Coherent Var-type Measures

If f is a risk measure, the diversification benefit of aggregating portfolio’s A and B is defined to be (1) f(A) + f(B)− f(A + B) When using full revaluation VaR as the methodology for computing a risk measure, its quite possible to get negative diversification. Pathological examples are possible, but the following example is not absurd: Suppose one has a portfolio that is made up by a Trader A and Trader B. Trader A has a portfolio consisting of a put that is far out of the money, and has one day to expiry. Trader B has a portfolio that consists of a call that is also far out of the money, and also has one day to expiry. Using any historical VaR approach, say we find that each option has a probability of 4% of ending up in the money. Trader A and B each have a portfolio that has a 96% chance of not losing any money, so each has a 95% VaR of zero. However, the combined portfolio has only a 92% chance of not losing any money, so its VaR is non-trivial. Therefore we have a case where the risk of the combined portfolio is greater than the risks associated with the individual portfolios, i.e. negative diversification benefit if VaR is used to measure the diversification benefit. This example appears in [1].