Why ILF Calculations Need a Consistency Check
An increased limits factor (ILF) converts a premium or expected loss at a base limit to an indicated amount at a higher limit. The calculation is simple only after the coverage being measured is defined: the same loss basis, deductible treatment, and limit type must be used at both limits. A plausible-looking ratio can still be wrong if one of those items changes between numerator and denominator.
The checks in this tutorial focus on four recurring errors: reversing the ratio, mixing loss bases, applying a deductible twice, and treating an aggregate limit as though it applied separately to each claim. All numerical examples are hypothetical. They use dollar amounts in thousands where indicated; no interest or mortality assumptions are relevant to these claim-payment calculations.
For a ground-up loss \(X\geq 0\) with no deductible and a per-claim limit \(L\), the insurer payment is \(g_L(X)=\min(X,L)\). If instead there is a deductible \(d\) and \(L\) is the policy limit inclusive of the deductible, the payment is $$ g_{d,L}(X)=\min\big((X-d)^+,(L-d)^+\big), $$ where \(z^+=\max(z,0)\). This formula reflects that the insurer pays nothing below the deductible and, at most, the amount between the deductible and the policy limit. If the contract defines \(L\) as a limit above the deductible, its payment formula is different; the limit convention must be stated before calculating an ILF.
Two Checks That Catch Many Errors
First, check the direction of the ratio. With identical terms and a higher limit \(H\geq B\), every claim payment at \(H\) is at least the payment at \(B\). Thus \(g_H(X)\geq g_B(X)\) for every \(X\), and consequently \(E[g_H(X)]\geq E[g_B(X)]\). When the base expected payment is positive, the ILF must be at least 1. A result below 1 signals a reversed ratio or an inconsistency that needs investigation.
Second, check that the ratio compares like with like. To see why the expected-payment ratio is the appropriate factor, suppose the claim frequency per exposure is \(\lambda\) at both limits and severity follows the same distribution. Expected loss per exposure at limit \(L\) is \(\lambda E[g_L(X)]\). Therefore,
The common frequency cancels. If the numerator and denominator use different loss bases or different coverage terms, this cancellation does not justify the resulting ratio. Likewise, applying an ILF to a base premium is appropriate only when that premium represents the same underlying coverage basis and the adjustment is intended to change the limit, not to change other rating assumptions.
Worked Example: Correcting a Reversed ILF
A hypothetical policy has ground-up severity \(X\), in thousands of dollars, with the following distribution: \(X=50\) with probability 0.50, \(X=150\) with probability 0.30, and \(X=300\) with probability 0.20. There is no deductible. Calculate the ILF from a base per-claim limit of 100 to a higher limit of 250. A base-limit premium is $12,000. State the indicated premium at the higher limit.
Method. Calculate the expected limited payment at each limit using the same ground-up severity distribution, then divide the higher-limit amount by the base-limit amount. The severity distribution and parameters are given explicitly; interest and mortality do not apply.
At the 100 limit, the claim payments for severities 50, 150, and 300 are respectively \(50\), \(100\), and \(100\). Thus:
At the 250 limit, the payments are \(50\), \(150\), and \(250\). Thus:
The correct factor is \(120/75=1.6\). Applying it to the base premium gives \(12{,}000(1.6)=\$19{,}200\). The check is consistent with the limits: the higher limit has a greater expected payment, and the ILF exceeds 1.
Common mistake avoided. Reversing the ratio gives \(75/120=0.625\), which would imply that increasing the limit reduces the indicated premium to \(12{,}000(0.625)=\$7{,}500\). The payment monotonicity check immediately shows that this direction is wrong.
Keep the Loss Basis the Same
A loss basis describes which amount is being limited and what adjustments have already been made. For example, ground-up severity uses the full loss \(X\); a deductible-adjusted payment uses the portion payable after the deductible. An ILF can be calculated on either basis, but a numerator on one basis cannot be divided by a denominator on another and interpreted as an ILF for a single coverage change.
