Separating Inflation Effects from Deductible Changes

Why Separate the Two Effects?

A change in expected insurer payment between two periods can reflect more than one change. Claim severities may increase with inflation, while the deductible may also be adjusted. To understand the result, it is useful to calculate how much each change contributes. The calculation must keep the effects distinct: the impact of a deductible depends on the severity distribution to which it is applied.

This tutorial compares expected payment per claim under two conditions: a ground-up loss $X$ and deductible $d_0$ initially, then a proportional severity trend and deductible $d_1$. The deductible is an ordinary per-loss deductible, and there is no policy limit or coinsurance. All loss amounts and deductibles are in the same currency and at the same price level unless a trend factor is explicitly applied.

Definition: For ground-up loss $X$ and ordinary deductible $d\geq 0$, the insurer payment per loss is $Y^L=(X-d)_+=\max(X-d,0)$. If severity increases proportionally by a factor $c>0$, the trended loss is $cX$, and payment with deductible $d$ is $(cX-d)_+$. Define $g(c,d)=E[(cX-d)_+]$, the expected insurer payment per claim.

The subscript $+$ matters: a loss below the deductible produces zero insurer payment, not a negative payment. The starting expected payment is $g(1,d_0)$; the ending expected payment is $g(c,d_1)$. Their difference is the net change per claim.

Two Exact Ways to Decompose the Change

A decomposition is an accounting identity, not a claim that the two effects are independent. One approach applies the severity trend first while holding the deductible at its original level, then changes the deductible at the trended severity level. A second approach changes the deductible first and applies trend afterward.

\begin{aligned} g(c,d_1)-g(1,d_0) &=\underbrace{[g(c,d_0)-g(1,d_0)]}_{\text{trend effect at }d_0} +\underbrace{[g(c,d_1)-g(c,d_0)]}_{\text{deductible effect after trend}}\\ &=\underbrace{[g(1,d_1)-g(1,d_0)]}_{\text{deductible effect before trend}} +\underbrace{[g(c,d_1)-g(1,d_1)]}_{\text{trend effect at }d_1}. \end{aligned}

Each line adds and subtracts an intermediate expected payment. In the first line, the intermediate value is $g(c,d_0)$; in the second, it is $g(1,d_1)$. Thus each pair of effects must sum to the same net change. The individual effects can differ between the two decompositions because changing the deductible can alter which claims generate payment.

Calculate the starting payment.
Find $g(1,d_0)=E[(X-d_0)_+]$ under the original severity distribution and deductible.

Choose and state an order.
For trend-first, calculate $g(c,d_0)$. For deductible-first, calculate $g(1,d_1)$.

Calculate the ending payment.
Evaluate $g(c,d_1)$, applying the positive-part rule to every loss value or throughout the severity distribution.

Subtract and check.
Calculate the two effects using the selected path, then verify that their sum equals $g(c,d_1)-g(1,d_0)$.

For a discrete severity distribution with possible losses $x_j$ and probabilities $p_j$, where $\sum_j p_j=1$, each expectation is

g(c,d)=\sum_j p_j\max(cx_j-d,0).

This is also a practical computational method: trend each loss, subtract the deductible, replace any negative result by zero, and take the probability-weighted average. For a continuous severity distribution, the same expectation can be evaluated by integration or by a suitable distribution-specific formula.

Worked Calculations

Worked Example: Trend First, Then Change the Deductible

A hypothetical insurer models ground-up claim severity as $X=2{,}000$, $4{,}000$, or $6{,}000$ dollars with probabilities $0.5$, $0.3$, and $0.2$, respectively. Severity increases by 10%, so $c=1.10$. The deductible changes from $d_0=\$300$ to $d_1=\$800$. Find the net change in expected payment per claim and decompose it trend-first.

Method and assumption. Use the probability-weighted positive-part formula. The probabilities sum to $0.5+0.3+0.2=1$. The model changes severity and the deductible only; it does not change claim frequency.

Starting payment. The payments at the original deductible are $2{,}000-300=1{,}700$, $4{,}000-300=3{,}700$, and $6{,}000-300=5{,}700$. All exceed zero, so

g(1,300)=0.5(1{,}700)+0.3(3{,}700)+0.2(5{,}700) =850+1{,}110+1{,}140=\$3{,}100.

