.91466
[3] P. Chen, Markowitz’s mean-variance asset-liability management with regime switching: A continuous-time model,, Insurance: Mathematics and Economics, 43, 456, (2008) · Zbl 1152.91496
[4] Z. P. Chen, Time consistent policy of multi-period mean-variance problem in stochastic markets,, Journal of Industrial and Management Optimization, 12, 229, (2016) · Zbl 1318.91178
[5] C. H. Chiu, The premium of dynamic trading,, Quantitative Finance, 11, 115, (2011) · Zbl 1210.91119
[6] V. Chow, <em>Conditional Sharpe Ratios</em>,, Finance Research Letters, (2014)
[7] X. Y. Cui, Continuous-time mean-variance portfolio selection with finite transactions,, Stochastic analysis and applications to finance, 13, 77, (2012)
[8] X. Y. Cui, Optimal multi-period mean-variance policy under no-shorting constraint,, European Journal of Operational Research, 234, 459, (2014) · Zbl 1304.91185
[9] J. Cvitanic, Implications of the sharpe ratio as a performance measure in multi-period settings,, Journal of Economic Dynamics and Control, 32, 1622, (2008) · Zbl 1181.91330
[10] D. M. Danga, Better than pre-commitment mean-variance portfolio allocation strategies: A semi-self-financing Hamilton-Jacobi-Bellman equation approach,, European Journal of Operational Research, 250, 827, (2016) · Zbl 1348.91250
[11] V. DeMiguel, Optimal versus Naive Diversification: How ineficient is the 1/N portfolio strategy?,, Review of Financial Studies, 22, 1915, (2009)
[12] K. Dowd, Adjusting for risk: An improved Sharpe ratio,, International Review of Economics and Finance, 9, 209, (2000)
[13] W. H. Fleming, Controlled Markov processes and viscosity solutions,, 2ed. Springer, (2006) · Zbl 1105.60005
[14] D. Li, Optimal dynamic portfolio selection: Multiperiod mean-variance formulation,, Mathematical Finance, 10, 387, (2000) · Zbl 0997.91027
[15] X. Li, Dynamic mean-variance portfolio selection with no-shorting constraints,, SIAM Journal on Control and Optimization, 40, 1540, (2002) · Zbl 1027.91040
[16] H. Markowitz, Portfolio selection,, Journal of Finance, 7, 77, (1952)
[17] R. C. Merton, An analytic derivation of the efficient portfolio frontier,, Journal of Financial and Quantitative Analysis, 7, 1851, (1972)
[18] M. Schuster, A note on empirical Sharpe ratio dynamics,, Economics Letters, 116, 124, (2012) · Zbl 1253.91127
[19] W. F. Sharpe, <em>Capital asset prices: A theory of market equilibrium under conditions of risk</em>,, Journal of Finance, 19, 425, (1964)
[20] W. F. Sharpe, Mutual fund performance,, Journal of Business, 39, 119, (1966)
[21] W. F. Sharpe, The Sharpe ratio,, The Journal of Portfolio Management, 21, 49, (1994)
[22] A. D. Roy, Safety first and the holding of assets,, Econometrica, 20, 431, (1952) · Zbl 0047.38805
[23] Z. Wang, Multi-period mean-variance portfolio selection with fixed and proportional transaction costs,, Journal of Industrial and Management Optimization, 9, 643, (2013) · Zbl 1281.90080
[24] H. X. Yao, Continuous-time mean-variance portfolio selection with only risky assets,, Economic Modelling, 36, 244, (2014)
[25] H. X. Yao, Dynamic mean-variance asset allocation with stochastic interest rate and inflation rate,, Journal of Industrial and Management Optimization, 12, 187, (2016) · Zbl 1317.90248
[26] V. Zakamouline, Portfolio performance evaluation with generalized Sharpe ratios: Beyond the mean and variance,, Journal of Banking and Finance, 33, 1242, (2009)
[27] Y. Zeng, Robust equilibrium reinsurance-investment strategy for a mean-variance insurer in a model with jumps,, Insurance: Mathematics and Economics, 66, 138, (2016) · Zbl 1348.91192
[28] X. Y. Zhou, Continuous-time mean-variance portfolio selection: A stochastic LQ framework,, Applied Mathematics and Optimization, 42, 19, (2000) · Zbl 0998.91023
[29] S. S. Zhu, Risk control over bankruptcy in dynamic portfolio selection: A generalized mean-variance formulation,, IEEE Transactions on Automatic Control, 49, 447, (2004) · Zbl 1366.91150
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.