Hi Could you pls explain the answer to the question given on pg 13 of ch 36-Capital requirement?I did not understand how the equivalent TailVaR confidence level would need to be less than 99.5%.
Hi: if we consider the lower tail of a distribution, which contains the greatest losses, we can think of the VaR as being the loss that forms the boundary of that tail and TVaR as being the average of the losses in that tail. For example, if the VaR is a loss of 100 at a confidence level of say 99.5%, then the TVaR at the same confidence level of 99.5% would be the average of all losses which are worse than 100. So the TVaR must be > 100 at that level. In order for the TVaR to equal 100, which is what the question you refer to is asking about, then we would need to be taking the average of all losses which are worse than X where X<100. For example, we might be looking at 100 = average of all losses which are worse than 80. X is effectively a new VaR', but because this new VaR' must be less than the original VaR (since we must have X<100), the new VaR' has a lower confidence level, eg 95%. (In this example, we believe that 99.5% of our outcomes will be better than a 100 loss, and 95% will be better than a loss of 80.) Therefore, since we are trying to determine the level at which TVaR = 100, we are now at the 95% confidence level (ie with the boundary of 80) and hence a lower confidence level than what was needed to give VaR = 100. Hope that example helps.
