Skip to content

Derive the normal tail mean

For a standard normal Z, the upper-tail first moment above z is φ(z).

On this page

Start with the loss-side threshold

Let z be the standard normal quantile at confidence c. For a continuous normal law, the probability above z is 1−c and the probability at exactly z is zero. The upper-tail conditional average therefore uses the same mass as the selected worst-tail average.

Use the derivative of the density

The derivative of φ(x)=exp(−x²/2)/√(2π) is −xφ(x). Integrating xφ(x) from z to infinity gives φ(z), because the density tends to zero at the upper endpoint.

E[Z∣Z>z]=∫z∞xϕ(x) dx1−c=ϕ(z)1−c\mathbb{E}[Z\mid Z>z]=\frac{\int_z^\infty x\phi(x)\,dx}{1-c}=\frac{\phi(z)}{1-c}

Here z=Φ⁻¹(c); continuity removes threshold mass.

Restore location and scale

For L=μL+σL Z with σL>0, the selected upper-tail mean is μL+σL φ(z)/(1−c). Keep the density divided by the tail probability; φ(z) alone is not the expected shortfall multiplier.

Do not export the shortcut to every discrete model

A finite distribution can have positive probability at its quantile. Its exact-tail average may need fractional boundary inclusion. The existing discrete-loss resources explain why unqualified threshold conditional averages can then differ.

Further references