Time Consistent Risk Measure Under Stopping Time Framework and its Applications∗

For financial investment with a pre-specified investment target, the risk can be regarded as the uncertainty of the earliest target reaching time. As a refinement of this new perspective of risk measure, we introduce a new risk measure which is the extra time cost between the current time and the earliest time that the investment achieves the target or beats the benchmark at some confidence level. The new risk measure is based on a probabilistic constraint and can be seen as a generalization of the payback period method in corporate financial management. We show that the new risk measure has good properties such as normalization, monotonicity, positive homogeneity and weak time consistency. We also discuss the relationship between the new risk measure and variance and VaR. Then we consider two applications of the new measure, the theoretical application is the introduction of arbitrage with probability, which extends the usual definition of arbitrage opportunity and can help the investor differentiate markets which are arbitrage free; the practical application is the multi-period portfolio selection. For the chance constrained multi-period investment decision problem constructed under the new measure, we use the Chebyshev inequality ∗This research was supported by the National Natural Science Foundation of China (Grant Numbers 70971109 and 71371152). †The corresponding author: [email protected]