Paper: arXiv 2309.07488

Authors: Michael Preisel

Abstract

This paper studies the mean-variance optimal portfolio choice of an investor pre-committed to a deterministic investment policy in continuous time in a market with mean-reversion in the risk-free rate and the equity risk-premium. In the tradition of Markowitz, optimal policies are restricted to a subclass of factor exposures in which losses cannot exceed initial capital and it is shown that the optimal policy is characterized by an Euler-Lagrange equation derived by the method of Calculus of Variations. It is a main result, that the Euler-Lagrange equation can be recast into a matrix differential equation by an integral transformation of the factor exposure and that the solution to the characteristic equation can be parametrized by the eigenvalues of the associated lambda-matrix, hence, the optimization problem is equivalent to a spectral problem. Finally, explicit solutions to the optimal policy are provided by application of suitable boundary conditions and it is demonstrated that - if in fact the equity risk-premium is slowly mean-reverting - then investors committing to long investment horizons realize better risk-return trade-offs than investors with shorter investment horizons.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 2.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a sophisticated theoretical framework for long-term mean-variance optimization, recasting Euler-Lagrange equations into a spectral matrix differential problem. However, it is entirely analytical with limited empirical backtesting or real-world data validation provided in the summary.

Research Flowchart

  flowchart TD
    A[Research Goal: Long-Term Mean-Variance Portfolio Optimization] --> B[Inputs: Mean-Reverting Equity Risk-Premium & Risk-Free Rate]
    B --> C[Methodology: Restrict Factor Exposures & Apply Calculus of Variations]
    C --> D[Derive Euler-Lagrange Equation]
    D --> E[Transform to Matrix Differential Equation & Spectral Problem]
    E --> F[Apply Boundary Conditions to Solve for Optimal Policy]
    F --> G[Finding: Long-Horizon Investors Achieve Superior Risk-Return Trade-Offs]