Abstracts

The class of distortion risk measures is the one we should use if we stick to the law invariance and comonotonic additivity in addition to the four axioms of coherence, and is broad enough to fully express agents’ subjective assessment of risk (conservatism). If one is to apply this risk measure to practical financial risk management problems, it is necessary to pick one distortion function depending on his/her attitude towards risk.
In our previous work, we have shown that estimating distortion risk measure is possible based on general weakly dependent times series data. The next step in financial risk management is backtesting. The purpose of backtesting is twofold: to monitor the performance of the model and estimation methods for risk measurement, and to compare relative performance of the models and methods. It is a tool for the validation process which is indispensible for adequate risk management and is even required by the Basel 2 Capital Accord.
Statistically, it is just a form of cross validation; the ex ante risk measure forecast from the model is compared with the ex post realized portfolio loss. In the case of Value-at-Risk (VaR), a popular procedure for backtesting VaR depends on the number of VaR violations; we say that a VaR violation occurred when the loss exceeds VaR. Extending a backtesting procedure for the renowned expected shortfall (ES), we suggest, based on a simple observation, a backtesting procedure for distortion risk measures, and check its effectiveness in a simulation study using ES and also proportional odds distortion risk measure.
Many people claim that it is easier to backtest VaR than ES and other risk measures for the following reasons: (i) the existing tests for ES are based on parametric assumptions for the null distribution; (ii) asymptotic approximation is needed for the null distribution of the test statistics; (iii) testing an expectation is harder than testing a single quantile. Summing up these arguments, what they call ‘backtestability’of a risk measure means that it can be nonparametrically backtested with small samples.
It has recently been claimed that expected shortfall (and distortion risk measures) cannot be backtested because it fails to satisfy the so-called elicitability condition. Roughly speaking, a statistical functional is called elicitable if it is a unique minimizer of some expected loss function. While elicitability is useful when we wants to compare and rank several estimation procedures, there seems to be no clear connection with backtestability. We will try to illustrate this with simple examples including the expectiles.