Common Errors In Excess And Reinsurance Pricing

Why Small Definition Errors Matter

Excess and reinsurance pricing calculations can be arithmetically correct and still produce a wrong indication. The source is often a mismatch between the contract and the calculation: the attachment is mistaken for the limit, allocated loss adjustment expense is included when the contract excludes it, or an adjusted loss total is divided by an exposure denominator that measures something different.

A reliable calculation starts by stating the contract terms and the basis of every input. For historical pricing, that means identifying how losses have been adjusted and what exposure measure is being used. For a premium indication, it means separating fixed expenses from expenses expressed as a percentage of premium. Reinstatement calculations also require the exact amount and time provisions in the contract.

Definition: A layer written as $L$ excess of $A$ has attachment point $A$ and limit $L$. For a covered loss amount $X$, the recovery is the amount of loss above $A$, capped at $L$: $Y=\min(\max(X-A,0),L)$. The contract defines which components of loss belong in $X$.

Translate the Contract Before Calculating

For a layer with attachment $A$ and limit $L$, the recovery is zero when the covered loss does not exceed $A$, increases dollar for dollar above $A$, and cannot exceed $L$. Its maximum covered loss is $A+L$. These points provide a quick check on a proposed recovery.

Y = \begin{cases} 0, & X \le A,\\ X-A, & A < X < A+L,\\ L, & X \ge A+L. \end{cases}

This piecewise expression gives the standard formula directly. Below the attachment there is no loss in the layer. Between the attachment and the exhaustion point, the layer takes the portion $X-A$. At or above $A+L$, that portion reaches the limit, so recovery is capped at $L$. The formula applies only when $X$ is already measured on the loss basis specified by the contract.

For example, a contract might include allocated loss adjustment expense (ALAE) in covered loss, exclude it, or provide for it separately. A calculation must follow that wording rather than assume that all reported loss amounts are treated alike. Likewise, when an inuring recovery reduces the loss entering a layer, apply the specified inuring order before calculating the layer recovery; do not silently use gross loss in place of the contractually defined net loss.

Worked Example: Attachment, Limit, and ALAE Basis

A hypothetical treaty covers $2,000,000 excess of $3,000,000. It includes both indemnity and ALAE in covered loss, subject to the same attachment and limit. A claim has $3,450,000 of indemnity and $250,000 of ALAE. Calculate the recovery and identify two plausible errors.

Method. First calculate covered loss on the treaty’s stated basis:

X = \$3{,}450{,}000+\$250{,}000=\$3{,}700{,}000.

The attachment is $3,000,000 and the limit is $2,000,000. The loss above the attachment is $ \$3{,}700{,}000-\$3{,}000{,}000=\$700{,}000 $, which is below the limit. Therefore:

Y=\min(\$700{,}000,\$2{,}000{,}000)=\$700{,}000.

Error 1: treating the limit as the attachment. The incorrect calculation $\min(\$3{,}700{,}000-\$2{,}000{,}000,\$3{,}000{,}000)=\$1{,}700{,}000$ reverses the contract terms. The corrected recovery is $700,000.

Error 2: omitting included ALAE. Using indemnity alone gives $\$3{,}450{,}000-\$3{,}000{,}000=\$450{,}000$. That would be the recovery if ALAE were excluded, but it is not correct under the stated terms. The stated recovery is $700,000. Both answers are below the $2,000,000 limit, so neither is capped.

Check. Covered loss is above the $3,000,000 attachment, and the calculated $700,000 recovery is between zero and the $2,000,000 limit. The calculation avoids both a term reversal and an inconsistent loss basis.

Historical Adjustments and Exposure Denominators

A burning-cost indication relates adjusted historical layer losses to a compatible measure of exposure. Each loss adjustment should be applied to the loss it is intended to adjust. The denominator should represent the exposure corresponding to those same historical observations. If the adjusted losses are combined across years, the corresponding exposures should also be combined before calculating the rate.

The reason is that the aggregate rate is an exposure-weighted average of the individual-year rates, not generally their simple arithmetic average. If year $j$ has adjusted loss $L_j$ and exposure $E_j$, then:

\frac{\sum_j L_j}{\sum_j E_j} = \sum_j \left(\frac{E_j}{\sum_k E_k}\right)\left(\frac{L_j}{E_j}\right).

