Common Denominator and Censoring Errors in ILFs

Why Consistent Loss Bases Matter

An increased limits factor (ILF) compares expected loss under a higher limit with expected loss under a lower, base limit. The comparison is meaningful only when both amounts refer to the same loss basis and exposure. A denominator from a different base limit, a capped claim treated as uncensored, or a ground-up amount mixed with a limited amount can all produce a plausible-looking but biased factor.

For a ground-up loss $X$ and a policy limit $L$, the loss payment before any other coverage terms is $\min(X,L)$. This tutorial focuses on auditing ILF calculations for three specific errors: incompatible base limits, policy-limit censoring, and mixing ground-up and limited losses. The formulas assume comparable coverage terms and a consistent valuation basis; differences in development, trend, or coverage must be addressed before making the comparison.

Definition: For a common population and a fixed valuation basis, the ILF at target limit $L$ relative to base limit $B$, where $L\geq B>0$, is $$ \operatorname{ILF}(L;B)=\frac{E[\min(X,L)]}{E[\min(X,B)]}. $$ The denominator is the expected loss limited at $B$, not ground-up expected loss and not expected loss at some other base limit. Since $\min(X,L)\geq\min(X,B)$ for every loss when $L\geq B$, the factor is at least 1 when the denominator is positive.

Pooling Requires a Common Denominator

When a portfolio contains groups with different expected losses, its combined ILF is not generally the unweighted average of the groups’ ILFs. If group $j$ has expected base-limited loss $D_j$ and target-limited loss $N_j$, its factor is $N_j/D_j$. Provided the group amounts use the same base limit, target limit, and valuation basis, the portfolio factor is

\frac{\sum_j N_j}{\sum_j D_j} = \sum_j \left(\frac{D_j}{\sum_k D_k}\right)\frac{N_j}{D_j}.

Thus, the correct weights are each group’s shares of total expected base-limited loss. The expression follows by expanding the right-hand side: each term is $D_jN_j/(D_j\sum_k D_k)=N_j/\sum_k D_k$, and summing gives the left-hand side. Equal group weights are justified only when the groups have equal expected base-limited losses.

This pooling identity does not make factors with different base limits comparable. Before pooling, convert each group’s numerator and denominator to amounts on one common base. An exposure curve can provide the relationship between expected losses at two limits for a group; without a valid conversion, the factors do not share a denominator definition.

Worked Example: Pooling Groups With Different Base Limits

A hypothetical portfolio has two groups and a target limit of $150,000. Group A has expected losses of $100 thousand at its $50,000 base limit and $150 thousand at the target limit. Group B has expected losses of $990 thousand at its $100,000 base limit and $1,260 thousand at the target limit. For Group B, an exposure-curve estimate gives $900 thousand at $50,000. Estimate the portfolio ILF relative to $50,000.

Assumptions and method. The amounts are expected losses for the same valuation period, expressed in thousands of dollars. They require no assumed severity distribution because the expected amounts are supplied. Interest and mortality are not applicable; there is no discounting. First put both groups on the $50,000 base.

Group A is already expressed at that base. Its target factor is $150/100=1.5$.

Group B’s factor relative to the required $50,000 base is $1{,}260/900=1.4$. Its factor relative to its own $100,000 base would instead be $1{,}260/990=1.2727$ (rounded to four decimals). That is not the factor to pool with Group A’s $50,000-base factor.

The common-base expected losses are $100 thousand and $900 thousand, for a total base loss of $100+900=1{,}000$ thousand. The corresponding target losses total $150+1{,}260=1{,}410$ thousand. Therefore,

\operatorname{ILF}(150;50)=\frac{1{,}410}{1{,}000}=1.410.

As a check, Group A’s share of total base expected loss is $100/1{,}000=0.1$, or one-tenth; Group B’s share is $900/1{,}000=0.9$. The weighted calculation is $0.1(1.5)+0.9(1.4)=0.15+1.26=1.41$, matching the ratio of totals.

