Franchise Deductible¶

Definition¶

With a franchise deductible $d\ge0$$0$, the insurer pays the entire loss once it exceeds $d$: $Y^L=X\indicator_{\{X>d\}}$$X{\text{𝟙}}_{\{X>d\}}$. If $X\le d$$d$, payment is zero. Conditional on a payment, the paid amount has the law of $X\mid X>d$$d$.

Interpretation¶

The threshold decides whether the insurer pays, but is not deducted from a qualifying loss. This creates a jump in the payment rule at the threshold.

The figure draws the Examples case $d=100$$100$. Below the threshold both rules pay nothing, so the loss 80 gives payment 0. Above it the franchise payment $Y^L$ (red) is the whole loss, while the ordinary-deductible payment $(X-d)_+$ (dashed) is $d$ lower: the loss 150 gives 150 against 50. The open circle at $(100,100)$ marks the jump: under the strict rule a loss of exactly 100 is not paid, and no payment lies strictly between 0 and 100.

Key Results¶

• $\mathbb{E}[Y^L]=\mathbb{E}[(X-d)_+]+d\mathbb{P}(X>d)$ when finite.
• Payment probability is the same as for an ordinary deductible, but qualifying payments are larger by $d$.
• For $d>0$$0$, there are no payment amounts strictly between zero and $d$.
• Whether equality qualifies is a contract convention; here the threshold rule is strict.

Examples¶

At deductible 100, losses 80 and 150 produce franchise payments 0 and 150. The ordinary deductible would produce 0 and 50.

Prerequisites¶

• Ordinary Deductible

Related Concepts¶

• Stop-Loss Transform and Reinsurance
• Mixed Discrete and Continuous Loss Distribution

Sources¶

• Risk Insurance · Lecture 2

Aliases: franchise insurance, threshold deductible, franchise Keywords: Y = X 1{X > d}, E[Y^L] = E[(X - d)+] + d P(X > d), jump at the threshold, franchise versus ordinary deductible

Last updated: 16 October 2026 22:28