Paper: arXiv 2303.08521

Authors: Nicole Bäuerle, Antje Mahayni

Abstract

We consider the classical multi-asset Merton investment problem under drift uncertainty, i.e. the asset price dynamics are given by geometric Brownian motions with constant but unknown drift coefficients. The investor assumes a prior drift distribution and is able to learn by observing the asset prize realizations during the investment horizon. While the solution of an expected utility maximizing investor with constant relative risk aversion (CRRA) is well known, we consider the optimization problem under risk and ambiguity preferences by means of the KMM (Klibanoff et al. (2005)) approach. Here, the investor maximizes a double certainty equivalent. The inner certainty equivalent is for given drift coefficient, the outer is based on a drift distribution. Assuming also a CRRA type ambiguity function, it turns out that the optimal strategy can be stated in terms of the solution without ambiguity preferences but an adjusted drift distribution. To the best of our knowledge an explicit solution method in this setting is new. We rely on some duality theorems to prove our statements. Based on our theoretical results, we are able to shed light on the impact of the prior drift distribution as well as the consequences of ambiguity preferences via the transfer to an adjusted drift distribution, i.e. we are able to explain the interaction of risk and ambiguity preferences. We compare our results with the ones in a pre-commitment setup where the investor is restricted to deterministic strategies. It turns out that (under risk and ambiguity aversion) an infinite investment horizon implies in both cases a maximin decision rule, i.e. the investor follows the worst (best) Merton fraction (over all realizations of it) if she is more (less) risk averse than a log-investor. We illustrate our findings with an extensive numerical study.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a strong theoretical contribution by finding an explicit solution to the multi-asset Merton problem under ambiguity and learning using duality theorems and the KMM approach. However, it relies entirely on theoretical proofs and numerical simulations rather than real-world market data backtesting.

Research Flowchart

  flowchart TD
    A[Research Goal: Solve multi-asset Merton investment problem under drift uncertainty and ambiguity] --> B[Inputs: Prior drift distribution, asset prices, CRRA risk/ambiguity utility KMM model]
    B --> C[Methodology: Apply duality theorems to calculate inner & outer certainty equivalents]
    C --> D[Computational Process: Transfer ambiguity preferences into an adjusted drift distribution]
    D --> E[Comparison: Analyze adaptive optimal strategies vs. pre-commitment deterministic strategies]
    E --> F[Key Outcome: Explicit optimal strategy solved; infinite horizon yields a maximin decision rule]