Claim Frequency And Severity In ILF Ratios

Why Frequency and Severity Separate in an ILF

An increased limits factor (ILF) compares expected losses at two policy limits. A frequency-severity model makes the source of that comparison clear: expected loss is claim frequency multiplied by expected payment per claim. If choosing a higher limit changes only the payment on each claim, then the same frequency appears in both expected losses and cancels from the ratio. The ILF is therefore a ratio of expected limited severities.

That conclusion depends on the comparison being like-for-like. The exposure basis, claim population, claim definition, and coverage conditions must be held constant; the limit must be the only relevant change. If the frequency or the mix of claims differs between the two groups being compared, dividing limited severities alone does not describe the ratio of their expected losses.

Definition: Let $X$ be the ground-up loss for a covered claim and let $L$ be a per-claim limit. Under a standard cap with no deductible, the limited payment is $Y_L=\min(X,L)$. Its expected value, $E[Y_L]$, is the limited mean severity at limit $L$. For a base limit $b$, the pure-loss increased limits factor at limit $L$ is the ratio of expected limited losses at $L$ and $b$, on the same exposure and coverage basis.

Here, “severity” means average payment per claim, not average payment per policy or exposure. The per-claim limit caps each claim’s payment; it does not cap the number of claims. If the claim frequency per exposure unit is $\lambda$, expected loss per exposure unit is $\lambda E[Y_L]$, subject to the homogeneous frequency-severity assumptions stated below.

The Frequency-Severity Derivation

Consider a homogeneous portfolio on a fixed exposure basis. Let $N$ be the number of covered claims for that exposure unit, with $E[N]=\lambda$. Assume that each claim has the same severity distribution at either limit, that the only change is the per-claim limit, and that the expected limited payment per claim is not altered by the number of claims. No particular probability distribution for $N$ or $X$ is needed. For this expected-loss calculation, independence between $N$ and individual severities is not required if the conditional expected limited payment per claim is the same for each claim count.

S_L=\sum_{j=1}^{N}\min(X_j,L), \qquad E[S_L]=E[N]E[\min(X,L)]=\lambda E[Y_L].

To justify the result, condition first on the number of claims. With $N=n$, the aggregate payment is a sum of $n$ limited payments. By linearity of expectation, its conditional expected value is $nE[Y_L]$; independence among the payments is not necessary for this step. Taking expectations over $N$ gives $E[S_L]=E[N]E[Y_L]$ when the per-claim expected payment is the same for each claim count. Thus the expected aggregate loss factors into expected claim count and expected limited payment per claim.

\operatorname{ILF}(L;b) = \frac{E[S_L]}{E[S_b]} = \frac{\lambda E[Y_L]}{\lambda E[Y_b]} = \frac{E[Y_L]}{E[Y_b]}, \qquad \lambda>0,\quad E[Y_b]>0.

The condition $\lambda>0$ matters: if no claims are expected, both expected losses are zero and their ratio is undefined. The base-limit expected limited severity must also be positive. In practical use, the ratio is a pure severity ILF only when the compared expected claim counts are the same on the chosen exposure basis.

Result: When claim frequency, exposure, claim population, and coverage conditions are unchanged across limits, the pure-loss ILF is the ratio of limited mean severities, $E[Y_L]/E[Y_b]$. It is not the ratio of the policy limits, except in unusual circumstances where the loss model makes those ratios coincide.

Worked Applications

The examples below use hypothetical portfolios. No interest or mortality adjustment is made, and no particular severity distribution is assumed: the limited mean severities are supplied as expected values. The calculation concerns expected loss only, not expenses or other premium components.

Worked Example: From Frequency and Severity to an ILF

A hypothetical liability portfolio has 10,000 policy-years. It is expected to produce 0.08 claims per policy-year under either of two per-claim limits. The expected limited severity is $2,400 at the $50,000 limit and $3,120 at the $100,000 limit. Find the ILF and the expected aggregate loss at each limit.

