Frequency-Severity Factorization

Why Separate Claim Counts From Claim Amounts?

Expected loss per exposure can be described in two distinct ways: how often claims are expected to occur, and how much is expected to be paid for each claim. This distinction is useful in ratemaking because claim counts and claim amounts are different responses, measured on different units, and may require different data and model specifications. The frequency-severity factorization writes expected loss per exposure as the product of those two expected values.

The factorization is an accounting and expectation result under a specified model; it does not mean that frequency and severity are interchangeable, or that either component alone describes the expected loss. A frequency prediction is measured in claims per exposure. A severity prediction is measured in dollars per claim. Their product is dollars per exposure, the unit of a pure premium.

Definition: For a specified rating group and observation period, let $e$ be the amount of exposure, $N$ the number of claims, and $X_j$ the loss amount for claim $j$. Aggregate loss is $L=\sum_{j=1}^{N}X_j$. Expected claim frequency per exposure is $\lambda=E[N]/e$, and expected claim severity is $\mu=E[X_j]$ for a claim drawn from the specified claim population. The pure premium is expected aggregate loss per exposure, $E[L]/e$.

The Factorization and Its Assumptions

To obtain the basic product formula, assume a fixed positive exposure amount $e$, that claim amounts $X_1,X_2,\ldots$ have the same finite mean $\mu$, and that the claim amounts are independent of the claim count $N$. In particular, conditional on there being $n$ claims, the expected aggregate loss is $n\mu$. Under this model, expected aggregate loss per exposure is the expected claim count per exposure multiplied by the expected amount per claim.

E[L]=E[N]\mu \qquad\text{and therefore}\qquad \frac{E[L]}{e} = \frac{E[N]}{e}\mu = \lambda\mu.

The units confirm the interpretation:

\left(\frac{\text{claims}}{\text{exposure}}\right) \left(\frac{\text{dollars}}{\text{claim}}\right) = \frac{\text{dollars}}{\text{exposure}}.

The assumptions are important. The claim amount must refer to the same claim population and loss basis used to define the claim count. For example, the claim count and severity must both be based on the same coverage and claim definition. The simple derivation also uses a common claim severity mean and independence between claim count and claim amounts. More general conditions for multiplication are treated separately in the material on conditions for multiplying frequency and severity; the product should not be applied automatically when its assumptions do not hold.

Why the Product Formula Holds

The justification is the law of iterated expectation. First condition on the observed claim count. If $N=n$, aggregate loss is the sum of $n$ claim amounts. Each has expected amount $\mu$, so linearity of expectation gives conditional expected aggregate loss $n\mu$. Then average over the possible values of $N$.

Condition on the claim count.
When $N=n$, $L=X_1+\cdots+X_n$. Under the stated common-mean model, $E[L\mid N=n]=n\mu$.

Take expectation over the count.
The law of iterated expectation gives $E[L]=E[E[L\mid N]]=E[N\mu]$.

Use the fixed severity mean and divide by exposure.
Since $\mu$ is constant, $E[L]=\mu E[N]$. Dividing by $e$ gives $E[L]/e=(E[N]/e)\mu=\lambda\mu$.

This derivation explains the roles of the two components. Frequency uses information about the number of claims relative to exposure. Severity uses information about the amount of loss per claim. Neither component is itself the pure premium, and multiplying them produces an expected value, not a guarantee about the loss on any individual policy.

Worked Calculations

Worked Example: Calculating Frequency and Severity From Aggregate Experience

A hypothetical portfolio has 240 policy-years of exposure. During the observation period it reports 72 claims with total covered loss of $540,000. For this calculation, treat the observed frequency and average severity as the respective component estimates. Find the frequency, severity, pure premium per policy-year, and the implied aggregate loss estimate.

Method and common mistake avoided. Divide the claim count by policy-years to obtain frequency, and divide total loss by claim count to obtain severity. Do not divide total loss by exposure and then call that result severity; that quotient is already the pure premium.

Step 1: Frequency. There are 72 claims over 240 policy-years:

\widehat{\lambda}=\frac{72\text{ claims}}{240\text{ policy-years}} =0.30\text{ claims per policy-year}.

Step 2: Severity. The average loss per claim is:

\widehat{\mu}=\frac{\$540{,}000}{72\text{ claims}} =\$7{,}500\text{ per claim}.

Step 3: Pure premium. Multiply the components:

\widehat{\lambda}\widehat{\mu} =0.30\frac{\text{claims}}{\text{policy-year}} \times \$7{,}500\frac{1}{\text{claim}} =\$2{,}250\text{ per policy-year}.

Step 4: Check. The implied aggregate loss is $240\times \$2{,}250=\$540{,}000$, equal to the stated total. The component units cancel claims and leave dollars per policy-year, as required.

Worked Example: Comparing Two Rating Groups

A hypothetical insurer evaluates two rating groups. Group A has 80 policy-years, expected frequency 0.25 claims per policy-year, and expected severity of $4,000 per claim. Group B has 120 policy-years, expected frequency 0.50 claims per policy-year, and expected severity of $7,000 per claim. Find each group’s expected aggregate loss and the portfolio pure premium per policy-year.

Method and common mistake avoided. Calculate expected loss by group using exposure times frequency times severity, then divide total expected loss by total exposure. A portfolio rate is exposure-weighted; taking an unweighted average of the two group pure premiums would give the wrong result because the groups have different exposure amounts.

Group A. Expected claims are $80\times0.25=20$. Expected aggregate loss is $20\times \$4{,}000=\$80{,}000$. Equivalently, its pure premium is $0.25\times \$4{,}000=\$1{,}000$ per policy-year, and $80\times \$1{,}000=\$80{,}000$.

