Backtesting Conditional Expected Shortfall

Abstract

In recent years, the question of whether expected shortfall is backtestable has been hot topic of discussion after the ndings in 2011 that expected shortfall lacks the elicitability property in mathematics. However, current literature has indicated that expected shortfall is backtestable and that it does not have to be very di cult. Several researches on risk measures have revealed that Value-at- Risk (VaR) is not a coherent risk measure while Expected Shortfall is coherent. Due to this weakness of VaR, Expected Shortfall has been popular and conse- quently in 2012 the Basel committee suggest that bank or nancial institutions should move from VaR to ES as a measure of risk for minimum capital cover for potential loss. Models are backtest to establish whether their predictions are concurrent with the actual realized values. The backtesting of VaR is simple, direct and well establish in many nancial literature. That of Expected Short- fall is not well explored and widely unknown. In this work the Extreme Value Theory and GARCH model are combined to estimate conditional quantile and conditional expected shortfall so as to estimate risk of assets more accurately. This hybrid model provides a robust risk measure for the Nairobi 20 Share index by combining two well known facts about return time series: volatility cluster- ing, and non-normality leading to fat tailedness of the return distribution. First the GARCH model with di erent innovations is tted to our return data using pseudo maximum likelihood to estimate the current volatility and then the GPD- approximation proposed by EVT to model the tail of the innovation distribution of the GARCH-model. The estimates are then backtested. In backtesting VaR, three methods are used: Unconditional coverage test, independent test and con- ditional coverage test whereas for Expected Shortfall two methods were used: bootstrap method and V-test.