Common mistake avoided. An unweighted average of the common-base group factors gives $(1.5+1.4)/2=1.45$, which incorrectly gives each group half the weight. Averaging the factors relative to each group’s own base, $(1.5+1.2727)/2=1.3864$, is also invalid because those factors have different denominators. The denominator for the correct portfolio factor is the total expected loss at the same $50,000 base.

Recognizing Policy-Limit Censoring

A recorded claim payment may be capped by the policy limit. If the ground-up claim is $X$ and the policy limit is $C$, the observed payment is $Y=\min(X,C)$. To estimate the payment at target limit $L$, the relevant quantity is $\min(X,L)$, not automatically $\min(Y,L)$.

When $C\geq L$, the observed payment identifies the target-limit payment: $\min(Y,L)=\min(X,L)$. When $C<L$ and the claim reaches the cap, $Y=C$, and the ground-up loss is unknown above $C$. The target payment can then range from $C$ to $L$. The record identifies the payment at a base limit no greater than $C$, but it does not by itself identify the payment at the higher target. If the observed payment is below $C$, the claim was not capped and the ground-up loss is observed, so its target payment can be calculated.

This distinction matters when estimating an exposure curve from policy records. Simply treating each capped payment as the ground-up loss understates the target payment for claims censored below the target. Excluding such records is not automatically a solution: the remaining records may represent a different risk mix. A correction requires a defensible source of information, such as suitable uncensored observations or a justified model, and must account for how the data were selected.

Worked Example: A Capped Claim Below the Target Limit

A hypothetical sample contains four equally weighted risks. Their ground-up losses, used here to check the arithmetic, are $140, $120, $180, and $300. Their policy limits are $100, $200, $400, and $400, respectively. Estimate the sample ILF for a $300 target relative to a $100 base, and show what happens if a capped payment is mistaken for ground-up loss.

Assumptions and method. The empirical distribution assigns probability $1/4$ to each of the four losses; there is no parametric severity distribution. The values are in dollars for one common valuation period, with no discounting; interest and mortality are not applicable. Calculate the base- and target-limited payments claim by claim.

The observed payments $Y=\min(X,C)$ are $100, $120, $180, and $300. The first claim is censored at $100, which is below the $300 target. The other three policy limits are at least $300, so their target-limited payments are identified.

At the $100 base limit, the payments are $100,100,100,100$. The total is $400$, so the empirical expected base-limited loss is $400/4=\$100$.

Using the ground-up values for this validation calculation, the true target-limited payments are $140,120,180,300$. Their total is $740$, so the empirical expected target-limited loss is $740/4=\$185$. The factor is therefore $185/100=1.850$.

If the first observed payment of $100 is incorrectly treated as the ground-up loss, the purported target payments become $100,120,180,300$. Their total is $700$, their average is $700/4=\$175$, and the resulting factor is $175/100=1.750$. This understates the factor by $1.850-1.750=0.100$.

Check and limitation. The correctly calculated target-limited payment for every claim is at least its base-limited payment, and the sample factor exceeds 1. But the true first loss of $140 was supplied only to verify the example. In an actual capped record with $Y=C=100$, that value would not be known. The capped record alone cannot establish the $300 target payment.

Keep Ground-Up and Limited Losses Separate

A ground-up loss includes the full loss before applying a policy limit. A limited loss is the amount payable under a specified limit. An ILF compares two limited expected losses; replacing either one with a ground-up expected loss changes the quantity being measured. This error can inflate or otherwise distort a factor even when all claims are uncensored and the arithmetic is correct.

For a random loss $X$, the base and target quantities are $E[\min(X,B)]$ and $E[\min(X,L)]$. In general, neither equals $E[X]$. The values happen to agree with ground-up expected loss only when the limit is high enough that it never truncates any loss in the distribution.