Group B. Expected claims are $120\times0.50=60$. Expected aggregate loss is $60\times \$7{,}000=\$420{,}000$. Equivalently, its pure premium is $0.50\times \$7{,}000=\$3{,}500$ per policy-year, and $120\times \$3{,}500=\$420{,}000$.

Portfolio total and rate. Total expected loss is $\$80{,}000+\$420{,}000=\$500{,}000$, and total exposure is $80+120=200$ policy-years. Therefore:

\text{Portfolio pure premium} = \frac{\$500{,}000}{200\text{ policy-years}} = \$2{,}500\text{ per policy-year}.

Check. The portfolio frequency is $80(0.25)+120(0.50)=80$ expected claims divided by 200 policy-years, or 0.40 claims per policy-year. The exposure-weighted average severity is $\$500{,}000/80=\$6{,}250$ per claim. Their product, $0.40\times \$6{,}250=\$2{,}500$, agrees with the portfolio calculation. This check uses the aggregate claim and loss totals, not an unweighted average of group rates.

Worked Example: Converting Partial-Year Exposure Correctly

A hypothetical book contains 160 policies, each insured for nine months. A frequency estimate is 0.18 claims per policy-year, and the expected severity is $5,500 per claim. Find total exposure in policy-years, expected claim count, expected aggregate loss, and pure premium per policy-year.

Method and common mistake avoided. Convert the exposure period to years before using a frequency stated per policy-year. Multiplying 160 policies by 0.18 without converting the nine-month term would treat every policy as a full policy-year.

Step 1: Exposure. Nine months is $9/12=0.75$ year per policy. Thus total exposure is $160\times0.75=120$ policy-years.

Step 2: Expected claim count.

E[N]=120\text{ policy-years}\times 0.18\frac{\text{claims}}{\text{policy-year}} =21.6\text{ claims}.

An expected count need not be an integer; 21.6 is an expectation, not a claim count that must occur in one realization.

Step 3: Expected aggregate loss.

E[L]=21.6\text{ claims}\times \$5{,}500\text{ per claim} =\$118{,}800.

Step 4: Pure premium and check. The per-policy-year pure premium is $0.18\times \$5{,}500=\$990$. Multiplying by 120 policy-years gives $120\times \$990=\$118{,}800$, matching the expected aggregate loss.

What Each Component Represents

In a frequency-severity approach, the components answer different actuarial questions. The frequency component describes expected claim counts relative to exposure. Its underlying response is a claim count, and its denominator is exposure. The severity component describes expected loss conditional on a claim being in the defined claim population. Its underlying response is a claim amount, and its denominator is the number of claims. The product converts these into expected loss per exposure.

Component	Quantity summarized	Typical unit	Primary data basis
Frequency	Expected number of claims relative to exposure	Claims per policy-year	Claim counts and exposure
Severity	Expected loss amount per claim	Dollars per claim	Loss amounts for the defined claims
Pure premium	Expected aggregate loss relative to exposure	Dollars per policy-year	Frequency and severity, or aggregate loss and exposure

The claim definition matters. A severity mean calculated from one set of claims cannot be combined casually with a frequency count defined on another basis. Likewise, the exposure unit must match the one used to state frequency. If frequency is per vehicle-year, multiplying it by severity gives loss per vehicle-year—not automatically loss per policy-year. These are unit checks, but they also help reveal mismatched coverage definitions or inconsistent data preparation.

Common Exam Trap: Averaging Rates Without Weighting

Suppose two groups have pure premiums of $1,000 and $3,500 per policy-year. Their simple average is $2,250. That number is not generally the portfolio pure premium. In the worked portfolio example, Group A has 80 policy-years and Group B has 120, so the correct total is:

\frac{80(\$1{,}000)+120(\$3{,}500)}{80+120} = \frac{\$500{,}000}{200} = \$2{,}500\text{ per policy-year}.

The incorrect calculation, $(\$1{,}000+\$3{,}500)/2=\$2{,}250$, assigns equal weight to groups despite their unequal exposures. It would be correct only if their exposures were equal. More generally, add expected losses across groups and divide by their combined exposure.

Key takeaway: Under the stated common-mean, count-independent model, pure premium per exposure is expected claims per exposure multiplied by expected loss per claim. Always identify the claim population, exposure unit, component units, and weighting basis before interpreting or combining the results.

Check Your Understanding

For each problem, keep claim and exposure units explicit and check the final result against expected aggregate loss where possible.

A hypothetical class has 500 policy-years, an expected claim count of 60, and expected severity of $2,800 per claim. Find frequency and pure premium per policy-year. (Answer: 0.12 claims per policy-year; $336 per policy-year.)
For a hypothetical segment, exposure is 300 policy-years, expected claim count is 45, and expected aggregate loss is $315,000. Find expected severity and pure premium. (Answer: $7,000 per claim; $1,050 per policy-year.)
Group A has 60 policy-years, frequency 0.20, and severity $4,500. Group B has 90 policy-years, frequency 0.40, and severity $3,000. Find the portfolio pure premium. (Answer: $1,080 per policy-year.)
A hypothetical book has 240 policies insured for six months each. Its frequency estimate is 0.25 claims per policy-year and expected severity is $6,400. Find total exposure, expected claims, expected aggregate loss, and pure premium per policy-year. (Answer: 120 policy-years; 30 expected claims; $192,000; $1,600 per policy-year.)
A hypothetical rating group has expected frequency of 30 claims per 100 exposure units and expected severity of $3,900 per claim. State the frequency per exposure unit and the pure premium per exposure unit. (Answer: 0.30 claims per exposure unit; $1,170 per exposure unit.)

What You'll Learn