Leírás
The Basel Accords require banks to set aside capital in line with their levels of risk. Currently, Value-at-Risk (VaR) is the applied risk measure of the potential loss in the value of a portfolio. In particular, 99% VaR is utilized, which is the loss that is likely to be exceeded only 1% of the time. While VaR is widely used and easy to compute, it has no information on the magnitude of the biggest 1% of losses. Moreover, it is not a coherent risk measure. Indeed, it is not subadditive, which means that VaR of a portfolio can be higher than the sum of the VaRs of the individual assets in the portfolio. The Fundamental Review of the Trading Book is expected to make a complete revision of the approach to calculating risk-based capital requirements for investments. The 99% VaR is supposed to be replaced by 97.5% Expected Shortfall, which is the average of VaR(x) for x between 0.975 and 1. Li and Wang [2] studied the effect of this proposed change. Similarly to their work, we are examining higher-order Expected Shortfalls as potential alternative risk measures. The n-th-order Expected Shortfall is similar to the classical one (which is a special case for n=1), the difference is that instead of a simple average, it is a weighted average of the VaR values, weighted by a function that depends on n. We define PELVE(n), which basically tells us what level n-th-order Expected Shortfall corresponds to a certain level VaR. We investigate its properties and calculate PELVE(2) for some important distributions including ones with heavy tail. Moreover, for PELVE(2), we present some simulation results along with real data analysis.
[1] Barczy, M., K. Nedényi, F. and Sütő, L. (2022). Probability equivalent level of Value at Risk and higher-order Expected Shortfalls. ArXiv: https://arxiv.org/abs/2202.09770
[2] Li, H. and Wang, R. (2019). PELVE: Probability Equivalent Level of VaR and ES.
To appear in Journal of Econometrics. Available also at SSRN. 30(2) 325-341. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3489566