Value-at-Risk vs. Expected Shortfall : Beware of the Aggregation Currency!

This note highlights the impact of changing the aggregation currency when testing for capital adequacy. This is exemplified using the two most prominent capital adequacy tests used in the financial industry: Value-at-Risk and Expected Shortfall. The test consists in establishing whether the Value-at-Risk or, respectively, the Expected Shortfall of the capital position is negative or not. We show that the sign of the Expected Shortfall of a capital position -- and, hence, the corresponding capital adequacy test -- critically depends on the aggregation currency. This is in contrast to Value-at-Risk, whose sign is independent of the chosen aggregation currency. This implies that due caution is needed when using Expected Shortfall to measure capital adequacy in a multi-currency environment