Two-fund separation under hyperbolically distributed returns and concave utility functions

Paper: arXiv 2410.04459

Abstract

Portfolio selection problems that optimize expected utility are usually difficult to solve. If the number of assets in the portfolio is large, such expected utility maximization problems become even harder to solve numerically. Therefore, analytical expressions for optimal portfolios are always preferred. In our work, we study portfolio optimization problems under the expected utility criterion for a wide range of utility functions, assuming return vectors follow hyperbolic distributions. Our main result demonstrates that under this setup, the two-fund monetary separation holds. Specifically, an individual with any utility function from this broad class will always choose to hold the same portfolio of risky assets, only adjusting the mix between this portfolio and a riskless asset based on their initial wealth and the specific utility function used for decision making. We provide explicit expressions for this mutual fund of risky assets. As a result, in our economic model, an individual’s optimal portfolio is expressed in closed form as a linear combination of the riskless asset and the mutual fund of risky assets. Additionally, we discuss expected utility maximization problems under exponential utility functions over any domain of the portfolio set. In this part of our work, we show that the optimal portfolio in any given convex domain of the portfolio set either lies on the boundary of the domain or is the unique globally optimal portfolio within the entire domain.

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper employs advanced stochastic modeling, multivariate distributions, and stochastic calculus, which demands high mathematical sophistication. However, the work is purely theoretical with closed-form solutions, lacking any backtesting, dataset analysis, or implementation details for live trading.

Research Flowchart

  flowchart TD
  A["Research Goal: Find analytical optimal portfolios<br>under expected utility maximization"] --> B["Assumptions: Hyperbolic Distribution for Returns<br>& Concave Utility Functions"]
  B --> C{"Analysis"}
  C --> D["Derive closed-form solution for<br>optimal risky asset portfolio"]
  C --> E["Analyze exponential utility case<br>for convex domains"]
  D --> F["Outcome: Two-Fund Monetary Separation<br>holds: Riskless asset + Mutual Fund of Risky Assets"]
  E --> G["Outcome: Optimal portfolio lies on domain boundary<br>or is unique global optimum"]
  F --> H["Key Implication: Closed-form optimal portfolio<br>found for any wealth/utility"]
  G --> H

Cited References (47)

1. Exponential utility maximization in small/large financial markets (2022) — link
2. When do two- or three-fund separation theorems hold? (2021) — link
3. Evolution of the Arrow–Pratt measure of risk-tolerance for predictable forward utility processes (2020) — link
4. Portfolio optimization under the generalized hyperbolic distribution: optimal allocation, performance and tail behavior (2020) — link
5. Robust utility maximisation in markets with transaction costs (2019) — link
6. A stein type lemma for the multivariate generalized hyperbolic distribution (2017) — link
7. Multi-Period Investment Strategies under Cumulative Prospect Theory (2016) — link
8. Portfolio optimization under a generalized hyperbolic skewed t distribution and exponential utility (2016) — link

0 of 47 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.