Method and assumptions. Use the frequency-severity model, with the same claim frequency and claim population at both limits. The supplied expected severities are limited means; no underlying distribution is needed. The common mistake to avoid is applying the limit ratio, $100{,}000/50{,}000$, as the ILF.

First calculate the expected number of claims:

$$ E[N]=10{,}000(0.08)=800. $$

At the $50,000 limit, expected aggregate loss is $800(\$2{,}400)=\$1{,}920{,}000$. At the $100,000 limit, it is $800(\$3{,}120)=\$2{,}496{,}000$. Therefore,

\operatorname{ILF}(100{,}000;50{,}000) = \frac{\$2{,}496{,}000}{\$1{,}920{,}000} = \frac{\$3{,}120}{\$2{,}400} = 1.300.

Check. Multiplying the base-limit expected loss by 1.300 gives $\$1{,}920{,}000(1.300)=\$2{,}496{,}000$, the higher-limit expected loss. The limit ratio is 2.000, not the ILF: it incorrectly assumes that expected severity doubles when the limit doubles.

Worked Example: Convert a Claim Frequency Rate to Expected Loss per Policy-Year

For a hypothetical coverage, the expected claim frequency is 0.06 claims per policy-year at both limits. The limited mean severity is $1,800 at the $25,000 limit and $2,520 at the $75,000 limit. Find the expected loss per policy-year and the ILF.

Method and assumptions. Multiply claim frequency by limited mean severity to obtain expected loss per policy-year. Frequency and coverage terms are unchanged. There is no need to assign a distribution to claim counts or losses because the required expected severities are given. The common mistake to avoid is using the 3-to-1 limit ratio instead of the ratio of expected payments.

At the $25,000 limit, expected loss per policy-year is $0.06(\$1{,}800)=\$108.00$. At the $75,000 limit, it is $0.06(\$2{,}520)=\$151.20$. The ILF is

\operatorname{ILF}(75{,}000;25{,}000) = \frac{\$151.20}{\$108.00} = \frac{\$2{,}520}{\$1{,}800} = 1.400.

Check. The higher-limit expected loss is $\$108.00(1.400)=\$151.20$ per policy-year. Frequency cancels because it is 0.06 at both limits.

Worked Example: Combine Classes with Different Frequencies

A hypothetical book has two classes. Class A has 4,000 exposure units and a claim frequency of 0.10 per unit. Its limited mean severities are $1,000 at the base limit and $1,500 at the higher limit. Class B has 6,000 exposure units and a claim frequency of 0.05 per unit. Its corresponding limited mean severities are $2,000 and $2,600. Find the book-level ILF, assuming each class’s frequency is unchanged between limits.

Method and assumptions. Calculate each class’s expected number of claims, then aggregate its expected losses across classes at each limit. The class frequencies differ, so averaging the two class ILFs without weights would be a mistake. There is no assumed severity distribution; the limited means are provided.

Class A has $4{,}000(0.10)=400$ expected claims. Its base-limit expected loss is $400(\$1{,}000)=\$400{,}000$, and its higher-limit expected loss is $400(\$1{,}500)=\$600{,}000$.

Class B has $6{,}000(0.05)=300$ expected claims. Its base-limit expected loss is $300(\$2{,}000)=\$600{,}000$, and its higher-limit expected loss is $300(\$2{,}600)=\$780{,}000$.

Adding the class amounts gives base-limit expected loss of $\$400{,}000+\$600{,}000=\$1{,}000{,}000$, and higher-limit expected loss of $\$600{,}000+\$780{,}000=\$1{,}380{,}000$. Therefore,

\operatorname{ILF}_{\text{book}} = \frac{\$1{,}380{,}000}{\$1{,}000{,}000} = 1.380.

