The Cumulative Distribution Function of Severity

What the Severity CDF Measures

A severity model describes the amount of one claim. Once a model is specified, an insurer or actuary may need to answer a threshold question: what is the probability that a claim is no greater than a stated dollar amount? The cumulative distribution function, or CDF, gives that probability directly.

For example, a hypothetical insurer might want to assess how often an individual claim is at most $10,000. The question concerns a single claim amount, not the number of claims in a period, the total amount of all claims, or the payment after applying a policy feature. Those are different quantities. Here, $X$ denotes the amount of an individual claim, measured in a stated unit.

Definition: The cumulative distribution function of the severity random variable $X$ is $$ F_X(x)=P(X\leq x). $$ For a specified amount $x$, $F_X(x)$ is the probability that the individual claim amount $X$ does not exceed $x$. The inequality includes equality: a claim exactly equal to $x$ satisfies $X\leq x$.

The CDF takes an amount as its input and returns a probability. Thus, if $X$ is measured in dollars, the input $x$ must also be in dollars; if $X$ is measured in thousands of dollars, $x$ must be expressed in thousands. The output is a probability, not a dollar amount or a percentage of the claim.

A CDF can be given directly as a function, or it can be obtained from a specified probability model for $X$. The model and its parameters matter: the threshold alone does not determine the answer. For example, a $10,000 claim threshold can have different probabilities under different severity distributions.

Deriving a CDF for a Uniform Severity Model

A simple way to see how a CDF is formed is to consider a hypothetical claim amount uniformly distributed between zero and a fixed maximum. Under this model, the probability is proportional to the length of the part of the range being considered. For $0\leq x\leq b$, the amounts from zero through $x$ occupy length $x$ of the full range of length $b$. Therefore,

F_X(x)=P(X\leq x)=\frac{x}{b}, \qquad 0\leq x\leq b.

This is a specific result for the stated uniform model, not a general formula for every severity distribution. Its practical use is straightforward: confirm that the threshold lies within the model’s range, then substitute the threshold and the model’s upper endpoint in the same units.

Worked Example: A Uniform Claim Amount

A hypothetical equipment insurer models one repair claim $X$ as uniform on the interval from $0 to $50,000. Find the probability that a claim does not exceed $32,000.

Model and parameters. $X\sim\operatorname{Uniform}(0,50{,}000)$, with amounts measured in dollars. This is a severity model for one claim; no interest or mortality assumption is involved.

Method. For a uniform severity on $[0,b]$, $F_X(x)=x/b$ within the range. Here, $b=50{,}000$ and $x=32{,}000$, so

F_X(32{,}000)=P(X\leq 32{,}000) =\frac{32{,}000}{50{,}000} =0.64.

The probability, rounded to four decimal places, is 0.6400, or 64.00%. The threshold is within the specified range, and the ratio is dimensionless because both amounts are in dollars.

Check and trap avoided. The threshold is $32/50=0.64$ of the modeled range, so the computed CDF value is consistent with the uniform assumption. A common error is to use $50{,}000/32{,}000$; that reverses the ratio and produces a value greater than 1, which cannot be a probability.

Evaluating a CDF from Other Severity Models

The uniform formula is not universal. For another distribution, use that model’s CDF and its stated parameterization. A useful habit is to write down the model and units before substituting. This makes it easier to detect a parameter that is expressed per dollar when the claim amount is entered in thousands, or vice versa.

For an exponential severity model with mean $\theta$, the CDF is $F_X(x)=1-e^{-x/\theta}$ for $x\geq 0$. Equivalently, its rate parameter is $\lambda=1/\theta$, and the CDF is $1-e^{-\lambda x}$. The rate and amount must use reciprocal units: if $x$ is in dollars, $\lambda$ is per dollar.

Worked Example: An Exponential Claim Amount

A hypothetical small-business insurer models the amount of one reported property claim as exponential with mean $12,000. Find the probability that the claim does not exceed $18,000.

Model and parameters. $X$ is exponential with mean $\theta=\$12{,}000$, or rate $\lambda=1/12{,}000$ per dollar. The model concerns one claim amount only, so interest and mortality are not applicable.

Method. Apply the exponential CDF at $x=\$18{,}000$:

\begin{aligned} F_X(18{,}000) &=1-e^{-18{,}000/12{,}000}\\ &=1-e^{-1.5}\\ &=1-0.223130\ldots\\ &=0.776869\ldots \end{aligned}

Rounded to four decimal places, the requested probability is 0.7769, or 77.69%. The exponent is dimensionless: the dollar threshold is divided by the dollar mean.

