Truncated Claim Sizes and Severity Credibility

Why Truncation Matters for Severity Credibility

A severity credibility calculation concerns the variability of the claim amounts being averaged. Those amounts must match the insurer’s actual measure: ground-up losses, payments after a deductible, or payments subject to a policy limit. A limit can compress the upper tail; a deductible can create a mass of zero payments. Both can change the mean and variance, and therefore the coefficient of variation (CV) used in a severity credibility standard.

A key first step is to identify the observation being averaged. Payment per loss assigns a payment, including zero, to every ground-up claim. Payment per payment describes only claims for which the insurer makes a positive payment. These have different distributions when a deductible applies, so their moments and credibility implications must not be interchanged.

Definition: Let $X$ be a nonnegative ground-up claim amount with survival function $S_X(x)=P(X>x)$. A payment limit $u$ produces payment $Y=\min(X,u)$. A deductible $d$ produces payment per loss $Y=(X-d)_+=\max(X-d,0)$. If the deductible payment is also limited to $L$, the payment per loss is $Y=\min((X-d)_+,L)$. For any positive-mean severity $Y$, its coefficient of variation is $\operatorname{CV}(Y)=\sqrt{\operatorname{Var}(Y)}/E[Y]$.

Moment Formulas for Limited Payments

For a nonnegative random variable, moments can be obtained by integrating its survival probabilities. This is especially useful for limited payments because the payment is bounded even when the ground-up loss is not.

E[\min(X,u)^r]=r\int_0^u t^{r-1}S_X(t)\,dt,\qquad r>0.

In particular, the first two raw moments are

E[Y]=\int_0^u S_X(t)\,dt,\qquad E[Y^2]=2\int_0^u tS_X(t)\,dt,

and $\operatorname{Var}(Y)=E[Y^2]-(E[Y])^2$. To see why the formula holds, for any nonnegative payment $Y$, write $Y^r=\int_0^Y r t^{r-1}\,dt$. Taking expectations and reversing the order of integration gives $E[Y^r]=\int_0^\infty r t^{r-1}P(Y>t)\,dt$. When $Y=\min(X,u)$, $P(Y>t)=S_X(t)$ for $0\le t<u$, and it is zero for $t\ge u$. This gives the stated integral.

A cap also changes the shape of the distribution: for a continuous ground-up distribution, the capped payment has probability mass at $u$, equal to $P(X\ge u)$. It is not simply the original continuous distribution cut off and renormalized below $u$. A deductible has a different effect: payment per loss has a mass at zero, equal to $P(X\le d)$.

Worked Example: Exponential Claims Capped at the Mean

Suppose hypothetical ground-up claims have an exponential distribution with mean $\theta=\$1{,}000$. The insurer pays at most $u=\$1{,}000$ per claim, so $Y=\min(X,1{,}000)$. Find the payment mean, second moment, and CV.

Method. For an exponential distribution with mean $\theta$, $S_X(t)=e^{-t/\theta}$. Set $a=u/\theta=1$. The limited-payment moment formulas reduce to

E[Y]=\theta(1-e^{-a}),\qquad E[Y^2]=2\theta^2[1-(1+a)e^{-a}].

Mean. Since $e^{-1}=0.3678794$,

E[Y]=1{,}000(1-0.3678794)=\$632.12.

Second moment.

E[Y^2]=2(1{,}000)^2[1-2(0.3678794)] =2{,}000{,}000(0.2642411) =\$528{,}482.24.

Variance and CV. Squaring the unrounded mean gives $E[Y]^2\approx399{,}576.40$. Thus $\operatorname{Var}(Y)=528{,}482.24-399{,}576.40=\$128{,}905.84$, the standard deviation is approximately $\$359.04$, and $\operatorname{CV}(Y)=359.04/632.12=0.5680$. The result is dimensionless. The cap has reduced the CV from the ground-up exponential CV of 1.

Check. The second moment exceeds the squared mean, as required for a nonnegative variance. The cap is included in the payment distribution, including its probability mass at $\$1{,}000$.