Worked Example: Detecting a Ground-Up Numerator

A hypothetical severity model has $X=\$50$ with probability $0.5$, $X=\$150$ with probability $0.3$, and $X=\$350$ with probability $0.2$. Calculate the ILF at a $300 target relative to a $100 base, and compare it with the result from mistakenly using ground-up expected loss in the numerator.

Assumptions and method. The distribution is discrete with the probabilities and loss amounts stated; the probabilities sum to $0.5+0.3+0.2=1.0$. Losses are measured in dollars on the same valuation basis, with no discounting. Interest and mortality are not applicable.

At the $100 base, the limited losses are $50, $100, and $100. Thus,

E[\min(X,100)]=0.5(50)+0.3(100)+0.2(100)=25+30+20=\$75.

At the $300 target, the limited losses are $50, $150, and $300. Thus,

E[\min(X,300)]=0.5(50)+0.3(150)+0.2(300)=25+45+60=\$130.

The correct ILF is $130/75=26/15=1.7333$, rounded to four decimals. For comparison, the ground-up expected loss is $E[X]=0.5(50)+0.3(150)+0.2(350)=25+45+70=\$140$. Substituting this ground-up amount for the target-limited numerator produces $140/75=1.8667$, not the ILF. The $350 loss is limited to $300 under the target policy, so its extra $50 must not be included in that numerator.

Check. The target limit exceeds the base limit, and the correct factor $1.7333$ is greater than 1. The numerator uses $130 of expected target-limited loss, not $140 of ground-up loss.

Audit Sequence and Exam Traps

A reliable ILF review checks the loss basis before accepting a ratio. The following sequence separates denominator errors from censoring and loss-basis errors.

State the limits being compared.
Write the target limit and the single base limit explicitly. Verify that the target is at least as high as the base.

Define both loss amounts.
Confirm that the numerator is expected loss limited at the target and the denominator is expected loss limited at the base, on the same valuation and coverage basis.

Check censoring record by record.
For target-limit calculations, identify observations whose policy caps are below the target and whose payments reach those caps. Do not treat those capped payments as known ground-up losses.

Align before pooling.
Convert group amounts to the same base and target limits. Then pool loss amounts, or equivalently weight group factors by their shares of total base-limited expected loss.

A quick diagnostic is that, for a common population and consistent data, an ILF at a higher limit should not be below 1. A factor below 1 may signal a mismatched base, inconsistent coverage, a data-selection issue, or an error in the calculation; it is a warning to investigate, not proof of a particular cause. Conversely, a factor above 1 does not establish that the calculation is valid: mismatched denominators and ground-up substitutions can still produce plausible values.

Key takeaway: An ILF is a ratio of target-limited expected loss to base-limited expected loss on one consistent basis. Align denominators before pooling, recognize that a cap below the target can conceal the needed payment, and never substitute ground-up loss for a limited loss.

Check Your Understanding

For each problem, identify the appropriate common loss basis and check whether the available records support the requested calculation.

Two groups have expected losses at a common base of $80 and $720. Their target-to-base factors are 1.25 and 1.50. Calculate the combined ILF. (Answer: 1.475)
A hypothetical severity distribution has $X=\$20$ with probability 0.40, $X=\$100$ with probability 0.35, and $X=\$400$ with probability 0.25. Calculate the ILF at limit $200 relative to limit $50. (Answer: $93/38=2.4474$, rounded to four decimals)
Three claims have observed payments and policy limits of ($80, $80), ($140, $200), and ($250, $300). For a $250 target limit, determine whether the exact average target-limited payment is identifiable from these records alone. Give the range consistent with the records. (Answer: Not identifiable; the average can range from $470/3=\$156.67$ to $640/3=\$213.33$, rounded to cents)
A severity distribution has $X=\$100$ with probability 0.5 and $X=\$300$ with probability 0.5. Calculate the ILF at target limit $250 relative to base limit $150, and state the result of incorrectly using ground-up expected loss in the numerator. (Answer: Correct ILF $175/125=1.400$; incorrect mixed-basis ratio $200/125=1.600$)

What You'll Learn