Trend effect at the original deductible. Trended losses are $1.10(2{,}000)=2{,}200$, $1.10(4{,}000)=4{,}400$, and $1.10(6{,}000)=6{,}600$. After the $\$300$ deductible, payments are $\$1{,}900$, $\$4{,}100$, and $\$6{,}300$. Therefore,

g(1.10,300)=0.5(1{,}900)+0.3(4{,}100)+0.2(6{,}300) =950+1{,}230+1{,}260=\$3{,}440.

The trend effect is $\$3{,}440-\$3{,}100=+\$340$ per claim.

Deductible effect after trend. With the trended losses and the new $\$800$ deductible, payments are $\$1{,}400$, $\$3{,}600$, and $\$5{,}800$. Thus

g(1.10,800)=0.5(1{,}400)+0.3(3{,}600)+0.2(5{,}800) =700+1{,}080+1{,}160=\$2{,}940.

The deductible effect is $\$2{,}940-\$3{,}440=-\$500$. The net change is $+\$340-\$500=-\$160$ per claim. Checking directly, $\$2{,}940-\$3{,}100=-\$160$. The decomposition reconciles.

Common mistake avoided: Treating the deductible change as a separate fixed dollar reduction without specifying the severity level. Here it reduces expected payment by $\$500$ after trend; that is the effect for this distribution and this path.

Worked Example: A Claim Crosses the Deductible

In a hypothetical portfolio, $X$ is $\$400$, $\$1{,}200$, or $\$3{,}000$ with probabilities $0.25$, $0.50$, and $0.25$. Severity increases by 50%, and the deductible rises from $\$500$ to $\$1{,}000$. Find the net change and a trend-first decomposition.

Starting payment. At $d_0=\$500$, the three payments are $\max(400-500,0)=0$, $\$700$, and $\$2{,}500$. Hence

g(1,500)=0.25(0)+0.50(700)+0.25(2{,}500) =0+350+625=\$975.

Trend effect at $d_0$. The trended losses are $\$600$, $\$1{,}800$, and $\$4{,}500$. Their payments after the $\$500$ deductible are $\$100$, $\$1{,}300$, and $\$4{,}000$. Therefore,

g(1.50,500)=0.25(100)+0.50(1{,}300)+0.25(4{,}000) =25+650+1{,}000=\$1{,}675.

Trend increases expected payment by $\$1{,}675-\$975=+\$700$.

Deductible effect after trend. With $d_1=\$1{,}000$, the trended $\$600$ loss produces no payment. The other payments are $\$800$ and $\$3{,}500$. Thus

g(1.50,1{,}000)=0.25(0)+0.50(800)+0.25(3{,}500) =0+400+875=\$1{,}275.

The deductible effect after trend is $\$1{,}275-\$1{,}675=-\$400$. The net change is $+\$700-\$400=+\$300$. Directly, $\$1{,}275-\$975=+\$300$.

Common mistake avoided: Subtracting the deductible from every loss without applying the zero floor. The $\$600$ trended loss at the new deductible creates zero payment, not a negative $\$400$ payment.

Worked Example: The Allocation Depends on the Order

A hypothetical severity model assigns probability $0.5$ to each of $\$1{,}600$ and $\$2{,}400$. Severity trends by 20%; the deductible rises from $\$500$ to $\$1{,}500$. Calculate both decompositions and explain why the allocated effects differ.

Starting payment. At the original deductible, payments are $\$1{,}100$ and $\$1{,}900$, so $g(1,500)=0.5(1{,}100)+0.5(1{,}900)=\$1{,}500$.

Trend-first path. Trended losses are $1.20(1{,}600)=\$1{,}920$ and $1.20(2{,}400)=\$2{,}880$. At $d_0=\$500$, payments are $\$1{,}420$ and $\$2{,}380$, giving $g(1.20,500)=0.5(1{,}420)+0.5(2{,}380)=\$1{,}900$. The trend effect is $+\$400$.

