Journal of Optimization Theory and Applications

, Volume 150, Issue 3, pp 615–634 | Cite as

Degeneracy Resolution for Bilinear Utility Functions

Article
First Online:

Abstract

Loss-aversion is a phenomenon where investors are particularly sensitive to losses and eager to avoid them. An efficient method to solve the portfolio optimization problem of maximizing the bilinear utility function is given by Best et al. (Loss-Aversion with Kinked Linear Utility Functions, CORR 2010-04, University of Waterloo, 2010). This method is useful because it performs its computations only using asset related quantities rather than much higher dimensional quantities of the LP formulation. However, a difficulty with this method is that it requires a nondegeneracy assumption which may not be satisfied. This paper implements Bland’s least-index rules to the method in such a way that the efficiency of the method is retained. Then we describe the numerical results of applying our algorithm to a series of six asset problems in which the degree of loss-aversion is increased.

Keywords

Loss-aversion Degeneracy Bland’s least-index rules Active set method 
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References

  1. 1.
    Kahneman, D., Tversky, A.: Prospect theory: an analysis of decision under risk. Econometrica 47, 363–391 (1979) CrossRefGoogle Scholar
  2. 2.
    Bowman, D., Minehart, D., Rabin, M.: Loss aversion in a consumption-savings model. J. Econ. Behav. Organ. 38, 155–178 (1999) CrossRefGoogle Scholar
  3. 3.
    Barberis, N., Huang, M.: Mental accounting, loss aversion and individual stock returns. J. Finance 56, 1247–1292 (2001) CrossRefGoogle Scholar
  4. 4.
    Siegmann, A.: Optimal financial decision making under loss averse preferences. PhD thesis, University of Amsterdam (2002) Google Scholar
  5. 5.
    Siegmann, A.: Shortfall allowed: loss aversion and habit formation. Working paper 741/0321, The Netherlands Central Bank (2003) Google Scholar
  6. 6.
    Berkelaar, A.B., Kouwenberg, R., Post, T.: Optimal portfolio choice under loss aversion. Rev. Econ. Stat. 86, 973–987 (2004) CrossRefGoogle Scholar
  7. 7.
    Cremers, J.H., Kritzman, M., Page, S.: Optimal hedge fund allocations: do higher moments matter? J. Portf. Manag. 31, 135–155 (2005) Google Scholar
  8. 8.
    Maggi, M.A.: Loss aversion and perceptual risk aversion. J. Math. Psychol. 50, 426–430 (2006) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Abdellaoui, M., Bleichrodt, H., Paraschiv, C.: Loss aversion under prospect theory: a parameter-free measurement. Manag. Sci. 53, 1659–1674 (2007) CrossRefGoogle Scholar
  10. 10.
    Apesteguia, J., Ballester, M.A.: A theory of reference-dependent behavior. Econom. Theory 40, 427–455 (2009) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Zank, H.: On probabilities and loss aversion. Theory Decis. 68, 243–261 (2010) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Markowitz, H.M.: Portfolio selection. J. Finance 7, 77–91 (1952) CrossRefGoogle Scholar
  13. 13.
    Markowitz, H.M.: Portfolio Selection: Efficient Diversification of Investments. Wiley, New York (1959) Google Scholar
  14. 14.
    Levy, H., Markowitz, H.M.: Approximating expected utility by a function of mean and variance. Am. Econ. Rev. 69, 308–317 (1979) Google Scholar
  15. 15.
    Markowitz, H.M.: Mean-Variance Analysis in Portfolio choice and Capital Markets. Blackwell, Oxford (1987) MATHGoogle Scholar
  16. 16.
    Adler, T., Kritzman, M.: Mean-variance versus full-scale optimization: in and out of sample. J. Asset Manag. 7, 302–311 (2007) CrossRefGoogle Scholar
  17. 17.
    Hagströmer, B., Anderson, R.G., Binner, J.M., Elger, T., Nilsson, B.: Mean-variance vs full-scale optimization: broad evidence for the UK. Working paper 2007-016B, Federal Reserve Bank of St. Louis, Research division (2007) Google Scholar
  18. 18.
    Best, M.J., Hlouskova, J.: An algorithm for portfolio optimization with transaction costs. Manag. Sci. 51, 1676–1688 (2005) CrossRefGoogle Scholar
  19. 19.
    Best, M.J., Hlouskova, J.: An algorithm for portfolio optimization with variable transaction costs, Part 1: theory. J. Optim. Theory Appl. 135, 563–581 (2007) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Best, M.J., Hlouskova, J.: An algorithm for portfolio optimization with variable transaction costs, Part 2: computational analysis. J. Optim. Theory Appl. 135, 531–547 (2007) MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Best, M.J., Grauer, R.R., Hlouskova, J., Zhang, X.: Loss-Aversion with Kinked Linear Utility Functions. CORR 2010-04, University of Waterloo (2010) Google Scholar
  22. 22.
    Bland, R.G.: New finite pivoting rules for the simplex method. Math. Oper. Res. 2, 103–107 (1977) MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Best, M.J., Ritter, K.: Linear Programming: Active Set Analysis and Computer Programms. Prentice Hall, Englewood Cliffs (1985) Google Scholar
  24. 24.
    Rockafellar, R.T., Uryasev, S.: Optimization of conditional value-at-risk. J. Risk 2, 21–42 (2000) Google Scholar
  25. 25.
    Grauer, R.R.: Beta in linear risk tolerance economies. Manag. Sci. 31, 1390–1402 (1985) MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Combinatorics and Optimization, Faculty of MathematicsUniversity of WaterlooWaterlooCanada
  2. 2.School of Business AdministrationSouth China University of TechnologyGuangzhouChina

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