Why the definition of ES matters

(a) Show that for a normal loss $\operatorname{ES}_{0.975} = 2.3378\sigma$ and $\operatorname{VaR}_{0.99} = 2.3263\sigma$ (zero mean). (b) Prove that $\alpha \mapsto \operatorname{ES}_\alpha(L)$ is non-decreasing. (c) For the two independent bonds of Proposition 14.18 (each loses 100 with probability 0.04), compute the "tail conditional expectation" $\operatorname{TCE}_{0.95}(L) = \mathbb E[L \mid L \ge \operatorname{VaR}_{0.95}(L)]$ for one bond and for the pair, and show that TCE is not subadditive. Explain which step of the proof of Proposition 14.20 fails for TCE.