Upper limits of financial risk measures of various degrees of catastrophicity

Abstract

The question of assessing the magnitude of risks using certain risk measures presents one of the most important problems of modern finance. However, many modern risk measures require considerable effort at times and, in practice, the investor would have sufficient knowledge of the upper limits of those risks. Comparing them with their risk appetite, an investor, in the case when the upper limits of risk measures would fit into their risk appetite, could assess this risk as acceptable to themselves. Only if the upper limit of the appropriate risk measure exceeded their risk appetite would there be a need for a detailed assessment of the appropriate risk measure. The aim of this paper is to consider upper limits first for known risk measures such as value at risk, VaR, and expected deficit or notional value at risk of ES. Next, upper limits are obtained for the risk measures VaR to the degree of t, VaR^{(t )} and ES to the degree of t, ES^{(t )} introduced by the author into scientific use. Also, using the results of V. Hürlimann, representations for maximum values of risk measures VaR^{(t )} and ES^{(t )} . The method of obtaining the described results is the application of certain representations of all these risk measures, the application of P. Chebyshev’s inequalities, as well as the results of V. Hürlimann. As a result of the study, descriptions have been proposed for the upper limits, expressing them only after a few first moments of the loss distribution law. The author concludes that the study of the upper limits of important risk measures of scientific interest has practical value for the express assessment of relevant risks.