Risk management under incomplete information: Exact upper and lower bounds for the Value at Risk

A key problem in financial and actuarial research, and particularly in the field of risk management, is the choice of models so as to avoid systematic biases in the measurement of risk. An alternative consists of working with incomplete information, by fixing only a number of parameters instead of a complete distribution, which results in bounds instead of unique results. In the present contribution, we derive upper and lower bounds for the Value at Risk , in case the information about the underlying distribution is restricted to successive moments, and possibly the mode. These bounds are obtained by means of a transformation of similar results about tail probabilities.