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

TL;DR

This paper demonstrates that under hyperbolic return distributions and broad utility functions, the optimal portfolio follows a two-fund separation principle, simplifying portfolio choice to a linear combination of a riskless asset and a mutual fund of risky assets.

Contribution

It establishes the two-fund separation theorem for hyperbolic distributions and broad utility functions, providing explicit expressions for the mutual fund of risky assets.

Findings

Two-fund separation holds under hyperbolic distributions.

Explicit formulas for the mutual fund of risky assets.

Optimal portfolios either lie on the boundary or are unique within convex domains.

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.