Worked Example: Diagnosing a Mixed Loss Basis
A hypothetical severity distribution, in thousands of dollars, is \(X=20\) with probability 0.40, \(X=60\) with probability 0.35, and \(X=120\) with probability 0.25. The policy has a deductible of 20 and a policy limit inclusive of the deductible. Find the ILF from a 50 limit to a 100 limit on the deductible-adjusted basis. Then assess a proposed calculation that divides the 100-limit deductible-adjusted expected payment by the 50-limit ground-up expected payment.
Method. Apply the deductible and inclusive limit once to each ground-up loss. The distribution probabilities sum to \(0.40+0.35+0.25=1.00\); interest and mortality are not involved.
At the 50 policy limit, the maximum insurer payment is \(50-20=30\). The payments are 0 for a loss of 20, 30 for a loss of 60, and 30 for a loss of 120. Therefore:
At the 100 policy limit, the maximum insurer payment is \(100-20=80\). The payments are 0, 40, and 80. Therefore:
The deductible-adjusted ILF is \(34/18=1.8889\) to four decimal places. As a separate basis check, the 50-limit ground-up payments are 20, 50, and 50, giving
The proposed mixed-basis ratio is \(34/38=0.8947\) to four decimal places. That is not the ILF just calculated: the numerator deducts 20 from covered losses while the denominator does not. Its value below 1 is a warning, not evidence that raising the limit reduces payment.
Common mistake avoided. Do not compare a net-of-deductible expected payment with a ground-up expected payment and call the result an ILF. If a ground-up-basis ILF is required instead, calculate the ground-up expected payment at both limits. At 100, it is \(0.40(20)+0.35(60)+0.25(100)=54\), so the ground-up ILF is \(54/38=1.4211\), not \(34/38\).
Apply a Deductible Once
A deductible changes which part of a loss the insurer pays. If the input data are ground-up losses, apply the deductible in the payment function. If the data already contain insurer payments after the deductible, do not subtract the deductible again. Before calculating, identify whether each claim amount is a ground-up loss or an already-adjusted payment.
Worked Example: Finding a Double-Applied Deductible
A hypothetical portfolio has ground-up severity \(X=40\), \(80\), or \(160\) thousand dollars, with respective probabilities 0.25, 0.50, and 0.25. There is a deductible of 30 and policy limits inclusive of the deductible. Calculate the ILF from a 100 limit to a 200 limit. Then quantify the effect of mistakenly subtracting the deductible again from the insurer payments.
Method. First calculate each policy payment from the ground-up severity with \(g_{30,L}(X)=\min((X-30)^+,(L-30)^+)\). The probabilities total \(0.25+0.50+0.25=1.00\); interest and mortality do not apply.
At the 100 limit, the insurer-payment cap is \(100-30=70\). The payments for losses 40, 80, and 160 are \(10\), \(50\), and \(70\). Hence:
At the 200 limit, the cap is \(200-30=170\). The payments are \(10\), \(50\), and \(130\), so:
The correct ILF is \(60/45=1.3333\) to four decimal places. As a check, the higher limit increases the payment for the 160 loss from 70 to 130 and does not reduce any other payment.
Suppose an analyst mistakenly subtracts 30 again from the already-calculated insurer payments. At the 100 limit, the resulting incorrect amounts are \((10-30)^+=0\), \((50-30)^+=20\), and \((70-30)^+=40\), with expected value \(0.25(0)+0.50(20)+0.25(40)=20\). At the 200 limit they are 0, 20, and 100, with expected value \(0.25(0)+0.50(20)+0.25(100)=35\). The resulting ratio \(35/20=1.75\) is not the policy ILF; the deductible has been charged twice. The correction is to use the original ground-up losses and apply the deductible once, producing \(60/45\).
Do Not Confuse Aggregate and Per-Claim Limits
A per-claim limit caps the insurer payment on each claim separately. An aggregate limit caps total payments for a policy period or other defined group of claims. The calculation must follow the contract structure: in general, applying a limit to each claim and summing is not the same as summing claims and then applying an aggregate cap.
