Constructing Empirical Claim Severity Distributions

Why the Observation Unit Matters

A severity distribution describes the amount of an individual claim, not the total cost of every claim made by a policyholder. This distinction matters whenever a policy can generate multiple claims: each claim is a separate severity observation, while the sum of those claims is a policy-level total. Mixing the two changes the distribution being estimated and can change the calculated probabilities substantially.

Here, assume the available records contain complete individual claim amounts. Each recorded amount is one entire claim amount in the stated measurement basis; no claim is missing part of its amount. The empirical distribution assigns probability according to the observed claim records. It is a direct summary of those records, not by itself a guarantee that a future set of claims will have the same distribution.

Definition: Given $n$ complete individual claim amounts $x_1,x_2,\ldots,x_n$, the empirical claim severity distribution assigns probability $1/n$ to each claim record. Repeated amounts are retained as repeated observations. Its empirical cumulative distribution function is $$ F_n(x)=\frac{1}{n}\sum_{j=1}^{n} I(x_j\le x), $$ where $I(\text{statement})$ equals 1 when the statement is true and 0 otherwise. Thus, $F_n(x)$ is the fraction of recorded claims with amounts at most $x$.

The denominator is the number of claim records, not the number of distinct dollar amounts, policies, or policy-years. For example, if three of eight claims have the same amount, all three contribute to the empirical distribution. The observed probability at that amount includes all three records.

Interval Probabilities and Exceedances

Once the empirical distribution has been constructed, a probability is a count divided by the number of claim observations. It is important to specify whether each endpoint is included. In particular, “exceeds $t$” means strictly greater than $t$, whereas “at least $t$” includes claims equal to $t$.

P_n(a<X\le b)=F_n(b)-F_n(a) =\frac{\#\{j:a<x_j\le b\}}{n}

This formula follows because $F_n(b)$ counts every observation at or below $b$, while $F_n(a)$ counts the observations at or below $a$. Subtracting removes all observations at or below $a$, leaving exactly those greater than $a$ and at most $b$. For an interval that includes its lower endpoint, use a strict inequality in the subtracted count:

P_n(a\le X\le b) =\frac{\#\{j:a\le x_j\le b\}}{n}.

For a threshold $t$, the empirical exceedance probability and observed exceedance count are

P_n(X>t)=\frac{\#\{j:x_j>t\}}{n}=1-F_n(t), \qquad P_n(X\ge t)=\frac{\#\{j:x_j\ge t\}}{n}.

The count is a number of claims; the proportion is a probability under the empirical distribution. If a question asks for an observed frequency “per 100 claims,” multiply the proportion by 100. That is still a claim-based frequency. It is not a rate per policy or per year unless the denominator is changed to the relevant policy or exposure units.

Worked Example: Interval Probability and Strict Exceedance

A hypothetical insurer has six complete claim records, in dollars: $400, 650, 650, 900, 1{,}250,$ and $2{,}150$. Using the empirical distribution, find (a) the probability that a randomly selected recorded claim is greater than $600$ and at most $1{,}250$, and (b) the probability that it strictly exceeds $1{,}000$. Also give the corresponding counts.

Method. Treat each of the six claim records as one equally weighted observation, retaining both claims of $650$. The denominator for both probabilities is $n=6$.

(a) Interval. The claims satisfying $600<X\le1{,}250$ are $650,650,900,$ and $1{,}250$. The count is 4, so

P_n(600<X\le1{,}250)=\frac{4}{6}=\frac{2}{3}=0.6667.

(b) Strict exceedance. The claims greater than $1{,}000$ are $1{,}250$ and $2{,}150$. The count is 2, so

P_n(X>1{,}000)=\frac{2}{6}=\frac{1}{3}=0.3333.

Check. Four claims lie in the stated interval, and two exceed $1{,}000$; each count is out of the six original claim records. Probabilities are rounded to four decimal places. A common mistake would be to count $1{,}000$ as an exceedance if it appeared in the data. The condition here is $X>1{,}000$, not $X\ge1{,}000$.

Why Each Claim Gets Its Own Place in the Distribution

The empirical distribution can be understood by choosing one record uniformly at random from the observed claims. Each record then has probability $1/n$ of being selected. For any threshold $x$, the event that the selected amount is at most $x$ occurs for exactly as many records as satisfy $x_j\le x$. Therefore its probability is that count divided by $n$, which is precisely $F_n(x)$.

This justification also explains why tied amounts are not collapsed into one equally weighted value. If an amount appears three times, there are three records with that amount, and the chance of selecting one of them is $3/n$. Keeping those repetitions preserves the observed claim-level composition.

Key distinction: A claim severity observation is one individual claim amount. A policy-level total is the sum of all claim amounts associated with a policy over a defined period. The empirical distribution of individual claims uses claim records as observations; the empirical distribution of policy totals uses policies as observations. They answer different questions and generally have different probabilities.

Worked Example: Claims Versus Policy Totals

A hypothetical five-policy portfolio has the following complete claims during a specified observation period: Policy A has claims of $\$300$ and $\$700$; Policy B has a claim of $\$1{,}100$; Policy C has claims of $\$250,\$250,$ and $\$1{,}500$; Policy D has no claims; and Policy E has claims of $\$600$ and $\$1{,}800$. Find (a) the empirical claim-level probability of a claim exceeding $\$1{,}000$, and (b) the fraction of policies whose total claims exceed $\$1{,}000$.

Method. First use individual claim records for the severity probability. Then, separately, add claims within each policy and count policies. These calculations have different observation units and denominators.

