Mean of the upper tail

Idea. The density of the normal has the property $z\phi (z)=-{\phi }^{\prime }(z)$, so the partial mean above a threshold is just the density at the threshold.

Step 1: the partial mean. $${\int }_1^{\infty }z\phi (z)dz=[-\phi (z){]}_1^{\infty }=\phi (1).$$

Step 2: condition. $$\begin{array}{rl}\mathbb{E}[Z\mid Z>1]& =\frac{\phi (1)}{1-\Phi (1)}\\ & \approx \frac{0.241971}{0.158655}\approx \mathrm{1.525.}\end{array}$$

In general. $\mathbb{E}[Z\mid Z>a]=\frac{\phi (a)}{1-\Phi (a)}$, the inverse Mills ratio. At $a=0$ it is $\sqrt{2/\pi }\approx 0.80$, and for large $a$ it is about $a+\frac{1}{a}$: the tail beyond a high threshold sits just above the threshold. This is the calculation behind expected shortfall for normal returns.