Regarding translation invariance (equivariance) property of risk measure While studying some literatures for stochastic programming and risk averse optimization, I read that risk measure is called as coherent, when it satisfies:


*

*Convexity

*Monotonicity

*TRANSLATIONAL INVARIANCE

*Positive homogeneity.


To point out my question regarding 3., let $\rho[]$ be a risk measure. We say that $\rho[]$ is translational invariant, if for any $a\in\mathbb{R}$, we have $\rho[Z+a] = \rho[Z] + a$.
However, if we remember the definition of translational invariance of measure theory, let $\lambda$ be Lebesgue measure defined on $\mathbb{R}$. Then, given a set $I$ (let $I = (c,d) \in \mathcal{F(\mathbb{R}})$, if we define $I+a = [c,d] + a = [a+c, d+c]$, we have $\lambda(I) = \lambda(I+a) = d-c$, which makes Lebesgue measure translation invariant.
In this regard, clearly the definition of translational invariance for risk measure is, indeed, TRANSLATIONAL VARIANCE. Since this variation is linear (quantifiable given input), we might want to call it as Translational EQUIVARIANT, but not invariant. Am I wrong here...?
I don't get why the name translation invariance appears in risk measure literatures...
Anybody has answer to this question?

This is not quite right as stated. We think of adding $a$ as adding cash to the portfolio. It seems to me that most commonly this property is called cash-additivity. This stipulates risk reduction by the amount of cash thrown into the portfolio ($\rho(Z+a) = \rho(Z)-a$).
I agree that the name translation invariance is unfortunate in this context.
