Finding Bayesian Optimal Portfolios with Skew-Normal Returns

48 Pages Posted: 24 Jul 2020 Last revised: 16 Aug 2021

Date Written: August 1, 2021


In the presence of asset returns’ non-normal behavior, optimal portfolio selection techniques should consider higher-moment risks. In this paper, we extend the Black-Litterman (BL) asset allocation model (Black & Litterman, 1990) by applying the hidden truncation skew-normal distribution (Arnold & Beaver, 2000). Based on Simaan’s approach (Simaan, 1993), we also develop a three-moment BL model considering skewness. Using the skew-normal returns, the skew-normal BL model provides optimal portfolios with the same expected return but less risk compared to an optimal portfolio of the classical BL model. We find that the portfolios become more negatively skewed as the expected returns of portfolios increase at any level of N (non-spherical variance), which suggests that the investors trade a negative skewness for a higher expected return. We also find the negative relation between portfolio volatility and portfolio skewness. In other words, the investors trade a lower volatility for a higher skewness or vice versa reflecting that stocks with big price declines are more volatile.

Keywords: Bayesian Asset Allocation, Black-Litterman Model, Skew-Normal Distribution, Non-Normal Distribution, Mean-Variance-Skewness Optimization

JEL Classification: C61, G11, G12

Suggested Citation

Park, Jungjun, Finding Bayesian Optimal Portfolios with Skew-Normal Returns (August 1, 2021). Available at SSRN: or

Jungjun Park (Contact Author)

University of Pittsburgh ( email )

4700 Wesley W. Posvar Hall
Pittsburgh, PA Pennsylvania 15260
United States
15260 (Fax)

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