Deductibles: Payment per Loss and Payment per Payment

For a deductible $d$, the insurer’s payment per loss is zero when $X\le d$ and $X-d$ when $X>d$. Its moments can be written using the ground-up survival function:

E[(X-d)_+^r]=r\int_0^\infty t^{r-1}S_X(d+t)\,dt,\qquad r>0.

With a maximum payment $L$ above the deductible, replace the upper integration limit by $L$. For payment per payment, condition on a positive payment: divide the per-loss raw moment by $P(X>d)=S_X(d)$. The same division must be applied to both the first and second raw moments before calculating the conditional variance.

For exponential ground-up claims, the memoryless property makes one conditional calculation especially simple: given $X>d$, the excess $X-d$ is again exponential with the same mean as $X$. This is a property of the exponential model, not a general rule for claim distributions. In other models, the conditional excess distribution must be derived from $S_X(d+t)/S_X(d)$.

Worked Example: Deductible and Limit with Exponential Claims

Suppose hypothetical ground-up claims are exponential with mean $\$1{,}000$. The policy has a $\$200$ deductible and pays at most $\$800$ above the deductible. Find the probability of a positive payment, the mean payment per payment, and the mean payment per loss.

Probability of a positive payment. A payment occurs exactly when $X>200$, so

P(X>200)=e^{-200/1{,}000}=e^{-0.2}=0.8187.

Conditional payment distribution. Given $X>200$, the memoryless property gives $X-200$ an exponential distribution with mean $\$1{,}000$. The payment per payment is therefore an exponential amount with that mean, capped at $\$800$. Here $a=800/1{,}000=0.8$, so

E[Y\mid Y>0]=1{,}000(1-e^{-0.8}) =1{,}000(1-0.4493290) =\$550.67.

Payment per loss. A ground-up claim produces zero with probability $1-e^{-0.2}$; otherwise its expected payment is the conditional mean just calculated. Thus

E[Y]=P(X>200)E[Y\mid Y>0] =e^{-0.2}\,1{,}000(1-e^{-0.8}) =1{,}000(e^{-0.2}-e^{-1}) =\$450.85.

Check. The payment-per-loss mean is lower than the payment-per-payment mean because the former includes the zero payments. Using $\$550.67$ as the mean per ground-up claim would overstate expected insurer payments.

Using the Adjusted CV in a Severity Credibility Standard

For independent, identically distributed payments with positive mean $\mu_Y$, finite variance, and CV $c_Y$, the sample mean of $n$ payments has standard error $\sigma_Y/\sqrt{n}$. Under the usual normal approximation, requiring the sample mean to be within a relative error $k$ of $\mu_Y$ with confidence level $p$ gives the full-credibility standard

n\ge \left(\frac{z_p c_Y}{k}\right)^2,

where $z_p$ is the standard normal quantile for the chosen two-sided confidence level. This follows by standardizing the sample mean: its approximate error in units of its mean is $z_p\sigma_Y/(\sqrt{n}\mu_Y)=z_pc_Y/\sqrt{n}$. Requiring that expression to be no greater than $k$ and squaring both sides gives the formula. It is a claim-count standard for the stated severity measure, not an exposure standard. The normal approximation may be poor for very small $n$ or highly skewed payments.

The CV in this calculation must correspond to the payments being averaged. If the data consist of positive payments only, use the payment-per-payment CV. If zero payments are included for every ground-up claim, use the payment-per-loss CV. The sample size and the severity definition must use the same basis.

Worked Example: A Policy Limit Changes the Required Claim Count

A hypothetical ground-up claim amount is uniformly distributed from $\$0$ to $\$2{,}000$. Compare the full-credibility claim counts for the ground-up severity and for payment per loss with a $\$1{,}000$ cap. Use a two-sided 90% confidence level, $z_p=1.6449$, and relative error $k=0.10$.

