Sensitivity Of ILFs To The Severity Tail

Why Severity-Tail Assumptions Matter

An increased limits factor (ILF) compares expected losses at different policy limits. The comparison depends on the severity distribution: a model that assigns more probability to large losses can produce a different amount of expected loss above a given limit. But an ILF is a ratio, not simply the expected loss at the higher limit. A tail assumption can affect both the numerator and the base-limit denominator, so a heavier tail does not automatically imply a higher ILF.

To isolate this effect, compare models using the same loss definition, policy terms, base limit, and increased limit. Where useful, also hold the mean severity fixed. Differences in the resulting ILFs then reflect the models’ different allocations of probability across the severity range, rather than a change in average severity or coverage basis.

Definition: For nonnegative ground-up claim severity $X$, define the limited loss at limit $L$ as $\min(X,L)$. On a common loss basis, the ILF for increased limit $L$ relative to base limit $b$ is $$ \operatorname{ILF}(L;b)=\frac{E[\min(X,L)]}{E[\min(X,b)]}, \qquad L\ge b>0. $$ The ratio assumes the same severity population and coverage basis at both limits, and a positive base-limit expected loss. It compares expected loss only; expenses, risk loads, and other premium components require separate treatment unless explicitly included in the modeled loss.

This definition applies to a per-claim limit and a severity variable defined on the same basis in both terms. It does not by itself handle an aggregate limit, a deductible, or a change in coverage terms. Those changes must be reflected in the payment variable before making the comparison. A consistent loss basis is essential: comparing gross losses at one limit with net losses at another does not measure sensitivity to the severity tail alone.

A Direct Calculation for Limited Expected Loss

For a continuous severity distribution with density $f_X(x)$, distribution function $F_X(x)$, and survival function $S_X(x)=1-F_X(x)$, partition claims according to whether they exceed the limit. Claims at or below $L$ pay $X$; claims above $L$ pay $L$. Therefore,

E[\min(X,L)] =\int_0^L x f_X(x)\,dx+L\,S_X(L).

This identity follows directly from the payment rule: the first term averages the actual loss over claims below the limit, and the second adds the capped payment for claims above it. It is useful here because it lets us compute the same limited loss under different severity assumptions. The formula as written assumes a continuous distribution; if there is probability mass exactly at the limit, the payment definition still applies, but the calculation should account for that mass explicitly.

Model conditions: The comparisons below assume nonnegative claim severities, no discounting, and no mortality or survival-contingent payments. Thus no interest or mortality assumption enters the calculations. Any stated dollar amounts are hypothetical. Severity parameters and limits use the same units within each example.

Worked Comparisons Across Severity Models

Worked Example: Exponential Versus Pareto at Common Limits

A hypothetical insurer wants to compare two severity assumptions for the same coverage. Both models have mean severity of $100,000. Calculate the ILF for a $200,000 limit relative to a $50,000 base limit under (a) an exponential model and (b) a Pareto Type I model. All amounts below are in thousands of dollars. There is no discounting, and no mortality assumption applies.

Method and assumptions. Under the exponential model, let $X$ have mean $\mu=100$, so $f_X(x)=\frac{1}{100}e^{-x/100}$ for $x\ge0$. Under the Pareto Type I model, let the scale be $\theta=50$ and shape be $\alpha=2$, so $S_X(x)=1$ for $0\le x<50$ and $S_X(x)=(50/x)^2$ for $x\ge50$. Its mean is $\alpha\theta/(\alpha-1)=100$.

For the exponential model, the direct limited-loss calculation gives $E[\min(X,L)]=\mu(1-e^{-L/\mu})$. At the base limit, $$ E[\min(X,50)]=100(1-e^{-0.5})=100(1-0.60653066)=39.346934. $$ At the increased limit, $$ E[\min(X,200)]=100(1-e^{-2})=100(1-0.13533528)=86.466472. $$ Therefore, the exponential-model ILF is $86.466472/39.346934=2.1975$, to four decimal places.

