Mixed value-at-risk and its numerical investigation

We study an extension of value-at-risk (VaR) measure, named as Mixed VaR, a weighted sum of multiple VaRs quantified at different confidence levels. Classical VaR or single VaR computed at a fixed confidence level corresponds to a single percentile of distribution and therefore, is unable to reveal much information of risk involved in it. As a remedy to this, we propose to investigate the role of Mixed VaR and its deviation version in risk management of extreme events in the portfolio selection problem. We analyze the computational performance of portfolios from optimization models minimizing single VaR and Mixed VaR (and their deviation variants) for different combinations of confidence levels over historical as well as on simulated data in various financial performance parameters including mean value, risk measures (VaR and CVaR values quantified at multiple confidence levels), and risk-reward measures (Sharpe ratio, Sortino ratio, Sharpe with VaR, and Sharpe with CVaR). We also study the numerical comparison between Mixed VaR with its most crucial counterpart, the Mixed conditional value-at-risk (Mixed CVaR). We find that the performance of portfolios from Mixed VaR model is lying between the performance of its single VaR counterparts and rarely yield any worst value in any of the considered performance parameters. A similar observation is concluded for the deviation version of the models. Further, we find that Mixed VaR outperforms Mixed CVaR with respect to risk-reward measures considered in the study on both of the data sets, historical as well as on simulated. (C) 2019 Elsevier B.V. All rights reserved.