Combining Frequency and Severity Relativities

Why Frequency and Severity Relativities Must Be Combined

A pure-premium indication estimates expected claim cost per unit of exposure. That cost can change because a risk is expected to have more or fewer claims, because its claims are expected to be larger or smaller, or both. A frequency relativity measures the first change; a severity relativity measures the second. Neither component alone describes the change in expected loss cost when both components differ.

The central result is simple: for a matched pair of rating profiles, the pure-premium relativity is the frequency relativity multiplied by the severity relativity. This product is valid only when the frequency and severity estimates refer to compatible profiles, exposure units, claim definitions, and coverage. The product is not a general rule for multiplying any two factors labeled “frequency” and “severity.”

Definition: Let $Z$ denote a rating profile and $e$ its exposure. Let $N$ be the number of claims, and let $Y_j$ be the loss amount for claim $j$. The expected claim frequency per exposure unit is $\lambda(Z)=E[N\mid Z]/e$. The expected severity is $m(Z)=E[Y_j\mid Z,\text{claim}]$, with the claim population and loss basis specified. The pure premium is the expected aggregate loss per exposure unit. Under the conditions developed below, it is $P(Z)=\lambda(Z)m(Z)$.

A frequency model commonly estimates expected claim counts using exposure as an offset, while a severity model estimates average loss conditional on a claim. In either case, the fitted values must be interpreted on the scales and units used by the models. A frequency of 0.06 claims per policy-year multiplied by a severity of $2,500 per claim gives $150 per policy-year, not $150 per claim.

Deriving the Product of the Relativities

Consider one rating profile $Z$ with exposure $e$. Write aggregate loss as $L=\sum_{j=1}^{N}Y_j$, with $L=0$ if $N=0$. Suppose that, conditional on the profile, each claim has expected severity $m(Z)$, including conditional on the value of $N$. In particular, this condition holds when claim severities are independent of claim count and have the same conditional mean $m(Z)$. Also suppose the expected claim count is finite.

Conditioning first on the number of claims makes the expected aggregate loss clear. Given $N=n$, the expected sum of the $n$ claim amounts is $n\,m(Z)$. Averaging over the possible values of $N$ gives:

\begin{aligned} E[L\mid Z] &=\sum_{n=0}^{\infty}P(N=n\mid Z)\,E[L\mid N=n,Z]\\ &=\sum_{n=0}^{\infty}P(N=n\mid Z)\,n\,m(Z)\\ &=E[N\mid Z]\,m(Z). \end{aligned}

Dividing by exposure $e$ yields $P(Z)=\lambda(Z)m(Z)$. This is an expected-cost identity: it does not require the claim count to be Poisson. A Poisson frequency model is one possible model for estimating the mean count, but the identity follows from the stated conditional-mean condition. If the expected severity per claim depends on the number of claims, the product of the marginal expected frequency and the marginal expected severity need not give the expected aggregate loss.

For two profiles, a reference profile $Z_0$ and a comparison profile $Z_1$, define the frequency and severity relativities as ratios to their respective reference values. The corresponding pure-premium ratio is:

R_{\text{PP}} =\frac{P(Z_1)}{P(Z_0)} =\frac{\lambda(Z_1)m(Z_1)}{\lambda(Z_0)m(Z_0)} =\underbrace{\frac{\lambda(Z_1)}{\lambda(Z_0)}}_{R_F} \underbrace{\frac{m(Z_1)}{m(Z_0)}}_{R_S} =R_F R_S.

The equation compares the same two profiles in both component models. A factor of 1.20 means a 20% increase relative to the relevant model’s reference value; it does not mean an increase of 20 percentage points. If fitted log-link effects are used, their exponentiated effects are multiplicative relativities. On the log scale, multiplication of compatible relativities corresponds to addition of their log effects.

Conditions for the product: The frequency estimate must be expected claims per the stated exposure unit, and the severity estimate must be expected loss per claim on the same loss and coverage basis. Both relativities must compare the same reference and target profiles. The expected aggregate loss must factor as expected claim count times the applicable expected severity; for example, claim severities may be conditionally independent of claim count with a common conditional mean. Any interactions or profile differences represented in either fitted model must be included in the comparison.

Worked Applications

Worked Example: Combining One Frequency and One Severity Factor

A hypothetical insurer uses a policy-year exposure unit. For a reference profile, fitted frequency is 0.08 claims per policy-year and fitted average severity is $2,500 per claim. A comparison profile has a frequency relativity of 1.25 and a severity relativity of 0.92. Find both pure premiums and the pure-premium relativity.

