The Frequency-Severity Decomposition

Why Decompose Expected Loss?

A total loss estimate can be understood through two separate components: how often claims occur and how costly they are on average. This decomposition is useful because claim counts and claim amounts can change for different reasons. It also makes clear how the size of a portfolio affects the total dollars expected: the same expected loss per exposure produces different aggregate losses when applied to different exposure volumes.

The calculation requires a consistent definition of a claim, a consistent loss amount for each claim, and a specified exposure base. For example, the claim count and loss amounts might both be for claims occurring in a period, while exposure is measured in policy-years during that period. The details of selecting the exposure denominator are a separate decision; here, treat the denominator as given and use it consistently.

Definition: For a portfolio with $C$ claims, $E$ exposures, and total losses $L$, the observed claim frequency is $C/E$, the observed average severity is $L/C$ when $C>0$, and the observed loss per exposure is $L/E$. The product of frequency and average severity equals loss per exposure: $ (C/E)(L/C)=L/E $.

Frequency is a count per exposure, not a percentage of exposures that necessarily have a claim. An exposure may generate more than one claim, so a frequency can exceed one. Severity is an average amount per claim, not an amount per exposure. Their product is an amount per exposure, often called the pure premium or expected loss cost per exposure when used as an estimate of expected losses.

From Claim-Level Losses to Expected Loss

To explain the expectation behind the decomposition, let $N$ be the number of claims associated with one exposure, and let $X_j$ be the loss amount for claim $j$. The aggregate loss for that exposure is $S=\sum_{j=1}^{N}X_j$, with $S=0$ when $N=0$. Suppose the claim severities have a common finite mean $\mu_X=E[X_j]$, and, conditional on the count, their mean remains $\mu_X$. In particular, this holds when claim severities are independent of the claim count and identically distributed. Then:

E[S] =E\!\left[E[S\mid N]\right] =E[N\mu_X] =E[N]E[X_j].

The first equality is the law of iterated expectations. Given $N=n$, there are $n$ claim amounts, each with mean $\mu_X$, so $E[S\mid N=n]=n\mu_X$. Taking expectations over $N$ gives $E[S]=E[N]\mu_X$. If expected claim count per exposure is $\lambda$ and expected severity is $\mu_X$, expected loss per exposure is therefore $\lambda\mu_X$. For $E$ comparable exposures, linearity of expectation gives aggregate expected loss $E\lambda\mu_X$, provided the assumptions apply across those exposures.

Model and conditions: The frequency-severity product estimates expected loss per exposure when frequency estimates expected claim count per exposure and severity estimates the mean loss per claim on the same loss basis. The expected aggregate loss is exposure volume multiplied by that expected loss per exposure. The product does not require the claim count to follow a particular count distribution, but the simple mean-severity argument assumes that expected severity is not altered by the claim count. If severity depends on count, or the claim definition and loss basis differ between the two components, the product may not represent the intended expected loss.

In experience data, the identity between observed loss per exposure and observed frequency times observed severity is exact when both components use the same claims and loss amounts. Using experience averages to project future expected losses adds a modeling assumption: the selected frequency and severity are suitable estimates for the exposures being priced or projected. Observed values can vary from expected values, especially when the number of claims is small.

Estimate frequency.
Divide the relevant claim count by the number of exposures. Keep the stated exposure unit, such as exposure-years, attached to the result.

Estimate severity.
Divide losses for the same claims by that claim count. Confirm that the loss amounts use the intended valuation and payment basis.

Calculate loss per exposure.
Multiply the frequency by the average severity. The units should reduce to dollars per exposure.

Scale to the portfolio.
Multiply loss per exposure by the number of applicable exposures to estimate aggregate expected losses.

Worked Applications

Worked Example: Estimating Loss Cost From Pooled Experience

A hypothetical small commercial program recorded 80 claims and $480,000 in losses over 400 exposure-years. Estimate its observed frequency, average severity, and loss per exposure. Then apply the estimated loss per exposure to a projection of 250 exposure-years.

Method. Use the same 80 claims and the same $480,000 loss total for both components. This ensures that the frequency-severity product reconciles to observed loss per exposure.

Frequency: $80/400=0.20$ claims per exposure-year.

Average severity: $\$480{,}000/80=\$6{,}000$ per claim.

Loss per exposure: $0.20\times \$6{,}000=\$1{,}200$ per exposure-year. As a check, direct division gives $\$480{,}000/400=\$1{,}200$ per exposure-year.

Projected aggregate expected losses: $250\times \$1{,}200=\$300{,}000$. The projection uses 250 exposure-years, not 250 claims.

Common mistake avoided: Multiplying the projected exposures by average severity alone would give $250\times \$6{,}000=\$1{,}500{,}000$, incorrectly treating every exposure as a claim. Frequency must first convert exposures into expected claims.

Worked Example: Calculating and Combining Segment Estimates

A hypothetical insurer analyzes two groups separately. Group U has 4 claims per 100 exposures, average severity of $300, and 20,000 projected exposures. Group V has 3 claims per 100 exposures, average severity of $100, and 12,000 projected exposures. Estimate expected losses for each group and in total.

Group U frequency: $4/100=0.04$ claims per exposure. Expected claims are $0.04\times20{,}000=800$. Expected losses are $800\times \$300=\$240{,}000$.

