Dependence Between Frequency and Severity

Why the Portfolio Average May Not Be a Product

A frequency–severity model often expresses expected loss per policy as expected claim frequency multiplied by expected claim severity. That factorization can be useful for a single risk class. But a portfolio may contain classes with different frequencies and severities, and those class characteristics may move together. If higher-frequency classes also have higher severities, multiplying the portfolio’s average frequency by its policy-weighted average severity can understate expected cost. If high frequency is paired with low severity, it can overstate expected cost.

The key is to average the expected cost at the risk-class level. The difference between that portfolio pure premium and the product of the two policy-weighted averages is the covariance of class frequency and severity, using the same policy or exposure weights for both averages. This is a statement about association across risk classes; it does not by itself describe every possible dependence between claim count and claim sizes within a class.

Definition: Suppose a portfolio is divided into risk classes $k=1,\ldots,K$, with policy or exposure weights $w_k$, where $w_k\geq 0$ and $\sum_k w_k=1$. Let $\lambda_k$ be the expected number of claims per policy or exposure unit in class $k$, and let $\mu_k$ be the expected payment per claim in that class. Under the frequency–severity factorization, the portfolio pure premium is $\sum_k w_k\lambda_k\mu_k$. The policy-weighted average frequency is $\bar{\lambda}=\sum_k w_k\lambda_k$, and the policy-weighted average severity is $\bar{\mu}=\sum_k w_k\mu_k$.

All quantities must use a consistent unit. For example, if $\lambda_k$ is claims per policy-year, the weights should represent shares of policy-years, and the resulting pure premium is dollars per policy-year. For policies with unequal exposure, weight by exposure rather than treating each policy as a full year. No discounting is used in the examples below. They require only the stated class-level expected frequencies and mean severities; no particular claim-count or severity distribution is assumed.

The Covariance Identity

Define the weighted covariance between class frequency and class severity by $ \operatorname{Cov}_w(\lambda,\mu)=\sum_k w_k(\lambda_k-\bar{\lambda})(\mu_k-\bar{\mu}) $. Using the same weights in the averages and covariance gives the identity $ \sum_k w_k\lambda_k\mu_k=\bar{\lambda}\bar{\mu}+\operatorname{Cov}_w(\lambda,\mu) $. The covariance therefore measures exactly how much the product of policy-weighted averages differs from the portfolio pure premium.

\begin{aligned} \operatorname{Cov}_w(\lambda,\mu) &=\sum_k w_k(\lambda_k-\bar{\lambda})(\mu_k-\bar{\mu})\\ &=\sum_k w_k\lambda_k\mu_k -\bar{\mu}\sum_k w_k\lambda_k -\bar{\lambda}\sum_k w_k\mu_k +\bar{\lambda}\bar{\mu}\sum_k w_k\\ &=\sum_k w_k\lambda_k\mu_k-\bar{\lambda}\bar{\mu}. \end{aligned}

The last equality follows because the weights sum to one, while their weighted averages satisfy $\sum_k w_k\lambda_k=\bar{\lambda}$ and $\sum_k w_k\mu_k=\bar{\mu}$. Rearranging proves the identity. Positive covariance means higher-frequency classes tend to have higher severity, so the class-level pure premium exceeds the product of the two averages. Negative covariance means higher frequency tends to be paired with lower severity, so the class-level pure premium is below that product. Zero covariance makes the two calculations equal, even if the class values are not all identical.

From Class-Level Expected Loss to Portfolio Cost

For a policy in class $k$, let $N$ be its claim count and let $X_j$ be the payment for claim $j$. Its aggregate claim payment is $S=\sum_{j=1}^{N}X_j$. If the expected payment per claim in class $k$ is $\mu_k$, and the expected payment per claim does not change with the claim count within that class, then $E[S\mid k]=E[N\mid k]\mu_k=\lambda_k\mu_k$. Averaging these conditional expected payments over the portfolio gives $\sum_k w_k\lambda_k\mu_k$. The condition matters: if claim sizes within a class have a mean that depends on the claim count, the simple class-level product may not give that class’s expected aggregate payment. The severity mean used in the factorization must correspond to the same risk population and payment definition as the frequency.

Set the unit and weights.
Use policy weights for a portfolio of equal-length policies, or exposure weights when policies contribute different amounts of exposure.

Calculate each class cost.
For every class, multiply its expected frequency by its expected payment per claim to obtain expected loss per unit.

Average class costs.
Weight each class cost by its share of the portfolio. This is the portfolio pure premium under the stated factorization.

Compare with the shortcut.
Calculate the two policy-weighted averages and their product. The difference, true pure premium minus shortcut, must equal the weighted covariance.

