Inflation and Reinsurance on Trended Claims

Why the Order of Operations Matters

A severity model describes ground-up claim amounts, before a policy deductible, payment limit, or other coverage term is applied. If claim costs increase, the insurer must first decide what is being trended: the underlying losses, the contract terms, or both. These choices can produce different expected payments. In particular, applying an inflation factor to an expected payment calculated under unchanged terms generally does not give the expected payment on trended losses.

Consider a hypothetical policy with a fixed deductible and a fixed maximum payment. Inflation can move a claim that was below the deductible above it, or move a claim into the portion of the loss above the maximum payment. A reliable calculation therefore follows the contract’s order: trend the ground-up loss, apply the applicable policy terms, and then calculate any reinsurance share of the resulting covered payment.

Definition: Let $X$ be a nonnegative ground-up claim severity, and let $g>0$ be a deterministic trend factor. The trended severity is $gX$. For a deductible $d$ and maximum payment $u$, both nonnegative, define the policy payment per claim by $h_{d,u}(x)=\min((x-d)_+,u)$, where $(z)_+=\max(z,0)$. If terms remain unchanged, the payment on a trended claim is $h_{d,u}(gX)$. If both monetary terms are indexed by $g$, it is $h_{gd,gu}(gX)$.

Here, $u$ is the maximum amount the policy pays per claim, not necessarily a cap on the ground-up loss. Under this payment map, losses at or below $d$ produce no payment; losses between $d$ and $d+u$ produce the loss above the deductible; and losses at or above $d+u$ produce the maximum payment $u$. The expected payment per claim is the expectation of the payment map under the severity distribution.

Trend First, Then Apply the Contract

For a discrete severity distribution, calculate $gX$ separately for each possible claim amount, apply the payment map to each trended amount, and weight the payments by their probabilities. For a continuous distribution, use the distribution of $gX$ or integrate the payment map against its density. The order is the same in either case.

Worked Example: Fixed Deductible and Indexed Payment Maximum

A hypothetical claim severity $X$ is $200$ dollars with probability $0.40$ and $600$ dollars with probability $0.60$. Claim amounts are trended by $g=1.10$. The policy keeps its $150$-dollar deductible, while its original $400$-dollar maximum payment is indexed to $440$ dollars. Find the expected payment per claim after trending.

Method. Trend each ground-up amount first, then apply the $150$-dollar deductible and $440$-dollar maximum payment. The payment map is $h_{150,440}(x)=\min((x-150)_+,440)$.

For the $200$-dollar claim, the trended severity is $1.10(200)=220$ dollars. Its payment is $\min(220-150,440)=70$ dollars.

For the $600$-dollar claim, the trended severity is $1.10(600)=660$ dollars. Its payment is $\min(660-150,440)=440$ dollars.

The expected payment is $0.40(70)+0.60(440)=28+264=292$ dollars per claim. The answer is \$292.00.

Check and common mistake. The second claim reaches the indexed maximum payment, so its payment cannot exceed $440$ dollars. A tempting but incorrect shortcut is to multiply the expected payment under the original $150$-dollar deductible and $400$-dollar maximum by $1.10$. That original expectation is $0.40(50)+0.60(400)=260$, and $1.10(260)=286$, not $292$. The shortcut fails because the deductible remains fixed while the ground-up claims are trended.

A useful comparison is between $E[h_{d,u}(gX)]$, the expected payment after trending under unchanged terms, and $gE[h_{d,u}(X)]$, the original expected payment multiplied by the trend factor. These are different operations. The first evaluates the contract at new loss amounts; the second scales already-calculated payments. A deductible or cap makes the payment map nonlinear, so the operations need not agree.

