• Bernoulli
  • Volume 15, Number 2 (2009), 464-474.

Portfolio optimization when expected stock returns are determined by exposure to risk

Carl Lindberg

Full-text: Open access


It is widely recognized that when classical optimal strategies are applied with parameters estimated from data, the resulting portfolio weights are remarkably volatile and unstable over time. The predominant explanation for this is the difficulty of estimating expected returns accurately. In this paper, we modify the n stock Black–Scholes model by introducing a new parametrization of the drift rates. We solve Markowitz’ continuous time portfolio problem in this framework. The optimal portfolio weights correspond to keeping 1/n of the wealth invested in stocks in each of the n Brownian motions. The strategy is applied out-of-sample to a large data set. The portfolio weights are stable over time and obtain a significantly higher Sharpe ratio than the classical 1/n strategy.

Article information

Bernoulli Volume 15, Number 2 (2009), 464-474.

First available in Project Euclid: 4 May 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

1/n strategy Black–Scholes model expected stock returns Markowitz’ problem portfolio optimization ranks


Lindberg, Carl. Portfolio optimization when expected stock returns are determined by exposure to risk. Bernoulli 15 (2009), no. 2, 464--474. doi:10.3150/08-BEJ163.

Export citation


  • [1] Bielecki, T.R., Jin, H., Pliska, S.R. and Zhou, X.Y. (2005). Continuous-time mean-variance portfolio selection with bankruptcy prohibition., Math. Finance 15 213–244.
  • [2] Black, F. and Litterman, R. (1992). Global portfolio optimization., Financial Analysts Journal 48 28–43.
  • [3] Fernholz, R. (2002)., Stochastic Portfolio Theory. New York: Springer.
  • [4] Korn, R. (1997)., Optimal Portfolios. Singapore: World Scientific.
  • [5] Korn, R. and Trautmann, S. (1995). Continuous-time portfolio optimization under terminal wealth constraints., ZOR 42 69–93.
  • [6] Li, X., Zhou, X.Y. and Lim, A.E.B. (2001). Dynamic mean-variance portfolio selection with no-shorting constraints., SIAM J. Control Optim. 40 1540–1555.
  • [7] Lim, A.E.B. and Zhou, X.Y. (2002). Mean-variance portfolio selection with random parameters., Math. Oper. Res. 27 101–120.
  • [8] Markowitz, H. (1952). Portfolio selection., J. Finance 7 77–91.
  • [9] Merton, R. (1969). Lifetime portfolio selection under uncertainty: The continuous time case., Rev. Econom. Statist. 51 247–257.
  • [10] Merton, R. (1971). Optimum consumption and portfolio rules in a continuous time model., J. Econom. Theory 3 373–413; Erratum. J. Econom. Theory 6 213–214.
  • [11] Ross, S.A. (1976). The arbitrage theory of capital asset pricing., J. Econom. Theory 13 341–360.
  • [12] Zhou, X.Y. and Li, D. (2000). Continuous time mean-variance portfolio selection: A stochastic LQ framework., Appl. Math. Optim. 42 19–33.
  • [13] Zhou, X.Y. and Yin, G. (2003). Markowitz’ mean-variance portfolio selection with regime switching: A continuous time model., SIAM J. Control Optim. 42 1466–1482.