Live model
Value only the portion of a normal outcome above a threshold
The unconditional normal mean is not enough for a deductible, strike, service credit, or excess-demand decision. This page evaluates the closed-form expected positive part, its conditional size when triggered, and a linear portfolio payout.
| Stage | Reference value | Probability | Excess / payout value | Meaning |
|---|
Current calculation process
Formula, substitution, intermediate quantities, and check
z=(K-mu)/sigma; p=1-Phi(z); E[(X-K)+]=sigmaphi(z)+(mu-K)p; E[X-K | X>K]=E[(X-K)+]/p
The positive-part formula integrates only the area-weighted distance above K. Outcomes below K contribute zero. Multiplying expected excess by unit value and exposure count is linear and does not impose a cap or fixed trigger payment.
Use the calculator
Five steps for a threshold-value decision
- Define the outcome and normal model. mu and sigma must describe one exposure in the same unit as K.
- Enter the contractual threshold. Confirm whether value begins strictly above K and whether a deductible, strike, or service floor applies.
- Enter marginal unit value. Use the amount paid or saved for each additional unit above K.
- Count additive exposures. Portfolio multiplication assumes expected values can be summed and uses no cap.
- Separate frequency from severity. Read exceedance probability, unconditional expected excess, and conditional excess together.
Five foundations
Why threshold expected value is more than mu
1. The payoff has a floor at zero
Positive part (X-K)+ equals X-K above K and zero below, producing a nonlinear transformation of a normal outcome.
2. Probability alone misses magnitude
Two models can have the same exceedance chance but different expected excess because one has a wider or more distant tail.
3. Conditional and unconditional differ
Expected excess averages zeros from non-triggered exposures; conditional excess averages only triggered cases.
4. Spread creates option value
Even when mu is below K, positive upper-tail outcomes create non-zero expected excess. Larger sigma can raise that value.
5. Linearity enables portfolio totals
Expected values add across exposures even without independence, but reserve and tail-risk calculations generally do not.
Calculation anatomy
Symbols, payoff units, and default substitution
| Symbol | Meaning | Unit |
|---|---|---|
| X | Normally distributed outcome | outcome units |
| K | Threshold | same outcome units |
| (X-K)+ | Positive excess above K | outcome units |
| p | P(X>K) | 0-1 |
| v | Value per excess unit | currency/outcome unit |
| N | Number of exposures | whole count |
Defaults standardize z=(125-110)/18. Expected excess is 18phi(z)+(110-125)[1-Phi(z)]. The portfolio expected payout is that excess multiplied by $45 and 30 exposures, with no intermediate display rounding.
Deep analysis
Three threshold-value perspectives
Trigger frequency
P(X>K) helps plan how often a payout or overflow workflow activates, but it does not state average cost.
Triggered severity
Conditional excess estimates average overage among activated cases and can support per-case handling capacity.
Portfolio expectation
Unconditional excess x unit value x N supports average funding. A high-confidence reserve still needs dependence and payout-distribution analysis.
Decision cases
Overflow and far-tail examples
Demand above contracted capacity
Daily demand is modeled N(110,18^2), contracted capacity is 125, and overflow costs $45 per unit. Thirty days of expected overflow are valued without charging days that remain below capacity.
Threshold far above the mean
As K moves many sigma above mu, exceedance and expected excess approach zero. Conditional excess may remain meaningful but becomes numerically sensitive because it divides by a tiny probability.
Terms
Threshold-value vocabulary
- Positive part
- max(X-K,0), the amount above threshold only.
- Exceedance probability
- Chance that X lies above K.
- Expected excess
- Unconditional mean positive part per exposure.
- Conditional excess
- Mean overage among exposures that cross K.
- Marginal unit value
- Linear value assigned to one additional excess unit.
- Stop-loss expectation
- Another name for expected positive excess above a deductible or retention.
FAQ
Questions about normal threshold value
Why is this not just mu?
The payoff uses only max(X-K,0), not the full outcome.
What does expected excess mean?
Long-run average overage including zero for non-exceeding outcomes.
How does conditional excess differ?
It averages only cases that cross the threshold.
Can value be non-currency?
Yes, if one consistent linear value scale is used.
What if payout is capped?
A second threshold or piecewise payoff model is required.
What if K is far below mu?
Expected excess approaches mu-K as exceedance approaches certainty.
Limits and evidence
Valuation boundaries
- The underlying outcome is assumed normal with fixed mu and sigma.
- The payoff is linear above one threshold, zero below, and has no cap.
- Parameter uncertainty, discounting, fixed trigger fees, and dependence are excluded.
- Portfolio expectation does not provide a percentile reserve or worst-case loss.
- Conditional excess is set to zero when exceedance is numerically negligible; do not overinterpret remote tails.
Sources and related tools