VaR Coherence - Sub-additivity

[sn_riskrookie]

David,

In T1.Chapter1-Jorion, page 15, a True or False question, you have asked:

Is Var Coherent. Your answer states, False. Sub-additivity is a coherence criteria. Var is not Sub-additive.

What does your explanation mean?

Thanks,

 

[David Harper CFA FRM]

Hi sn_riskrookie,

It's a classic (and testable) weakness of VaR as a distributional quantile. By not subadditive, VaR allows for a counter-intuitive, "anti-diversifying" outcome: VaR(X + Y ) can be > VaR(X) + VaR(Y).
Dowd's super-clear example is a $100 bond with PD = 4% (LGD = 100%): it has a 95% VaR of zero.
But a portfolio of two bonds, each with PD = 4%, has a VaR of $100 (why?), such that the quantile allows for a puzzling outcome: VaR(bond) + VaR(bond) < VaR (two bonds); i.e., 0 + 0 < $100
Dowd says it's not academic, that it has real & practical disadvantaged. Hence, he favors coherent risk measures. Hope that helps, David