Ordinary Deductible¶

Definition¶

For a nonnegative ground-up loss $X$ (the full loss before any deductible or limit is applied) and deductible $d\ge0$$0$, the insurer's per-loss payment is $Y^L=(X-d)_+=\max(X-d,0)$$d,0)$. The insured retains $\min(X,d)$. The per-payment distribution is the conditional law of $X-d$ given $X>d$ (written $Y^P$), defined only when $\mathbb{P}(X>d)>0$$0$.

Interpretation¶

Every loss is included in the per-loss model, with a zero payment below the deductible. The per-payment model describes only cases generating a positive payment.

For $X\sim\operatorname{Exp}(0.01)$ and $d=100$$100$, the blue area is $\mathbb{P}(Y^L=0)=F_X(d)=1-\mathrm{e}^{-1}\approx0.632$$0.632$: these losses stay in the per-loss model with payment zero. The red area $\bar F_X(d)=\mathrm{e}^{-1}\approx0.368$ holds the losses that produce a payment $X-d$$d$; rescaling that tail by $1/\bar F_X(d)$ gives the per-payment law $Y^P$, which here is again exponential with mean 100. Hence $\mathbb{E}[Y^L]=100\mathrm{e}^{-1}\approx36.79$$36.79$, as in the example.

The survivor function $\bar F_X(d)=\mathbb{P}(X>d)$ gives the probability that the deductible is exceeded. Dividing the shifted tail by this probability normalizes the per-payment law over cases receiving a positive payment.

Key Results¶

• $\mathbb{P}(Y^L=0)=F_X(d)$ and $F_{Y^L}(y)=F_X(d+y)$ for $y\ge0$$0$.
• For $y\ge0$$0$, $\mathbb{P}(Y^P>y)=\bar F_X(d+y)/\bar F_X(d)$$y)/{\bar{F}}_X(d)$.
• $\mathbb{E}[Y^L]=\bar F_X(d)\mathbb{E}[Y^P]$ when finite.
• For continuous $X$, $f_{Y^P}(y)=f_X(d+y)/\bar F_X(d)$ for $y>0$$0$.

Examples¶

For $X\sim\operatorname{Exp}(0.01)$ and $d=100$$100$, the probability of payment is $\mathrm{e}^{-1}$. By the memoryless property, conditional payment has mean 100, so mean payment per loss is $100\mathrm{e}^{-1}\approx36.79$$36.79$.

Prerequisites¶

• Survivor Function
• Conditional Probability

Related Concepts¶

• Stop-Loss Transform and Reinsurance
• Franchise Deductible
• Loss Elimination Ratio
• Hunger for Bonus
• Mixed Discrete and Continuous Loss Distribution

Sources¶

• Risk Insurance · Lecture 1

Aliases: excess deductible, per-loss deductible payment, straight deductible Keywords: (X - d)+, per-loss variable Y^L, per-payment variable Y^P, ground-up loss, memoryless exponential deductible

Last updated: 16 October 2026 22:28