# Is this comment right about subadditivity?

I found this comment in a book I bought about risk management: Risk Management in Banking by Joel Bessis.

This is the well-known rule that states that the sum of individual risks is less than the risk of the sum, or, that risks should be sub-additive. Risks do not add up algebraically because of diversification.

Doesn't he really mean the opposite, namely that the risk of the sum is less than the sum of the individuals:

$$\rho(A+B) \leq \rho(A) + \rho(B)$$

Or is his wording just odd?

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Your interpretation is the correct one. VaR is not subadditive, but Expected Shortfall/CVaR is. –  John Jan 4 '13 at 22:30
lots of punch for a one-liner, nice ;-) –  Matt Wolf Jan 5 '13 at 3:08
many thanks John! –  user3544 Jan 5 '13 at 17:03

## 1 Answer

You're right, I hope he meant exactly the opposite, and the formula you provided is indeed part of the definition of a coherent risk measure.

In fact, I would say that the risk of the sum is less than or equal to the sum of the individuals as in some cases you would like your model to accept no diversification effect.

As John mentioned in his comment, Value at Risk typically is not coherent, just as Volatility, but other measures such as the Expected Shortfall are.

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many thanks SRKX and FREDDY. It's been bugging me for a while how he chose his wording but I am glad to have your confirmation –  user3544 Jan 5 '13 at 17:02
@user3544 you can accept the answer if you're satisfied then! –  SRKX Jan 5 '13 at 17:15