Why Coverage Bases Must Match
An experience rating comparison is meaningful only when its losses represent comparable insurance protection. If the historical policy had a lower limit or a larger deductible than the prospective policy, its recorded insurer payments may be lower even when the underlying losses and risk are unchanged. Treating that difference as improved experience can distort the rating result.
The task here is to align coverage terms: identify what was covered in the experience period, identify what will be covered prospectively, and express losses on one consistent basis. This is distinct from matching losses to earned exposure. It is also not always possible to recover an exact prospective-basis loss from a historical payment alone.
For the examples below, define \(X\) as ground-up loss, \(d\) as the per-loss deductible, and \(L\) as the maximum insurer payment per loss. The insurer payment is
Thus, a loss at or below the deductible produces no payment; a loss above the deductible produces payment up to \(L\). Here, \(L\) is the insurer-payment limit, not the total loss amount at which coverage ends. Policy wording may define limits differently, so the payment formula must reflect the actual contract before any adjustment is made.
Two Ways to Put Losses on a Common Basis
When ground-up losses are known, calculate each loss under the target terms. When only historical insurer payments are known, a distribution-based conversion may be appropriate, provided the severity model and relevant risk mix are suitable. These methods answer different data situations and should not be confused.
This factor is an expected-value adjustment, not a claim-by-claim reconstruction. It depends on the assumed severity distribution, coverage definitions, and risk mix. If those differ materially across segments, calculate conversion factors by segment and combine them using compatible exposure weights rather than applying one pooled factor indiscriminately.
Why the Expected-Payment Formula Holds
For each possible loss amount, payment can be viewed as accumulating one dollar for every layer of loss retained by the insurer: the layer begins above \(d\) and ends at \(d+L\). At a layer point \(x\) in that interval, the layer is reached exactly when \(X>x\). Its probability of being reached is \(S_X(x)\). Summing the expected contributions of all such layers gives the integral.
The identity also clarifies the limits of the method. The formula assumes the specified per-loss payment structure and a ground-up severity distribution suitable for the losses being adjusted. Aggregate limits, coinsurance, exclusions, and other terms may change the payment function; they should be represented explicitly rather than forced into this simple formula.
Worked Example: Convert a Portfolio to a Higher-Limit Basis
A hypothetical portfolio has ground-up loss severity \(X\) uniformly distributed from $0 to $100,000. Historical coverage has a $10,000 deductible and a $30,000 maximum insurer payment per loss. The prospective terms have a $20,000 deductible and a $50,000 maximum insurer payment. Historical aggregate insurer payments are $180,000. Estimate those payments on the prospective coverage basis, assuming the same severity distribution and mix.
Method. Because only the aggregate historical payments are given, use the expected-payment conversion factor. Work in thousands of dollars, so \(X\) is uniform from 0 to 100. Its survival function is \(S_X(x)=1-x/100\) for \(0\le x\le100\).
For the historical terms, the integration interval is from \(d_H=10\) to \(d_H+L_H=40\):
For the target terms, the interval is from \(d_T=20\) to \(d_T+L_T=70\):
The conversion factor is \(r=27.5/22.5=11/9=1.2222\ldots\). Therefore the estimated prospective-basis aggregate is
The result is $220,000. The factor is greater than one because, under the assumed severity model, expected payment per loss is greater on the target basis. Common mistake avoided: applying the ratio of limits, \(50/30\), would ignore both deductibles and the distribution of losses within the covered layers.
Reconstruct When Ground-Up Losses Are Available
If the ground-up amount for each claim is available, calculate the historical and target payments claim by claim. This is preferable to a portfolio conversion factor when the required data exist because it uses the actual observed losses instead of an assumed severity distribution.
Worked Example: Recalculate Individual Payments Under New Terms
A hypothetical account has ground-up losses of $3,000, $8,000, $20,000, $40,000, and $70,000. Historical coverage has a $5,000 deductible and a $25,000 maximum insurer payment per loss. Prospective coverage keeps the deductible but raises the maximum payment to $50,000. Find the total historical and prospective payments.
Method. Apply \(Y^L(d,L)=\min((X-d)_+,L)\) separately to each ground-up loss. Amounts below the $5,000 deductible pay zero.
| Ground-up loss | Historical payment: \(d=5{,}000,\ L=25{,}000\) | Prospective payment: \(d=5{,}000,\ L=50{,}000\) |
|---|---|---|
| $3,000 | \(\min(0,25{,}000)=\$0\) | \(\min(0,50{,}000)=\$0\) |
| $8,000 | \(\min(3{,}000,25{,}000)=\$3,000\) | \(\min(3{,}000,50{,}000)=\$3,000\) |
| $20,000 | \(\min(15{,}000,25{,}000)=\$15,000\) | \(\min(15{,}000,50{,}000)=\$15,000\) |
| $40,000 | \(\min(35{,}000,25{,}000)=\$25,000\) | \(\min(35{,}000,50{,}000)=\$35,000\) |
| $70,000 | \(\min(65{,}000,25{,}000)=\$25,000\) | \(\min(65{,}000,50{,}000)=\$50,000\) |
The historical total is \(0+3{,}000+15{,}000+25{,}000+25{,}000=\$68{,}000\). The prospective total is \(0+3{,}000+15{,}000+35{,}000+50{,}000=\$103{,}000\). The difference is $35,000, arising from the higher cap on the two losses that exceed the historical payment limit. Common mistake avoided: multiplying $68,000 by the ratio of limits would not reproduce the actual target payments. The individual ground-up amounts show exactly which claims are affected.
