Burning-Cost Calculations From Historical Losses

What a Historical Burning Cost Measures

A burning-cost calculation uses historical claims to measure the loss produced by a specified layer of insurance or reinsurance. For each claim, the analyst calculates the amount payable by the layer, adds those payments over the experience period, and divides by an exposure measure for that same period. The result is an observed layer loss cost per unit of exposure, not automatically a final rate.

The layer’s terms and the denominator must be explicit. A layer might be described as $50,000 excess of $100,000, meaning it attaches at $100,000 and has a $50,000 limit per occurrence. The burning cost can then be expressed per building-year, per vehicle-year, or as a ratio to subject premium, depending on the exposure measure supplied. These measures have different units and should not be compared without conversion.

Definition: For a per-occurrence layer attaching at deductible $d$ with layer limit $u$, the payment on a ground-up claim $X$ is $Y^L=\min(\max(X-d,0),u)$. For $n$ historical claims, the observed aggregate layer loss is $\sum_{j=1}^{n}Y^L_j$. The observed burning cost is that aggregate divided by the matching exposure measure.

Here, $d$ is the attachment point and $u$ is the amount of insurance available in the layer, not the upper attachment point. The layer’s upper attachment is $d+u$. A claim below attachment produces no layer payment; a claim above the upper attachment produces the full layer limit. The formula applies to the stated per-occurrence indemnity layer. It does not, by itself, calculate an annual aggregate layer, reinstatement premium, or other treaty feature that changes payment across occurrences.

A historical calculation is empirical: it does not require a parametric claim distribution. It does require claims and exposure that correspond to the same experience period and coverage basis. Unless adjustments are specifically part of the calculation, the result is unadjusted historical experience. In particular, development, trend, and portfolio changes are not silently included.

Calculation Procedure

The essential discipline is to apply the payment function to each occurrence before summing. Do not first add ground-up claims and then apply the per-occurrence layer. The latter would treat several separate occurrences as one claim and generally produce a different result.

Specify the layer.
Record the attachment $d$, the layer limit $u$, and whether the terms apply per occurrence. Convert all claim amounts and layer amounts to the same currency and units.

Calculate each payment.
For each occurrence, subtract the attachment only when the claim exceeds it, then cap the result at the layer limit: $Y^L_j=\min(\max(X_j-d,0),u)$.

Aggregate payments.
Add the calculated layer payments across all occurrences in the experience period. Claims with no layer payment contribute zero, but the exposure period still includes the corresponding covered exposure.

Divide by consistent exposure.
Use exposure for the same period and business basis. State the resulting units, such as dollars per vehicle-year or layer loss as a fraction of subject premium.

Worked Calculations

Worked Example: Per-Occurrence Payments and Premium-Based Burning Cost

A hypothetical insurer reviews five ground-up claims, measured in thousands of dollars: 20, 65, 90, 180, and 240. The layer is $100,000 excess of $50,000 per occurrence, and the matching subject premium for the experience period is $1,700,000. Calculate the observed burning cost as a percentage of subject premium. No interest rate or mortality assumption is relevant: there is no discounting or life-contingent exposure. Use the observed claims directly; assume no parametric loss distribution.

Method. The attachment is $d=50$ thousand dollars and the layer limit is $u=100$ thousand dollars. Thus, each payment, in thousands, is $\min(\max(X-50,0),100)$.

The five claim payments are:

Claim 20: $\min(\max(20-50,0),100)=\min(0,100)=0$.
Claim 65: $\min(\max(65-50,0),100)=\min(15,100)=15$.
Claim 90: $\min(\max(90-50,0),100)=\min(40,100)=40$.
Claim 180: $\min(\max(180-50,0),100)=\min(130,100)=100$.
Claim 240: $\min(\max(240-50,0),100)=\min(190,100)=100$.