Worked Example: Applying an Aggregate Limit to Two Claims
A hypothetical policy has exactly two claims in the period, each with a ground-up loss of $80,000. There is no deductible and no per-claim limit. Compare aggregate limits of $100,000 and $150,000, and calculate the factor for increasing the aggregate limit. The claim count and severities are deterministic: there are two claims of 80 each, with probability 1.00. Interest and mortality do not apply.
Method. Sum the claims first, then apply the aggregate limit. Total ground-up loss is \(80+80=160\) thousand dollars. At the 100 aggregate limit, total insurer payment is \(\min(160,100)=100\). At the 150 aggregate limit, it is \(\min(160,150)=150\). Thus the aggregate-limit factor is \(150/100=1.5\).
Common mistake avoided. Treating the aggregate limits as per-claim limits gives payments of \(80+80=160\) at both limits, because each claim is below even the 100 limit. That calculation produces a ratio of \(160/160=1\) and misses the change in aggregate coverage. The correct calculation applies the cap to the total of the two claims, yielding an increase from 100 to 150.
A Practical ILF Audit
Before accepting an indicated factor, work through the following checks in order. They do not replace a full review of the underlying data or coverage wording, but they help locate structural calculation errors.
Identify whether each input is a ground-up loss, a loss after deductible, or an insurer payment. Do not infer the basis from a column label alone.
Confirm whether the limit is inclusive of the deductible, above the deductible, per claim, or aggregate. Write the corresponding payment rule before evaluating claims.
Calculate the base and higher expected payments from the same severity distribution and with the same terms, changing only the limit under study.
Use higher-limit expected payment over base-limit expected payment. With a positive base expected payment and otherwise identical terms, the factor should be at least 1.
A factor below 1 is a useful diagnostic, but a factor above 1 is not proof that the calculation is correct. For example, a deductible applied twice can still produce a factor above 1 while distorting its size. Verify the payment amounts and their bases before interpreting the ratio or using it to adjust a premium.
Check Your Understanding
For each problem, identify the relevant payment basis and check whether the resulting factor is consistent with the limit change.
- A hypothetical ground-up severity distribution in thousands is \(X=40\) with probability 0.50, \(X=100\) with probability 0.30, and \(X=200\) with probability 0.20. There is no deductible. Calculate the per-claim ILF from limit 80 to limit 160. (Answer: \(82/60=1.3667\), to four decimal places.)
- A policy has a deductible of 10 and an inclusive limit of 60. A claim has ground-up loss 45. What is the insurer payment? What payment would result from incorrectly applying the deductible a second time to that insurer payment? (Answer: Correct payment \(=\min(45-10,60-10)=\$35\); the incorrect second application gives \((35-10)^+=\$25\).)
- A hypothetical severity distribution in thousands is \(X=30\) with probability 0.50 and \(X=90\) with probability 0.50. With no deductible, calculate the ground-up per-claim ILF from limit 40 to limit 80. (Answer: \(E[g_{40}(X)]=0.50(30)+0.50(40)=35\), \(E[g_{80}(X)]=0.50(30)+0.50(80)=55\), so the ILF is \(55/35=1.5714\), to four decimal places.)
- A hypothetical policy has two deterministic claims of $70,000 each and no per-claim limit. Calculate the factor for increasing the aggregate limit from $90,000 to $120,000. Why would applying those amounts separately to each claim be incorrect? (Answer: Aggregate payments are \(\min(140,90)=90\) and \(\min(140,120)=120\), in thousands, so the factor is \(120/90=1.3333\), to four decimal places; the limits cap total policy-period payments, not each claim.)
- A base premium of $8,400 corresponds to a base-limit expected payment of $42,000. A higher limit has expected payment of $63,000 on the same loss basis and with unchanged terms. What is the ILF and the indicated higher-limit premium? (Answer: ILF \(=63{,}000/42{,}000=1.5\); indicated premium \(=\$8{,}400(1.5)=\$12{,}600\).)