Hostname: page-component-745bb68f8f-kw2vx Total loading time: 0 Render date: 2025-01-18T19:52:05.932Z Has data issue: false hasContentIssue false
  • English
  • Français

Portfolio Selection in a Lognormal Market When the Investor Has a Power Utility Function

Published online by Cambridge University Press:  19 October 2009

J. A. Ohlson  and
W. T. Ziemba
Get access Rights & Permissions [Opens in a new window]

Extract

Multiasset portfolio selection models stated in terms of the expected utility criterion generally require the evaluation of multiple integrals. This reality has severely hindered attempts towards the development of computation methods to determine optimal portfolio allocations when there are a large number of assets. Aside from special cases, expected utility is not convergent into a simple closed form; the complexity from the point of view of computation is then perhaps most easily appreciated if one realizes that every iteration in a nonlinear program demands the estimation of several integrals (see Ziemba [23] for details). Such calculations are extremely costly when the number of assets is large. It is, consequently, of interest to approximate the expected utility function by a function which is easier to optimize over the set of feasible portfolios.

Type
Research Article
Information
Journal of Financial and Quantitative Analysis , Volume 11 , Issue 1 , March 1976 , pp. 57 - 71
Copyright
Copyright © School of Business Administration, University of Washington 1976

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

*

University of California, Berkeley, and University of British Columbia. This research was partially supported by the National Research Council of Canada. The authors would like to thank C. E. Lemke, D. Murdoch, S. J. Press, and an anonymous referee for valuable comments on an earlier version of this paper.

References

REFERENCES

[1] Blattberg, R. C., and Gonedes, N. J.. “A Comparison of the Stable-Paretian And Student Distributions as Statistical Models for Stock Prices.” Journal of Business, Vol. 47 (1974), pp. 244–280.CrossRefGoogle Scholar
[2] Dexter, A. S.; Yu, J. N. W.; and Ziemba, W. T.. “Portfolio Selection in a Lognormal Market when the Investor Has a Power Utility Function: Computational Results.” In Proceedings of the International Conference on Stochastic Programming, edited by Dempster, M. A. H.. New York: Academic Press (1975), forthcoming.Google Scholar
[3] Fletcher, R., ed. Optimization. New York: Academic Press, Inc. (1969).Google Scholar
[4] Frank, M., and Wolfe, P.. “An Algorithm for Quadratic Programming.” Naval Research Logistics Quarterly, Vol. 3 (1956), pp. 95–110.CrossRefGoogle Scholar
[5] Hakansson, N. H. “Multi-Period Mean Variance Analysis: Toward a General Theory of Portfolio Choice.” Journal of Finance, Vol. 26 (1971), pp. 857–884.Google Scholar
[6] Hakansson, N. H. “On Optimal Myopic Portfolio Policies, with and without Serial Correlation of Yields.” Journal of Business (1972), pp. 324–334.Google Scholar
[7] Hogan, W. W., and Warren, J. M.. “Computation of the Efficient Boundary in the E-S Portfolio Selection Model.” Journal of Financial and Quantitative Analysis, Vol. 7 (1972), pp. 1881–1896.CrossRefGoogle Scholar
[8] Lintner, J. “Equilibrium in a Random Walk and Lognormal Securities Market.” Discussion Paper Number 235 (July 1972). Harvard Institute of Economic Research, Harvard University.Google Scholar
[9] Loeve, M. Probability Theory. Princeton: D. Van Nostrand and Co. (1963).Google Scholar
[10] Mangasarian, O. L. “Convexity, Pseudo-Convexity and Quasi-Convexity of Composite Functions.” Cahiers du Centre d'Etudes de Recherche Operationelle, Vol. 12 (1970), pp. 114–122.Google Scholar
[11] Marcus, M., and Ming, H.. A Survey of Matrix Theory and Matrix Inequalities. Boston: Allyn and Bacon Publishing Co., Inc. (1964).Google Scholar
[12] Merton, R. C. “Lifetime Portfolio Selection under Uncertainty: the Continuous-Time Case.” Review of Economics and Statistics, Vol. 51 (1969), pp. 247–257.CrossRefGoogle Scholar
[13] Merton, R. C. “Optimum Consumption and Portfolio Rules in a Continuous-Time Model.” Journal of Economic Theory, Vol. 3 (1971), pp. 373–413.CrossRefGoogle Scholar
[14] Merton, R. C. “An Intertemporal Capital Asset Pricing Model.” Econometrica, Vol. 41 (1973).CrossRefGoogle Scholar
[15] Ohlson, J. A. “Optimal Portfolio Selection in a Lognormal Market When the Investor's Utility Function is Logarithmic.” Research Paper No. 117, Graduate School of Business, Stanford University (September 1972).Google Scholar
[16] Ohlson, J. A. “The Asymptotic Validity of Quadratic Utility as the Period-Spacing Approaches Zero.” In Stochastic Optimization Models in Finance, edited by Ziemba, W. T. and Vickson, R. G.. New York: Academic Press (1975).Google Scholar
[17] Pye, G. “Portfolio Selection and Security Prices.” Review of Economics and Statistics, Vol. 49 (1967), pp. 111–115.CrossRefGoogle Scholar
[18] Rosenberg, B. “The Behavior of Random Variables with Non-Stationary Variance and the Distribution of Security Prices.” Unpublished Paper. University of California, Berkeley (1973).Google Scholar
[19] Samuelson, P. A. “The Fundamental Approximation Theorem of Portfolio Analysis in Terms of Means, Variances and Hiqher Moments.” Review of Economic Studies, Vol. 37 (1970), pp. 537–542.CrossRefGoogle Scholar
[20] Wagner, H. M. Fundamentals of Operations Research. Englewood Cliffs, N. J.: Prentice-Hall, Inc. (1969).Google Scholar
[21] Wolfe, P. “Convergence Theorv in Non-Linear Programming.” In Integer and Non-Linear Programming, edited by Abadie, J.. Amsterdam: North-Holland Publishing Co., Inc. (1971).Google Scholar
[22] Zangwill, W. I. Nonlinear Programming: A Unified Approach. Englewood Cliffs, N. J.: Prentice-Hall, Inc. (1969).Google Scholar
[23] Ziemba, W. T. “Solving Nonlinear Programming Problems with Stochastic Objective Functions.” Journal of Financial and Quantitative Analysis, Vol. 7 (1972), pp. 1809–1827.CrossRefGoogle Scholar