Burning Cost Pricing from Historical Losses

Purpose of a Historical Burning Cost

A historical burning cost estimates the treaty recoveries associated with a portfolio’s past experience, expressed per unit of exposure. It is a direct way to turn historical losses into a cost indication for a reinsurance layer. The calculation has two essential parts: apply the contract terms to each relevant historical loss, then divide the resulting recoveries by the exposure that produced those losses.

The order matters. Dividing gross losses by exposure does not measure the cost of a reinsurance layer: losses below the attachment produce no recovery, and losses above the layer limit cannot produce additional recovery under that occurrence layer. The result is an experience-based indication, not a complete reinsurance premium. This method is aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.

Definition: A historical burning cost is the total recovery calculated under specified treaty terms for a selected historical experience period, divided by the corresponding exposure for that period. If $R_j$ is the treaty recovery for historical loss $j$, and $E$ is the matching exposure, the indicated burning cost is $\sum_j R_j/E$, stated in currency per exposure unit.

Applying Occurrence-Layer Terms

For an occurrence excess-of-loss layer with attachment $A$ and limit $L$, a covered loss $X$ produces the recovery

R(X)=\min\bigl((X-A)^+,L\bigr), \qquad (z)^+=\max(z,0).

The attachment is the amount retained before the layer begins to pay; the limit is the maximum recovery for one occurrence. Thus a loss at or below $A$ generates no recovery. A loss between $A$ and $A+L$ generates the amount above $A$, and a loss at or above $A+L$ exhausts the layer. This formula assumes that $X$ is the loss subject to the treaty, that the occurrence definition and coverage terms have already been applied, and that no other contract feature changes the recovery. If the contract includes aggregate limits, reinstatements, exclusions, or other conditions, apply those terms as well before summing recoveries.

For a historical record of $n$ covered losses, calculate each recovery under the treaty and sum them:

R_{\mathrm{total}}=\sum_{j=1}^{n}R(X_j), \qquad BC=\frac{R_{\mathrm{total}}}{E_{\mathrm{total}}}.

Here $E_{\mathrm{total}}$ must be expressed in the chosen exposure unit, such as insured-unit-years, policy-years, or dollars of payroll. The resulting burning cost has units of currency per exposure unit. The exposure base should correspond to the business and loss experience included in the numerator; the denominator is not automatically the number of claims or the number of policies.

Worked Example: Recoveries from a Per-Occurrence Layer

A hypothetical treaty covers a layer of $120,000 excess of $80,000 per occurrence, with no aggregate limit or reinstatement complication. The historical covered losses are $65,000, $110,000, $260,000, and $340,000. They arise from 1,500 matching insured-unit-years. Calculate the burning cost per insured-unit-year.

Method. Apply the occurrence-layer recovery function to each loss, sum the recoveries, and divide by the matching exposure. The calculation assumes all four amounts are covered losses subject to the treaty and that the stated layer terms apply unchanged.

The layer attaches at $80,000 and has a $120,000 limit. Its exhaustion point is $80{,}000+120{,}000=\$200{,}000$.

For the $65,000 loss, $65{,}000\leq80{,}000$, so the recovery is $0.

For the $110,000 loss, the amount above attachment is $110{,}000-80{,}000=\$30{,}000$. This is below the $120,000 limit, so the recovery is $30,000.

For the $260,000 loss, the amount above attachment is $260{,}000-80{,}000=\$180{,}000$. The layer pays only its $120,000 limit.

For the $340,000 loss, the amount above attachment is $340{,}000-80{,}000=\$260{,}000$, also capped at $120,000.

Total recovery is $0+30{,}000+120{,}000+120{,}000=\$270{,}000$. Therefore,

BC=\frac{\$270{,}000}{1{,}500\text{ insured-unit-years}} =\$180\text{ per insured-unit-year}.

Check. The two largest losses each exhaust the layer, so together they contribute $240,000; adding the $30,000 partial recovery gives the stated $270,000 total. A common mistake is to count the full amount above attachment for an exhausted loss. That would overstate recovery because each occurrence is capped at $120,000.

