Why Combine Deductibles and Limits?
A rating plan may vary both the amount the insured retains through a deductible and the maximum amount the insurer pays through a policy limit. Each feature affects the insurer’s expected payment, and the effects must be evaluated on the same ground-up claim severity distribution. The central task is to calculate the payment for each possible loss under each plan, then average those payments using the stated severity probabilities.
The limit in this tutorial is a gross loss limit: it caps the covered loss before subtracting the deductible. We consider per-claim payments, with no coinsurance, expenses, taxes, or other coverage provisions. The calculations are undiscounted, so no interest rate is applied; mortality assumptions are not applicable. All severity distributions below are hypothetical.
For a finite gross limit, the insurer pays nothing when \(X\le d\), pays \(X-d\) when \(d<X<u\), and pays \(u-d\) when \(X\ge u\). In particular, the maximum insurer payment is \(u-d\), not \(u\). This distinction is essential when comparing a deductible-and-limit plan with another plan.
Calculate Payments Before Comparing Plans
For a discrete severity distribution, list each possible value of \(X\), determine the insurer payment at that value, and weight the payment by its probability. The resulting expected payment is the loss cost for that plan. Calculate each plan’s loss cost before forming a relative factor; a factor is a ratio of expected costs, not a ratio of policy limits.
Worked Example: Higher Limits with a Fixed Deductible
A hypothetical equipment policy has ground-up claim severity \(X\) with this distribution. Calculate loss costs at a deductible of $3 and gross limits of $12, $25, and unlimited. Then calculate the higher-limit factors relative to the $12 limit. The distribution is \(P(X=0)=0.20\), \(P(X=3)=0.25\), \(P(X=8)=0.25\), \(P(X=15)=0.20\), and \(P(X=30)=0.10\). There is no discounting and no mortality assumption applies.
Method. Apply \(Y^L(d,u)=\min((X-d)_+,u-d)\) to every severity value. With \(d=3\), the maximum insurer payments are $9 at limit $12 and $22 at limit $25.
Limit $12. For \(X=0,3,8,15,30\), the respective insurer payments are $0, $0, $5, $9, and $9. Therefore,
Limit $25. The respective payments are $0, $0, $5, $12, and $22. Thus,
Unlimited limit. The payments are $0, $0, $5, $12, and $27, so
The increased limits factors at this deductible are \(5.85/3.95=1.4810\) for limit $25 and \(6.35/3.95=1.6076\) for unlimited limit, with the $12 limit normalized to 1.0000. As a check, the loss costs rise as the gross limit rises, and each factor is the higher-limit loss cost divided by the $12-limit loss cost. The common mistake this method avoids is using the gross limit itself as the insurer’s maximum payment: for example, at limit $12 the insurer’s maximum payment is $12-$3=$9.
Why the Combined Formula Works
The combined payment can also be understood as the part of the loss lying above the deductible and below the gross limit. This gives a useful justification for the expected-payment formula. For each fixed loss \(X\), the interval of levels \(x\) satisfying \(d<x<\min(X,u)\) has length equal to the insurer payment. Equivalently, the payment is the accumulated indicator that the loss exceeds each level between \(d\) and \(u\).
Taking expectations gives the tail-integral expression below. The exchange of expectation and integration is valid because the integrand is nonnegative; for an unlimited limit, the expression requires a finite expected payment for the resulting value to be finite. For a discrete severity distribution, the same identity applies: it is an alternative way to sum the payment contributions over the loss distribution.
This relationship explains why increasing the limit adds expected payment from the layer above the old limit, while increasing the deductible removes expected payment from the layer below the new deductible. It does not mean that a deductible factor and an increased limits factor can always be multiplied as though they were independent: both depend on the underlying severity distribution and on the plan terms.
Check the Gross Limit and Deductible Together
A reliable calculation states whether the limit is gross or applies to the insurer’s payment. For a gross limit \(u\), the payment cap is \(u-d\). If instead a contract explicitly states that the insurer’s payment is capped at an amount \(c\) after the deductible, the formula is \(\min((X-d)_+,c)\); that is a different limit convention. Do not switch conventions midway through a calculation.
Worked Example: Comparing Deductibles and Correcting a Limit Error
A hypothetical machinery policy has \(P(X=0)=0.10\), \(P(X=6)=0.30\), \(P(X=10)=0.25\), \(P(X=18)=0.20\), and \(P(X=30)=0.15\). Calculate the loss cost for deductible $6 and gross limit $18, then for deductible $10 with the same gross limit. Find the deductible-plan factor for the $10 deductible relative to the $6 deductible. No discounting or mortality assumption applies.
Deductible $6, gross limit $18. The insurer payment cap is \(18-6=\$12\). For severity values \(0,6,10,18,30\), the payments are $0, $0, $4, $12, and $12. The expected payment is
Deductible $10, gross limit $18. The payment cap is now \(18-10=\$8\). The payments at the same severities are $0, $0, $0, $8, and $8. Hence,
The relative loss-cost factor is \(2.80/5.20=0.5385\), to four decimal places. This factor compares the two deductible options at the same gross limit; it is not an increased limits factor.
Common mistake and correction. If the $18 gross limit is incorrectly treated as a cap on insurer payment after the deductible, the payment for \(X=30\) is wrongly set to $18 rather than $12. That error gives
The corrected loss cost is $5.20 because the contract’s gross limit leaves at most \(18-6=\$12\) payable by the insurer. The higher deductible’s factor is checked by confirming that its loss cost, $2.80, is less than the $5.20 cost at the lower deductible.
