Left-Truncated Severity Samples Above Deductibles

Why the Deductible Changes the Likelihood

Suppose an insurer’s hypothetical data set records only claims that exceed a deductible. A claim below the deductible is absent, rather than recorded as a zero or as a known amount. The observed claims are therefore not a complete sample from the severity distribution: each recorded loss is known to have passed a selection threshold. Estimating a severity model as if these observations were an ordinary sample can distort the fitted parameters.

This situation is called left truncation. The key adjustment is to use the distribution of a loss conditional on exceeding the deductible. This tutorial focuses on fitting parametric severity models when the loss amounts above the thresholds are observed. It is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Definition: Let $X$ be a continuous loss with density $f_\theta(x)$ and survival function $S_\theta(x)=P(X>x)$, where $\theta$ denotes the model parameter or parameters. If a loss is observed only when it exceeds a known deductible $d$, its conditional density is $$ f_\theta(x\mid X>d)=\frac{f_\theta(x)}{S_\theta(d)},\qquad x>d. $$ The denominator is the probability that a loss enters the observed sample.

For $n$ independent observed losses $x_1,\ldots,x_n$, each selected by exceeding a known threshold $d_i$, the conditional likelihood is $$ L(\theta)=\prod_{i=1}^n \frac{f_\theta(x_i)}{S_\theta(d_i)},\qquad x_i>d_i. $$ When all observations share the same deductible $d$, set $d_i=d$ for every $i$. The conditional likelihood treats the number of observed claims as fixed. If the number of claims below the deductible, the total number of potential claims, or the total number of claims selected is also part of the data, that information may require additional likelihood terms.

Why Conditioning Produces the Correct Density

For a small interval of width $\Delta x$ around a value $x>d$, the probability of both exceeding the deductible and falling in that interval is approximately $f_\theta(x)\Delta x$. Among losses that are known to exceed the deductible, divide this joint probability by the probability of selection, $P(X>d)=S_\theta(d)$. Thus

P(x<X\leq x+\Delta x\mid X>d) \approx \frac{f_\theta(x)\Delta x}{S_\theta(d)}.

Dividing by $\Delta x$ and taking the limit gives $f_\theta(x\mid X>d)=f_\theta(x)/S_\theta(d)$. For independent observations selected at thresholds $d_i$, their conditional densities multiply, producing the likelihood above. The survival term is not optional: it corrects for the fact that losses which fail the selection rule cannot appear in the sample.

Specify the severity model.
Write its density $f_\theta$ and survival function $S_\theta$, including the parameterization and support.

Record each selection threshold.
For every observed loss $x_i$, identify the threshold $d_i$ that it had to exceed.

Form the conditional likelihood.
Use $f_\theta(x_i)/S_\theta(d_i)$ for each observation and multiply the contributions.

Maximize and check.
Estimate the parameters from the conditional likelihood, then check that the fitted model and resulting probabilities respect the thresholds and support.

Exponential Severity: The Excess Above the Deductible

Assume $X$ has an exponential distribution with mean parameter $\theta>0$. Its density and survival function are $f_\theta(x)=\theta^{-1}e^{-x/\theta}$ and $S_\theta(x)=e^{-x/\theta}$, for $x\geq 0$. Conditional on exceeding deductible $d$, $$ f_\theta(x\mid X>d)=\frac{1}{\theta}e^{-(x-d)/\theta},\qquad x>d. $$ Thus the excess $X-d$ among selected losses is itself exponential with mean $\theta$. This is the memoryless property, and it makes the conditional maximum likelihood estimate particularly direct.

Worked Example: Exponential Fit Above a Common Deductible

A hypothetical portfolio records only losses above a deductible of $200$ dollars. Four recorded losses are $250, 300, 400,$ and $450$ dollars. Assume independent losses from an exponential severity distribution with unknown mean $\theta$, and condition on the fact that each recorded loss exceeded $200$. Estimate $\theta$, then calculate the fitted probability that a recorded loss exceeds $500$ dollars, conditional on exceeding the deductible.

Method. Use the exponential conditional density, so the likelihood is based on the four excesses above $200$, not on the four loss amounts as a complete sample.

The excesses are $250-200=50$, $300-200=100$, $400-200=200$, and $450-200=250$ dollars. Their sum is $50+100+200+250=600$, and their mean is $600/4=150$ dollars. The log-likelihood, apart from the threshold-independent representation of the same conditional model, is

\ell(\theta)=-4\log\theta-\frac{600}{\theta}.

Differentiating gives $\ell'(\theta)=-4/\theta+600/\theta^2$. Setting this equal to zero yields $-4\theta+600=0$, so the maximum likelihood estimate is $\hat{\theta}=150$ dollars. The second derivative at this value is negative, confirming a maximum.

