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Risks 2018, 6(2), 53; https://doi.org/10.3390/risks6020053

A General Framework for Portfolio Theory—Part I: Theory and Various Models

1
Institut für Mathematik, RWTH Aachen University, Templergraben 55, 52062 Aachen, Germany
2
Department of Mathematics, Western Michigan University, 1903 West Michigan Avenue, Kalamazoo, MI 49008, USA
*
Author to whom correspondence should be addressed.
Received: 30 March 2018 / Revised: 23 April 2018 / Accepted: 1 May 2018 / Published: 8 May 2018
(This article belongs to the Special Issue Computational Methods for Risk Management in Economics and Finance)
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Abstract

Utility and risk are two often competing measurements on the investment success. We show that efficient trade-off between these two measurements for investment portfolios happens, in general, on a convex curve in the two-dimensional space of utility and risk. This is a rather general pattern. The modern portfolio theory of Markowitz (1959) and the capital market pricing model Sharpe (1964), are special cases of our general framework when the risk measure is taken to be the standard deviation and the utility function is the identity mapping. Using our general framework, we also recover and extend the results in Rockafellar et al. (2006), which were already an extension of the capital market pricing model to allow for the use of more general deviation measures. This generalized capital asset pricing model also applies to e.g., when an approximation of the maximum drawdown is considered as a risk measure. Furthermore, the consideration of a general utility function allows for going beyond the “additive” performance measure to a “multiplicative” one of cumulative returns by using the log utility. As a result, the growth optimal portfolio theory Lintner (1965) and the leverage space portfolio theory Vince (2009) can also be understood and enhanced under our general framework. Thus, this general framework allows a unification of several important existing portfolio theories and goes far beyond. For simplicity of presentation, we phrase all for a finite underlying probability space and a one period market model, but generalizations to more complex structures are straightforward. View Full-Text
Keywords: convex programming; financial mathematics; risk measure; utility functions; efficient frontier; Markowitz portfolio theory; capital market pricing model; growth optimal portfolio; fractional Kelly allocation convex programming; financial mathematics; risk measure; utility functions; efficient frontier; Markowitz portfolio theory; capital market pricing model; growth optimal portfolio; fractional Kelly allocation
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Maier-Paape, S.; Zhu, Q.J. A General Framework for Portfolio Theory—Part I: Theory and Various Models. Risks 2018, 6, 53.

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