E[h_{d,u}(gX)] \ne gE[h_{d,u}(X)] \quad\text{in general.}

When Indexed Terms Scale Exactly

There is an important special case. Suppose the deductible and maximum payment both increase by the same factor as severity, from $d,u$ to $gd,gu$. For $g>0$, the positive-part operation and the minimum operation both scale by $g$: $(gx-gd)_+=g(x-d)_+$, and $\min(ga,gb)=g\min(a,b)$. Consequently, $h_{gd,gu}(gx)=g h_{d,u}(x)$ for every $x\geq 0$. Taking expectations gives the corresponding identity for expected payments.

h_{gd,gu}(gx) =\min((gx-gd)_+,gu) =g\min((x-d)_+,u) =g h_{d,u}(x), \qquad E[h_{gd,gu}(gX)]=gE[h_{d,u}(X)].

This identity requires the terms in the payment map to scale consistently. If only the maximum payment is indexed, or only the deductible is indexed, the identity does not generally hold. It also assumes that the trend is a common deterministic factor and that the payment map has no other unindexed monetary thresholds.

Worked Example: Trend Applied to a Continuous Severity

A hypothetical ground-up severity has a uniform distribution from $0$ to $500$ dollars, written $X\sim\operatorname{Uniform}(0,500)$. Thus its density is $1/500$ on that interval. Claims are trended by $g=1.10$. The policy retains a fixed $100$-dollar deductible and a fixed $200$-dollar maximum payment. Find the expected payment per claim after trending, and compare it with the scaled original expected payment.

Method. Since $X$ is uniform on $[0,500]$, the trended severity $Z=1.10X$ is uniform on $[0,550]$, with density $1/550$. Under the fixed terms, the payment is zero for $Z\leq100$, equals $Z-100$ for $100<Z<300$, and equals $200$ for $Z\geq300$.

The integral of the payment over the middle interval is the area of a triangle with base $200$ and height $200$: $\int_{100}^{300}(z-100)\,dz=\tfrac12(200)(200)=20{,}000$. For $300\leq z\leq550$, the payment is $200$; this interval has length $250$, contributing $200(250)=50{,}000$. Dividing the total by the interval length $550$ gives $E[h_{100,200}(1.10X)]=(20{,}000+50{,}000)/550=127.2727\ldots$ dollars. To the nearest cent, the expected payment is \$127.27.

For comparison, before trending, $X$ is uniform on $[0,500]$. The middle-interval contribution remains $20{,}000$, and the maximum payment applies over $300\leq X\leq500$, contributing $200(200)=40{,}000$. Therefore, $E[h_{100,200}(X)]=(20{,}000+40{,}000)/500=120$ dollars, and multiplying this original expectation by $1.10$ gives $132$ dollars.

The trend-first result, $\$127.27$, is less than $\$132.00$. Under the fixed maximum payment, some of the trended claims have reached the cap; the cap prevents those payments from increasing in proportion to severity. The calculation avoids the mistake of scaling the entire original expected payment despite the unchanged limit.

Proportional Reinsurance on Trended Covered Payments

A proportional quota-share arrangement assigns a fixed percentage of each covered policy payment to the reinsurer. Let $q$, with $0\leq q\leq1$, be the ceded share. If the primary policy payment on a trended claim is $h_{d,u}(gX)$, the reinsurer’s payment is $q h_{d,u}(gX)$, and the insurer’s retained payment is $(1-q)h_{d,u}(gX)$. By linearity of expectation, their expected payments are $qE[h_{d,u}(gX)]$ and $(1-q)E[h_{d,u}(gX)]$. This allocation applies to covered policy payments; it does not mean that the reinsurer receives the same percentage of the ground-up loss when the policy pays less than the full loss.

Worked Example: Trend, Policy Terms, and Quota Share

A hypothetical ground-up claim severity $X$ is $80$ dollars with probability $0.25$, $180$ dollars with probability $0.50$, and $380$ dollars with probability $0.25$. Severities are trended by $g=1.25$. The primary policy has a fixed $100$-dollar deductible and a fixed $250$-dollar maximum payment. A quota-share treaty cedes $40\%$ of each primary policy payment. Calculate the expected primary payment, expected reinsurer payment, and expected insurer-retained payment per claim.

Method. Trend each severity, apply the policy payment map $h_{100,250}$, and then split that payment using $q=0.40$.