Check and trap avoided. The threshold is 1.5 times the mean, and the CDF calculation gives a probability below 1. The value $e^{-1.5}=0.2231$ is not the probability that a claim does not exceed $18,000; using it as the answer confuses the requested CDF probability with a complement. The question asks for $F_X(18{,}000)$, so the appropriate calculation is $1-e^{-1.5}$.

A Weibull model gives another example of why the distribution and parameters must be explicit. Using scale parameter $\eta>0$ and shape parameter $k>0$, one standard parameterization has

F_X(x)=1-\exp\!\left[-\left(\frac{x}{\eta}\right)^k\right], \qquad x\geq 0.

The ratio $x/\eta$ must use consistent units. The shape parameter changes how that ratio enters the exponent; it is not a dollar amount.

Worked Example: A Weibull Claim Amount in Thousands

A hypothetical insurer models one machinery claim using a Weibull severity distribution with scale $\eta=10$ thousand dollars and shape $k=2$. Calculate the probability that the claim does not exceed 15 thousand dollars, and then evaluate the CDF at 5 thousand dollars as a check on the scale of the results.

Model and parameters. $X$ is Weibull with $\eta=10$ and $k=2$, with $X$ and both thresholds measured in thousands of dollars. This is a one-claim severity calculation; no interest or mortality assumption applies.

Method. At 15 thousand dollars, the ratio $x/\eta$ is $15/10=1.5$, and its square is $2.25$:

\begin{aligned} F_X(15) &=1-\exp\!\left[-\left(\frac{15}{10}\right)^2\right]\\ &=1-e^{-2.25}\\ &=1-0.105399\ldots\\ &=0.894600\ldots \end{aligned}

The probability at 15 thousand dollars is 0.8946, rounded to four decimal places. At 5 thousand dollars, $x/\eta=5/10=0.5$, so

\begin{aligned} F_X(5) &=1-\exp\!\left[-\left(\frac{5}{10}\right)^2\right]\\ &=1-e^{-0.25}\\ &=1-0.778801\ldots\\ &=0.221199\ldots \end{aligned}

Thus, the probability at 5 thousand dollars is 0.2212, rounded to four decimal places. The two calculated CDF values are consistent with the thresholds: the larger threshold gives the larger probability of not exceeding it.

Check and trap avoided. Because the scale and thresholds are both in thousands of dollars, their ratios are $15/10$ and $5/10$. Substituting 15,000 while leaving the scale as 10 would mix dollars and thousands, producing an invalid exponent. Convert both quantities to dollars or keep both in thousands.

Reading the Question Before Calculating

A threshold question can look simple, but careful interpretation prevents several common errors. First, identify the random variable: in this tutorial, $X$ is the amount of one claim. Second, identify the event requested. “Does not exceed $x$” means $X\leq x$, including a claim equal to the threshold. Third, identify the correct CDF and units before evaluating it.

The CDF is not itself a deductible, limit, or payment amount. A statement such as “the probability of a claim not exceeding $10,000” concerns the underlying severity $X$, unless the problem explicitly defines a different random variable. Do not silently replace claim amount with insurer payment or with a modified amount.

Another trap is to report the input threshold as the answer. For example, evaluating $F_X(18{,}000)$ does not yield $18,000; it yields a number between 0 and 1 that describes a probability under the stated model. If a result is intended as a percentage, convert the probability only after calculating it.

Key takeaway: To find the probability that one claim does not exceed a specified amount, identify the severity model, express the threshold and parameters in consistent units, and evaluate $F_X(x)=P(X\leq x)$. The model determines the numerical CDF value.

Check Your Understanding

For each hypothetical one-claim severity model, identify the requested CDF value and evaluate it. Give probabilities to four decimal places.

A claim amount is uniform from $0 to $80,000. Find the probability that it does not exceed $52,000. (Answer: 0.6500)
A claim amount is exponential with mean $5,000. Find the probability that it does not exceed $10,000. (Answer: 0.8647)
A claim amount is Weibull with scale 20 thousand dollars and shape 2. Find the probability that it does not exceed 10 thousand dollars. (Answer: 0.2212)
A hypothetical severity model is specified by $F_X(x)=(x/50{,}000)^3$ for thresholds from $0 through $50,000. Find the probability that a claim does not exceed $30,000. (Answer: 0.2160)
A claim amount is exponential with mean $7,500. An analyst reports $e^{-2}$ as the probability that the claim does not exceed $15,000. Calculate the requested probability and identify the analyst’s error. (Answer: 0.8647; the analyst used $e^{-2}=0.1353$ rather than $1-e^{-2}$, confusing the requested CDF value with its complement.)

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