At the trended severities and $d_1=\$1{,}500$, payments are $\$420$ and $\$1{,}380$, giving $g(1.20,1{,}500)=0.5(420)+0.5(1{,}380)=\$900$. The deductible effect after trend is $\$900-\$1{,}900=-\$1{,}000$. The net change is $+\$400-\$1{,}000=-\$600$.

Deductible-first path. Before trend, the payments at $d_1=\$1{,}500$ are $\$100$ and $\$900$. Thus $g(1,1{,}500)=0.5(100)+0.5(900)=\$500$, and the deductible effect is $\$500-\$1{,}500=-\$1{,}000$. Applying trend at that deductible gives $g(1.20,1{,}500)=\$900$, so the trend effect is $\$900-\$500=+\$400$. This path also gives $-\$1{,}000+\$400=-\$600$.

Why the allocations match here. All losses remain above both deductibles before and after trend. In this case, raising the deductible reduces each payment by exactly $\$1{,}000$, regardless of the order. In general, crossing the deductible can make the two allocations differ. The final net change must still agree.

Interpretation and Exam Checks

The trend effect is not always a simple percentage of the starting expected payment. With an ordinary deductible, the positive-part function makes payment nonlinear in severity: some losses may move above the deductible after trend, while others may remain below it. Similarly, increasing the deductible can eliminate payment on claims that were previously covered. These threshold effects make the selected decomposition path important when reporting separate contributions.

Unless a problem specifies otherwise, state the path used rather than presenting the two effects as unique quantities. A useful check is to calculate the starting and ending expected payments independently and confirm that their difference equals the sum of the stated effects. If claim frequency is unchanged, multiplying both per-claim expectations by the same expected claim count gives the corresponding expected total payments; this does not change the decomposition per claim.

Apply the trend to ground-up severity before applying the deductible: use $(cX-d)_+$, not $c(X-d)_+$.
Use the positive-part rule for every loss, including losses below the deductible.
Identify whether the deductible is held fixed in original dollars or changes between periods.
State whether the reported effects are trend-first or deductible-first.
Reconcile the effects to the total change in expected payment.

Key takeaway: The net change from $g(1,d_0)$ to $g(c,d_1)$ is exact and does not depend on decomposition order. The separate trend and deductible effects do depend on the chosen order when the deductible interacts with the severity distribution.

Check Your Understanding

For each problem, use ordinary per-loss deductibles, apply the positive-part rule, and report the requested effects per claim unless an annual total is requested.

A hypothetical discrete severity is $\$1{,}000$ with probability $0.4$ and $\$2{,}000$ with probability $0.6$. Severity trends by 10%, and the deductible rises from $\$200$ to $\$500$. Find the trend-first effects and net change. (Answer: trend $+\$160$; deductible $-\$300$; net change $-\$140$.)
Ground-up severity is uniform on $[0,\$4{,}000]$. It trends by 25%, and the deductible changes from $\$500$ to $\$1{,}000$. Use the expected payment for a uniform severity to find the trend-first effects and net change. (Answer: trend $+\$493.75$; deductible $-\$425$; net change $+\$68.75$.)
A hypothetical severity is $\$300$, $\$1{,}000$, or $\$2{,}500$, with probabilities $0.2$, $0.5$, and $0.3$. Severity trends by 50%, and the deductible rises from $\$400$ to $\$900$. Find the trend-first effects and net change, applying the zero floor. (Answer: trend $+\$635$; deductible $-\$410$; net change $+\$225$.)
A severity model assigns equal probability to $\$1{,}800$ and $\$2{,}600$. Trend is 10%, and the deductible rises from $\$300$ to $\$700$. Calculate the trend-first decomposition and check it against the direct change. (Answer: trend $+\$220$; deductible $-\$400$; net change $-\$180$.)
A hypothetical insurer expects 120 claims, with unchanged claim frequency. Each claim is $\$1{,}000$ or $\$3{,}000$ with equal probability. Severity trends by 20%, and the deductible changes from $\$400$ to $\$1{,}000$. Find the trend-first per-claim effects and the net change in expected total payment. (Answer: trend $+\$400$ per claim; deductible $-\$600$ per claim; net $-\$200$ per claim, or $-\$24{,}000$ in expected total payment.)

What You'll Learn