A NOTE ON THE QUANTILE FORMULATION

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13 Citations (Scopus)

Abstract

Many investment models in discrete or continuous-time settings boil down to maximizing an objective of the quantile function of the decision variable. This quantile optimization problem is known as the quantile formulation of the original investment problem. Under certain monotonicity assumptions, several schemes to solve such quantile optimization problems have been proposed in the literature. In this paper, we propose a change-of-variable and relaxation method to solve the quantile optimization problems without using the calculus of variations or making any monotonicity assumptions. The method is demonstrated through a portfolio choice problem under rank-dependent utility theory (RDUT). We show that this problem is equivalent to a classical Merton's portfolio choice problem under expected utility theory with the same utility function but a different pricing kernel explicitly determined by the given pricing kernel and probability weighting function. With this result, the feasibility, well-posedness, attainability, and uniqueness issues for the portfolio choice problem under RDUT are solved. It is also shown that solving functional optimization problems may reduce to solving probabilistic optimization problems. The method is applicable to general models with law-invariant preference measures including portfolio choice models under cumulative prospect theory (CPT) or RDUT, Yaari's dual model, Lopes' SP/A model, and optimal stopping models under CPT or RDUT.
Original languageEnglish
Pages (from-to)589-601
Number of pages13
JournalMathematical Finance
Volume26
Issue number3
DOIs
Publication statusPublished - 1 Jul 2016

Keywords

  • atomic
  • atomless/nonatomic
  • behavioral finance
  • calculus of variations
  • change-of-variable
  • CPT
  • functional optimization problem
  • law-invariant
  • portfolio choice/selection
  • probability weighting/distortion function
  • quantile formulation
  • RDUT
  • relaxation method
  • time consistency

ASJC Scopus subject areas

  • Accounting
  • Social Sciences (miscellaneous)
  • Finance
  • Economics and Econometrics
  • Applied Mathematics

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