For the Pareto model, the base limit equals the minimum possible severity, so every claim reaches the $50,000 cap. Thus $E[\min(X,50)]=50$. For $L\ge\theta$, integrating the payment below the limit and adding the capped payment above it gives $$ E[\min(X,L)]=\theta+\int_\theta^L (50/x)^2\,dx =50+50\left(1-\frac{50}{L}\right). $$ At $L=200$, this is $50+50(1-50/200)=50+37.5=87.5$. The Pareto ILF is $87.5/50=1.7500$, to four decimal places.

Check and interpretation. Both models have mean 100, and both limited expected losses increase from the base to the higher limit. Yet their ILFs differ: 2.1975 versus 1.7500. The Pareto model has a higher limited expected loss at $200,000 (87.5 versus 86.466472), but it also has a higher base-limit limited expected loss (50 versus 39.346934). The denominator effect is why its ILF is lower. The common mistake is to infer the ILF ranking from the higher-limit expected losses alone.

Worked Example: Changing the Pareto Shape Parameter

Compare two Pareto Type I models with the same mean severity of $100,000. Model A has shape $\alpha=2$; Model B has shape $\alpha=3$. For each, set the scale to produce the stated mean, then calculate the ILF for a $240,000 limit relative to an $80,000 base limit. Amounts are in thousands of dollars. There is no discounting or mortality assumption.

Method and parameter check. For a Pareto Type I distribution with $\alpha>1$, the mean is $\alpha\theta/(\alpha-1)$. Thus Model A uses $\theta=50$, because $2(50)/(2-1)=100$. Model B uses $\theta=200/3=66.666667$, because $3(200/3)/(3-1)=100$. Both models therefore have the same mean.

When $L\ge\theta$, the direct limited-loss calculation is $E[\min(X,L)]=\alpha\theta/(\alpha-1)-\theta^\alpha L^{1-\alpha}/(\alpha-1)$. For Model A, this reduces to $100-2500/L$. Hence the base limited loss is $100-2500/80=68.75$, and the higher-limit limited loss is $100-2500/240=89.583333$. Its ILF is $89.583333/68.75=1.3030$, to four decimal places.

For Model B, the formula is $100-\theta^3/(2L^2)$, with $\theta^3=(200/3)^3=296296.296296$. At the base limit, the limited loss is $100-296296.296296/(2\cdot80^2)=100-23.148148=76.851852$. At the higher limit, it is $100-296296.296296/(2\cdot240^2)=100-2.572016=97.427984$. The ILF is $97.427984/76.851852=1.2677$, to four decimal places.

Check and interpretation. Model A’s ILF exceeds Model B’s, even though Model A has the heavier Pareto tail in this equal-mean comparison. Model A also has a lower limited expected loss at each of these two limits. The base-limit denominator grows enough under Model B to reduce its ratio. A common mistake is to treat a change in the tail parameter as if it affects only losses above the higher limit; in fact, it changes the full severity distribution and both limited expected losses.

Worked Example: A Tail Extension in a Discrete Severity Model

Two hypothetical portfolios each have mean claim severity of $90,000. In Model A, severity is $50,000 with probability 0.80 and $250,000 with probability 0.20. In Model B, severity is $50,000 with probability 0.90 and $450,000 with probability 0.10. Calculate each ILF for a $300,000 limit relative to a $100,000 base limit. Amounts are in thousands of dollars. No discounting or mortality assumption applies.

Method and distribution check. First verify the means: Model A has $0.80(50)+0.20(250)=40+50=90$. Model B has $0.90(50)+0.10(450)=45+45=90$. For Model A, the base limited loss is $0.80(50)+0.20(100)=40+20=60$. At the $300,000 limit, neither possible loss is capped below its amount, so the limited loss is $0.80(50)+0.20(250)=40+50=90$. Its ILF is $90/60=1.5000$.

For Model B, the base limited loss is $0.90(50)+0.10(100)=45+10=55$. At the $300,000 limit, the $450,000 loss is capped, so the limited loss is $0.90(50)+0.10(300)=45+30=75$. Its ILF is $75/55=1.3636$, to four decimal places.

Check and interpretation. Model B places probability on a more extreme loss, but its ILF is lower. At the higher limit its expected limited loss is also lower, while the relative increase from its own base limit is smaller. The models cross in their tail probabilities, so this is a tail-extension sensitivity test, not a claim that one distribution is uniformly heavier at every severity. The trap is to assume that the most extreme possible claim determines the ILF without checking the probability-weighted base and increased limited losses.