For a loss already known to exceed $200$, exceeding $500$ requires an excess greater than $500-200=300$. The fitted conditional probability is

P(X>500\mid X>200)=e^{-(500-200)/150}=e^{-2}=0.1353.

The probability is rounded to four decimals. Check: the estimate is the mean observed excess, $150$ dollars, and the probability is between zero and one. The common mistake avoided is using the mean of the original losses, $350$ dollars, as the exponential mean despite observing only losses above $200$.

Pareto Severity: Conditioning Changes the Tail Likelihood

For a Pareto Type I severity model with known lower bound $x_m$ and shape parameter $\alpha>0$, the survival function is $S(x)=(x_m/x)^\alpha$ for $x\geq x_m$. If the deductible $d$ is at least $x_m$, then for $x>d$, $$ P(X>x\mid X>d)=\frac{S(x)}{S(d)}=\left(\frac{d}{x}\right)^\alpha. $$ The conditional density is $\alpha d^\alpha x^{-(\alpha+1)}$. The original lower bound cancels from the conditional model; the relevant scale for the selected sample is its deductible.

Worked Example: Pareto Shape Estimate and a Conditional Tail Probability

A hypothetical data set includes only losses above a deductible of $100$ dollars. The underlying severity is assumed to follow a Pareto Type I distribution with known lower bound $x_m=50$ dollars and unknown shape $\alpha$. The three observed losses are $200, 400,$ and $800$ dollars. Estimate $\alpha$ by conditional maximum likelihood, then find the fitted probability that a recorded loss exceeds $400$ dollars.

Method. Since selection is at $d=100$, use the Pareto density conditional on $X>100$. The conditional likelihood is

L(\alpha)=\prod_{i=1}^3 \alpha\,100^\alpha x_i^{-(\alpha+1)}.

Equivalently, the conditional log-likelihood is $3\log\alpha+3\alpha\log100-(\alpha+1)\sum_i\log x_i$. Rearranging its derivative gives the estimate $$ \hat{\alpha}=\frac{3}{\sum_{i=1}^3\log(x_i/100)}. $$

The three ratios to the deductible are $200/100=2$, $400/100=4$, and $800/100=8$. Therefore, $$ \sum_{i=1}^3\log(x_i/100)=\log2+\log4+\log8=\log64=4.1589, \qquad \hat{\alpha}=\frac{3}{\log64}=0.7213. $$

The requested conditional probability is $$ P(X>400\mid X>100)=\left(\frac{100}{400}\right)^{\hat{\alpha}} =0.25^{0.7213}\approx 0.3679. $$

The probability is rounded to four decimals and the shape estimate to four decimals. Using the unrounded estimate $3/\log64$, the exponent calculation is exactly $\exp(-1)$, because $(3/\log64)\log4=1$. Check: $400$ dollars is four times the deductible, so the conditional Pareto tail probability is $4^{-\hat{\alpha}}$, which is less than one. A common arithmetic trap is to report $0.6065$ for this power; direct evaluation gives $0.25^{0.7213}\approx0.3679$, not $0.6065$. Another modeling trap is to maximize the ordinary, unconditioned likelihood for losses above $x_m=50$: that incorrectly treats the absent losses between $50$ and $100$ as if they could have appeared in the sample.

Weibull Severity With Different Thresholds

A conditional likelihood also works when different observations have different known deductibles. Suppose $X$ has a Weibull distribution with known shape $k=2$ and unknown scale $\lambda>0$, with survival $S(x)=e^{-(x/\lambda)^2}$ and density $f(x)=2x\lambda^{-2}e^{-(x/\lambda)^2}$ for $x>0$. For an observation $x_i>d_i$, its conditional density is $$ f(x_i\mid X>d_i)=\frac{2x_i}{\lambda^2} \exp\left(-\frac{x_i^2-d_i^2}{\lambda^2}\right). $$ The difference $x_i^2-d_i^2$ replaces the ordinary squared loss contribution because each observation is conditioned on passing its own threshold.

Worked Example: Weibull Scale Estimate With Varying Deductibles

A hypothetical sample comes from a Weibull severity model with known shape $k=2$ and unknown scale $\lambda$. Claim-specific deductibles are $3,4,$ and $5$ in the same monetary units as the losses. The recorded losses are $5,8,$ and $9$, respectively. Each observation is included only if it exceeds its stated deductible. Estimate $\lambda$, and calculate the fitted probability that a loss selected above deductible $5$ exceeds $10$.

