The super-efficiency data envelopment analysis model is innovative in evaluating the performance of crude oil prices’ volatility forecasting models. This multidimensional ranking, which takes account of multiple criteria, gives rise to a unified decision as to which model performs best. However, the rankings are unreliable because some efficiency scores are infeasible solutions in nature. What’s more, the desirability of indexes is worth discussing so as to avoid incorrect rankings. Hence, herein we introduce four models, which address the issue of undesirable characteristics of indexes and infeasibility of the super efficiency models. The empirical results reveal that the new rankings are more robust and quite different from the existing results.

This paper investigates the generalized multi-period mean-variance investment-reinsurance optimization model in a discrete-time framework for a general insurance company that contains a reinsurer and an insurer. The intertemporal restrictions and the common interests of the reinsurer and the insurer are considered. The common goal of the reinsurer and the insurer is to maximize the expectation of the weighted sum of their wealth processes and minimize the corresponding variance. Based on the game method, we obtain the Nash equilibrium investment-reinsurance strategies for the above-proposed model and find out the equilibrium strategies when unilateral interest is considered. In addition, the Nash equilibrium investment-reinsurance strategies are deduced under two special premium calculated principles (i.e., the expected value premium principle and the variance value premium principle). We theoretically study the effect of the intertemporal restrictions on Nash equilibrium investment-reinsurance strategies and find the effect depends on the value of some parameters, which differs from the previous researches that generally believed that intertemporal restrictions would make investors avoid risks. Finally, we perform corresponding numerical analyses to verify our theoretical results.

Most of the existing literature on optimal investment-reinsurance only studies from the perspective of insurers and also treats the investment-reinsurance decision as a continuous process. However, in practice, the benefits of reinsurers cannot be ignored, nor can decision-makers engage in continuous trading. Under the discrete-time framework, we first propose a multi-period investment-reinsurance optimization problem considering the joint interests of the insurer and the reinsurer, among which their performance is measured by two generalized mean-variance criteria. We derive the time-consistent investment-reinsurance strategies for the proposed model by maximizing the weighted sum of the insurer’s and the reinsurer’s mean-variance objectives. We discuss the time-consistent investment-reinsurance strategies under two special premium principles. Finally, we provide some numerical simulations to show the impact of the intertemporal restrictions on the time-consistent investment-reinsurance strategies. These results indicate that the intertemporal restrictions will urge the insurer and the reinsurer to shrink the position invested in the risky asset; however, for the time-consistent reinsurance strategy, the impact of the intertemporal restrictions depends on who is the leader in the proposed model.

In this paper, we propose a generalized multiperiod mean-variance portfolio optimization based on consideration of benchmark orientation and intertemporal restrictions, in which the investors not only focus on their own performance but also tend to compare the performance gap between themselves and the benchmark. We aim to find the time-consistent strategy under the generalized mean-variance criterion, such that their relative performance is maximized. We derive the time-consistent strategy for the proposed model with and without a risk-free asset by using the backward induction approach. The results show that, in the case that there exists a risk-free asset, the time-consistent strategy is a feedback strategy about the benchmark process. However, in the other case, the time-consistent strategy is a double feedback strategy on both the benchmark process and the wealth process. Finally, we carry out some numerical simulations to show the evolution process of the time-consistent strategy. These simulations indicate that the proposed strategy can not only reduce the risk of investment existed in the intermediate time period but also imitate the return of the benchmark process.