Insurance Payments: Deductibles, Coinsurance, Benefit Limits, and Inflation

A policy has a $500 deductible, 80% coinsurance, and a $50,000 cap. Misordering these three features changes the premium by thousands. Every Exam P insurance problem tests whether you can translate policy terms into payment functions, and whether you apply them in the right sequence.

Let $X$ denote the ground-up loss, the total loss before any policy modifications.

All policy features transform $X$. You apply them in a fixed order: deductible, then coinsurance, then benefit limit.

With an ordinary deductible $d$, the insurer pays nothing below $d$ and pays the excess above $d$:

$$Y^L=(X-d{)}_{+}=\{\begin{array}{ll}0,& X\le d\\ X-d,& X>d\end{array}$$

The expected payment per loss:

$$E[Y^L]=E[X]-E[X\wedge d]$$

HIGH-FREQUENCY: The limited expected value $E[X\wedge d]$ is the fundamental building block. It represents the expected cost if losses were capped at $d$:

Common mistakes

• Confusing payment per loss with payment per payment. For an exponential with $\theta =1000$ and $d=200$, the per-loss answer is 818.73 and the per-payment answer is 1000. Reporting one when asked for the other is the most common error. Trap: 1000 when the problem asks per loss.
• Using $E[X\wedge u]$ instead of $E[X\wedge (d+u)]$. The upper LEV bound is the maximum covered loss $d+u$, not $u$ alone. With $d=1000$ and $u=2400$: the correct bound is 3400, not 2400. Trap: $E[X\wedge 2400]-E[X\wedge 1000]=360$ instead of the correct 1344.
• Applying coinsurance to the full loss instead of the excess. With $d=1000$ and $c=0.80$, the payment is $0.80(X-1000{)}_{+}$, not $0.80X-1000$. For a $3,000 loss: correct is $0.80\times 2000=1600$; wrong is $2400-1000=1400$. Trap: 1400.

Bottom line

• Payment with deductible $d$ and limit $u$: $E[Y^L]=E[X\wedge (d+u)]-E[X\wedge d]$; the upper LEV bound is the maximum covered loss $d+u$, not $u$
• Coinsurance $c$: scales the expected payment linearly by $c$; apply deductible first, then coinsurance, then limit, always in this order
• Per loss vs. per payment: per payment divides by $S(d)$; per loss includes the zero payments below the deductible
• Exponential memoryless: expected payment per payment always equals $\theta$ regardless of the deductible

Exam shortcut

Before computing anything, write the payment function $Y^L$ as a piecewise formula. Label three regions: zero (below deductible), variable (between $d$ and $d+u$), capped (above $d+u$). "LEV at d+u minus LEV at d." This single formula handles deductible-plus-limit problems for any distribution. "Per loss: nothing. Per payment: divide by survival." The word "per payment" is your trigger to divide by $S(d)$.

The full lesson (about 3,845 words, 26 min read) adds 5 worked examples, all 8 common mistakes, a self-check, free in the app.

Learning objectives

• 2e

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