Method and assumptions. Treat the fitted frequency as the mean of a Poisson claim count per policy-year, and the fitted severity as the mean of an independent exponential claim-severity distribution. The exponential mean is $2,500 for the reference profile; its target mean is obtained from the severity relativity. No investment income or present-value discounting is included (effective annual interest $i=0\%$); mortality is not applicable.

Step 1: Reference pure premium. Multiply the expected frequency by expected severity:

P_0=0.08\times \$2{,}500=\$200.00\text{ per policy-year}.

Step 2: Target frequency and severity. The target frequency is $0.08\times1.25=0.10$ claims per policy-year. The target severity is $\$2{,}500\times0.92=\$2{,}300.00$ per claim.

Step 3: Target pure premium.

P_1=0.10\times \$2{,}300.00=\$230.00\text{ per policy-year}.

Step 4: Check the indicated change. The direct pure-premium ratio is $\$230/\$200=1.15$. The component product is $1.25\times0.92=1.15$, so the indication is a 15% increase. This check confirms that the target values and relativities agree. The common mistake is to use only the frequency factor and report a 25% increase; that ignores the 8% lower severity.

Worked Example: Combining Relativities Across Two Rating Variables

For a hypothetical rating plan, the reference profile has fitted frequency 0.06 claims per policy-year and fitted mean severity $4,000 per claim. A comparison profile changes both territory and vehicle group. The fitted frequency relativities are 1.30 for territory and 0.80 for vehicle group; the fitted severity relativities are 1.25 and 1.10, respectively. Assume the fitted specification has no interaction between these variables. Calculate the pure-premium relativity and the comparison profile’s pure premium.

Method and assumptions. Use multiplicative factorization with no frequency or severity interaction between territory and vehicle group. The count is modeled as Poisson with the stated per-policy-year mean; claim severity is exponential with its stated mean and independent of count. The reference exponential mean is $4,000. No discounting is applied ($i=0\%$); mortality is not applicable.

Step 1: Combine the frequency relativities. Under the stated no-interaction specification, multiply the two frequency factors:

R_F=1.30\times0.80=1.04.

The comparison frequency is $0.06\times1.04=0.0624$ claims per policy-year.

Step 2: Combine the severity relativities.

R_S=1.25\times1.10=1.375.

The comparison mean severity is $\$4{,}000\times1.375=\$5{,}500.00$ per claim.

Step 3: Calculate the pure-premium relativity and amount.

R_{\text{PP}}=1.04\times1.375=1.43, \qquad P_1=0.0624\times\$5{,}500.00=\$343.20\text{ per policy-year}.

The reference pure premium is $0.06\times\$4{,}000=\$240.00$, and $\$343.20/\$240.00=1.43$. Thus the direct calculation checks the product. A common mistake is to multiply all four factors without grouping or checking what each applies to; here that happens to give the same arithmetic, but the grouped calculation makes clear that frequency factors affect frequency and severity factors affect severity.

Worked Example: Keeping Exposure Units Consistent

A hypothetical fleet segment has 42 expected claims in 1,000 vehicle-years. Its reference severity is modeled as exponential with mean $1,300 per claim. For a comparison segment, the fitted frequency relativity is 1.10 and the fitted severity relativity is 1.20. Find the pure premiums per vehicle-year, the comparison expected aggregate loss for 1,000 vehicle-years, and the pure-premium relativity.

Method and assumptions. The count for 1,000 vehicle-years is Poisson with mean 42; equivalently, the frequency is 0.042 claims per vehicle-year. Claim severities are exponential with the stated reference mean and are independent of claim count. No discounting is applied ($i=0\%$); mortality is not applicable.

Step 1: Convert the observed scale to the exposure-unit scale.

\lambda_0=\frac{42\text{ claims}}{1{,}000\text{ vehicle-years}} =0.042\text{ claims per vehicle-year}.

Step 2: Calculate the reference pure premium.

P_0=0.042\times\$1{,}300=\$54.60\text{ per vehicle-year}.

Step 3: Apply each comparison relativity. The comparison frequency is $0.042\times1.10=0.0462$ claims per vehicle-year. The comparison severity is $\$1{,}300\times1.20=\$1{,}560.00$ per claim. Therefore:

P_1=0.0462\times\$1{,}560.00=\$72.072\text{ per vehicle-year} \approx \$72.07.

The pure-premium relativity is $1.10\times1.20=1.32$, a 32% increase. For 1,000 vehicle-years, the comparison expected aggregate loss is $1{,}000\times\$72.072=\$72{,}072.00$. The reference aggregate is $1{,}000\times\$54.60=\$54{,}600.00$; their ratio is $72{,}072/54{,}600=1.32$. The common mistake is to treat 42 claims as a frequency per vehicle-year rather than per 1,000 vehicle-years. That would overstate the pure premium by a factor of 1,000.