The three trended severities are $1.25(80)=100$, $1.25(180)=225$, and $1.25(380)=475$ dollars. The corresponding primary payments are $h_{100,250}(100)=0$, $h_{100,250}(225)=\min(125,250)=125$, and $h_{100,250}(475)=\min(375,250)=250$ dollars.

The expected primary payment is $0.25(0)+0.50(125)+0.25(250)=0+62.50+62.50=125$ dollars. The expected reinsurer payment is $0.40(125)=50$ dollars, and the expected insurer-retained payment is $0.60(125)=75$ dollars. The check $50+75=125$ confirms that the expected shares add to the expected primary payment.

Check and common mistake. The quota share applies after the primary policy terms, so it is a percentage of the $125$-dollar expected covered payment, not of the expected trended ground-up loss. The expected trended ground-up loss is $1.25[0.25(80)+0.50(180)+0.25(380)]=1.25(205)=256.25$ dollars. Ceding $40\%$ of that amount would incorrectly treat ground-up losses as though they were all covered by the primary policy.

A Reliable Calculation Sequence

In a multi-step claim problem, writing the sequence explicitly makes it easier to distinguish severity inflation from contract indexing and reinsurance allocation. Use the payment map appropriate to the terms in force at the trended valuation date; do not adjust a term unless the problem says it is indexed.

Specify the severity and trend.
Record the distribution of ground-up $X$ and the deterministic factor $g$. The trended severity is $gX$.

Identify the terms after trending.
State whether the deductible $d$, maximum payment $u$, or other monetary terms are fixed or indexed. This determines the payment map.

Calculate payments before taking expectations.
For discrete severities, evaluate the payment for each possible outcome. For continuous severities, integrate the payment map over the trended severity distribution.

Allocate the covered payment to reinsurance.
For a quota share, multiply the primary payment by the ceded percentage and the retained percentage. Check that the two shares sum to the primary payment.

The order matters because expected payment depends both on the severity distribution and on where that distribution falls relative to contract thresholds. Trending can change the probability of a positive payment and the probability that the maximum payment is reached. Consequently, a calculation based only on the original expected loss, without tracking the payment map, can miss changes in coverage.

Key takeaway: When severity is trended and policy terms are fixed, calculate $E[h_{d,u}(gX)]$ by applying the terms to trended losses. Scaling the original expected payment by $g$ is justified for this deductible-and-maximum-payment map when both monetary terms are indexed by the same factor; it is not generally valid when terms stay fixed or only some terms change.

Check Your Understanding

For each problem, trend the ground-up claim amounts before applying the stated terms. Answers involving money are rounded to the nearest cent.

A hypothetical severity is $100$ dollars with probability $0.50$ and $400$ dollars with probability $0.50$. Claims are trended by $g=1.10$. The deductible is fixed at $100$ dollars and the maximum payment is fixed at $250$ dollars. Find the expected policy payment per claim. (Answer: $\$130.00$)
A hypothetical severity is uniform on $[0,400]$ dollars. It is trended by $g=1.25$, while the deductible $100$ dollars and maximum payment $200$ dollars stay fixed. Find the expected payment. (Answer: $\$120.00$)
A hypothetical severity is $50$ dollars with probability $0.20$, $150$ dollars with probability $0.50$, and $350$ dollars with probability $0.30$. With $g=1.20$, a $100$-dollar fixed deductible, and a $300$-dollar fixed maximum payment, find the expected primary payment. (Answer: $\$130.00$)
A hypothetical severity is $100$ dollars with probability $0.40$ and $300$ dollars with probability $0.60$. Claims are trended by $g=1.10$. The policy has a fixed $120$-dollar deductible and a fixed $250$-dollar maximum payment, and a quota share cedes $25\%$ of each policy payment. Find the expected ceded and retained payments. (Answer: ceded $\$31.50$; retained $\$94.50$)
A hypothetical severity is uniform on $[0,200]$ dollars. Its original contract has a $20$-dollar deductible and a $60$-dollar maximum payment. After trending by $g=1.50$, first find the expected payment if both terms remain fixed; then find it if both terms are indexed by $g$. (Answer: fixed terms $\$50.00$; indexed terms $\$67.50$)

What You'll Learn