Portfolio optimization with two coherent risk measures
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Portfolio optimization with two coherent risk measures Tahsin Deniz Aktürk1 · Çagın ˘ Ararat2 Received: 24 March 2019 / Accepted: 29 June 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We provide analytical results for a static portfolio optimization problem with two coherent risk measures. The use of two risk measures is motivated by joint decision-making for portfolio selection where the risk perception of the portfolio manager is of primary concern, hence, it appears in the objective function, and the risk perception of an external authority needs to be taken into account as well, which appears in the form of a risk constraint. The problem covers the risk minimization problem with an expected return constraint and the expected return maximization problem with a risk constraint, as special cases. For the general case of an arbitrary joint distribution for the asset returns, under certain conditions, we characterize the optimal portfolio as the optimal Lagrange multiplier associated to an equality-constrained dual problem. Then, we consider the special case of Gaussian returns for which it is possible to identify all cases where an optimal solution exists and to give an explicit formula for the optimal portfolio whenever it exists. Keywords Portfolio optimization · Coherent risk measure · Mean-risk problem · Markowitz problem Mathematics Subject Classification 90C11 · 90C20 · 90C90 · 91B30 · 91G10
1 Introduction The mean-variance portfolio selection problem introduced in the seminal work Markowitz [12] is one of the most well-studied optimization problems. In the basic static version of the problem, one considers multiple correlated assets with known expected returns and covariances, and looks for an allocation of these assets. Considering the trade-off between the linear expected return and the quadratic variance, the problem can be formulated as a biobjective optimization problem whose efficient solutions form the so-called efficient frontier on the
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Ça˘gın Ararat [email protected] Tahsin Deniz Aktürk [email protected]
1
Booth School of Business, The University of Chicago, Chicago, IL, USA
2
Department of Industrial Engineering, Bilkent University, Ankara, Turkey
123
Journal of Global Optimization
mean-variance (or mean-standard deviation) plot of all portolios. Merton [13] provides an analytical derivation of the efficient frontier for the general case of n ≥ 2 assets. The biobjective mean-variance problem can also be studied in terms of a parametric family of scalar (single-objective) problems. Among the popular scalarizations are the ones where one minimizes variance over the set of all portfolios at a given expected return level, which is used as the parameter of the scalar problem. Analogously, one can impose a constraint on the variance using an upper bound parameter and maximizes expected return. Quite naturally, both approaches can be used to verify the analytical results in Merton [13]. Started with Artzner et al. [1], the theory of coherent ri
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