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

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.

💡 Research Summary

The paper extends the classic two‑fund separation theorem to a setting where asset returns follow a multivariate Normal‑Mean‑Variance Mixture (NMVM) distribution and investors have a broad class of concave, bounded‑above utility functions. An NMVM random vector is defined as X = μ + γ Z + √Z A N_d, where Z is a non‑negative mixing variable independent of a standard normal vector N_d, μ is the location vector, γ captures skewness, and Σ = AAᵀ is the dispersion matrix. This construction encompasses a wide variety of heavy‑tailed and asymmetric distributions (e.g., generalized hyperbolic, skew‑t, variance‑gamma), thereby addressing empirical stylized facts such as tail dependence and skewness that normal or elliptical models cannot capture.

The authors consider a one‑period portfolio problem: an investor with initial wealth W₀ chooses a weight vector x on the d risky assets (the (d+1)‑th asset is risk‑free with rate r_f) to maximize expected utility E