Method. Use the product of the Weibull conditional densities. Let $q=\lambda^2$. The log-likelihood, omitting terms that do not involve $q$, is $$ \ell(q)=-3\log q-\frac{(5^2-3^2)+(8^2-4^2)+(9^2-5^2)}{q}. $$

The three threshold-adjusted squared differences are $25-9=16$, $64-16=48$, and $81-25=56$. Their sum is $16+48+56=120$. Differentiating gives $\ell'(q)=-3/q+120/q^2$. Setting this to zero gives $q=120/3=40$, so $\hat{\lambda}=\sqrt{40}=6.3249$.

For the third threshold, the conditional probability is $$ P(X>10\mid X>5)=\frac{S(10)}{S(5)} =\exp\left(-\frac{10^2-5^2}{40}\right) =e^{-1.875}=0.1534. $$

The scale is rounded to four decimals and the probability to four decimals. Check: the probability uses the survival ratio at $10$ and $5$, not $S(10)$ alone, because the loss is already known to exceed $5$. The common mistake avoided is pooling the raw squares $5^2,8^2,9^2$ without subtracting the corresponding threshold squares.

Modeling Checks and Exam Traps

A deductible-based data set is informative about the severity distribution conditional on the inclusion rule. It does not, by itself, reveal how many losses fell below the deductible. Before maximizing a likelihood, check what the data collection process actually reports: just the selected losses, or also counts and exposure for the full population at risk. The likelihood must reflect that sampling design.

Do not omit the survival denominator. For a selected loss, use $f_\theta(x)/S_\theta(d)$, not just $f_\theta(x)$.
Keep thresholds paired with observations. With varying deductibles, the contribution for $x_i$ uses $d_i$, not an average deductible.
Do not confuse truncation with censoring. Under truncation, losses that fail the threshold are absent. A censored loss is observed in some form but its exact value is not known; the likelihood contribution is different.
Respect the model support. A conditional density requires $x_i>d_i$, and the assumed severity distribution must assign positive probability above each threshold.
Check the target probability’s condition. If the question asks about a recorded loss, use a conditional probability such as $S(x)/S(d)$, not an unconditional tail probability without regard to selection.

The practical consequence is that two analysts can fit the same named severity family to the same recorded losses and obtain different estimates if one accounts for truncation and the other does not. The conditional likelihood is essential when the observation process deliberately excludes losses below known deductibles. The formulas above rely on known thresholds and selection based only on whether the loss exceeds them; if the selection mechanism depends on additional unmodeled variables, the likelihood needs to represent that mechanism as well.

Key takeaway: When only losses exceeding known deductibles are observed, model each observation with its conditional density $f_\theta(x_i)/S_\theta(d_i)$. Estimate parameters from the resulting conditional likelihood, and calculate probabilities consistently with the same selection condition.

Check Your Understanding

For each problem, treat the stated observations as independent and condition on their stated selection thresholds.

A hypothetical exponential severity sample has mean parameter $\theta$, deductible $50$, and recorded losses $80,110,$ and $170$. Find the conditional maximum likelihood estimate of $\theta$, then calculate $P(X>190\mid X>50)$. (Answer: $\hat{\theta}=70$; probability $e^{-2}=0.1353$.)
A Pareto Type I model has known lower bound $x_m=40$ and unknown shape $\alpha$. Only losses above deductible $100$ are recorded; the observations are $200,400,$ and $800$. Find the conditional maximum likelihood estimate of $\alpha$. (Answer: $\hat{\alpha}=3/\log64=0.7213$.)
A Weibull model has known shape $k=2$ and unknown scale $\lambda$. One loss of $5$ is observed above deductible $3$, and one loss of $8$ above deductible $4$. Find the maximum likelihood estimate of $\lambda^2$. (Answer: $\hat{\lambda}^2=[(25-9)+(64-16)]/2=32$, so $\hat{\lambda}=\sqrt{32}\approx5.6569$.)
An exponential model with unknown mean $\theta$ produces recorded losses $140,300,$ and $420$, observed above respective deductibles $100,200,$ and $300$. Estimate $\theta$ and calculate $P(X>500\mid X>300)$. (Answer: $\hat{\theta}=(40+100+120)/3=86.6667$; probability $e^{-200/86.6667}\approx0.0995$.)
A Pareto Type I model has unknown shape $\alpha$, known lower bound $x_m=20$, and deductible $50$. The observed losses are $100,200,400,$ and $800$. Estimate $\alpha$, then calculate the fitted probability that a selected loss exceeds $400$. (Answer: $\hat{\alpha}=4/\log(2\cdot4\cdot8\cdot16)=0.5771$; $P(X>400\mid X>50)=8^{-0.5771}\approx0.3010$.)

What You'll Learn