162. Standard Normal Loss Function
The standard normal loss function, denoted or , is the expected shortfall above a threshold for a standard normal random variable :
where is the standard normal PDF and .
162.1. Closed form
Integration by parts gives:
where is the standard normal CDF. Both and are tabulated; combining them gives .
Example
decreases rapidly with — large thresholds have negligible expected shortfall.
162.2. Table of values
For : (use symmetry of ).
162.3. 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 and the reorder point is :
So converts “how many sigma above the mean did I stock” into “expected units short per replenishment.” This drives:
- Fill rate: — see Fill Rate
- Safety stock: solve for , then
- Newsvendor expected lost sales: at the optimal critical-ratio quantile
- Newsvendor expected leftover:
162.4. Properties
- Non-increasing: decreases monotonically as increases (more inventory → less expected shortfall)
- Asymptotic: as ; as
- Derivative: (note: , confirming monotonicity)
- Convexity: is convex — second derivative