Robust Portfolio Optimization

Since the 2008 Global Financial Crisis, the financial market has become more unpredictable than ever before, and it seems set to remain so in the forseeable future. This means an investor faces unprecedented risks, hence the increasing need for robust portfolio optimization to protect them against uncertainty, which is potentially devastating if unattended yet ignored in the classical Markowitz model, whose another deficiency is the absence of higher moments in its assumption of the distribution of asset returns. We establish an equivalence between the Markowitz model and the portfolio return value-at-risk optimization problem under multivariate normality of asset returns, so that we can add these excluded features into the former implicitly by incorporating them into the latter. We also provide a probabilistic smoothing spline approximation method and a deterministic model within the location-scale framework under elliptical distribution of the asset returns to solve the robust portfolio return value-at-risk optimization problem. In particular for the deterministic model, we introduce a novel eigendecomposition uncertainty set which lives in the positive definite space for the scale matrix without compromising on the computational complexity and conservativeness of the optimization problem, invent a method to determine the size of the involved uncertainty sets, test it out on real data, and explore its diversification properties. Although the value-at-risk has been the standard risk measure adopted by the banking and insurance industry since the early nineties, it has since attracted many criticisms, in particular from McNeil et al. (2005) and the Basel Committee on Banking Supervision in 2012, also known as Basel 3.5. Basel 4 even suggests a move away from the ``what" value-at-risk to the ``what-if" conditional value-at-risk' measure. We shall see that the former may be replaced with the latter or even other risk measures in our formulations easily.

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