Why Allocation Is a Separate Calculation
When a policy-period aggregate cap applies to several claims, two questions must be answered. First, what is the total payment for all claims combined? Second, how is that total assigned to each claim? The aggregate cap answers the first question, but does not by itself answer the second.
For example, suppose the claim amounts eligible for payment before the aggregate cap are $5,000 and $3,000, and the cap is $6,000. The total payment is $6,000. That total could be allocated in different ways if the contract does not specify a rule. Under a pro-rata rule, each claim receives the same percentage of its eligible amount. Under a claim-priority rule, payments could instead be made in a specified order. The allocation rule matters because claim-level payments are not determined by the aggregate cap alone.
This tutorial focuses on identifying the total after the cap and applying an explicit allocation rule to obtain the individual claim payments. The scenarios and values are hypothetical. The methods are aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.
Pro-Rata Allocation
A common allocation rule is pro rata to the otherwise-payable amounts. Each claim receives the same fraction of its otherwise-payable amount. If the total otherwise payable is positive, the fraction is the capped total divided by the otherwise-payable total.
Here \(P_j\) is the allocated payment for claim \(j\). When the cap does not bind, each claim receives its otherwise-payable amount. When the cap binds, the common factor \(C/B\) is less than one, so every otherwise-payable amount is reduced by the same proportion.
This formula assumes that the contract or stated problem uses pro-rata allocation. It is not a universal consequence of having an aggregate cap. If a policy instead specifies claim priority, a different allocation rule must be followed. If no rule is stated, one should not claim that the aggregate cap alone uniquely determines each claim’s payment.
Why the Pro-Rata Formula Works
The rule is designed to distribute the capped total in proportion to the amounts otherwise payable. When the cap binds, the proportion assigned to each claim is \(C/B\). Adding those allocations gives:
Thus the allocations use exactly the aggregate cap. Also, for any two claims with positive otherwise-payable amounts, \(P_j/P_k=B_j/B_k\): their relative sizes are preserved. If the cap does not bind, \(P_j=B_j\) for every claim, so the allocations sum to \(B\). In either case, the sum of the individual allocations equals \(T=\min(B,C)\).
For each claim, apply the stated claim-level terms to find its amount otherwise payable before the aggregate cap.
Add the otherwise-payable amounts to get \(B\).
Calculate \(T=\min(B,C)\), and determine whether the cap binds.
For pro rata, multiply every otherwise-payable amount by \(T/B\) when \(B>0\).
Confirm that no claim receives more than its otherwise-payable amount and that the allocations sum to the total after the cap.
Worked Examples
Worked Example: Apply Claim Terms, Then Allocate Pro Rata
A hypothetical policy applies a $1,000 deductible and a $5,000 maximum payment per claim, followed by a $7,000 aggregate cap for the policy period. Covered losses are $8,000, $6,000, and $5,000. The contract specifies pro-rata allocation if the aggregate cap binds. Find the otherwise-payable amount and the final allocated payment for each claim.
Method. First calculate each amount otherwise payable under the per-claim terms. For a covered loss \(X_j\), the amount is the loss above the deductible, subject to the maximum payment: \(\min(\max(X_j-1{,}000,0),5{,}000)\).
For the $8,000 loss, the amount above the deductible is \(8{,}000-1{,}000=7{,}000\). Applying the $5,000 maximum gives \(B_1=\min(7{,}000,5{,}000)=\$5{,}000\).
For the $6,000 loss, the amount above the deductible is \(6{,}000-1{,}000=5{,}000\), so \(B_2=\min(5{,}000,5{,}000)=\$5{,}000\).
For the $5,000 loss, the amount above the deductible is \(5{,}000-1{,}000=4{,}000\), below the maximum. Thus \(B_3=\min(4{,}000,5{,}000)=\$4{,}000\).
The otherwise-payable total is \(B=5{,}000+5{,}000+4{,}000=\$14{,}000\). Since \(\$14{,}000>\$7{,}000\), the aggregate cap binds. The pro-rata factor is \(C/B=7{,}000/14{,}000=0.5\).
The allocated payments are \(P_1=5{,}000(0.5)=\$2{,}500\), \(P_2=5{,}000(0.5)=\$2{,}500\), and \(P_3=4{,}000(0.5)=\$2{,}000\).
Check and common mistake avoided. The allocations sum to \(2{,}500+2{,}500+2{,}000=\$7{,}000\), exactly the cap, and none exceeds its otherwise-payable amount. The common mistake is to stop at the otherwise-payable amounts or to allocate an incorrect total. The total after the cap is $7,000, and the pro-rata factor is \(7{,}000/14{,}000=0.5\).
