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 at 22:30
lots of punch for a one-liner, nice ;-) –  Matt Wolf Jan 5 at 3:08
many thanks John! –  user3544 Jan 5 at 17:03