Pooling Years and Weighting by Exposure

When experience spans several years, calculate the recovery total and exposure total across all included years. If year $t$ has recovery $R_t$ and exposure $E_t$, the pooled burning cost is

BC_{\mathrm{pooled}} =\frac{\sum_t R_t}{\sum_t E_t}.

This is an exposure-weighted average of the annual burning costs $R_t/E_t$, not generally a simple average. A year with more exposure contributes more to the pooled indication because it represents more of the experience base.

Worked Example: Pooling Two Years Correctly

A hypothetical $100,000 excess of $50,000 occurrence layer has no annual aggregate limit. In Year 1, the matching exposure is 800 unit-years and the covered losses are $30,000, $70,000, and $180,000. In Year 2, exposure is 2,400 unit-years and covered losses are $55,000, $120,000, and $300,000. Calculate the pooled burning cost and compare it with the unweighted average of the annual rates.

Method. First calculate each year’s recoveries using the same layer terms. Then pool the dollar recoveries and exposure units. This avoids giving the two years equal weight when their exposures differ.

In Year 1, the $30,000 loss is below the $50,000 attachment and recovers $0. The $70,000 loss recovers $70{,}000-50{,}000=\$20{,}000$. The $180,000 loss has $130,000 above attachment, capped at the $100,000 limit. Year 1 recovery is $0+20{,}000+100{,}000=\$120{,}000$, giving an annual rate of $120{,}000/800=\$150$ per unit-year.

In Year 2, the $55,000 loss recovers $5,000, the $120,000 loss recovers $70,000, and the $300,000 loss recovers the $100,000 limit. Year 2 recovery is $5{,}000+70{,}000+100{,}000=\$175{,}000$, giving an annual rate of $175{,}000/2{,}400=\$72.9167$ per unit-year.

Across both years, recovery is $\$120{,}000+\$175{,}000=\$295{,}000$, and exposure is $800+2{,}400=3{,}200$ unit-years. Thus,

BC_{\mathrm{pooled}} =\frac{\$295{,}000}{3{,}200} =\$92.1875\text{ per unit-year} \approx \$92.19\text{ per unit-year}.

The exposure-weighted annual-rate calculation gives the same result: \[ \frac{800(150)+2{,}400(72.9167)}{3{,}200} \approx \$92.19. \] The unweighted average is $(150+72.9167)/2\approx\$111.46$, which is not the pooled rate. Check. The pooled amount divided by pooled exposure gives the same value as the weighted calculation. The common mistake is to average annual rates without accounting for the amount of exposure in each year.

Choosing and Matching the Exposure Base

The exposure base is the denominator that makes the indication interpretable and links it to the experience being measured. Common choices include policy-years, insured-unit-years, vehicle-years, payroll, or another portfolio measure appropriate to the covered business. State both the unit and the period. For example, “$180 per insured-unit-year” is more informative than “180 per unit.”

If insured items have different periods of coverage, convert them to a consistent time basis before calculating the denominator. Forty insured units each covered for half a year represent $40(0.5)=20$ insured-unit-years, not 40 full unit-years. In general, exposure for item $k$ covered for $t_k$ years is $t_k$ units of annual exposure; total exposure is the sum over items. The numerator must include losses associated with the same portfolio and period.

The quotient also has a useful justification. Suppose each unit of exposure has the same expected treaty recovery rate $r$, and the historical exposure amount $E_{\mathrm{total}}$ is fixed. The expected total recovery is then $rE_{\mathrm{total}}$. Dividing by exposure gives

\frac{E[R_{\mathrm{total}}]}{E_{\mathrm{total}}} =\frac{rE_{\mathrm{total}}}{E_{\mathrm{total}}} =r.

This explains why recoveries per exposure unit estimate a cost rate under a stable, comparable exposure mix. It does not guarantee that a finite historical estimate equals the future rate. Changes in risk mix, coverage, claim reporting, or loss experience can make the historical portfolio a poor guide to the prospective one.

Worked Example: Converting Partial-Year Exposure

A hypothetical $200,000 excess of $75,000 occurrence layer has no aggregate limit. The covered losses during the experience period are $50,000, $100,000, $190,000, and $350,000. The portfolio contains 40 insured risks covered for six months each and 30 risks covered for a full year. Calculate the burning cost per insured-risk-year.