If the historical file contains only the capped payment of $25,000 for a claim, its ground-up loss could be $30,000, $70,000, or another amount above the cap threshold. The target payment cannot then be determined exactly from that payment alone. A distribution-based estimate may be possible, but it requires an explicit model and assumptions; it does not recover the missing ground-up amount.
Preserve Segment Mix in a Model-Based Adjustment
A single conversion factor can be misleading when severity differs across risk segments. The expected payment for a coverage layer depends on the severity distribution: a higher-limit layer may contribute much more for a segment with heavier losses than for one with lighter losses. Calculate historical and target expected payments using the relevant segment distributions, then weight them using consistent segment exposure proportions.
Worked Example: Weight Conversion Across Two Severity Segments
A hypothetical portfolio has two segments: 75% of its exposure is in Segment A and 25% in Segment B. In each segment, ground-up severity is exponential. Segment A has mean severity \(\theta_A=\$20,000\); Segment B has mean severity \(\theta_B=\$80,000\). Historical terms are \(d_H=0\) and \(L_H=\$20,000\). Target terms are \(d_T=0\) and \(L_T=\$40,000\). Historical aggregate payments are $1,000,000. Estimate the target-basis aggregate under the assumed segment mix.
Method. For an exponential severity with mean \(\theta\), \(S_X(x)=e^{-x/\theta}\), so
For Segment A, historical expected payment is \(20(1-e^{-1})=12.6424\) thousand dollars per exposure unit. Target expected payment is \(20(1-e^{-2})=17.2933\) thousand dollars per exposure unit.
For Segment B, historical expected payment is \(80(1-e^{-0.25})=17.6959\) thousand dollars per exposure unit. Target expected payment is \(80(1-e^{-0.5})=31.4776\) thousand dollars per exposure unit.
Weight each basis using the same exposure proportions:
Thus \(r=20.8394/13.9058\approx1.4986\), and the estimated target-basis aggregate is \(\$1{,}000{,}000(1.4986)\approx\$1{,}498{,}600\), to the nearest $100. The adjustment is not the simple average of the two segment factors: the segments have different historical expected payments, so the pooled conversion must reflect their weighted expected amounts on both bases. Common mistake avoided: using one segment's factor, or changing the segment weights between the two coverage bases without a reason, can confound coverage differences with a change in portfolio composition.
Assumptions, Checks, and Common Traps
Before applying a coverage adjustment, verify that the historical and target payment definitions are comparable. The adjustment should address the change in coverage terms, not silently absorb changes in the insured population, exposure mix, claim definition, or other risk characteristics. The model-based factor is an estimate of expected payment on a new basis; it is not a claim-specific fact.
Specify the deductible, what the limit caps, and any other relevant terms for the historical and target policies.
Use ground-up losses for exact claim-level recalculation. If only historical payments are available, determine whether a defensible severity model and compatible segment data support an expected-value adjustment.
Apply the target payment rule to ground-up losses, or calculate target-to-historical expected payment factors under stated assumptions. Preserve appropriate segment weights.
Ask whether the direction and size of the adjustment are plausible given the coverage change. State clearly whether the result is reconstructed or estimated.
- Do not scale by limits alone. Deductibles and the severity distribution affect how much loss falls within each layer.
- Do not apply a target deductible to an already-net historical payment as though it were ground-up loss. The historical payment has already been altered by its original coverage terms.
- Do not treat capped payments as exact ground-up losses. A payment at the cap only establishes that the relevant loss reached the cap threshold; it does not reveal how far beyond it the ground-up loss went.
- Do not conceal unsupported assumptions. If the severity model, segment mix, or policy wording is inadequate for conversion, report that the historical data do not identify a reliable target-basis amount.
Check Your Understanding
For each problem, state the method and keep the historical and target payment rules distinct.
- A hypothetical portfolio has \(X\) uniform from $0 to $50,000. Historical terms are a $5,000 deductible and a $20,000 insurer-payment limit; target terms are a $10,000 deductible and a $30,000 insurer-payment limit. Historical aggregate payments are $280,000. Find the expected-payment conversion factor and estimated target-basis aggregate. (Answer: factor \(15/14=1.0714\); target-basis aggregate $300,000.)
- Ground-up losses are $12,000, $38,000, and $90,000. Historical coverage has an $8,000 deductible and a $25,000 payment limit; target coverage has a $15,000 deductible and a $50,000 payment limit. Find both payment totals. (Answer: historical total $54,000; target total $73,000.)
- For exponential severity with mean \(\theta=\$30,000\), historical terms are \(d=0,L=\$30,000\) and target terms are \(d=\$10,000,L=\$20,000\). Historical aggregate payments are $200,000. Find the conversion factor and target-basis estimate. (Answer: factor approximately 0.5516; target-basis estimate approximately $110,312.)
- A portfolio has two segments with exposure weights 60% and 40%. Expected historical payments per exposure unit are $10 and $20; expected target payments per exposure unit are $12 and $15. Historical aggregate payments are $350,000. Calculate the pooled conversion factor and target-basis estimate. (Answer: factor \(13.2/14=0.9429\); target-basis estimate $330,000.)
- A historical file shows a claim payment exactly equal to the historical per-loss limit. Can the exact payment under a higher prospective limit be calculated from that amount alone? Explain what additional information or method would be needed. (Answer: No. The ground-up loss is not identified above the historical cap; obtain ground-up information or use a stated, suitable severity model for an expected estimate.)