Worked Example: Find the Pro-Rata Factor From Unequal Claim Amounts
A hypothetical insurer has three covered claims with otherwise-payable amounts of $12,500, $7,500, and $5,000 after all claim-level terms. Its policy-period aggregate cap is $15,000, and the contract requires pro-rata allocation. Find each final payment.
Method. Add the otherwise-payable amounts: \(B=12{,}500+7{,}500+5{,}000=\$25{,}000\). The cap binds because \(\$25{,}000>\$15{,}000\), so the total after the cap is \(T=\$15{,}000\).
The common pro-rata factor is \(15{,}000/25{,}000=0.6\). Therefore, the first claim receives \(12{,}500(0.6)=\$7{,}500\); the second receives \(7{,}500(0.6)=\$4{,}500\); and the third receives \(5{,}000(0.6)=\$3{,}000\).
Check and common mistake avoided. The allocations total \(7{,}500+4{,}500+3{,}000=\$15{,}000\). They preserve the ratio \(12{,}500:7{,}500:5{,}000\). Dividing the aggregate cap equally among the three claims would give $5,000 each, which is not pro rata to the otherwise-payable amounts.
Worked Example: Allocate Exactly and Handle Cent Rounding
A hypothetical policy has otherwise-payable amounts of $9,000, $6,000, and $3,000 for three claims. A $10,000 aggregate cap applies, with pro-rata allocation. Find the payments to the nearest cent and verify the total.
Method. The otherwise-payable total is \(B=9{,}000+6{,}000+3{,}000=\$18{,}000\). Since the cap is lower than this total, the total payment is $10,000. The pro-rata factor is \(10{,}000/18{,}000=5/9\).
The first allocation is \(9{,}000(5/9)=\$5{,}000.00\). The second is \(6{,}000(5/9)=\$3{,}333.333\ldots\), which rounds to $3,333.33. The third is \(3{,}000(5/9)=\$1{,}666.666\ldots\), which rounds to $1,666.67.
Check and common mistake avoided. After rounding to cents, the allocations sum to \(5{,}000.00+3{,}333.33+1{,}666.67=\$10{,}000.00\). Round only after calculating each unrounded allocation, then check the rounded total. Here ordinary rounding already preserves the cap exactly. In other cases, cent rounding may require assigning a one-cent adjustment according to the stated accounting convention so that the recorded payments add to the capped total.
Common Allocation Traps
- Confusing the capped total with a claim payment. The aggregate cap limits the combined payment across the relevant policy period; it is not automatically the payment for each claim.
- Applying the cap before finding eligible amounts. First apply the stated claim-level terms to find each \(B_j\). Then compare their sum with the aggregate cap.
- Assuming an allocation rule that was not given. Pro-rata allocation is appropriate when specified or expressly assumed. A different contractual priority rule can produce different claim payments while keeping the same total.
- Dividing the cap equally among claims. Equal division is not pro rata unless the otherwise-payable amounts are equal. Pro rata preserves the relative sizes of those amounts.
- Failing to check the sum. Under any allocation of the capped total, the claim payments must add to \(T\), subject to the stated rounding convention.
Check Your Understanding
For each hypothetical policy, calculate the total after the aggregate cap and the claim-level allocations using the stated rule.
- A policy has otherwise-payable amounts of $4,200, $2,800, and $3,000, with a $7,500 aggregate cap and pro-rata allocation. Find the allocated payment for each claim. (Answer: $3,150, $2,100, and $2,250; total $7,500.)
- Three covered losses are $6,000, $5,000, and $4,000. Each has a $1,000 deductible and a $4,000 maximum payment; a $8,800 aggregate cap then applies, with pro-rata allocation. Find each final payment. (Answer: Otherwise payable $4,000, $4,000, and $3,000; allocated payments $3,200, $3,200, and $2,400.)
- Otherwise-payable amounts are $12,500, $7,500, and $5,000. The aggregate cap is $15,000, and the policy specifies pro-rata allocation. Find the common reduction factor and each claim payment. (Answer: Factor 0.6; payments $7,500, $4,500, and $3,000.)
- Otherwise-payable amounts are $3,900, $2,600, and $1,500, and the aggregate cap is $9,000. The policy specifies pro-rata allocation if the cap binds. Does it bind, and what are the final payments? (Answer: No; total $8,000, so payments remain $3,900, $2,600, and $1,500.)
- Otherwise-payable amounts are $10,000, $7,000, and $3,000. The aggregate cap is $12,000, with pro-rata allocation. Find each payment and verify the total. (Answer: $6,000, $4,200, and $1,800; total $12,000.)