The weights are the shares of total exposure and sum to one. A simple average of annual rates gives equal weight to each year, even if one year has much less exposure than another. That is usually not the rate represented by total adjusted losses divided by total exposure.

Worked Example: Adjusted Losses and a Consistent Denominator

A hypothetical treaty has three historical years of layer losses. For year 1, the reported layer loss is $100,000, the development factor is 1.20, the trend factor is 1.10, and exposure is 80 units. For year 2, the corresponding values are $120,000, 1.10, 1.00, and 100 units. For year 3, they are $150,000, 1.00, 1.00, and 120 units. Calculate the aggregate adjusted loss cost per exposure unit and the indicated loss cost for a current portfolio with 150 units.

Method. Apply the stated development and trend factors to each reported layer loss:

\begin{aligned} L_1 &= \$100{,}000(1.20)(1.10)=\$132{,}000,\\ L_2 &= \$120{,}000(1.10)(1.00)=\$132{,}000,\\ L_3 &= \$150{,}000(1.00)(1.00)=\$150{,}000. \end{aligned}

The adjusted loss total is $\$132{,}000+\$132{,}000+\$150{,}000=\$414{,}000$. The matching exposure total is $80+100+120=300$ units. Thus the aggregate adjusted loss cost is $\$414{,}000/300=\$1{,}380$ per unit. Applied to 150 current units, the indicated loss cost is $150(\$1{,}380)=\$207{,}000$.

Common mistake. The individual-year rates are $132,000/80 = $1,650, $132,000/100 = $1,320, and $150,000/120 = $1,250 per unit. Their simple mean is $1,406.67, which would imply $211,000 for 150 units. That calculation gives each year equal weight. The aggregate method correctly weights each annual rate by that year’s share of the 300 total exposure units.

Check. The exposure-weighted rate is $(80/300)(\$1{,}650)+(100/300)(\$1{,}320)+(120/300)(\$1{,}250)=\$1{,}380$ per unit. It agrees with the aggregate calculation.

Adjustments also need to match the loss measure. A factor intended to develop reported historical layer losses should not be applied a second time to losses that are already developed. A trend factor should correspond to the intended cost level and time period. When historical policy terms differ from current terms, the analyst must determine whether the losses have been restated to a consistent layer basis; simply trending an amount does not correct a difference in attachment or limit.

Gross Up Premium-Based Expenses

A premium indication can include expected loss cost, fixed expenses, and variable provisions stated as percentages of premium. The key distinction is that a premium-based expense changes when the premium changes. It is therefore not generally correct to calculate that expense as a percentage of the loss cost and add it once.

If expected losses are $L$, fixed expenses are $F$, and the combined premium-based expense and margin rate is $v$, the indicated premium $P$ satisfies $P=L+F+vP$. Rearranging gives the gross-up formula, provided $v<1$:

P=\frac{L+F}{1-v}.

The condition $v<1$ is necessary for this finite positive solution when $L+F>0$. This formula assumes the stated percentages are all percentages of the same premium base. If a contract or rating procedure defines an expense on a different base, use that definition rather than this formula unchanged.

Worked Example: Grossing Up a Layer Premium

A hypothetical excess layer has an indicated expected loss cost of $240,000 and fixed expenses of $12,000. Brokerage and other variable expenses total 8% of premium, and a margin provision is 5% of premium. Calculate the premium before rounding and identify the error in a common shortcut.

Method. The total premium-based proportion is $v=0.08+0.05=0.13$. The amount required before those premium-based provisions is $\$240{,}000+\$12{,}000=\$252{,}000$. Therefore:

P=\frac{\$252{,}000}{1-0.13} =\frac{\$252{,}000}{0.87} =\$289{,}655.1724\ldots

Rounded to the nearest cent, the indicated premium is $289,655.17. The variable expenses are $0.13(\$289{,}655.1724\ldots)=\$37{,}655.1724\ldots$. Adding those to $252,000 gives $289,655.1724…, which checks the result.

Common mistake. Applying 13% to $252,000 and adding it once gives $\$252{,}000+0.13(\$252{,}000)=\$284{,}760$. But 13% of that proposed premium is only $37,018.80, so the total of loss, fixed expense, and premium-based provisions would be $289,018.80—not $284,760. The shortcut underfunds the required premium because the provisions themselves are based on premium.

