关键词: 广义凸风险度量, 场景选择, 市场宏观状况, 高阶信息, 分布鲁棒

Abstract: The concept of financial risk consists of two main components: the possibility of negative outcomes, namely loss; and the variability of expected results, known as bias. With modern financial theory, risk measure has been the most important basis of risk management and an important tool to quantify the size of risk. Early risk measures mainly focus on the degree of random fluctuation or dispersion of investment returns deviating from its mean, such as variance, lower half moment and deviation measure. Since value at risk was proposed in 1997, almost all risk measures have been constructed based on the concept of loss of risk. One of the important reasons is that a large number of empirical studies show that the random returns of financial assets do not follow the normal distribution, but have obvious bias. Therefore, in order to ensure proper risk control, most scholars have turned their attention to tail risk measure, and an important task in this respect has been to propose a coherent risk measure. Since then, some scholars have replaced the sub-additivity and positive homogeneity of the coherent risk measures with a weaker convexity, and proposed a broader set of risk measures, called the convex risk measure. Compared with coherent risk measures, convex risk measures can better reflect the change of risk after the expansion of asset size, and explain liquidity risk. However, at present, the research on convex risk measures rather than coherent risk measures mostly stays in the more abstract level, and there are just a few cases in which they are successfully applied to real financial investment problems. Considering that monotonicity and convexity are the main properties of risk measures which are accepted by both academia and industry, some scholars have put forward the concept of generalized convex risk measures recently. Therefore, an important issue worthy of attention is how to integrate the information of high-order moments in random returns, the investor’s risk preference and the change in market macro conditions into the construction of new risk measures.
Considering these issues and the pros and cons of current risk measures, we obtain the following three achievements: (1)We propose a class of market regime-based non-linearly weighted convex measures by combining the generalized convex risk measure with the market regime selection for the first time. Then, we demonstrate its theoretical properties and design a practical algorithm for market regime classification by using the idea of Markov chain. (2)Through considering different collections of market regime sets, we empirically demonstrate that the new risk measure has better performance than relevant measures and can flexibly reflect the influences of different market regimes. The two-dimensional regime model will be more suitable for the financial market when the macro situation is good, while the three-dimensional regime model will be more suitable for the poor macro condition of the financial market. And whether the financial market is in a stable state or an extreme condition, both are applicable.(3)We establish a corresponding portfolio selection model based on the new risk measure, and a series of empirical tests show the superior performance of the optimal portfolio in the new risk measure with respect to typical performance ratios. And the empirical results in and out of the sample indicate the practicality, effectiveness, and stability of the proposed new risk measures and corresponding portfolio selection models. Therefore, our new risk measures can help investors make efficient and robust investment decisions.
On the basis of the work in this paper, we can also select other newly proposed coherent risk measures or (generalized) convex risk measures to construct new risk measures based on the market regime selection method, by modeling after the previous paper. Furthermore, we can also consider the broadening model of new risk measures proposed in this paper in the multi-period situation.

Key words: generalized convex risk measure, regime selection, market macro conditions, high-order information, distributionally robust