Coherence Properties of Risk Measures

[Bruno]

Hi David,
Regarding the Coherence Properties of Risk Measures (page 112), I would like you to double check the following properties:
(i) Monotonicity: If expected value of Y is greater than X, then risk of Y is LESS than X. Is it "LESS than" or "GREATER than"?
(ii) Translation Invariance: For constant=c, ρ(X+c)=ρ(X)-c. Is it "minus c" or "plus c"?
Thanks,

 

[David Harper CFA FRM]

Hi Bruno,

Yes, these both look correct. In (i), higher expected value implies, ceteris paribus, less risk and (ii) that's correct for invariance.

I realize my (i) is different than the source reading (Wilmott's) but if you compare his (1) and (2), they don't reconcile; i.e., his monotonicity is reversed. I think the source has a slight error here.

To subtract invariance is maybe counter-intuitive, but the risk measure (rho) refers to the cash needed to make the position acceptable. Or, alternatively, you can think of additional guaranteed return reduces the absolute VaR (loss relative to zero)

David