Ground-up severity. For $X\sim\operatorname{Uniform}(0,2{,}000)$, the mean is $\$1{,}000$ and the standard deviation is $2{,}000/\sqrt{12}=\$577.35$. Thus $c_X=577.35/1{,}000=1/\sqrt{3}=0.57735$. The claim-count standard is

n_X\ge\left(\frac{1.6449(0.57735)}{0.10}\right)^2 =90.19.

The minimum whole-number count meeting this standard is 91 claims.

Capped payment severity. Let $Y=\min(X,1{,}000)$. Half the ground-up claims exceed the cap, so the payment distribution has probability $0.5$ at $\$1{,}000$. Its first raw moment is

E[Y]=\frac{1}{2{,}000}\int_0^{1{,}000}x\,dx +\frac{1}{2}(1{,}000) =250+500=\$750.

Its second raw moment is

E[Y^2]=\frac{1}{2{,}000}\int_0^{1{,}000}x^2\,dx +\frac{1}{2}(1{,}000)^2 =166{,}666.67+500{,}000 =\$666{,}666.67.

Therefore, $\operatorname{Var}(Y)=666{,}666.67-750^2=\$104{,}166.67$, and $c_Y=\sqrt{104{,}166.67}/750=0.43033$. The capped-payment claim-count standard is

n_Y\ge\left(\frac{1.6449(0.43033)}{0.10}\right)^2 =50.11.

The minimum whole-number count is 51 claims. The cap reduces the CV and hence the required count, but the result is based on payment severity; it does not say that 51 claims meet the ground-up severity standard.

Check. The capped payment mean is less than the ground-up mean, and its CV is also smaller. The count calculation uses each severity’s own CV, rather than carrying the ground-up CV into the capped calculation.

Common Mistake and Practical Checks

A frequent error is to use the ground-up CV after a limit or deductible has changed the payment distribution. In the uniform example, using $0.57735$ for capped payments gives 90.19 as the required count. That is incorrect for the capped severity: its CV is $0.43033$, giving 50.11 before rounding up to a whole claim. Conversely, using a payment-per-payment CV when the data include zero payments also uses the wrong distribution.

Before applying a severity credibility standard, check the following:

Payment definition: Is each observation a ground-up loss, payment per loss, or payment per payment?
Contract terms: Is the stated limit a maximum insurer payment, or a limit on the ground-up loss? State which amount the cap applies to.
Moments: Calculate the variance as the second raw moment minus the squared mean. Do not treat the second raw moment as the variance.
Credibility basis: Use the CV of the exact severity measure being averaged, with the stated confidence level and relative error.

Truncation does not guarantee that every chosen measure of variability falls. A cap often reduces the CV, as in the examples, but a deductible can change the CV differently depending on the severity distribution and whether zero payments are included. This is why moments should be recomputed from the contract-adjusted distribution rather than adjusted by intuition alone. Correctly matching the payment distribution to the credibility calculation is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Check Your Understanding

For each problem, identify the relevant payment basis before calculating.

Ground-up claims are uniform on $\$0$ to $\$1{,}000$, and payment is capped at $\$500$. Find the capped payment mean and CV. (Answer: mean $\$375.00$; CV $0.4303$)
Ground-up claims are exponential with mean $\$500$, and the deductible is $\$100$, with no payment limit. Find the probability of a positive payment and the mean payment per payment. (Answer: $0.8187$; $\$500.00$)
Ground-up claims are exponential with mean $\$400$, and payment is capped at $\$400$. Find the mean capped payment. (Answer: $\$252.85$)
Ground-up severity is uniform on $\$0$ to $\$2{,}000$, with payment capped at $\$1{,}000$. For a two-sided 90% confidence level with $z_p=1.6449$ and a 10% relative error, find the minimum whole-number claim count for capped payment severity. (Answer: 51 claims)
Ground-up claims are exponential with mean $\$1{,}000$, the deductible is $\$300$, and there is no limit. Find the expected payment per ground-up loss. (Answer: $\$740.82$)

What You'll Learn