Worked Calculations

Worked Example: Negative Association Across Two Classes

A hypothetical insurer has two equally weighted policy classes. Class A has expected frequency $0.10$ claims per policy-year and mean severity $$2{,}000$. Class B has expected frequency $0.30$ and mean severity $$1{,}000$. Calculate the portfolio pure premium, the product of the policy-weighted average frequency and severity, the weighted covariance, and the claim-weighted mean severity. No discounting or particular probability distribution is assumed.

Method. Calculate expected cost in each class first, then take the policy-weighted average. This avoids assuming that the product of separate averages equals the average of the products.

Class A expected loss is $0.10\times \$2{,}000=\$200$ per policy-year. Class B expected loss is $0.30\times \$1{,}000=\$300$ per policy-year. Since each class has weight $0.50$, the portfolio pure premium is $0.50(\$200)+0.50(\$300)=\$100+\$150=\$250$ per policy-year.

The policy-weighted average frequency is $0.50(0.10)+0.50(0.30)=0.20$ claims per policy-year. The policy-weighted average severity is $0.50(\$2{,}000)+0.50(\$1{,}000)=\$1{,}000+\$500=\$1{,}500$. Their product is $0.20\times \$1{,}500=\$300$ per policy-year.

The weighted covariance is the true pure premium minus the product of the averages: $\$250-\$300=-\$50$ per policy-year. This is negative because the higher-frequency class has the lower severity. The expected number of claims per policy-year is $0.20$, so the claim-weighted mean severity is $\$250/0.20=\$1{,}250$ per claim. The check is $0.20\times \$1{,}250=\$250$, matching the portfolio pure premium.

Common mistake avoided. The equal policy weights do not make the class expected costs equal. Averaging the two costs gives $\$250$, not $\$200$; multiplying the policy-weighted averages gives $\$300$, not the pure premium.

Worked Example: Positive Association Across Classes

A hypothetical portfolio has 25% of its policy-years in class C and 75% in class D. Class C has expected frequency $0.10$ and mean severity $$1{,}000$; class D has expected frequency $0.30$ and mean severity $$3{,}000$. Find the portfolio pure premium, the product of policy-weighted averages, the covariance, and the claim-weighted mean severity. No discounting or distributional assumption beyond the given means is needed.

Method. Use the policy-year shares to weight class-level expected costs, then compare with the shortcut.

Class C expected loss is $0.10\times \$1{,}000=\$100$ per policy-year. Class D expected loss is $0.30\times \$3{,}000=\$900$. Thus the portfolio pure premium is $0.25(\$100)+0.75(\$900)=\$25+\$675=\$700$ per policy-year.

The average frequency is $0.25(0.10)+0.75(0.30)=0.025+0.225=0.25$. The average severity is $0.25(\$1{,}000)+0.75(\$3{,}000)=\$250+\$2{,}250=\$2{,}500$. Their product is $0.25\times \$2{,}500=\$625$ per policy-year. Therefore the weighted covariance is $\$700-\$625=\$75$ per policy-year, which is positive as both frequency and severity are higher in class D.

The expected claim count per policy-year is $0.25$. The claim-weighted mean severity is $\$700/0.25=\$2{,}800$. Checking, $0.25\times \$2{,}800=\$700$, as required.

Common mistake avoided. The policy-weighted average severity of $$2{,}500$ is not the mean severity of a randomly selected claim. Claims are more likely to come from the higher-frequency class, which raises the claim-weighted mean to $$2{,}800$.

Worked Example: Unequal Exposure Shares and a Claim-Weighted Mean

A hypothetical book has 600 policy-years in class E and 400 policy-years in class F. Class E has expected frequency $0.08$ claims per policy-year and mean severity $$1{,}500$. Class F has expected frequency $0.20$ and mean severity $$900$. Calculate expected claims, expected losses, portfolio pure premium, the product of the portfolio-average frequency and policy-weighted severity, and the claim-weighted severity. Assume the stated frequencies and severities apply over the same period and coverage basis; no discounting or specific claim distributions are required.

Method. The weights are $600/1{,}000=0.60$ and $400/1{,}000=0.40$. Calculate total expected losses and divide by total policy-years.

Class E has $600\times0.08=48$ expected claims and expected loss $48\times \$1{,}500=\$72{,}000$. Class F has $400\times0.20=80$ expected claims and expected loss $80\times \$900=\$72{,}000$. Across the 1,000 policy-years, expected claims total $48+80=128$, and expected losses total $$72{,}000+\$72{,}000=\$144{,}000$. The pure premium is $\$144{,}000/1{,}000=\$144$ per policy-year.

The portfolio-average frequency is $128/1{,}000=0.128$. The policy-weighted severity is $0.60(\$1{,}500)+0.40(\$900)=\$900+\$360=\$1{,}260$. Their product is $0.128\times \$1{,}260=\$161.28$ per policy-year, which is not the portfolio pure premium. The covariance is $\$144-\$161.28=-\$17.28$ per policy-year.

