Standard Normal Loss Function

164. Standard Normal Loss Function

The standard normal loss function, denoted $𝐿 (𝑧)$ or $𝐺 (𝑧)$ , is the expected shortfall above a threshold $𝑧$ for a standard normal random variable $𝑍 \sim 𝒩︀ (0, 1)$ :

𝐿 (𝑧) = 𝐸 [{(𝑍 - 𝑧)}^{+}] = \int_{𝑧}^{\infty} (𝑡 - 𝑧) 𝜑 (𝑡) 𝑑 𝑡

where $𝜑$ is the standard normal PDF and ${(𝑥)}^{+} = max (𝑥, 0)$ .

164.1. Closed form

Integration by parts gives:

𝐿 (𝑧) = 𝜑 (𝑧) - 𝑧 [1 - Φ (𝑧)]

where $Φ$ is the standard normal CDF. Both $𝜑 (𝑧)$ and $Φ (𝑧)$ are tabulated; combining them gives $𝐿 (𝑧)$ .

Example

\begin{matrix} 𝐿 (0) = 𝜑 (0) - 0 = \frac{1}{\sqrt{2 𝜋}} \approx 0.3989 \\ 𝐿 (1) = 𝜑 (1) - 1 \cdot [1 - Φ (1)] = 0.2420 - 0.1587 \approx 0.0833 \\ 𝐿 (2) = 𝜑 (2) - 2 \cdot [1 - Φ (2)] = 0.0540 - 2 \cdot 0.0228 \approx 0.0085 \end{matrix}

$𝐿 (𝑧)$ decreases rapidly with $𝑧$ — large thresholds have negligible expected shortfall.

164.2. Derivation: tail minus threshold

Where does the closed form come from? Split the excess $(𝑡 - 𝑧)$ accumulated over the tail into total tail value minus threshold counted across the tail.

Quantity $𝐴$ — tail value. The full value $𝑡$ over the tail, weighted by likelihood. Using $𝑡 𝜑 (𝑡) = - 𝜑^{'} (𝑡)$ :

𝐴 = \int_{𝑧}^{\infty} 𝑡 𝜑 (𝑡) 𝑑 𝑡 = 𝜑 (𝑧)

Quantity $𝐵$ — threshold across the tail. Count $𝑧$ once per tail outcome:

𝐵 = \int_{𝑧}^{\infty} 𝑧 𝜑 (𝑡) 𝑑 𝑡 = 𝑧 [1 - Φ (𝑧)]

Subtract. Each tail outcome contributes $𝑡$ to $𝐴$ and $𝑧$ to $𝐵$ , so $𝐴 - 𝐵$ strips the baseline and leaves exactly the excess $(𝑡 - 𝑧)$ :

𝐿 (𝑧) = 𝐴 - 𝐵 = \underset{total tail value}{\underset{⏟}{\int_{𝑧}^{\infty} 𝑡 𝜑 (𝑡) 𝑑 𝑡}} - \underset{threshold across tail}{\underset{⏟}{\int_{𝑧}^{\infty} 𝑧 𝜑 (𝑡) 𝑑 𝑡}} = 𝜑 (𝑧) - 𝑧 [1 - Φ (𝑧)]

The shaded tail below is the region both integrals run over; $𝐿 (𝑧)$ is the per-tail gap between the two.

$𝐿 (𝑧)$ itself decreases monotonically and convexly toward 0:

164.3. Table of values

$𝑧$	$𝐿 (𝑧)$	$𝑧$	$𝐿 (𝑧)$	$𝑧$	$𝐿 (𝑧)$
$0.0$	$0.3989$	$1.0$	$0.0833$	$2.0$	$0.0085$
$0.1$	$0.3509$	$1.1$	$0.0686$	$2.1$	$0.0065$
$0.2$	$0.3069$	$1.2$	$0.0561$	$2.2$	$0.0049$
$0.3$	$0.2668$	$1.3$	$0.0455$	$2.3$	$0.0037$
$0.4$	$0.2304$	$1.4$	$0.0367$	$2.4$	$0.0027$
$0.5$	$0.1978$	$1.5$	$0.0293$	$2.5$	$0.0020$
$0.6$	$0.1687$	$1.6$	$0.0232$	$2.6$	$0.0015$
$0.7$	$0.1429$	$1.7$	$0.0183$	$2.7$	$0.0011$
$0.8$	$0.1202$	$1.8$	$0.0143$	$2.8$	$0.0008$
$0.9$	$0.1004$	$1.9$	$0.0111$	$2.9$	$0.0005$

For $𝑧 < 0$ : $𝐿 (𝑧) = 𝐿 (- 𝑧) + (- 𝑧) = 𝐿 (| 𝑧 |) - 𝑧$ (use symmetry of $𝜑$ ).

164.4. Use case: safety-stock derivations

The reason this function is everywhere in supply-chain math: the expected shortfall (in stock units) per replenishment cycle, when demand during lead time is $𝒩︀ (𝜇_{𝐿}, 𝜎_{𝐿}^{2})$ and the reorder point is $𝑟 = 𝜇_{𝐿} + 𝑧 𝜎_{𝐿}$ :

E[shortfall per cycle] = 𝜎_{𝐿} 𝐿 (𝑧)

So $𝐿 (𝑧)$ converts “how many sigma above the mean did I stock” into “expected units short per replenishment.” This drives:

Fill rate: $𝛽 = 1 - \frac{𝜎_{𝐿} 𝐿 (𝑧)}{𝑄}$ — see Fill Rate
Safety stock: solve $𝐿 (𝑧) = \frac{(1 - 𝛽) 𝑄}{𝜎_{𝐿}}$ for $𝑧$ , then $SS = 𝑧 𝜎_{𝐿}$
Newsvendor expected lost sales: $𝜎 𝐿 (𝑧^{*})$ at the optimal critical-ratio quantile
Newsvendor expected leftover: $𝜎 [𝐿 (𝑧) + 𝑧]$

164.5. Properties

Non-increasing: $𝐿 (𝑧)$ decreases monotonically as $𝑧$ increases (more inventory → less expected shortfall)
Asymptotic: $𝐿 (𝑧) \to 0$ as $𝑧 \to \infty$ ; $𝐿 (𝑧) \sim - 𝑧$ as $𝑧 \to - \infty$
Derivative: $𝐿^{'} (𝑧) = - [1 - Φ (𝑧)] = - 𝑃 (𝑍 > 𝑧)$ (note: $𝐿^{'} (𝑧) \leq 0$ , confirming monotonicity)
Convexity: $𝐿 (𝑧)$ is convex — second derivative $𝐿^{″} (𝑧) = 𝜑 (𝑧) \geq 0$

164.6. Connection to standard normal PDF / CDF

𝐿 (𝑧) = 𝜑 (𝑧) - 𝑧 Φ (𝑧), Φ (𝑧) = 1 - Φ (𝑧)

— see PDF, CDF, SF.