Check. The gross-up premium is greater than the $252,000 amount needed for losses and fixed expenses, and its 13% variable provisions leave the required $252,000 balance.

Reinstatement Calculations Depend on the Stated Basis

A reinstatement restores some or all of an exhausted treaty limit, subject to the wording and any reinstatement count. A reinstatement premium may be calculated pro rata as to the amount of limit restored, the time remaining, or both. The pricing method must reflect the actual clause. The original premium alone does not determine the reinstatement premium.

Worked Example: Amount and Time Pro Rata Reinstatement

A hypothetical treaty has one paid reinstatement. Its original annual premium is $120,000. The clause charges reinstatement premium pro rata as to both amount and time, with time measured by the fraction of the treaty year remaining when the restored limit is used. A covered event exhausts the original $1,000,000 limit exactly halfway through the year, and the full limit is reinstated. Calculate the reinstatement premium.

Method. The amount restored is the full limit, so the amount proportion is $ \$1{,}000{,}000/\$1{,}000{,}000=1$. Half of the treaty year remains, so the time proportion is $6/12=0.5$. The premium is:

\$120{,}000(1)(0.5)=\$60{,}000.

The reinstatement premium is $60,000. Charging $120,000 would incorrectly omit the time-pro-rata provision. Conversely, charging $30,000 would incorrectly apply a half-limit amount proportion even though the full limit was restored.

Check. The treaty restores 100% of its limit for 50% of the year, so the charge is 50% of the original annual premium under the stated two-part pro-rata basis.

For multiple events, check whether the contract permits further reinstatements and how much limit remains available after each recovery. Do not assume that every loss creates a new full limit. The calculation must respect the treaty’s aggregate limit, reinstatement count, and premium basis.

A Practical Error-Checking Sequence

Before relying on an excess or reinsurance indication, check the calculation in the following order. This sequence is useful because an error in the contract interpretation can make later arithmetic irrelevant.

Write the contract terms.
Identify attachment, limit, aggregate constraints, reinstatements, and the stated loss adjustment expense basis.

Define the loss entering the layer.
Determine whether the calculation uses gross or net loss and which loss components are included. Apply any specified inuring recoveries in the required order.

Reconcile loss adjustments and exposure.
Apply development, trend, or other stated adjustments consistently, then use the exposure measure corresponding to those adjusted losses.

Separate premium bases.
Distinguish fixed-dollar expenses from provisions stated as percentages of premium, and gross up the latter when required.

Check contractual caps and proportions.
Confirm that recoveries do not exceed the available limit and that reinstatement charges follow the stated amount and time bases.

These checks are especially useful when a result seems plausible: a mistake in a layer definition or denominator can produce a reasonable-looking number. A useful final review asks whether the result obeys the contract limits and whether every numerator and denominator refers to the same basis. These pricing checks are aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.

Check Your Understanding

For each problem, identify the relevant contract or pricing basis before calculating.

A hypothetical layer is $1,500,000 excess of $2,000,000. Covered loss, including all expenses specified by the contract, is $2,350,000. What is the recovery? (Answer: $350,000.)
Three years of adjusted layer losses are $90,000, $150,000, and $250,000. Their matching exposures are 50, 75, and 125 units. Find the aggregate loss cost per unit and the indicated loss cost for 130 current units. (Answer: $1,960 per unit; $254,800 for 130 units.)
A hypothetical layer has $300,000 expected loss cost and $15,000 fixed expenses. Premium-based expenses are 10% and the premium-based margin is 4%. Find the indicated premium to the nearest cent. (Answer: $366,279.07.)
An annual treaty premium is $80,000. A reinstatement clause charges pro rata as to amount and time. The restored limit is 50% of the original limit, and 3 months remain in the 12-month treaty year. Find the reinstatement premium. (Answer: $10,000.)
A hypothetical layer is $1,000,000 excess of $500,000. Indemnity is $1,200,000 and ALAE is $100,000. Calculate the recovery if ALAE is included in covered loss, then if it is excluded. (Answer: $800,000 if included; $700,000 if excluded.)

What You'll Learn