ILFs From Discrete Severity Distributions

Why Discrete Severity Tables Are Useful

An increased limits factor (ILF) compares expected losses at two per-claim policy limits. When claim severity is given as a discrete distribution, the calculation is direct: for each possible severity, apply the limit, multiply the resulting payment by its probability, and add the weighted payments. The central exam skill is applying the cap to each severity outcome, rather than applying it to the average severity or to the probabilities.

This method is useful when a hypothetical portfolio is described by a severity table instead of a continuous probability distribution. It also gives a transparent check on the effect of a limit change: a severity below the limit is unchanged, while a severity above it contributes only the limit.

Definition: Let $X$ be the nonnegative ground-up loss on one claim, and let $u$ be a per-claim limit. The payment per loss at that limit is $Y^L(u)=\min(X,u)$. If $X$ takes values $x_1,\ldots,x_n$, with probabilities $p_1,\ldots,p_n$, then the limited expected loss per claim is $$ E[Y^L(u)]=\sum_{j=1}^{n}p_j\min(x_j,u), $$ where $p_j\geq 0$ and $\sum_{j=1}^{n}p_j=1$. The ILF at limit $u$, relative to a positive base limit $u_0$, is $$ \operatorname{ILF}(u;u_0)=\frac{E[Y^L(u)]}{E[Y^L(u_0)]}, $$ provided the denominator is positive.

These formulas assume a severity distribution for individual ground-up claims and a per-claim limit, with no deductible or other coverage modification included. They calculate expected claim payments, not expenses, profit provisions, or a final charged premium. A discrete table must describe mutually exclusive severity outcomes and their probabilities; check that its probabilities sum to one before calculating.

From Limited Severity to Expected Loss

The table calculation gives expected payment per claim. If claim frequency is also supplied, expected loss per exposure is the expected number of claims per exposure multiplied by expected payment per claim. Let $\lambda$ be the expected claim count per exposure, and assume claim severities have the stated distribution, with expected limited payment not varying by claim count. Then

E[\text{limited loss per exposure at }u] =\lambda E[Y^L(u)].

For one portfolio with the same frequency and severity basis at both limits, $\lambda$ cancels from the ratio. If the frequency differs between risk classes, calculate each class's expected loss at each limit and combine those expected losses using the exposure mix. A simple average of class ILFs generally does not give the portfolio ILF.

Validate the table.
Confirm that every listed severity has a probability and that all probabilities sum to one.

Apply the limit outcome by outcome.
For each severity $x_j$, calculate $\min(x_j,u)$. Do this separately for every limit being evaluated.

Weight and add.
Multiply each limited payment by its severity probability and sum to obtain expected payment per claim.

Form the ratio on a consistent basis.
Divide the expected loss at the selected limit by the expected loss at the base limit. Include frequency in both amounts if reporting per-exposure losses.

Why the Discrete Formula Holds

The severity outcomes partition the possible values of $X$. On the event $X=x_j$, the payment is exactly $\min(x_j,u)$, and that event has probability $p_j$. The expectation of a discrete random variable is the sum of each possible value times its probability. Therefore,

E[Y^L(u)] =\sum_{j=1}^{n}P(X=x_j)Y^L(u)\big|_{X=x_j} =\sum_{j=1}^{n}p_j\min(x_j,u).

If an exposure has $N$ claims, its total limited loss is the sum of the limited payments for those claims. Under the stated frequency-severity assumption, conditional expected total payment is $N E[Y^L(u)]$. Taking expectations over $N$ gives $E[N]E[Y^L(u)]$. Thus, when $E[N]=\lambda$, expected loss per exposure is $\lambda E[Y^L(u)]$. With the same positive $\lambda$ at both limits, division gives

\frac{\lambda E[Y^L(u)]}{\lambda E[Y^L(u_0)]} =\frac{E[Y^L(u)]}{E[Y^L(u_0)]}.

This cancellation is not a reason to ignore frequency when comparing different classes or different exposure mixes. It applies when the same frequency basis multiplies both expected severities.

Worked Example: Two Limits and a Frequency Conversion

A hypothetical liability portfolio has a claim frequency of $0.12$ claims per exposure-year. Conditional claim severity is $8,000 with probability $0.40$, $25,000 with probability $0.35$, and $60,000 with probability $0.25$. Calculate expected losses per exposure-year at limits of $15,000 and $40,000, and calculate the ILF at $40,000 relative to $15,000. Use the same frequency at both limits.

Method and table check. The probabilities sum to $0.40+0.35+0.25=1.00$. Apply each limit to each severity. All amounts below are dollars per claim.

