# Incorporating Black-Litterman Views in Portfolio Construction when Stock Returns are a Mixture of Normals

## TL;DR

This paper develops a method for portfolio construction that combines market equilibrium insights with a mixture of normals return model, improving risk management while maintaining competitive returns.

## Contribution

It introduces a convex programming approach to CVaR minimization under a mixture of normals and integrates Black-Litterman views via inverse optimization.

## Key findings

- The proposed method effectively reduces portfolio risk compared to market-only strategies.
- The mixture of normals model accurately captures return distributions.
- Empirical results show improved risk profiles with comparable returns.

## Abstract

In this paper, we consider the basic problem of portfolio construction in financial engineering, and analyze how market-based and analytical approaches can be combined to obtain efficient portfolios. As a first step in our analysis, we model the asset returns as a random variable distributed according to a mixture of normal random variables. We then discuss how to construct portfolios that minimize the Conditional Value-at-Risk (CVaR) under this probabilistic model via a convex program. We also construct a second-order cone representable approximation of the CVaR under the mixture model, and demonstrate its theoretical and empirical accuracy. Furthermore, we incorporate the market equilibrium information into this procedure through the well-known Black-Litterman approach via an inverse optimization framework by utilizing the proposed approximation. Our computational experiments on a real dataset show that this approach with an emphasis on the market equilibrium typically yields less risky portfolios than a purely market-based portfolio while producing similar returns on average.