Types of Coverage Modifications

A warehouse burns down. The loss is $4.8 million. But the insurer's payout is not $4.8 million, a $250,000 deductible, an 80% coinsurance clause, and a $3 million limit each slash the payment. Stack all three, and the policyholder collects far less than the loss.

An ordinary deductible (straight deductible) of amount $d$ means you pay nothing on losses below $d$ and the excess above $d$ for larger losses. The insurer's payment is:

$$Y^L=(X-d{)}_{+}=\max (X-d,0)$$

KEY: This is the per-loss variable. It includes zeros for losses below the deductible.

The per-payment variable conditions on a payment actually being made:

$$Y^P=X-d\mid X>d$$

The per-payment variable is never zero. The two are linked by:

$$E[Y^L]=E[Y^P]\cdot \mathrm{Pr}(X>d)$$

A franchise deductible works differently. Below $d$, the insurer pays nothing, same as ordinary. But once the loss reaches $d$, the insurer pays the entire loss:

Common mistakes

• Confusing ordinary and franchise deductibles. With a $500 ordinary deductible and a $600 loss, the payment is $100. With a $500 franchise deductible and a $600 loss, the payment is $600. Trap: choosing $100 when the question specifies franchise.
• Getting the maximum covered loss wrong with coinsurance. With $d=500$, $\alpha =0.80$, $u=2000$: the correct MCL is $500+2000/0.80=3000$, not $500+2000=2500$. Trap: computing $E[X\wedge 2500]-E[X\wedge 500]$ and then multiplying by 0.80.
• Forgetting to subtract $E[X\wedge d]$. The expected excess payment is $E[X]-E[X\wedge d]$, not $E[X]-d$. For uniform on (0, 1000) with $d=200$: correct is $500-180=320$, not $500-200=300$. Trap: $300.

Bottom line

• Ordinary deductible pays $(X-d{)}_{+}$; franchise deductible pays full $X$ once $X\ge d$
• Policy limit $u$ caps the insurer's payment; maximum covered loss with deductible is $d+u$
• Coinsurance $\alpha$ scales the excess; with all three, maximum covered loss becomes $d+u/\alpha$
• Order matters: deductible first, then coinsurance, then limit (DCL)

Exam shortcut

When a problem combines deductible, coinsurance, and limit, write down the MCL $d+u/\alpha$ immediately. Then the expected payment is $\alpha \times [E[X\wedge (d+u/\alpha )]-E[X\wedge d]]$. This single formula handles the vast majority of coverage modification questions. Remember: DCL. Deductible, Coinsurance, Limit. Apply in that order. The MCL is always $d+u/\alpha$. "Limited minus limited equals layer."

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Learning objectives

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