A generalized tail mean-variance model for optimal capital allocation

Capital allocation is a core task in financial and actuarial risk management. Some well-known capital allocation principles, such as the “Euler principle” and the “haircut principle”, have been widely used in the banking and insurance industry. The partitions of allocated capital not only serve as a buffer against potential losses but also provide certain risk pricing and performance measurement to the underlying risks. Dhaene et al. (2012) proposed a unified distance-minimizing capital allocation framework. Their objective function in the optimization only considers the magnitude of the loss function but not the variability. In this paper, we propose a general tail mean-variance (GTMV) model, which employs the Bregman divergences to construct distance-minimizing functions, and takes both the magnitude and the variability into account. We prove the existence and uniqueness of the optimal allocation and provide the general system of equations that characterizes the optimal solution. In this context, we further introduce the Mahalanobis tail mean-variance (MTMV) model and provide explicit distribution-free optimal allocation formulas, which cover many existing results as special cases. In particular, we derive the parametric analytical solutions for multivariate generalized hyperbolic distributed risks. For multivariate log-generalized hyperbolic distributed non-negative risks, we use the convex approximation method to obtain explicit solutions. We present two numerical examples showing the good performance of our optimal capital allocation rules. The first one analyzes the market risk of S&P 500 industry sector indices. We show that our optimal capital allocation framework is applicable to various scenario analyses and provides a performance measure for the indices and the financial market. The other example is based on insurance claims from an Australian insurance company, showing our approximate formulas are both robust and accurate.