Advertisement

A Utility Equivalence Theorem for Concave Functions

  • Anand Bhalgat
  • Sanjeev Khanna
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8494)

Abstract

Given any two sets of independent non-negative random variables and a non-decreasing concave utility function, we identify sufficient conditions under which the expected utility of sum of these two sets of variables is (almost) equal. We use this result to design a polynomial-time approximation scheme (PTAS) for utility maximization in a variety of risk-averse settings where the risk is modeled by a concave utility function. In particular, we obtain a PTAS for the asset allocation problem for a risk-averse investor as well as the risk-averse portfolio allocation problem.

Keywords

Utility Maximization Optimal Portfolio Concave Function Stochastic Dominance Utility Maximization Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arrow, K.J.: The theory of risk aversion. Aspects of the Theory of Risk Bearing (1965)Google Scholar
  2. 2.
    Barahona, F., Pulleyblank, W.: Exact arborescences, matchings and cycles. Discrete Applied Mathematics (1987)Google Scholar
  3. 3.
    Bhalgat, A., Chakraborty, T., Khanna, S.: Mechanism design for a risk-averse seller. In: Goldberg, P.W. (ed.) WINE 2012. LNCS, vol. 7695, pp. 198–211. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  4. 4.
    Carraway, R.L., Schmidt, R.L., Weatherford, L.R.: An algorithm for maximizing target achievement in the stochastic knapsack problem with normal returns. Naval Res. Logist. 40, 161–173 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Fishburn, P., Porter, B.: Optimal portfolios with one safe and one risky asset: Effects of changes in rate of return and risk. Management Science (1976)Google Scholar
  6. 6.
    Geetha, S., Nair, K.: On stochastic spanning tree problem. Networks (1993)Google Scholar
  7. 7.
    Shiode, S., Ishii, H., Nishida Yoshikazu, T.: Stochastic spanning tree problem. Discrete Applied Mathematics (1981)Google Scholar
  8. 8.
    Hadar, J., Seo, T.: Asset proportions in optimal portfolios. Review of Economics Studies (1988)Google Scholar
  9. 9.
    Henig, M.: Risk criteria in stochastic knapsack. Oper. Res. 38(5), 820–825 (1990)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Kijima, M., Ohnishi, M.: Portfolio selection problems via the bivariate characterization of stochastic dominance relations. Journal Mathematical Finance (1996)Google Scholar
  11. 11.
    Landsberger, M., Meilijson, I.: Demand for risky financial assets: A portfolio analysis. Journal of Economic Theory (1990)Google Scholar
  12. 12.
    Lappan, H., Hennessey, D.: Symmetry and order in the portfolio allocation problem. Economic Theory (2002)Google Scholar
  13. 13.
    Li, J., Deshpande, A.: Maximizing expected utility for stochastic combinatorial optimization problems. In: FOCS (2011)Google Scholar
  14. 14.
    Pratt, J.W.: Risk aversion in the small and in the large. Econometrica 32(1-2), 122–136 (1964)CrossRefzbMATHGoogle Scholar
  15. 15.
    Sneidovik, M.: Preference order stochastic knapsack problems: Methodological issues. J. Oper. Res Soc. 31(11), 1025–1032 (1980)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Steinberg, E., Parks, M.S.: Preference order dynamic program for a knapsack problem with stochastic rewards. J. Oper. Res Soc. 30(2), 141–147 (1979)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Anand Bhalgat
    • 1
  • Sanjeev Khanna
    • 2
  1. 1.Facebook Inc.USA
  2. 2.University of PennsylvaniaUSA

Personalised recommendations