Abstract
Classical optimal strategies are notorious for producing remarkably volatile portfolio weights over time when applied with parameters estimated from data. This is predominantly explained by the difficulty to estimate expected returns accurately. In Lindberg (Bernoulli 15:464–474, 2009), a new parameterization of the drift rates was proposed with the aim to circumventing this difficulty, and a continuous time mean–variance optimal portfolio problem was solved. This approach was further developed in Alp and Korn (Decis Econ Finance 34:21–40, 2011a) to a jump-diffusion setting. In the present paper, we solve a different portfolio problem under the market parameterization in Lindberg (Bernoulli 15:464–474, 2009). Here, the admissible investment strategies are given as the amounts of money to be held in each stock and are allowed to be adapted stochastic processes. In the references above, the admissible strategies are the deterministic and bounded fractions of the total wealth. The optimal strategy we derive is not the same as in Lindberg (Bernoulli 15:464–474, 2009), but it can still be viewed as investing equally in each of the n Brownian motions in the model. As a consequence of the problem assumptions, the optimal final wealth can become non-negative. The present portfolio problem is solved also in Alp and Korn (Submitted, 2011b), using the L 2-projection approach of Schweizer (Ann Probab 22:1536–1575, 1995). However, our method of proof is direct and much easier accessible.
Similar content being viewed by others
References
Alp Ö.S., Korn R.: Continuous-time mean–variance portfolio optimization in a jump-diffusion market. Decis. Econ. Finance 34, 21–40 (2011a)
Alp, Ö.S., Korn, R.: Continuous-time mean–variance portfolios: a comparison. Submitted, (2011b)
Bielecki T.R., Jin H., Pliska S.R., Zhou X.Y.: Continuous-time mean–variance portfolio selection with bankruptcy prohibition. Math. Finance 15, 213–244 (2005)
Black F., Litterman R.: Global portfolio optimization. Financial Anal. J. 48, 28–43 (1992)
Karatzas I., Shreve S.: Methods of Mathematical Finance. Springer, New York (1998)
Korn R.: Optimal Portfolios. World Scientific, Singapore (1997)
Korn R., Trautmann S.: Continuous-time portfolio optimization under terminal wealth constraints. ZOR 42, 69–93 (1995)
Li X., Zhou X.Y., Lim A.E.B.: Dynamic mean–variance portfolio selection with no-shorting constraints. SIAM J. Control. Optim. 40, 1540–1555 (2001)
Lim A.E.B., Zhou X.Y.: Mean–variance portfolio selection with random parameters. Math. Oper. Res. 27, 101–120 (2002)
Lindberg C.: Portfolio optimization when expected returns are determined by exposure to risk. Bernoulli 15, 464–474 (2009)
Markowitz H.: Portfolio selection. J. Finance 7, 77–91 (1952)
Merton R.: Lifetime portfolio selection under uncertainty: the continuous time case. Rev. Econ. Stat. 51, 247–257 (1969)
Merton, R.: Optimum consumption and portfolio rules in a continuous time model. J. Econ. Theory 3, 373–413; Erratum, J. Econ. Theory 6, 213–214 (1971)
Ross S.A: The arbitrage theory of capital asset pricing. J. Econ. Theory 13, 341–360 (1976)
Schweizer M.: Approximation of random variables by stochastic integrals. Ann. Probab. 22, 1536–1575 (1995)
Zhou X.Y., Li D.: Continuous time mean–variance portfolio selection: a stochastic LQ framework. Appl. Math. Optim. 42, 19–33 (2000)
Zhou X.Y., Yin G.: Markowitz’ mean–variance portfolio selection with regime switching: a continuous time model. SIAM J. Control. Optim. 42, 1466–1482 (2003)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lindberg, C. Investing equally in risk. Decisions Econ Finan 36, 39–46 (2013). https://doi.org/10.1007/s10203-011-0121-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10203-011-0121-3
Keywords
- 1/n strategy
- Black–Scholes model
- Expected stock returns
- Markowitz’ problem
- Mean–variance
- Portfolio optimization