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Dynamically optimal portfolios for monotone mean--variance preferences

Aleš Černý, Johannes Ruf, Martin Schweizer

Abstract

Monotone mean-variance (MMV) utility is the minimal modification of the classical Markowitz utility that respects rational ordering of investment opportunities. This paper provides, for the first time, a complete characterization of optimal dynamic portfolio choice for the MMV utility in asset price models with independent returns. The task is performed under minimal assumptions, weaker than the existence of an equivalent martingale measure and with no restrictions on the moments of asset returns. We interpret the maximal MMV utility in terms of the monotone Sharpe ratio (MSR) and show that the global squared MSR arises as the nominal yield from continuously compounding at the rate equal to the maximal local squared MSR. The paper gives simple necessary and sufficient conditions for mean-variance (MV) efficient portfolios to be MMV efficient. Several illustrative examples contrasting the MV and MMV criteria are provided.

Dynamically optimal portfolios for monotone mean--variance preferences

Abstract

Monotone mean-variance (MMV) utility is the minimal modification of the classical Markowitz utility that respects rational ordering of investment opportunities. This paper provides, for the first time, a complete characterization of optimal dynamic portfolio choice for the MMV utility in asset price models with independent returns. The task is performed under minimal assumptions, weaker than the existence of an equivalent martingale measure and with no restrictions on the moments of asset returns. We interpret the maximal MMV utility in terms of the monotone Sharpe ratio (MSR) and show that the global squared MSR arises as the nominal yield from continuously compounding at the rate equal to the maximal local squared MSR. The paper gives simple necessary and sufficient conditions for mean-variance (MV) efficient portfolios to be MMV efficient. Several illustrative examples contrasting the MV and MMV criteria are provided.

Paper Structure

This paper contains 28 sections, 29 theorems, 177 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Dynamically optimal portfolio selection
  3. Asset prices and admissibility
  4. First steps
  5. Parametrization of trading strategies
  6. Optimal dollar investment per unit of local risk tolerance
  7. Optimal local expected utility strategy integrates R
  8. Optimal monotone mean--variance portfolios
  9. Duality
  10. Economic implications
  11. Classical mean--variance analysis
  12. Monotone mean--variance analysis
  13. Examples
  14. Related studies
  15. Monotone mean--variance preferences
  16. ...and 13 more sections

Key Result

Proposition 2.3

The following are equivalent. Furthermore, if either of the two conditions holds, then and eq:vMMV admits an optimizer for $x=0$ if and only if eq:uMMV does so for $x=0$, in which case eq:vMMV admits an optimizer for each $x\in\mathbb{R}$ and

Figures (1)

  • Figure 1: Quadratic utility and its monotonization.

Theorems & Definitions (73)

  • Definition 2.2
  • Proposition 2.3
  • Remark 2.4
  • Remark 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Remark 2.8
  • Definition 2.9
  • Proposition 2.10
  • Remark 2.11
  • ...and 63 more