Split Limits And Coverage-Specific ILFs

Why Split Limits Need Their Own Loss Basis

A split-limit liability policy places more than one cap on covered losses. In a bodily injury (BI) limit written as, for example, $75,000 per person and $150,000 per accident, the per-person cap applies to each injured person's covered loss. The per-accident cap then applies to the sum of those capped payments for one accident. The two limits work together; neither one can be interpreted in isolation.

Increased limits factors (ILFs) compare expected covered losses at different policy limits. With split limits, the expected loss must be calculated using the actual terms for that coverage. A BI limit structure acts on the BI claims arising from an accident. Property damage (PD), if insured under its own limit, has a separate covered-loss basis. Combining those bases too early can produce an incorrect indication.

Definition: Let $X_j$ be the covered BI loss for injured person $j$ in one accident, and let $P$ and $A$ denote the per-person and per-accident limits. The insurer's payment for that accident is $$ L(P,A)=\min\left(\sum_{j=1}^{N}\min(X_j,P),\,A\right), $$ where $N$ is the number of injured people with covered losses. For a specified accident-loss model, the split-limit ILF relative to base limits $(P_0,A_0)$ is $$ \operatorname{ILF}(P,A;P_0,A_0) =\frac{E[L(P,A)]}{E[L(P_0,A_0)]}, $$ provided the denominator is positive and both expected losses use the same population, loss basis, and valuation assumptions.

The formula describes the order of operations: first cap each person's covered loss at $P$, then add those capped amounts, and finally cap the accident total at $A$. It does not imply that losses must be independent. If claimant losses or claim counts are dependent, their joint distribution must be represented in the accident model.

Calculating Payments and the Split-Limit ILF

For a finite set of possible accident outcomes, assign each outcome its probability, calculate the payment under each limit schedule, and take the probability-weighted average. The following derivation explains why this average is the required expected loss.

Apply each limit to an outcome.
For outcome $k$, with probability $q_k$ and claimant losses $x_{k1},\ldots,x_{kN_k}$, calculate $l_k(P,A)=\min(\sum_j\min(x_{kj},P),A)$.

Average over outcomes.
By the definition of expectation for a discrete random variable, $E[L(P,A)]=\sum_k q_k l_k(P,A)$, where $\sum_k q_k=1$.

Compare like with like.
Apply the same calculation to the base limits. The ratio of the two averages compares expected covered losses under the two schedules while holding the accident model constant.

This method also explains a key limitation: in general, an ILF is not the ratio of the new limits to the old limits. The payment is capped and therefore does not increase in direct proportion to either limit. Nor is a split-limit ILF generally the product of a per-person factor and an accident-limit factor. The two caps act on the same set of claims, so changing one can affect how much of the other cap is used.

Worked Example: Calculating a Split-Limit BI Factor

A hypothetical insurer models BI accidents using the following distribution: with probability 0.50, one person has a covered loss of $40,000; with probability 0.30, two people each have a covered loss of $80,000; and with probability 0.20, two people each have a covered loss of $140,000. Calculate the ILF for limits increasing from $50,000 per person/$75,000 per accident to $100,000 per person/$150,000 per accident.

Assumptions and method. These are the complete accident outcomes and probabilities for the hypothetical model; no other outcomes are included. There is no discounting (effective annual interest rate $i=0$) and mortality is not relevant to this liability model. For each outcome, apply the per-person cap first and the per-accident cap second.

Base limits, $50,000/$75,000. In the first outcome, the payment is $\min(40,50,75)=\$40,000$, where amounts in this calculation are in thousands of dollars. In the second outcome, the per-person payments are $50+50=100$, then the accident cap reduces the payment to $\min(100,75)=\$75,000$. In the third outcome, the per-person payments are again $50+50=100$, so the accident payment is also $75,000. Thus,

E[L(50,75)] =0.50(40)+0.30(75)+0.20(75) =20+22.5+15 =57.5\text{ thousand dollars}.

Higher limits, $100,000/$150,000. The first outcome pays $40,000. The second pays $80+80=160$ before the accident cap, so it pays $150,000. The third pays $100+100=200$ before the accident cap, so it also pays $150,000. Therefore,

E[L(100,150)] =0.50(40)+0.30(150)+0.20(150) =20+45+30 =95\text{ thousand dollars}.

The ILF is $95/57.5=1.6521739\ldots$, or 1.652 to three decimal places. The calculation explicitly applies both caps in each outcome. A mistake to avoid is treating the two claims in an accident as if each had a separate $75,000 accident limit.

Coverage-Specific Factors and Premium Indications

An ILF is meaningful only relative to a stated loss basis and base limit. For BI, that basis may be the capped total of BI payments per accident under the split schedule. For PD, it may be the covered PD loss for an occurrence under a separate limit. Even when the same accident can produce both BI and PD losses, an increase in the BI limits does not directly change the PD payment cap, and vice versa.

When the two coverages are rated separately, calculate an ILF for each coverage using its own modeled losses and limit terms. If a combined indication is needed, apply each factor to its corresponding base expected loss or base premium, then add the indicated components. The combined factor is a base-weighted result, not generally the simple average of the component factors.