The aggregate layer loss is $0+15+40+100+100=255$ thousand dollars, or $255,000. Divide by the matching subject premium:

\text{Burning cost} =\frac{\$255{,}000}{\$1{,}700{,}000} =0.15 =15\%.

The observed layer loss is therefore $15 per $100 of subject premium. As a check, the result is less than the premium denominator, and both numerator and denominator cover the same experience period. The common mistake avoided here is treating $u=100$ as the upper attachment: the upper attachment is $d+u=150$ thousand dollars.

Worked Example: Burning Cost Per Exposure Unit Across Two Years

A hypothetical property program has a per-occurrence layer attaching at $100,000 with a $250,000 limit. Year 1 has 12 insured building-years and claims of $80,000, $140,000, $310,000, and $520,000. Year 2 has 18 insured building-years and claims of $210,000, $355,000, and $90,000. Calculate the combined observed burning cost per building-year. No interest rate, mortality assumption, or parametric loss distribution is used; the calculation is based on the listed historical claims.

Method. Here $d=100$ thousand dollars and $u=250$ thousand dollars, so the upper attachment is $100+250=350$ thousand dollars. Apply the payment function to each claim.

For Year 1, the payments, in thousands, are:

$80,000: $\min(\max(80-100,0),250)=0$.
$140,000: $\min(\max(140-100,0),250)=40$.
$310,000: $\min(\max(310-100,0),250)=210$.
$520,000: $\min(\max(520-100,0),250)=250$, capped at the layer limit.

Year 1 aggregate layer loss is $0+40+210+250=500$ thousand dollars.

For Year 2, the payments, in thousands, are:

$210,000: $\min(\max(210-100,0),250)=110$.
$355,000: $\min(\max(355-100,0),250)=250$.
$90,000: $\min(\max(90-100,0),250)=0$.

Year 2 aggregate layer loss is $110+250+0=360$ thousand dollars. The total is $500+360=860$ thousand dollars, and total exposure is $12+18=30$ building-years. Therefore:

\text{Burning cost} =\frac{\$860{,}000}{30\text{ building-years}} =\$28{,}666.67\text{ per building-year}.

The displayed result is rounded to the nearest cent. It is also $860/30=28.6667$ thousand dollars per building-year before rounding. The check is that the combined denominator is 30 building-years, not an unweighted average of two annual burning costs. The common mistake avoided is dividing the combined losses by only one year’s exposure.

Worked Example: Avoiding an Attachment-versus-Limit Error

A hypothetical equipment portfolio has four ground-up claims, in thousands of dollars: 150, 275, 500, and 900. The per-occurrence layer attaches at $200,000 and has a $400,000 limit. Subject premium for the same period is $2,500,000. Calculate the burning cost as a percentage of premium and identify the error in using $\min(X,u)-d$. No interest rate, mortality assumption, or parametric distribution is needed; use the observed claims as given.

Method. The payment function is $\min(\max(X-200,0),400)$, with claim amounts in thousands. The upper attachment is $200+400=600$ thousand dollars.

Claim 150: $\min(\max(150-200,0),400)=0$.
Claim 275: $\min(\max(275-200,0),400)=75$.
Claim 500: $\min(\max(500-200,0),400)=300$.
Claim 900: $\min(\max(900-200,0),400)=400$.

The aggregate is $0+75+300+400=775$ thousand dollars, or $775,000. The burning cost is:

\frac{\$775{,}000}{\$2{,}500{,}000} =0.31 =31\%,

equivalent to $31 per $100 of subject premium. The common wrong calculation uses $\min(X,u)-d$. For the $500,000 claim, it gives $\min(500,400)-200=200$ thousand dollars, but the correct payment is $\min(500-200,400)=300$ thousand dollars. The incorrect expression caps the claim before subtracting the attachment, so it confuses the layer limit with an upper attachment. The correct result also passes a cap check: no occurrence can pay more than $400,000.

