Abstract: In the financial risk management, research on the method of risk measures has always been an important topic. We use probability metric and convex function to construct a new risk measure. Since the risk measure does not satisfy tonicity and the idea of downside risk, we use convex function to construct new risk measures based on the probability semi-metric. It is found that the new risk measurement method includes many common risk measures, for example, semi-variance, semi-absolute deviation, the lower partial moment and ES. We show that the risk measure not only satisfies the convexity but also satisfies tonicity. Considering the importance of convexity and tonicity in portfolio selection and risk management, the proposed risk measure has certain theoretical value and practical significance.

Key words: risk, risk measures, axiom, probability metric, stochastic ordering