Build and Apply a Selected Schedule
A schedule is easiest to interpret when every plan is compared with one explicitly stated base plan. For a base plan with deductible \(d_0\) and gross limit \(u_0\), define the combined plan factor as
The base plan’s factor is 1.000. Factors below 1 indicate lower expected insurer loss cost than the base, and factors above 1 indicate higher expected insurer loss cost. When a base rate is a loss-cost rate and all other rating provisions are held constant, multiply that rate by the factor to obtain a plan’s indicated loss-cost rate. If the rate includes expenses or other provisions that do not vary proportionally with losses, a loss-cost factor alone does not determine the full premium.
Worked Example: Selecting a Full Deductible-and-Limit Schedule
A hypothetical service contract uses severity \(X\) with \(P(X=0)=0.20\), \(P(X=5)=0.30\), \(P(X=10)=0.25\), \(P(X=20)=0.15\), and \(P(X=40)=0.10\). Calculate a schedule for deductibles $0 and $5 and gross limits $10, $20, and unlimited. Use deductible $0 and gross limit $10 as the base plan. If the base loss-cost rate is $120 per exposure, find the indicated loss-cost rates. No discounting or mortality assumption applies.
Deductible $0. At gross limit $10, the payments for \(X=0,5,10,20,40\) are $0, $5, $10, $10, and $10. The loss cost is
At gross limit $20, the payments are $0, $5, $10, $20, and $20, giving
With unlimited limit, the payments are $0, $5, $10, $20, and $40, so
Deductible $5. At gross limit $10, the insurer-payment cap is $5. The payments are $0, $0, $5, $5, and $5, giving
At gross limit $20, the cap is $15. Payments are $0, $0, $5, $15, and $15; the loss cost is
With unlimited limit, payments are $0, $0, $5, $15, and $35, so
Divide each loss cost by the base cost of $6.50. The resulting schedule, with factors rounded to three decimals and rates rounded to cents, is:
| Deductible | Gross limit | Loss cost | Factor to base | Indicated rate |
|---|---|---|---|---|
| $0 | $10 | $6.50 | 1.000 | $120.00 |
| $0 | $20 | $9.00 | 1.385 | $166.15 |
| $0 | Unlimited | $11.00 | 1.692 | $203.08 |
| $5 | $10 | $2.50 | 0.385 | $46.15 |
| $5 | $20 | $5.00 | 0.769 | $92.31 |
| $5 | Unlimited | $7.00 | 1.077 | $129.23 |
For instance, the $5 deductible and $20 limit factor is \(5.00/6.50=0.769230\ldots\), and its indicated rate is \(120(5.00/6.50)=\$92.31\). The unlimited-limit factor with a $5 deductible exceeds 1 because its cost of $7.00 is greater than the base plan’s $6.50; a larger deductible does not guarantee a lower cost than every other plan when limits also differ. The check is to compare each factor with its own base loss cost and verify that the indicated rate equals $120 times the unrounded ratio, rounded only at the final step.
Practical Checks and Exam Traps
A combined schedule should satisfy basic consistency checks when the underlying severity distribution and coverage terms are otherwise unchanged. Increasing a gross limit cannot reduce the insurer’s payment for any claim, so its expected loss cost cannot decrease. Increasing a deductible cannot increase the payment for any claim, so its expected loss cost cannot increase. These are checks on the calculated loss costs, not a substitute for calculating them.
- Confirm the limit convention. For a gross limit \(u\), cap insurer payment at \(u-d\). A limit on payment after the deductible is a different contract.
- Use one severity basis. Compare plans using the same distribution of ground-up losses; do not mix capped losses from one plan with ground-up losses from another.
- Normalize to the stated base. A factor is the plan’s loss cost divided by the base plan’s loss cost. State the base before interpreting a factor.
- Do not assume factors multiply. A deductible factor and a limit factor calculated from different bases need not combine by multiplication. Calculate the combined plan’s expected payment or use a justified method that preserves the same underlying distribution.
- Keep units consistent. If losses are in dollars, the weighted expected payment is in dollars per claim. Do not relabel a value as thousands unless all inputs and results use that scale.
Integrated deductible-and-limit calculations connect contract terms to expected insurer loss cost and provide a coherent basis for rating-plan comparisons. They are aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.
Check Your Understanding
For each hypothetical problem, state the relevant gross-limit convention and calculate before rounding. No discounting or mortality assumption applies.
- A policy has \(P(X=0)=0.20\), \(P(X=4)=0.30\), \(P(X=9)=0.30\), and \(P(X=20)=0.20\). With deductible $4, find the loss costs at gross limits $14 and $20 and the $20-limit factor relative to $14. (Answer: loss costs $3.50 and $4.70; factor 1.3429.)
- A hypothetical repair policy has \(P(X=5)=0.40\), \(P(X=12)=0.40\), and \(P(X=25)=0.20\). Find the loss cost with deductible $5 and gross limit $15. (Answer: $4.80 per claim.)
- For \(P(X=0)=0.25\), \(P(X=10)=0.50\), and \(P(X=20)=0.25\), compare deductible $0 and deductible $10 at gross limit $20. Find the loss costs and the $10 deductible factor relative to deductible $0. (Answer: $10.00 and $2.50; factor 0.2500.)
- A contract has \(P(X=0)=0.20\), \(P(X=8)=0.30\), \(P(X=16)=0.30\), and \(P(X=40)=0.20\). With deductible $8, calculate costs at gross limits $16 and $24, then the higher-limit factor. (Answer: loss costs $4.00 and $5.60; factor 1.4000.)
- Using the severity distribution in question 4, take deductible $8 and gross limit $24 as the base plan, with a base loss-cost rate of $240 per exposure. Find the indicated loss-cost rate for deductible $8 and gross limit $16. (Answer: factor \(4.00/5.60=0.7143\); indicated rate $171.43 per exposure.)