Asymmetric Change-of-Probability Measures for Tail Risk Management

Large (extreme, high-impact) events can occur in complex systems as a result of fat-tailed distributions of the system behavior. Decision-making in the presence of a potential high-impact event where the law of large numbers (LLN) cannot be applied, is not as straightforward as decision-making in a normal scenario where the LLN is used. In this paper, a general framework is introduced for decision-making in the presence of a potential high-impact event using change-of-probability measures. The idea of the proposed framework is to weigh the high negative consequences of non-LLN decision-making problems as a tail risk management strategy. The proposed approach is named asymmetric change-of-probability measures (ACM) as the right and the left tails of the distributions are treated asymmetrically. A key to this approach is to define and satisfy required properties so that the change-of-measure operation is performed in a principled way. An important property is ensuring upper bounds for the relative entropy between the distributions. We first introduce asymmetric bounded expectation (ABE), as a special case of the general approach. We then extend the proposed asymmetric method to the general change-of-measure. Benefiting from the same properties as the symmetric change-of-measure, we show that the asymmetric approach can be potentially a promising method for decision-making under non-LLN risk management circumstances in complex systems. Through a practical example from venture capital (VC) in finance, and in comparison to the symmetric change-of-measure, we show that considering tail risk management will result in a different decision-making outcome where the VC is required to invest in more startups to avoid a loss.