Optimal trade-off portfolio selection between total risk and maximum relative marginal risk

Abstract

In this paper, we first introduce the concept of maximum relative marginal risk (MRMR) to measure the effect of risk diversification under the mean–variance portfolio selection. Then we develop an optimal trade-off model between the total risk and the MRMR. Since the MRMR is described as a maximum fractional quadratic functions, we convert it into a quadratic function in terms of auxiliary variables and transform it as a constraint. Then it turns to a non-convex quadratically constrained quadratic problems (QCQP). To solve this non-convex QCQP, we relax and approximate the optimal trade-off model, respectively, and obtain both second-order cone relaxation and inner approximation. Both of them provide a lower bound and an upper bound for the original non-convex QCQP. Furthermore, we solve the QCQP by using an efficient and globally convergence branch-and-bound solution algorithm proposed by Li et al. [Active allocation of systematic risk and control of risk sensitivity in portfolio optimization, Eur. J. Oper. Res. 228 (2013), pp. 556–570]. In the process of solution we only branch the variables related to the MRMR and reduce the numbers of branch largely. Finally, our model is illustrated and demonstrated by empirical analysis. The numerical results show that our model outperforms the existing models to deal with the risks between the total risk and the relative marginal risk in portfolio selection.