Derivation and Practical Interpretation

For a continuous exponential severity with mean $\mu$, the density and survival function are $f_X(x)=\mu^{-1}e^{-x/\mu}$ and $S_X(x)=e^{-x/\mu}$. Applying the direct partition formula gives

\begin{aligned} E[\min(X,L)] &=\int_0^L x\frac{1}{\mu}e^{-x/\mu}\,dx+Le^{-L/\mu}\\ &=\mu(1-e^{-L/\mu})-Le^{-L/\mu}+Le^{-L/\mu}\\ &=\mu(1-e^{-L/\mu}). \end{aligned}

The integral is evaluated by integration by parts. The two terms involving $Le^{-L/\mu}$ cancel, leaving the stated limited expected loss. This derivation shows why the formula depends on the limit relative to the severity scale $\mu$, rather than on the limit alone.

More generally, the ILF’s sensitivity is governed by how much each model changes the numerator relative to its effect on the denominator. A useful comparison table should therefore show both limited expected losses, not just the final ratio.

Modeling choice	What to hold fixed	What the comparison reveals
Alternative distribution families	Mean, loss basis, and both limits where possible	Sensitivity to different overall severity shapes
Alternative Pareto shape parameters	Mean and both limits	Sensitivity to tail shape after adjusting the scale
Alternative tail extensions	Mean and coverage limits	How additional extreme-loss probability affects both parts of the ratio

For ratemaking, tail sensitivity matters most when the increased limit reaches into a range with sparse or censored observations. A model may fit the observed body of the severity distribution reasonably well while assigning quite different probabilities to losses beyond the data. The implied ILF can consequently be sensitive to an assumption that the data do not strongly determine. A comparison is most informative when it reports the model, parameters, limited expected loss at each limit, and resulting ILF together.

Sensitivity analysis is not a substitute for choosing a defensible severity model. It helps identify which assumptions matter and whether a pricing indication depends heavily on tail extrapolation. If the ILF changes materially across plausible models, that uncertainty should be recognized in the actuarial review rather than hidden by reporting a single ratio.

Common Traps

Ranking ILFs by the higher-limit loss alone. The denominator is model-dependent too. Calculate both limited expected losses before taking their ratio.
Changing the mean while claiming to isolate tail shape. When comparing shape assumptions, adjust scale parameters if the intended comparison holds mean severity fixed.
Using inconsistent loss bases. The limits must be applied to the same ground-up or otherwise defined severity basis. A change in deductible or coverage terms is not merely a tail change.
Assuming a universal direction. Tail changes can affect probability at several ranges, and ILFs are ratios. No general rule says that every heavier-tail assumption must increase the ILF for every pair of limits.
Ignoring model support and parameter conditions. For the Pareto Type I mean formula used here, $\alpha>1$ is required. Also check whether a limit lies below or above the scale before applying a formula derived for $L\ge\theta$.

Key takeaway: An ILF is sensitive to the severity distribution through both its increased-limit and base-limit expected losses. Compare plausible tail assumptions at the same limits, verify the loss basis and parameters, and interpret the ratio alongside its two component limited expected losses.

Check Your Understanding

For each problem, use the stated severity model and limits. Round only the final ILF to four decimal places.

A hypothetical exponential severity has mean $120,000. Find the ILF for a $180,000 limit relative to a $60,000 base limit. Assume no discounting or mortality. (Answer: 1.9744)
A Pareto Type I severity has shape $\alpha=2$ and scale $\theta=40$, in thousands of dollars. Find the ILF for a $160,000 limit relative to an $80,000 base limit. (Answer: 1.1667)
A severity is $40,000 with probability 0.75 and $160,000 with probability 0.25. Find the ILF for a $200,000 limit relative to a $75,000 base limit. (Answer: 1.4359)
A Pareto Type I severity has shape $\alpha=3$ and scale $\theta=60$, in thousands of dollars. Find the ILF for a $180,000 limit relative to a $60,000 base limit. (Answer: 1.4444)

What You'll Learn