Translation equivariance of AV@R (average value at risk), proof

Question

I am trying to prove that the average value at risk is translation equivariant: $$AV@R_\alpha[Z+\tau] = AV@R_\alpha[Z] + \tau$$

where $$AV@R_\alpha[Z] := \inf_{t\in \mathbb{R}} \{t+\alpha^{-1} \mathbb{E}[(Z-t)_+] \}.$$

I started by plugging in the definition, but I don't know how to get $\tau$ out of the expected value operator.

$$AV@R_\alpha[Z+\tau] = \inf_{t\in \mathbb{R}} \{t+\alpha^{-1} \mathbb{E}[(Z+\tau-t)_+] \}$$

Any help would be appreciated.

Answer 1

Note that $\inf_t \phi(t) = \inf_t \phi(t-\tau)$.