Worked Example: Keeping BI and PD Bases Separate

A hypothetical insurer has a BI model with these accident outcomes: probability 0.40 of one $30,000 BI loss; probability 0.40 of two BI losses of $50,000 each; and probability 0.20 of two BI losses of $120,000 and $40,000. The BI limits change from $40,000 per person/$70,000 per accident to $80,000 per person/$120,000 per accident. Separately, PD claim severity is $20,000 with probability 0.50 and $60,000 with probability 0.50; its limit changes from $40,000 to $80,000. Find each ILF and the combined factor, using the base expected losses as the component weights.

Assumptions and method. The stated probabilities form the full distribution for each coverage model. There is no discounting ($i=0$); mortality is not applicable. Calculate the BI factor on BI accident payments and the PD factor on PD claim payments, then combine the indicated expected losses.

BI at base limits. The three payments are $30$, $\min(40+40,70)=70$, and $\min(40+40,70)=70$, in thousands of dollars. The last outcome has per-person losses $120$ and $40$, capped to $40$ and $40$. Therefore the base BI expected loss is $0.40(30)+0.40(70)+0.20(70)=12+28+14=\$54,000$.

BI at higher limits. The payments are $30$, $\min(50+50,120)=100$, and $\min(80+40,120)=120$. The higher-limit BI expected loss is $0.40(30)+0.40(100)+0.20(120)=12+40+24=\$76,000$. Thus the BI ILF is $76/54=1.407407\ldots$, or 1.407 to three decimal places.

PD factor. At the $40,000 limit, the two payments are $20,000 and $40,000, giving $0.50(20)+0.50(40)=\$30,000$. At the $80,000 limit, they are $20,000 and $60,000, giving $0.50(20)+0.50(60)=\$40,000$. The PD ILF is $40/30=1.333333\ldots$, or 1.333 to three decimal places.

Combined factor. The combined base expected loss is $54+30=\$84,000$, and the combined higher-limit expected loss is $76+40=\$116,000$. The combined factor is $116/84=1.380952\ldots$, or 1.381 to three decimal places. As a check, weighting the component factors by their base expected losses gives $(54(1.407407\ldots)+30(1.333333\ldots))/84=116/84$. Taking the unweighted average of the two factors would give the wrong combined indication because BI and PD have different base expected losses.

Common Traps and Practical Checks

A split-limit calculation is especially vulnerable to errors of order, basis, and interpretation. The limits are not two interchangeable descriptions of one cap. Each step has a defined role, and the loss being limited must match the coverage being rated.

Applying the accident cap before the per-person cap. First cap each individual's loss, then sum those payments and apply the accident limit.
Using the per-person limit as the total accident limit. With multiple injured people, the sum of their payments can exceed the per-person cap, subject to the separate accident cap.
Applying a BI factor to PD losses. Use the limit terms and severity basis for each coverage. A BI factor does not automatically describe PD limit changes.
Multiplying two marginal factors. Separate factors for the per-person and per-accident changes need not combine multiplicatively. Recalculate expected losses under the complete new schedule unless the model supplies a justified decomposition.
Comparing different bases. The numerator and denominator of an ILF must use the same population, coverage definition, and valuation assumptions. A change in exposure or loss trend is not a limit-only effect.

For example, in the first worked example, a mistaken base-limit calculation might add two $50,000 per-person payments and record $100,000, ignoring the $75,000 per-accident cap. The correct payment is $\min(50+50,75)=\$75,000$. Since this error overstates the base expected loss, it understates the calculated ILF. Always check that each outcome's final payment is no greater than either the applicable per-person cap for each individual or the accident cap for the occurrence.

Split-limit ILFs matter because the purchased coverage is defined by interacting caps, while pricing often needs a limit-specific expected loss or premium indication. A transparent outcome-by-outcome calculation preserves the policy terms, keeps coverage bases distinct, and makes the result auditable. This treatment is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.

Check Your Understanding

For each problem, state the loss basis and apply the stated limits in the order defined above. Dollar amounts are in thousands unless otherwise specified.

A BI claim has severity $25,000 with probability 0.40, $70,000 with probability 0.40, and $150,000 with probability 0.20. Find the ILF from a $40,000 per-person limit to a $100,000 per-person limit, with no accident cap. (Answer: $58/34=1.7059$, to four decimal places.)
A BI accident has probability 0.60 of one $30,000 loss, probability 0.25 of two $60,000 losses, and probability 0.15 of losses of $120,000 and $30,000. Calculate the ILF from $40,000/$80,000 to $80,000/$120,000 per person/per accident. (Answer: $64.5/48.5=1.3299$, to four decimal places.)
Base expected losses are $90,000 for BI and $30,000 for PD. Their respective coverage-specific ILFs are 1.40 and 1.20. Find the indicated combined expected loss and combined ILF. (Answer: $162,000 and $162/120=1.3500$.)
For a single accident, three BI losses are $90,000, $70,000, and $20,000. Calculate the payment under limits of $75,000 per person and $125,000 per accident. Explain which cap changes the uncapped total. (Answer: $\min(75+70+20,125)=\$125,000$; both the per-person cap and the accident cap reduce the uncapped $180,000 total.)
A hypothetical model has two equally likely BI accident outcomes: one $40,000 loss, or two $100,000 losses. Calculate the expected payment under $60,000/$90,000 limits and under $120,000/$180,000 limits, then find the ILF. (Answer: base $0.50(40)+0.50(90)=\$65,000$; higher $0.50(40)+0.50(180)=\$110,000$; ILF $110/65=1.6923$, to four decimal places.)

What You'll Learn