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.
Highlights
The modern portfolio theory of Markowitz (1959) pioneered the quantitative analysis of financial economics
The most important idea proposed in this theory is that one should focus on the trade-off between expected return and the risk measured by the standard deviation
The goal and main results of this paper are to extend the modern portfolio theory into a general framework under which one can analyze efficient portfolios that trade-off between a convex risk measure and a reward captured by a concave expected utility
Summary
The modern portfolio theory of Markowitz (1959) pioneered the quantitative analysis of financial economics. When a deviation measure (Rockafellar et al 2006) is used as risk measure, which happens e.g., when an approximation of the current drawdown is considered (see Maier-Paape and Zhu (2017)), and the expected return is used to gauge the performance, we show that the affine structure of the efficient solution in the classical capital market pricing model is preserved (cf Theorem 10 and Corollary 3), recovering and extending especially the results in Rockafellar et al (2006). The dual problem projects the efficient trade-off path into a concave curve in the risk-expected log utility space parallel to the role of Markowitz bullet in the modern portfolio theory and the capital market line in the capital asset pricing model.
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