The loss is not the payment
So far in this rung we built a clean two-part machine: a frequency model for how many claims arrive and a severity model for how big each one is. But there is a quiet gap in that story. The number we modelled — the ground-up loss X, the full size of the accident — is almost never the number the insurer actually writes a cheque for. A contract sits in between, and it bends the loss before it reaches the insurer's books.
Three contract features do almost all of this bending. A deductible lops off the bottom of each loss — the policyholder eats the first slice. A policy limit caps the top — the insurer refuses to pay beyond a ceiling. And coinsurance scales the whole thing down — the insurer pays only a fixed percentage of what remains. Each one is a simple idea, yet together they decide what the severity distribution looks like after it passes through the policy, which is the only distribution that matters for the insurer's wallet.
Deductibles: cutting off the bottom
The everyday kind is the ordinary deductible d: the insurer pays the part of the loss above d, and nothing below it. In symbols the payment per loss is max(X − d, 0). With a 500 deductible, a 300 loss pays nothing, a 1,200 loss pays 700, and a huge 50,000 loss still has its first 500 subtracted, paying 49,500. The deductible is *always* removed — it does not vanish on big claims. That last point trips up beginners constantly.
There is a sharper-edged cousin, the franchise deductible. It is all-or-nothing: below the threshold you get zero, but the instant the loss crosses d, the insurer pays the *entire* loss, deductible and all. With a 500 franchise, a 400 loss pays 0 while a 600 loss pays the full 600 — not 100. That jump right at the threshold is its signature, and it is common in marine cargo. Honestly, it also breeds a problem: a near-miss loss has a strong incentive to be nudged over the line, since crossing it flips the payout from nothing to everything.
Here is the subtle part the title promised. A deductible changes *two* things at once. It shrinks the size of every payment (each cheque is smaller by d), and it also shrinks the number of payments — every loss below d now pays zero and never becomes a claim at all. So a deductible cuts both the severity and the effective frequency the insurer sees. If 30% of losses fall under the deductible, the insurer's claim count drops by nearly a third before a single cheque shrinks. Forgetting the frequency effect is one of the most common mistakes in pricing a deductible.
Limits, coinsurance, and how they stack
If a deductible trims the bottom, a policy limit u chops off the top. The insurer pays the loss up to a ceiling and no further: the payment is min(X, u). Under a 300,000 limit, a 200,000 loss is paid in full, but a 450,000 loss pays only 300,000 — the policyholder swallows the excess 150,000. The quantity min(X, u) is called the *limited loss*, and its average, the limited expected value, is the building block actuaries use to price layers and to compute increased limits factors. Be honest about what a limit does: it does not make the big loss disappear, it just hands the part above u back to the insured.
Coinsurance is different again — it does not cut the top or the bottom, it scales everything by a factor alpha (a percentage like 80%). The insurer pays alpha times whatever is left, leaving the policyholder the other 20% of every loss, big or small. With pure 80% coinsurance, a 10,000 loss pays 8,000. In a fully layered real policy these features stack in a definite order: take the loss, subtract the deductible, apply coinsurance to what remains, then cap at the limit. That gives a per-loss payment of alpha × (min(X, u) − d) for losses above d.
loss X = 80,000 deductible d = 1,000 coinsurance a = 0.80 limit u = 50,000 step 1 subtract deductible: 80,000 - 1,000 = 79,000 step 2 apply coinsurance: 0.80 * 79,000 = 63,200 step 3 cap at the limit: min(63,200, 50,000) = 50,000 insurer pays 50,000 policyholder bears 80,000 - 50,000 = 30,000
The loss elimination ratio
When an insurer is deciding whether to add a deductible, it wants one honest number: what fraction of the losses we used to pay does this deductible save us? That number is the loss elimination ratio, or LER. The amount a deductible d removes from any single loss is min(X, d) — you keep all of a small loss below d, and exactly d of any larger one. The LER is then the expected amount eliminated divided by the expected loss with no deductible: E[min(X, d)] ÷ E[X].
A small worked example. Suppose average losses are 2,000, and a 500 deductible eliminates an average of 360 per loss (small losses give back their whole size, larger ones give back 500, and you average across the distribution). Then the LER is 360 ÷ 2,000 = 0.18. The deductible removes 18% of expected losses, so the insurer can cut the *loss part* of the pure premium by roughly 18%. That is exactly the kind of clean answer a pricing committee can act on.
Two ways to read a payment
There is one last distinction that separates people who really understand deductibles from people who can only recite the formula: the per-loss versus per-payment view. The *per-loss* variable looks at every loss event, including the small ones that paid zero — it is max(X − d, 0), zeros and all. The *per-payment* variable conditions on a cheque actually being written: it looks only at losses above d, so it is (X − d) given that X exceeds d. Same modification, two different averages.
The per-payment average is always the larger of the two, because the per-loss average is diluted by all those zero payments. If 30% of losses fall below the deductible and pay nothing, the per-loss mean is only 70% of the corresponding per-payment mean — the two are linked precisely through the probability that a payment is made. Which one you want depends on the question: per-loss feeds the total cost and the pure premium, while per-payment describes the average cheque the claims department actually cuts.
This distinction carries a real data trap. Claim data are usually recorded *per payment* and left-truncated: losses below the deductible were never reported, so they are simply missing from your file, not sitting there as zeros. If you fit a severity distribution to that data as though it were a complete picture of all losses, you silently throw away the small end and bias every estimate. Knowing whether your data are per-loss or per-payment is not pedantry — it decides whether your severity model is honest.