At the $15,000 limit, the limited payments are $\min(8{,}000,15{,}000)=8{,}000$, $\min(25{,}000,15{,}000)=15{,}000$, and $\min(60{,}000,15{,}000)=15{,}000$. Thus,

E[Y^L(15{,}000)] =0.40(8{,}000)+0.35(15{,}000)+0.25(15{,}000) =3{,}200+5{,}250+3{,}750 =\$12{,}200.

At the $40,000 limit, the limited payments are $8,000, $25,000, and $40,000. Therefore,

E[Y^L(40{,}000)] =0.40(8{,}000)+0.35(25{,}000)+0.25(40{,}000) =3{,}200+8{,}750+10{,}000 =\$21{,}950.

Multiply each expected payment per claim by the frequency. The $15,000-limit expected loss is $0.12(\$12{,}200)=\$1{,}464$ per exposure-year. The $40,000-limit expected loss is $0.12(\$21{,}950)=\$2{,}634$ per exposure-year. Hence,

\operatorname{ILF}(40{,}000;15{,}000) =\frac{\$2{,}634}{\$1{,}464} =\frac{\$21{,}950}{\$12{,}200} =1.799180\ldots \approx 1.7992.

Check. The higher limit produces higher expected loss, and the per-exposure ratio agrees with the per-claim ratio because the same frequency multiplies both amounts. The common mistake avoided here is applying the cap only after averaging severity; the cap must be applied to each outcome first.

Worked Example: Several Candidate Limits From One Table

A hypothetical equipment policy has claim severity, in thousands of dollars, of 5 with probability $0.25$, 18 with probability $0.30$, 35 with probability $0.30$, and 90 with probability $0.15$. Calculate limited expected loss per claim at limits of $10,000, $30,000, and $75,000. Then calculate the ILFs at $30,000 and $75,000 relative to $10,000.

Method and table check. The probabilities sum to $0.25+0.30+0.30+0.15=1.00$. Work in thousands of dollars per claim. For a $10,000 limit, the capped severity values are 5, 10, 10, and 10:

$$ E[Y^L(10)] =0.25(5)+0.30(10)+0.30(10)+0.15(10) =1.25+3.00+3.00+1.50 =8.75. $$

For a $30,000 limit, the capped values are 5, 18, 30, and 30:

$$ E[Y^L(30)] =0.25(5)+0.30(18)+0.30(30)+0.15(30) =1.25+5.40+9.00+4.50 =20.15. $$

For a $75,000 limit, the capped values are 5, 18, 35, and 75:

$$ E[Y^L(75)] =0.25(5)+0.30(18)+0.30(35)+0.15(75) =1.25+5.40+10.50+11.25 =28.40. $$

The two ILFs are $20.15/8.75=2.302857\ldots$, or $2.3029$, and $28.40/8.75=3.245714\ldots$, or $3.2457$, each rounded to four decimal places.

Check. The limited expected losses increase from $8,750 to $20,150 to $28,400 per claim as the limit rises. The larger limit has the larger ILF relative to the same base. The common mistake avoided is treating a severity above a limit as fully paid; for example, the $90,000 outcome contributes $75,000, not $90,000, at the $75,000 limit.

Worked Example: Combining Two Risk Classes

A hypothetical book has 60% of its exposure-years in Class A and 40% in Class B. Class A has frequency $0.06$ claims per exposure-year and severity $10,000 with probability $0.50$, $30,000 with probability $0.30$, and $80,000 with probability $0.20$. Class B has frequency $0.10$ and severity $6,000 with probability $0.40$, $20,000 with probability $0.40$, and $50,000 with probability $0.20$. Calculate the portfolio ILF at $60,000 relative to $20,000.

Method and table check. Each class's probabilities sum to one. First calculate expected limited severity by class, then multiply by class frequency to get expected loss per exposure-year. Finally apply the 60%/40% exposure mix.

For Class A, at $20,000 the capped values are 10, 20, and 20, in thousands:

$$ E[Y^L_A(20)]=0.50(10)+0.30(20)+0.20(20)=5+6+4=15 $$

thousand dollars per claim, so its expected loss is $0.06(\$15{,}000)=\$900$ per exposure-year. At $60,000, the capped values are 10, 30, and 60:

$$ E[Y^L_A(60)]=0.50(10)+0.30(30)+0.20(60)=5+9+12=26 $$

thousand dollars per claim, giving $0.06(\$26{,}000)=\$1{,}560$ per exposure-year.

