Why Small Definition Errors Change the Answer
Deductible and limit calculations often use simple arithmetic, but the result depends on what each amount represents. A ground-up loss is the loss before policy terms; the insurer’s payment is the amount remaining after the deductible and any applicable limit. Confusing those quantities can produce a plausible-looking but incorrect factor or loss cost.
This tutorial focuses on checking definitions, formulas, denominators, and weights before calculating. The examples use hypothetical losses and policy terms. The distinctions are aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.
The phrase “limit of \(u\)” is incomplete unless the limit’s basis is clear. If \(u\) caps the insurer’s payment, payment cannot exceed \(u\). If \(u\) caps the covered ground-up amount, the deductible is applied to the capped amount, so payment is \((\min(X,u)-d)_+\). These are different contracts, and they need not produce the same payment.
Worked Example: Deductible Before an Insurer-Payment Limit
Worked Example: Deductible Before an Insurer-Payment Limit
A hypothetical policy has a deductible of $100 and a limit of $400 on the insurer’s payment. Ground-up loss \(X\) takes values $0, $150, $350, and $700 with probabilities 0.20, 0.30, 0.30, and 0.20, respectively. Find the expected insurer payment. Then compare it with the result from incorrectly capping ground-up loss at $400 before applying the deductible.
Method. For a limit on insurer payment, first subtract the deductible, flooring the result at zero, and then apply the payment cap: \(Y^L=\min((X-100)_+,400)\).
At \(X=0\), payment is \(\min(0,400)=0\). At \(X=150\), it is \(\min(50,400)=50\). At \(X=350\), it is \(\min(250,400)=250\). At \(X=700\), it is \(\min(600,400)=400\).
The expected payment is \(0.20(0)+0.30(50)+0.30(250)+0.20(400)=0+15+75+80=\$170\).
The incorrect calculation first caps ground-up loss at $400 and then subtracts the $100 deductible. Its payments are $0, $50, $250, and $300, giving \(0.20(0)+0.30(50)+0.30(250)+0.20(300)=\$150\). That describes a $400 cap on ground-up covered loss, not a $400 cap on insurer payment.
Check. Under the stated terms, no payment exceeds $400; the largest payment is exactly $400. The expected payment of $170 lies between the minimum and maximum possible payments. The common mistake avoided is treating the two different limit definitions as interchangeable.
Factor Denominators Must Match the Intended Comparison
A deductible factor expresses a ratio of expected losses under specified payment terms. One common definition compares expected payment after a deductible with expected ground-up loss:
This definition answers: what proportion of expected ground-up loss is expected to be paid after applying the deductible? A different denominator answers a different question. For example, dividing by expected loss already subject to a policy limit measures payment relative to limited loss, not relative to ground-up loss.
Worked Example: Choosing the Ground-Up Denominator
For a hypothetical severity distribution, \(X\) is $100 with probability 0.50, $300 with probability 0.30, and $700 with probability 0.20. Calculate the factor for a $100 deductible under the definition above. Also calculate the ratio that would result from incorrectly using expected loss capped at $500 as the denominator.
Method. Calculate expected ground-up loss and expected payment separately. Then divide expected payment by the denominator required by the factor definition.
The expected ground-up loss is \(E[X]=0.50(100)+0.30(300)+0.20(700)=50+90+140=\$280\).
The payments after a $100 deductible are $0, $200, and $600. Therefore, \(E[(X-100)_+]=0.50(0)+0.30(200)+0.20(600)=0+60+120=\$180\).
The deductible factor relative to ground-up loss is \(180/280=0.642857\ldots\), or 0.6429 to four decimal places.
If ground-up losses are first capped at $500, their expected value is \(0.50(100)+0.30(300)+0.20(500)=50+90+100=\$240\). Using that denominator gives \(180/240=0.75\). The arithmetic is correct, but 0.75 is not the defined ground-up deductible factor: its denominator represents capped loss.
Check. The expected payment, $180, is less than the expected ground-up loss, $280, so the factor is below 1. The mistake avoided is changing the denominator without changing the name and interpretation of the ratio.
Keep Claim Weights Consistent
When observations have different weights, apply the same weights to both the payment total and the ground-up loss total. A weight might represent a frequency count or another explicitly stated weighting basis. Do not combine a weighted payment total with an unweighted loss total, or average claim-level ratios when the intended quantity is a ratio of aggregate losses.
Worked Example: Weighted Aggregate Losses
A hypothetical claims extract contains three ground-up losses: $100 with weight 2, $200 with weight 1, and $1,000 with weight 1. The weights are frequency weights: a weight of 2 means that the observation represents two claims. Find the empirical deductible factor for a $100 deductible, defined as weighted total payment divided by weighted total ground-up loss. Compare it with two tempting but inconsistent calculations.
Method. Apply the deductible to each observed claim amount. Multiply each resulting payment and each ground-up loss by that observation’s weight.
The payments are \((100-100)_+=\$0\), \((200-100)_+=\$100\), and \((1{,}000-100)_+=\$900\). The weighted payment total is \(2(0)+1(100)+1(900)=\$1{,}000\).