Interactions, Reference Profiles, and When Not to Multiply

When a fitted model has a multiplicative structure, multiple fitted effects can be combined by multiplying the relevant factors. But the factors must describe the actual comparison. If a model includes a territory-by-vehicle interaction, the comparison may require the interaction effect as well as the two main effects. Multiplying only the main-effect relativities would omit part of the fitted change. The same principle applies if an effect is conditional on another rating variable: use the relativity for the relevant condition, not a factor taken from a different profile.

Frequency and severity relativities also need compatible reference definitions. A frequency ratio comparing one territory with a base territory cannot automatically be combined with a severity ratio comparing a different pair of territories. Nor can a severity estimate conditional on one claim definition be multiplied by frequency for a different claim definition without a valid mapping between them. Check whether both components use the same policy period, exposure unit, coverage, claim inclusion rules, and loss basis.

A particularly important distinction is between an individual profile and a portfolio average. The product $R_F R_S$ gives the ratio for the matched profiles whose frequency and severity means appear in the ratios. It does not generally give an aggregate portfolio ratio by multiplying an average frequency relativity by an average severity relativity. Different profiles can have different claim frequencies, severities, and exposure weights. For portfolio totals, calculate expected aggregate loss for each relevant profile, sum those expected losses, and then form the portfolio ratio.

Common exam trap: Suppose a target profile has frequency relativity 1.20 and severity relativity 0.90. The incorrect calculation $1.20+0.90=2.10$ treats the two changes as additive, while $1.20-1+0.90-1=0.10$ also incorrectly reports a 10% increase. The correct calculation is $1.20\times0.90=1.08$, an 8% pure-premium increase. Multiplication follows because expected loss per exposure is the product of expected claims per exposure and expected loss per claim.

Practical Checklist for an Indication

Before reporting a combined indication, work through the following checks. They prevent unit errors and ensure the arithmetic represents the fitted model rather than an unintended comparison.

Identify the target and reference profiles.
State exactly which risks are being compared, including the rating-variable levels and any relevant interactions.

Verify the frequency basis.
Confirm that frequency is expected claims per exposure unit, not a raw count or a rate on a different time or exposure scale.

Verify the severity basis.
Confirm that severity is expected loss per claim and corresponds to the same coverage, claim definition, and loss basis as the frequency.

Combine compatible factors.
Multiply the frequency relativity by the severity relativity, including any fitted interactions required for the target profile.

Check against fitted means.
When available, calculate each profile’s frequency times severity directly and verify that their ratio equals the relativity product.

The result is a pure-premium indication, not automatically a final premium change. Other pricing components and constraints may affect a final rate, but they do not change the expected-loss identity derived here. The key actuarial judgment is to ensure the two fitted components represent a consistent decomposition of expected claim cost.

Check Your Understanding

For each problem, use the stated exposure, claim, and model assumptions. Round dollar amounts to the nearest cent where needed.

A reference risk has frequency 0.10 claims per policy-year and severity $2,200 per claim. A target risk has frequency relativity 0.80 and severity relativity 1.15. Find the target pure premium and its relativity to the reference. (Answer: $202.40 per policy-year; relativity 0.92.)
A reference profile has frequency 0.05 and mean severity $2,000. Two no-interaction rating effects have frequency relativities 1.20 and 0.90, and severity relativities 1.10 and 1.25. Find the combined pure-premium relativity and target pure premium. (Answer: 1.485; $148.50 per exposure unit.)
A segment has 30 expected claims in 600 policy-years and mean severity $1,800. Its target frequency relativity is 1.30 and severity relativity is 0.80. Find the reference and target pure premiums and the combined relativity. (Answer: $90.00 and $93.60 per policy-year; relativity 1.04.)
A fitted frequency model gives a target-to-reference main-effect product of $1.10\times0.90$, and an applicable frequency interaction relativity of 1.20. Severity relativity is 0.80. What is the correct combined pure-premium relativity, and what would be omitted by ignoring the interaction? (Answer: $1.10\times0.90\times1.20\times0.80=0.9504$; ignoring the interaction gives 0.792.)
Two portfolio classes have exposures of 80 and 20 units. Their respective fitted frequencies are 0.02 and 0.10 claims per unit, and severities are $1,000 and $3,000 per claim. Find the portfolio pure premium. (Answer: $(80\times0.02\times\$1{,}000+20\times0.10\times\$3{,}000)/100=\$76.00$ per exposure unit.)

What You'll Learn