Why the Ratio Is an Empirical Loss Cost

Suppose a period has exposure $E$ measured in a chosen unit and observed layer payments $Y^L_1,\ldots,Y^L_n$. The total observed payment is $\sum_{j=1}^{n}Y^L_j$. Dividing by $E$ allocates that total across the exposure that generated the experience:

\widehat{c} =\frac{\sum_{j=1}^{n}Y^L_j}{E}.

This ratio is an empirical cost per exposure unit because multiplying it by the observed exposure returns the aggregate layer loss:

\widehat{c}E =\left(\frac{\sum_{j=1}^{n}Y^L_j}{E}\right)E =\sum_{j=1}^{n}Y^L_j.

That identity explains why the denominator’s definition matters. If $E$ is earned premium, $\widehat{c}$ is a loss ratio for the layer relative to that premium. If $E$ is a count of insured units over time, $\widehat{c}$ is a loss cost per unit-time. A percentage and a dollar-per-unit measure are not interchangeable, even when both are informally called burning costs.

The calculation describes the experience as observed. It is not, by itself, a forecast of the next period’s loss cost or a complete premium indication. For the observed ratio to inform a pricing decision, the analyst must consider whether the historical claims, coverage, exposure, and layer terms are appropriate for the intended application. Development and trend adjustments, portfolio changes, and credibility treatment require additional methods; they are not included in the unadjusted calculation here.

Common Traps and Checks

Applying the layer to an aggregate of claims. For a per-occurrence layer, calculate one payment per claim, then sum. Aggregating ground-up claims first changes the coverage calculation.
Misreading the limit. A layer attaching at $d$ with limit $u$ reaches its upper attachment at $d+u$. The maximum payment on one occurrence is $u$.
Using mismatched units. Convert claims and layer terms to the same currency and scale before applying the payment function. A claim listed in thousands cannot be compared directly with an attachment stated in dollars.
Using an inconsistent denominator. Match the exposure period to the losses, and state whether the denominator is premium, policies, vehicle-years, building-years, or another measure.
Omitting zero-payment claims from the coverage logic. A claim below attachment contributes zero to the numerator. Its existence does not mean the exposure period should be reduced.
Calling an unadjusted historical ratio a prospective rate. The basic calculation records historical layer experience; it does not automatically account for changes in claim values, exposure, or portfolio composition.

Key takeaway: A burning-cost calculation has three linked parts: the correct per-occurrence layer payment, the aggregate of those payments over a defined experience period, and an exposure denominator measured on a consistent basis. Check all three before interpreting the result.

Check Your Understanding

For each problem, apply the stated per-occurrence layer to individual claims before aggregating. Amounts stated in thousands are in thousands of dollars.

A layer attaches at $40,000 and has a $80,000 limit. Claims are $25,000, $65,000, and $130,000; exposure is 7 fleet-years. Find the burning cost per fleet-year. (Answer: $15,000 per fleet-year.)
A layer attaches at $100,000 and has a $150,000 limit. Claims are $90,000, $160,000, $240,000, and $410,000. Matching subject premium is $1,400,000. Find the burning cost as a percentage of premium. (Answer: 25%, or $25 per $100 of premium.)
A layer attaches at $200,000 and has a $300,000 limit. Claims are $150,000, $260,000, $480,000, and $900,000. There are 16,000 vehicle-years of exposure. Find the burning cost per vehicle-year. (Answer: $40 per vehicle-year.)
A layer attaches at $25,000 and has a $50,000 limit. Across two years, claims are $30,000, $75,000, $110,000, and $200,000. Subject premium is $500,000 in Year 1 and $750,000 in Year 2. Find the combined burning cost as a percentage of premium. (Answer: 12.4%, or $12.40 per $100 of premium.)
A layer attaches at $80,000 and has a $120,000 limit. Year 1 has claims of $70,000, $130,000, and $240,000 with 10 exposure units. Year 2 has claims of $100,000 and $180,000 with 15 exposure units. Find the combined burning cost per exposure unit. (Answer: $11,600 per exposure unit.)

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