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Identify the original standard threshold
For confidence 95%, z is about 1.644854 and the upper-tail probability is 0.05. The positive-dispersion normal law has zero probability mass at that exact threshold.
Compute the unnormalised first moment
Since φ′(x)=−xφ(x), the integral of xφ(x) from z to infinity is φ(z). At the selected z this is approximately 0.1031356404. It is a probability-weighted first moment, not an already normalised tail mean.
Reconcile the three scales
| Quantity | Value |
|---|---|
| Upper-tail probability | 0.05 |
| Truncated standard first moment | 0.1031356404 approximately |
| Standard tail mean | 2.0627128075 approximately |
| Loss-tail mean for μ=0, σ=10 | 20.6271280751 approximately |
Keep density height and tail area separate
The equality between the first-moment integral and φ(z) follows from this density’s derivative. It does not make φ(z) the upper-tail probability. The expected-shortfall expression still divides by 1−confidence.