Portfolio Analysis with General Deviation Measures

U of Florida, Industrial and Systems Engineering Research Report No. 2003-8

36 Pages Posted: 17 Jul 2003

See all articles by R. Tyrrell Rockafellar

R. Tyrrell Rockafellar

University of Washington - Department of Mathmatics

Stanislav P. Uryasev

University of Florida

Michael Zabarankin

Stevens Institute of Technology - Department of Mathematical Sciences

Date Written: June 3, 2003

Abstract

Generalized measures of deviation, as substitutes for standard deviation, are considered in a framework like that of classical portfolio theory for coping with the uncertainty inherent in achieving rates of return beyond the risk-free rate. Such measures, associated for example with conditional value-at-risk and its variants, can reflect the different attitudes of different classes of investors. They lead nonetheless to generalized one-fund theorems as well as to covariance relations which resemble those commonly used in capital asset pricing models (CAPM), but have wider interpretations. A more customized version of portfolio optimization is the aim, rather than the idea that a single "master fund" might arise from market equilibrium and serve the interests of all investors.

The results cover discrete distributions along with continuous distributions, and therefore are applicable in particular to financial models involving finitely many future states, whether introduced directly or for purposes of numerical approximation. Through techniques of convex analysis, they deal rigorously with a number of features that have not been given much attention in this subject, such as solution nonuniqueness, or nonexistence, and a potential lack of differentiability of the deviation expression with respect to the portfolio weights. Moreover they address in detail the previously neglected phenomenon that, if the risk-free rate lies above a certain threshold, a master fund of the usual type will fail to exist and need to be replaced by one of an alternative type, representing a "net short position" instead of a "net long position" in the risky instruments.

Keywords: deviation measures, risk measures, value-at-risk, conditional value-at-risk, portfolio optimization, one-fund theorems, master funds, efficient frontiers, CAPM, convex analysis

JEL Classification: C0, C2, C6

Suggested Citation

Rockafellar, R. Tyrrell and Uryasev, Stanislav P. and Zabarankin, Michael, Portfolio Analysis with General Deviation Measures (June 3, 2003). U of Florida, Industrial and Systems Engineering Research Report No. 2003-8, Available at SSRN: https://ssrn.com/abstract=414268 or http://dx.doi.org/10.2139/ssrn.414268

R. Tyrrell Rockafellar

University of Washington - Department of Mathmatics ( email )

Box 354350
Seattle, WA 98195-4350
United States

Stanislav P. Uryasev (Contact Author)

University of Florida ( email )

303 Weil Hall
Gainesville, FL 32611-6595
United States
352-392-3091 (Phone)
352-392-3537 (Fax)

HOME PAGE: http://www.ise.ufl.edu/uryasev/

Michael Zabarankin

Stevens Institute of Technology - Department of Mathematical Sciences ( email )

Hoboken, NJ 07030
United States

HOME PAGE: http://personal.stevens.edu/~mzabaran/

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