On page 18, chapter 7, we have: "This general family of risk measures ... [ i.e. risk measures dependent on weightings ] ... satisfies the condition of coherence although the condition of translational invariance may not always be satisfied" However, it doesn't seem to me that this is true. As a counterexample, consider VaR. I think that it is possible to formulate VaR (or, rather, Excess VaR) as a weight-based measure. In a Monte-Carlo framework, we would weight the single simulation corresponding to the percentile of interest, and give all other simulations a weight of zero. In a continuous framework, our weighting function would be a Dirac delta function centred on the percentile of interest. Using this would give us an "Excess VaR" XVaR equal to VaR minus the mean of the untransformed distribution. Now, we know that VaR is not sub-additive. Therefore, XVaR as defined above will also not be sub-additive since it differs from VaR just by the mean of the original distribution (and will be equal to VaR for distributions with means of zero). Therefore, we have an example of an "outcome specific charge" measure that is not sub-additive and hence not coherent. Is this correct? What is the source for the assertion on page 18?
Thank you for your thoughts on this. The item you have questioned is part of the Core Reading so I have brought this to the attention of the Institute and will let you know when I have a response. From the point of view of the exam, mathematics is not the focus of SA3 so don’t spend too much time on this. Best wishes Duncan
