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

Burak Kocuk (bkocuk***at***andrew.cmu.edu)
Gérard Cornuéjols (gc0v***at***andrew.cmu.edu)

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. Since the CVaR measure does not have a closed form expression in this case, we propose an algorithm which can numerically compute this measure and solve the resulting convex program. We also construct a second-order cone representable approximation of the CVaR under the mixture model. 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.

Keywords: portfolio optimization, the Black-Litterman model, conditional value-at-risk, mixture of normals

Category 1: Applications -- OR and Management Sciences

Category 2: Nonlinear Optimization


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Entry Submitted: 06/10/2017
Entry Accepted: 06/11/2017
Entry Last Modified: 07/18/2017

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