(a) Claim-level calculation. The total number of claims is $2+1+3+0+2=8$. The amounts exceeding $\$1{,}000$ are $\$1{,}100,\$1{,}500,$ and $\$1{,}800$, for a count of 3. Thus,

P_n(X>1{,}000)=\frac{3}{8}=0.3750.

(b) Policy-level calculation. The policy totals are

\begin{aligned} A &: 300+700=1{,}000,\\ B &: 1{,}100,\\ C &: 250+250+1{,}500=2{,}000,\\ D &: 0,\\ E &: 600+1{,}800=2{,}400. \end{aligned}

Policies B, C, and E have totals strictly greater than $\$1{,}000$. Policy A's total equals $\$1{,}000$, so it does not qualify under the strict condition. The fraction is therefore

\frac{3}{5}=0.6000.

Check. There are eight claims but five policies, and the two answers use those respective denominators. The claim-level exceedance probability is $0.3750$; the policy-level fraction is $0.6000$. A common mistake is to use the five policy totals as the severity sample. That would answer a question about policy-level losses, not the distribution of individual claim amounts.

Constructing a Useful Empirical Summary

A small frequency table can make the empirical distribution easier to inspect without changing its definition. List each observed amount and its number of occurrences; divide each count by $n$ for its empirical probability. The probabilities across all distinct observed amounts must sum to 1. Cumulative probabilities can then be obtained by adding the probabilities through a specified amount.

Worked Example: Building a Frequency Table and Checking Endpoint Rules

A hypothetical sample contains eight complete individual claims, in thousands of dollars: $450,450,800,1{,}000,1{,}000,1{,}300,2{,}100,$ and $3{,}100$. Construct the empirical probability at each observed amount, then calculate $P_n(X\le1{,}000)$, $P_n(X\ge1{,}000)$, and $P_n(800<X\le2{,}100)$.

Method. Count the occurrences at each distinct amount and divide by $n=8$. Retain the units as thousands of dollars throughout.

Claim amount (thousands of dollars)	Count	Empirical probability
450	2	2/8
800	1	1/8
1,000	2	2/8
1,300	1	1/8
2,100	1	1/8
3,100	1	1/8

Check the total probability. The counts sum to $2+1+2+1+1+1=8$, and the probabilities sum to $8/8=1$.

At most $\$1{,}000$. In thousands of dollars, the qualifying observations are $450,450,800,1{,}000,1{,}000$. There are 5, giving

P_n(X\le1{,}000)=\frac{5}{8}=0.6250.

At least $\$1{,}000$. The qualifying observations are $1{,}000,1{,}000,1{,}300,2{,}100,$ and $3{,}100$. There are 5, so

P_n(X\ge1{,}000)=\frac{5}{8}=0.6250.

Between the stated endpoints. For $800<X\le2{,}100$, count $1{,}000,1{,}000,1{,}300,$ and $2{,}100$. There are 4, hence

P_n(800<X\le2{,}100)=\frac{4}{8}=0.5000.

Check. The amount $800$ is excluded by the strict lower endpoint, while $2{,}100$ is included by the upper endpoint. Both claims of $1{,}000$ count in the inclusive threshold calculation. All probabilities are rounded to four decimal places. A common mistake is to treat “at least” as “greater than” and omit the two claims equal to $\$1{,}000$.

Interpretation and Practical Checks

An empirical severity distribution describes the observed claim records under their stated basis and observation period. If the records come from a selected group, cover only certain types of claims, or represent a period unlike the one of interest, the empirical proportions may not represent a broader or future population. The arithmetic can still be correct while the interpretation is too broad.

Before calculating, verify these points:

Observation unit: Are the rows individual claims or policy-level totals?
Completeness: Does each record contain the complete individual claim amount required by the problem?
Denominator: How many claim records are included in the empirical distribution?
Endpoint wording: Does the condition use $<$, $\le$, $>$, or $\ge$?
Units: Are amounts stated in dollars, thousands of dollars, or another consistent unit?

The empirical distribution is especially useful for direct questions about the observed sample: how many claims fall within an interval, what fraction exceed a stated amount, and how the observed amounts are distributed across their values. State the observation unit, threshold convention, and denominator explicitly; doing so prevents most errors in empirical severity calculations.

Check Your Understanding

For each problem, use the complete claim records exactly as given, keep repeated amounts, and state the relevant observation unit.

A hypothetical sample has five claim amounts: $\$200,\$500,\$500,\$900,\$1{,}600$. Find the empirical probability that $X$ is at least $\$500$ and at most $\$900$, and the probability that $X$ strictly exceeds $\$900$. (Answer: $3/5=0.6000$ and $1/5=0.2000$, respectively.)
Seven complete claims are $\$350,\$700,\$700,\$950,\$1{,}200,\$1{,}200,\$2{,}600$. Find $P_n(X\le\$950)$ and $P_n(X\ge\$1{,}200)$. (Answer: $4/7=0.5714$ and $3/7=0.4286$, respectively.)
Six policies have claims as follows: A has $\$400,\$700$; B has $\$1{,}500$; C has $\$200,\$200$; D has no claims; E has $\$900,\$1{,}800$; and F has $\$1{,}100$. Find the claim-level probability of a claim exceeding $\$1{,}000$ and the fraction of policies with total claims exceeding $\$1{,}000$. (Answer: claim-level $3/8=0.3750$; policy-level $4/6=0.6667$.)
Eight complete claims, in thousands of dollars, are $100,300,300,500,750,750,1{,}000,1{,}400$. Find $P_n(300<X\le1{,}000)$ and $P_n(X\ge750)$. (Answer: $4/8=0.5000$ and $4/8=0.5000$, respectively.)

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