For Class B, at $20,000 the capped values are 6, 20, and 20:

$$ E[Y^L_B(20)]=0.40(6)+0.40(20)+0.20(20)=2.4+8+4=14.4 $$

thousand dollars per claim, giving $0.10(\$14{,}400)=\$1{,}440$ per exposure-year. At $60,000, the capped values are 6, 20, and 50:

$$ E[Y^L_B(60)]=0.40(6)+0.40(20)+0.20(50)=2.4+8+10=20.4 $$

thousand dollars per claim, giving $0.10(\$20{,}400)=\$2{,}040$ per exposure-year.

Combine the class expected losses using the exposure mix. At $20,000 the portfolio expected loss is $0.60(\$900)+0.40(\$1{,}440)=\$540+\$576=\$1{,}116$ per exposure-year. At $60,000 it is $0.60(\$1{,}560)+0.40(\$2{,}040)=\$936+\$816=\$1{,}752$ per exposure-year. Therefore,

\operatorname{ILF}_{\text{portfolio}}(60{,}000;20{,}000) =\frac{\$1{,}752}{\$1{,}116} =1.569892\ldots \approx 1.5699.

Check. The portfolio amounts use the same exposure weights at both limits, and the higher-limit expected loss is greater. The common mistake avoided is averaging class ILFs directly: Class A's ILF is $1{,}560/900=1.7333$, and Class B's is $2{,}040/1{,}440=1.4167$; their exposure-weighted average is not the portfolio ILF because the classes have different base expected losses.

Common Errors and Practical Checks

A discrete table makes an ILF calculation auditable, but a small error in the cap or weighting can materially change the ratio. Before accepting a result, check each of the following:

Cap every outcome separately. The payment is $\min(x_j,u)$, not the severity itself whenever $x_j>u$.
Keep per-claim and per-exposure amounts distinct. Multiply by frequency only when converting expected payment per claim to expected loss per exposure.
Use the same basis in numerator and denominator. If frequency, exposure mix, or severity assumptions differ between limits, do not cancel or omit them.
Check units. If severities are stated in thousands, the weighted sum is also in thousands until converted to dollars.
Check the base amount. The base expected loss must be positive for the ILF ratio to be defined.

One especially tempting wrong calculation is to take the mean severity first and then cap it. For a severity of $8,000, $25,000, and $60,000 with probabilities $0.40$, $0.35$, and $0.25$, the mean ground-up severity is $0.40(8{,}000)+0.35(25{,}000)+0.25(60{,}000)=\$28{,}950$. Capping that mean at $15,000 gives $15,000, which is not the limited expected loss. The correct calculation caps each outcome first and gives $0.40(8{,}000)+0.35(15{,}000)+0.25(15{,}000)=\$12{,}200$. In general, the order matters because $E[\min(X,u)]$ is not $\min(E[X],u)$.

Key takeaway: For a discrete severity distribution, an ILF is obtained by capping every severity at each candidate limit, taking the probability-weighted sum at each limit, and dividing by the corresponding base-limit expected loss. Keep the severity, frequency, and exposure-mix bases consistent.

Check Your Understanding

For each problem, apply the limit to each severity outcome before weighting. Report ILFs to four decimal places.

A hypothetical property portfolio has severity $4,000 with probability $0.20$, $12,000 with probability $0.50$, and $30,000 with probability $0.30$. Find limited expected losses per claim at $10,000 and $25,000, and the ILF at $25,000 relative to $10,000. (Answer: $8,800 and $14,300 per claim; ILF $=1.6250$.)
A hypothetical liability book has severity $15,000 with probability $0.40$, $40,000 with probability $0.35$, and $100,000 with probability $0.25$. Find expected losses per claim at limits of $25,000 and $70,000, then calculate the ILF. (Answer: $21,000 and $37,500 per claim; ILF $=1.7857$.)
A hypothetical coverage has frequency $0.08$ claims per exposure-year and severity $9,000 with probability $0.50$, $24,000 with probability $0.30$, and $55,000 with probability $0.20$. Find expected losses per exposure-year at $20,000 and $45,000 limits and the ILF. (Answer: $1,160 and $1,656 per exposure-year; ILF $=1.4276$.)
A hypothetical equipment portfolio has severity $2,000 with probability $0.20$, $10,000 with probability $0.50$, and $40,000 with probability $0.30$. Find limited expected losses at $8,000 and $25,000, then the ILF at $25,000 relative to $8,000. (Answer: $6,800 and $12,900 per claim; ILF $=1.8971$.)
A hypothetical policy has severity $50,000 with probability $0.60$ and $120,000 with probability $0.40$. Find the limited expected losses at $80,000 and $150,000, and the ILF at $150,000 relative to $80,000. (Answer: $62,000 and $78,000 per claim; ILF $=1.2581$.)

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