Group V frequency: $3/100=0.03$ claims per exposure. Expected claims are $0.03\times12{,}000=360$. Expected losses are $360\times \$100=\$36{,}000$.

Total expected losses: $\$240{,}000+\$36{,}000=\$276{,}000$. Total exposure is $20{,}000+12{,}000=32{,}000$, so the implied overall expected loss per exposure is $\$276{,}000/32{,}000=\$8.625$.

As a check, the weighted overall frequency is $(800+360)/32{,}000=0.03625$, and the weighted overall severity is $\$276{,}000/1{,}160=\$237.9310$ to four decimal places. Their product is approximately $\$8.625$ per exposure. The unrounded values give the same result.

Common mistake and correction: Dividing Group U's 4 claims by 400 rather than by the stated 100 exposures gives a mistaken frequency of $0.01$. That produces $(4/400)\times20{,}000\times\$300=\$60{,}000$. The correct denominator is 100, so the correct calculation is $(4/100)\times20{,}000\times\$300=\$240{,}000$. Always preserve the denominator attached to the reported frequency.

Worked Example: Comparing Two Portfolio Scenarios

A hypothetical product has 1,000 exposure-years, expected frequency of 0.08 claims per exposure-year, and expected severity of $5,000. A planned scenario has 1,200 exposure-years, frequency of 0.10, and severity of $5,500. Calculate the expected losses under each scenario and the ratio of scenario expected losses to the original expected losses.

Original scenario. Expected claims are $1{,}000\times0.08=80$. Expected losses are $80\times\$5{,}000=\$400{,}000$.

Planned scenario. Expected claims are $1{,}200\times0.10=120$. Expected losses are $120\times\$5{,}500=\$660{,}000$.

Ratio check. Exposure volume changes by a factor of $1{,}200/1{,}000=1.2$; frequency changes by $0.10/0.08=1.25$; and severity changes by $\$5{,}500/\$5{,}000=1.1$. Their product is $1.2\times1.25\times1.1=1.65$. Directly, $\$660{,}000/\$400{,}000=1.65$, so the scenario total is 65% higher.

Common mistake avoided: Comparing only frequency and severity would yield a factor of $1.25\times1.1=1.375$, omitting the 20% increase in exposure volume. That factor compares expected losses per exposure, not aggregate expected losses.

Segmentation, Interpretation, and Exam Traps

For multiple segments, calculate each segment's expected losses using its own frequency, severity, and exposure volume, then add the segment totals. This preserves the effect of differences in both risk mix and segment size. A simple unweighted average of segment frequencies or severities generally does not represent the overall portfolio. If an overall loss per exposure is required, divide total expected losses by total exposures, or calculate compatible portfolio-wide components using claim and exposure weights.

The product has useful units as a built-in check:

\frac{\text{claims}}{\text{exposure}} \times \frac{\text{dollars}}{\text{claim}} = \frac{\text{dollars}}{\text{exposure}}, \qquad \frac{\text{dollars}}{\text{exposure}} \times \text{exposures} = \text{dollars}.

A mismatch in units often exposes a calculation error. For instance, frequency stated per 100 exposures must either be converted to a per-exposure rate or applied with a matching number of groups of 100. Similarly, a severity calculated in thousands of dollars must not be multiplied as though it were a dollar amount. Keep full precision through intermediate calculations and round only the reported result, using the precision appropriate to the question.

The decomposition is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus because it connects claim counts and loss amounts to an expected loss cost. It is a loss estimate, not a complete rate: expenses, profit and contingencies, and other rating provisions are not included in frequency multiplied by severity.

Key takeaway: Expected aggregate losses equal exposure volume multiplied by expected claims per exposure and expected loss per claim, subject to a consistent claim and loss basis and suitable assumptions for the selected estimates. Check the claim denominator, the exposure unit, and the dollar basis before interpreting the result.

Check Your Understanding

For each problem, keep the stated claim count, loss basis, and exposure unit consistent.

A hypothetical portfolio has 75 claims, 500 exposure-years, and $450,000 in losses. Find the observed frequency, average severity, and loss per exposure-year. (Answer: 0.15 claims per exposure-year; $6,000 per claim; $900 per exposure-year.)
A hypothetical class has 6 claims per 200 exposures and average severity of $7,500. Estimate aggregate expected losses for 2,400 exposures. (Answer: frequency 0.03; expected claims 72; expected losses $540,000.)
Segment A has 2 claims per 50 exposures, average severity $2,500, and 5,000 exposures. Segment B has 5 claims per 100 exposures, average severity $4,000, and 3,000 exposures. Find expected losses by segment and in total. (Answer: A, $500,000; B, $600,000; total, $1,100,000.)
A portfolio has 1,000 exposures, frequency 0.08, and severity $5,000. A scenario has 1,200 exposures, frequency 0.10, and severity $5,500. Find both aggregate expected losses and the ratio of scenario to original. (Answer: original, $400,000; scenario, $660,000; ratio, 1.65.)
A hypothetical experience period has 160 claims, 800 exposure-years, and $960,000 in losses. Using its observed frequency and severity, estimate expected losses for 1,500 exposure-years. (Answer: frequency 0.20; severity $6,000; loss per exposure-year $1,200; expected losses $1,800,000.)

What You'll Learn