Method. Apply the per-occurrence layer to each covered loss, then convert the exposure to insured-risk-years before dividing. The common error to avoid is treating every risk as a full year of exposure merely because it appears in the policy count.

The layer attaches at $75,000 and is exhausted at $75{,}000+200{,}000=\$275{,}000$. The $50,000 loss recovers $0. The $100,000 loss recovers $100{,}000-75{,}000=\$25{,}000$. The $190,000 loss recovers $190{,}000-75{,}000=\$115{,}000$. The $350,000 loss exceeds the exhaustion point, so it recovers the $200,000 limit.

Total recovery is $0+25{,}000+115{,}000+200{,}000=\$340{,}000$. The partial-year risks contribute $40(0.5)=20$ insured-risk-years, while the full-year risks contribute $30(1)=30$ insured-risk-years. Total exposure is $20+30=50$ insured-risk-years. Therefore,

BC=\frac{\$340{,}000}{50\text{ insured-risk-years}} =\$6{,}800\text{ per insured-risk-year}.

Check. The 40 half-year risks contribute 20, not 40, exposure-years; adding the 30 full-year risks gives the stated denominator of 50. Dividing by the count of 70 risks would incorrectly treat partial-year and full-year coverage as equivalent.

Assumptions, Interpretation, and Common Traps

A historical burning cost is most useful when the historical losses and exposure represent the treaty being considered. Before interpreting it, verify the following:

Consistent loss basis: losses have been selected on a consistent basis, and the same coverage and occurrence definitions are used throughout.
Complete treaty application: attachment, limit, exclusions, and any relevant aggregate or reinstatement conditions are reflected in the recoveries.
Matched exposure: the denominator covers the same business and period as the losses in the numerator.
Comparable experience: historical and prospective exposure mix and coverage are sufficiently similar for the past rate to be informative.
Consistent loss valuation: the losses are measured on a consistent basis, such as paid or ultimate amounts. Claim development and changes in the level of losses may affect the indication; no adjustment for those effects is made in the basic calculation here.

The burning cost is a historical loss cost, not automatically the price charged for the treaty. It does not, by itself, include brokerage, expenses, profit, or a load for uncertainty. Nor is it the same as rate-on-line, which compares a premium with the layer limit. Keep the calculation in its stated units, retain cents where appropriate, and round only at the final presentation step.

Key takeaway: Apply the treaty to each historical loss before aggregating, and divide the resulting recoveries by the exposure that generated those losses. For multiple years, pool recoveries and exposure—or equivalently use an exposure-weighted average of annual rates.

Check Your Understanding

For each problem, calculate the requested historical burning cost and keep the monetary and exposure units consistent.

A hypothetical $80,000 excess of $40,000 occurrence layer has no aggregate limit. Covered losses are $20,000, $55,000, and $130,000, with 500 matching unit-years of exposure. Calculate the burning cost per unit-year. (Answer: $190.00 per unit-year)
A hypothetical $60,000 excess of $30,000 occurrence layer has no aggregate limit. Year A has 400 unit-years and losses of $30,000, $70,000, and $120,000. Year B has 600 unit-years and losses of $50,000 and $90,000. Calculate the pooled burning cost per unit-year. (Answer: $180.00 per unit-year)
A hypothetical $150,000 excess of $100,000 occurrence layer has no aggregate limit. Covered losses are $90,000, $180,000, and $400,000, with 700 matching unit-years. Calculate the burning cost per unit-year, rounded to the nearest cent. (Answer: $328.57 per unit-year)
A hypothetical $50,000 excess of $25,000 occurrence layer has no aggregate limit. The covered losses are $40,000, $90,000, and $100,000. The portfolio has 24 risks covered for six months each and 16 risks covered for a full year. Calculate the burning cost per insured-risk-year. (Answer: $4,107.14 per insured-risk-year)
A hypothetical $100,000 excess of $50,000 occurrence layer has no aggregate limit. The covered losses are $20,000, $80,000, and $200,000; a separate $500,000 loss is explicitly excluded from the treaty. Matching exposure is 1,200 unit-years. Calculate the burning cost per unit-year. (Answer: $108.33 per unit-year)

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