The weighted ground-up total is \(2(100)+1(200)+1(1{,}000)=\$1{,}400\). Thus the factor is \(1{,}000/1{,}400=5/7=0.714285\ldots\), or 0.7143 to four decimal places.
Ignoring the weights gives payment total \(0+100+900=\$1{,}000\) and loss total \(100+200+1{,}000=\$1{,}300\), producing 0.7692. This does not use the stated frequency weights. Another error is to average the claim-level ratios: \(2(0/100)+1(100/200)+1(900/1{,}000)\), divided by the total weight 4, equals \(1.4/4=0.35\). That is a weighted average of individual ratios, not the ratio of aggregate payment to aggregate loss.
Check. The weighted average ground-up loss per represented claim is \(1{,}400/4=\$350\), and the weighted average payment is \(1{,}000/4=\$250\). Their ratio is \(250/350=5/7\), confirming the factor. The mistake avoided is changing the weighting basis between numerator and denominator.
State What the Limit Caps
A limit definition should specify the amount that cannot be exceeded. For a deductible \(d\) and a limit \(u\) on insurer payment, the payment is \(\min((X-d)_+,u)\). For a limit \(u\) on covered ground-up loss, payment is \((\min(X,u)-d)_+\). When \(u>d\), the latter can also be written as \(\min((X-d)_+,u-d)\): the maximum insurer payment is \(u-d\), not \(u\).
Worked Example: Two Meanings of the Same Limit Amount
A hypothetical claim has ground-up loss $950 and deductible $100. Compare the payment if a stated $800 limit caps the insurer’s payment with the payment if it caps covered ground-up loss.
Method. Calculate each contract interpretation separately. Do not use the same formula for both definitions.
With a limit on insurer payment, the uncapped payment is \((950-100)_+=\$850\), so the capped payment is \(\min(850,800)=\$800\).
With a limit on covered ground-up loss, first cap the loss at $800. The payment is \((\min(950,800)-100)_+=(800-100)_+=\$700\). Equivalently, the maximum payment is \(800-100=\$700\).
Check. Both payments are below the $950 ground-up loss, but they differ by $100 because the limit caps a different amount. The common mistake avoided is interpreting a ground-up limit as though it were a limit on insurer payment.
A Practical Error-Checking Sequence
Use the following checks before accepting a deductible or limit calculation. They are especially useful when a problem supplies a factor table, claim data, or wording that could support more than one limit interpretation.
Confirm whether each value is ground-up loss, loss after a limit, or insurer payment. Do not apply a deductible to an amount whose contract adjustments are unknown.
State the deductible and whether the limit caps insurer payment or covered ground-up loss. Then apply the operations in the order specified by that formula.
Check whether the comparison is to ground-up loss, limited loss, or another stated base. A correct numerator divided by the wrong base is still the wrong factor.
Use the same observation weights for payment and ground-up loss when forming an aggregate ratio. Confirm that any supplied weights have the interpretation the calculation assumes.
Each payment must be nonnegative and no greater than the applicable maximum. Compare the aggregate result with the quantities it summarizes and explain what the factor’s denominator means.
Common Exam Traps
- Subtracting the deductible from a payment-limited loss. If the limit caps insurer payment, use \(\min((X-d)_+,u)\). Capping ground-up loss first instead describes a different contract.
- Using a convenient denominator. A ratio of payment to limited loss is not automatically a deductible factor relative to ground-up loss. State the denominator before calculating.
- Averaging individual ratios. The average of \(Y_j^L/X_j\) is generally not \(\sum_jY_j^L/\sum_jX_j\). Use the aggregate definition requested by the problem.
- Applying weights only once. If weights represent claim counts, apply them to both payment and ground-up loss totals. Otherwise the ratio does not compare like with like.
- Assuming every “limit” means the same thing. A policy-payment cap and a cap on covered ground-up loss can differ by the deductible amount.
Check Your Understanding
For each hypothetical situation, write the payment or factor formula that matches the stated terms before calculating.
- A loss is $0, $250, or $600 with probabilities 0.25, 0.50, and 0.25. There is a $150 deductible and no limit. Find expected payment. (Answer: $162.50)
- A severity is $200, $500, or $1,000 with probabilities 0.40, 0.40, and 0.20. Find the deductible factor for a $200 deductible relative to ground-up expected loss. (Answer: 0.5833)
- Losses are $200, $500, and $900 with probabilities 0.20, 0.50, and 0.30. A $100 deductible applies, followed by a $350 limit on insurer payment. Find expected payment. (Answer: $300.00)
- Three claims have losses $100, $200, and $1,000, with frequency weights 2, 1, and 1. Find the empirical factor for a $100 deductible as weighted total payment divided by weighted total ground-up loss. (Answer: 0.7143)
- A $1,100 ground-up loss has a $200 deductible and a stated limit of $900. Find payment under each interpretation: the limit caps insurer payment, or the limit caps covered ground-up loss. (Answer: $900 under the insurer-payment limit; $700 under the ground-up limit)