Check. The class ILFs are $1{,}500/1{,}000=1.500$ and $2{,}600/2{,}000=1.300$. Weighting them by their base-limit expected losses gives $(\$400{,}000(1.500)+\$600{,}000(1.300))/\$1{,}000{,}000=1.380$, matching the aggregate calculation. Their simple average, 1.400, would be wrong because the classes contribute different amounts of base-limit expected loss.

When Frequency Does Not Cancel

Frequency cancels only if the expected claim count is the same in the numerator and denominator on the same exposure basis. For example, if the two groups have different claim frequencies, the expected-loss ratio is

\frac{\lambda_L E[Y_L]}{\lambda_b E[Y_b]},

where $\lambda_L$ and $\lambda_b$ are the respective expected claim counts per exposure unit. This is not generally equal to the severity ratio $E[Y_L]/E[Y_b]$. A changed frequency can reflect a different population, exposure definition, claim reporting pattern, or other change in the comparison. It should not be attributed to the effect of the limit alone without examining the cause.

There is a similar issue with heterogeneous portfolios. If classes have different frequencies or severities, calculate expected losses by class and aggregate them before forming the book-level ratio. A single book-level frequency and severity can be used if they correctly represent the same claim-weighted population at both limits. Taking an unweighted average of class ILFs generally does not do that.

Common Exam Traps

Using the ratio of limits. A limit increase does not imply a proportional increase in expected payment. Use the ratio of expected limited severities.
Using a policy count as a claim count. Convert the exposure count to expected claims using the stated frequency rate before calculating aggregate expected loss.
Comparing different frequency bases. For instance, a rate per vehicle-year cannot be multiplied directly by policy-years unless the exposure units are reconciled.
Mixing class ILFs without appropriate weights. The book ILF is a ratio of total expected losses, not generally the simple average of class factors.
Calling a broader expected-loss ratio a pure ILF. If frequency, population, or coverage conditions also change, identify those changes; the limited-severity ratio alone no longer explains the full expected-loss ratio.

Key takeaway: Under a common exposure basis and unchanged claim frequency and coverage conditions, expected aggregate loss factors as frequency times limited mean severity. Frequency cancels in the ratio, leaving the higher-limit to base-limit ratio of expected limited payments per claim.

Check Your Understanding

For each problem, use the stated exposure basis and assume that no interest or mortality adjustment is required. A supplied limited mean severity is an expected value; no severity distribution is implied.

A hypothetical program has 2,000 policy-years and a frequency of 0.12 claims per policy-year at both limits. Limited mean severity is $500 at the base limit and $650 at the higher limit. Find the ILF and expected losses at both limits. (Answer: ILF = 1.300; expected losses = $120,000 and $156,000.)
A hypothetical policy has a claim frequency of 0.04 per policy-year at either limit. Its limited mean severities are $900 and $1,170. Find expected loss per policy-year at each limit and the ILF. (Answer: $36.00 and $46.80 per policy-year; ILF = 1.300.)
A hypothetical portfolio is expected to have 300 claims at either limit. Its limited mean severity is $2,400 at the base limit and $3,000 at the higher limit. Find aggregate expected loss at both limits and the ILF. (Answer: $720,000 and $900,000; ILF = 1.250.)
A hypothetical book has two classes. Class A has 100 exposure units, frequency 0.20, and limited means of $1,000 and $1,300. Class B has 200 exposure units, frequency 0.10, and limited means of $2,000 and $2,400. Frequencies are unchanged across limits. Find the book ILF. (Answer: base expected loss = $60,000; higher-limit expected loss = $74,000; ILF = 1.2333.)
For a hypothetical coverage, expected claim count is 500 at the base limit and 550 at the higher limit. Limited mean severity is $1,600 at the base limit and $2,000 at the higher limit. Find the limited-severity ratio and the ratio of aggregate expected losses. Explain why they differ. (Answer: severity ratio = 1.250; aggregate expected-loss ratio = $550(2{,}000)/[500(1{,}600)]=1.375$; frequency does not cancel because expected claim counts differ.)

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