The claim-weighted mean severity is total expected loss divided by expected claims: $\$144{,}000/128=\$1{,}125$ per claim. The check is $0.128\times \$1{,}125=\$144$ per policy-year. The result is consistent with the total-loss calculation.

Common mistake avoided. Using 600 and 400 as though they were equal class weights would be wrong. Here the exposure shares are 60% and 40%, and the claim-weighted severity uses expected claim counts, 48 and 80, rather than policy-years.

Which Severity Average Is Appropriate?

A policy-weighted severity answers: “What is the average of the class mean severities for a randomly selected policy or exposure unit?” A claim-weighted severity answers: “What is the expected severity for a randomly selected claim from the portfolio?” The second gives more weight to classes that produce more claims. If class $k$ contributes expected claim count proportional to $w_k\lambda_k$, the portfolio claim-weighted severity is $ \sum_k w_k\lambda_k\mu_k / \sum_k w_k\lambda_k $, provided the denominator is positive. Consequently, multiplying the portfolio-average frequency by claim-weighted severity reproduces the pure premium. Multiplying frequency by policy-weighted severity generally does not.

\bar{\lambda}\,\mu_{\text{claim}} = \left(\sum_k w_k\lambda_k\right) \left(\frac{\sum_k w_k\lambda_k\mu_k}{\sum_k w_k\lambda_k}\right) = \sum_k w_k\lambda_k\mu_k.

This identity is useful as a calculation check, not a reason to confuse the two severity definitions. A severity model fitted to claim records typically targets claim-level severity, while a class comparison may report mean severities with one record per policy or class. Before combining estimates, verify what population each mean represents and what weights were used. The frequency and severity must also refer to the same coverage, claim definition, time period, and exposure basis.

Common Exam Traps

Multiplying averages without checking association. The product of policy-weighted means omits the covariance term. It is valid when that weighted covariance is zero, not merely because frequency and severity are each correctly averaged.
Changing weights between calculations. To use the covariance identity, frequency, severity, and their joint product must use the same policy or exposure weights.
Using policy weights for claim severity. A randomly selected claim is not generally from a randomly selected policy. Claim weighting is proportional to both the policy share and its expected frequency.
Mixing exposure units. An annual frequency cannot be combined with an exposure denominator in months unless the frequency is converted to the matching time unit.
Assuming the covariance explains all dependence. The identity describes association across classes. Within-class dependence between claim count and severity can also matter and may invalidate the simple class-level factorization unless the severity mean is appropriate for the observed claim count.

Key takeaway: Under the frequency–severity factorization, calculate the portfolio pure premium as the weighted average of class-level frequency times class-level severity. The product of policy-weighted average frequency and severity differs by their weighted covariance. A claim-weighted severity, by contrast, combines with portfolio frequency to reproduce the pure premium.

Check Your Understanding

For each problem, use consistent policy or exposure weights and distinguish policy-weighted from claim-weighted severity.

A hypothetical portfolio has two equally weighted classes. Their frequency and severity pairs are $0.10$ claims per policy-year and $$800$, and $0.20$ claims per policy-year and $$1{,}600$. Find the portfolio pure premium, the product of policy-weighted averages, and the covariance. (Answer: pure premium $$200$ per policy-year; product $$180$; covariance $+\$20$ per policy-year.)
A portfolio is 25% class A and 75% class B by policy-year. Frequencies are $0.04$ and $0.12$; severities are $$2{,}500$ and $$1{,}000$, respectively. Find pure premium, product of policy-weighted averages, and claim-weighted severity. (Answer: pure premium $$115$ per policy-year; product $$137.50$; claim-weighted severity $$1{,}150$ per claim.)
A hypothetical book has 40% of exposure in class C and 60% in class D. The frequencies are $0.05$ and $0.15$, and the severities are $$3{,}000$ and $$1{,}200$. Calculate pure premium and the product of policy-weighted averages. (Answer: pure premium $$168$ per exposure unit; product $$211.20$.)
A portfolio has 500 policy-years in class E and 1,500 in class F. Their respective frequencies are $0.20$ and $0.08$, and their severities are $$500$ and $$2{,}500$. Find the total expected loss, pure premium, and claim-weighted severity. (Answer: expected loss $$350{,}000$; pure premium $$175$ per policy-year; claim-weighted severity $$1{,}590.91$ per claim.)
Two equally weighted classes have frequencies $0.05$ and $0.25$ claims per policy-year and severities $$1{,}000$ and $$2{,}000$, respectively. Find the true pure premium, the product of policy-weighted averages, and the claim-weighted mean severity; explain the sign of the covariance. (Answer: pure premium $$275$ per policy-year; product $$225$; claim-weighted severity $$1{,}833.33$ per claim; covariance $+\$50$ per policy-year because higher frequency coincides with higher severity.)

What You'll Learn