Comparing Frequency and Severity Components

Why Compare Frequency and Severity Components?

A ratemaking analysis may describe a risk through both its claim frequency and its claim severity. The two responses have different units: frequency is a count per exposure period, while severity is an amount of dollars per claim. Bühlmann variance components can be calculated for either response, but their numerical scales and interpretations are not interchangeable.

For each response, the components separate variation within a risk from variation in the risk’s expected value across the portfolio. The first is driven by random outcomes for a given risk; the second reflects persistent differences in expected outcomes among risks. Keeping the response and its observation unit explicit is essential when calculating or comparing these quantities.

Definition: Let \(\Theta\) represent a risk’s unknown or varying risk parameter, and let \(Y\) be an observation defined for a particular unit, such as claims per policy-year or dollars per claim. The expected process variance (EPV) is \(E[\operatorname{Var}(Y\mid\Theta)]\). The variance of hypothetical means (VHM) is \(\operatorname{Var}(E[Y\mid\Theta])\). Both components are defined for the same response and observation unit.

For frequency, \(Y\) might be a claim count during one policy-year. Its EPV and VHM are then measured in squared counts per policy-year. For severity, \(Y\) might be the payment on one claim; the corresponding components are measured in squared dollars per claim. The different units already warn against comparing their raw numerical sizes.

The Variance Split Behind the Components

The total variance of an observation combines its conditional variance within a risk and the variation in conditional means across risks. The law of total variance gives

\operatorname{Var}(Y) = E[\operatorname{Var}(Y\mid\Theta)] + \operatorname{Var}(E[Y\mid\Theta]) = \operatorname{EPV}+\operatorname{VHM}.

This identity is useful only when both terms refer to the same \(Y\) and the same observation unit. It does not say that the frequency EPV can be added to the severity VHM. Those terms describe different random variables and have different units.

Justification: Write \(Y-E[Y]\) as \((Y-E[Y\mid\Theta])+(E[Y\mid\Theta]-E[Y])\), then square and take expectations. The first squared term has expectation \(E[\operatorname{Var}(Y\mid\Theta)]\); the second has expectation \(\operatorname{Var}(E[Y\mid\Theta])\). The cross term is zero because \(E[Y-E[Y\mid\Theta]\mid\Theta]=0\). Thus total variance is the sum of EPV and VHM.

A frequency model commonly specifies a conditional claim-count distribution given a risk parameter, such as a claim probability \(p\) or a Poisson rate \(\lambda\). A severity model specifies a conditional distribution of claim payment given a risk-specific parameter, such as a mean severity \(\mu\). The parameters themselves vary across the portfolio, which creates VHM.

Worked Example: Bernoulli Frequency and Normal Severity

Consider a hypothetical portfolio with one policy-year as the frequency observation unit and one claim payment as the severity observation unit. For frequency, each risk has a claim probability \(p\), with \(p\sim\operatorname{Beta}(2,3)\), and its annual count \(N\mid p\sim\operatorname{Bernoulli}(p)\). For severity, a risk has mean payment \(\mu\), with \(\mu\sim N(1000,90000)\), and \(X\mid\mu\sim N(\mu,40000)\). Normal parameters are written as mean and variance. There is no interest or mortality assumption in this example.

Frequency EPV. A Bernoulli variable has conditional variance \(p(1-p)\). For a Beta distribution with parameters \(\alpha,\beta\), \(E[p(1-p)]=\alpha\beta/[(\alpha+\beta)(\alpha+\beta+1)]\). Therefore,

\operatorname{EPV}_N =E[p(1-p)] =\frac{2(3)}{(2+3)(2+3+1)} =\frac{6}{30} =0.20.

Frequency VHM. Since \(E[N\mid p]=p\), the VHM is \(\operatorname{Var}(p)\). For a Beta distribution, this variance is \(\alpha\beta/[(\alpha+\beta)^2(\alpha+\beta+1)]\), so

\operatorname{VHM}_N =\operatorname{Var}(p) =\frac{2(3)}{(2+3)^2(2+3+1)} =\frac{6}{150} =0.04.

The portfolio-average claim probability is \(E[p]=2/5=0.40\). As a check, the annual count variance is \(0.20+0.04=0.24\).

Severity EPV. The conditional variance of \(X\) is 40,000 dollars squared for every \(\mu\). Thus \(\operatorname{EPV}_X=40{,}000\) dollars squared per claim.

Severity VHM. Since \(E[X\mid\mu]=\mu\), the VHM is \(\operatorname{Var}(\mu)=90{,}000\) dollars squared per claim. The portfolio mean payment is \(E[\mu]=1{,}000\) dollars, and total severity variance is \(40{,}000+90{,}000=130{,}000\) dollars squared per claim.

The frequency components are variances of counts per policy-year; the severity components are variances of dollars per claim. The values \(0.20\) and \(40{,}000\), for example, cannot be ranked as if they were on one numerical scale.

Conditional Models Determine the Components

The component calculation follows directly from the conditional distribution and the distribution of its risk parameter. For a Poisson count \(N\mid\lambda\sim\operatorname{Poisson}(\lambda)\), both the conditional mean and variance are \(\lambda\). Consequently, frequency EPV is \(E[\lambda]\), while frequency VHM is \(\operatorname{Var}(\lambda)\). For severity, the conditional variance need not equal the conditional mean; use the variance specified by the severity model.

The following example uses a different conditional model for each response. The risk-parameter distribution is part of the assumptions: changing it can change the VHM even if the conditional model remains the same.

Worked Example: Poisson Frequency and Exponential Severity

For a hypothetical portfolio, annual claim count satisfies \(N\mid\lambda\sim\operatorname{Poisson}(\lambda)\), with \(\lambda\sim\operatorname{Gamma}(4,\text{ rate }2)\). Thus \(E[\lambda]=4/2=2\) and \(\operatorname{Var}(\lambda)=4/2^2=1\). Claim severity satisfies \(X\mid\Theta\sim\operatorname{Exponential}(\text{mean }\Theta)\), where \(\Theta\) is equally likely to be 600 or 1,000 dollars. There is no interest or mortality assumption.

Frequency. The conditional Poisson mean and variance are both \(\lambda\). Therefore,

\operatorname{EPV}_N=E[\lambda]=2,\qquad \operatorname{VHM}_N=\operatorname{Var}(\lambda)=1.

The total variance of the annual count is \(2+1=3\) counts squared per policy-year.

Severity EPV. An exponential variable with mean \(\Theta\) has variance \(\Theta^2\). The two possible conditional variances are \(600^2=360{,}000\) and \(1{,}000^2=1{,}000{,}000\). Hence

\operatorname{EPV}_X =E[\Theta^2] =\frac{360{,}000+1{,}000{,}000}{2} =680{,}000\text{ dollars}^2.

Severity VHM. The conditional mean is \(\Theta\). Its portfolio mean is \((600+1{,}000)/2=800\) dollars. The two deviations from that mean are \(-200\) and \(200\), so

\operatorname{VHM}_X =\operatorname{Var}(\Theta) =\frac{(-200)^2+200^2}{2} =40{,}000\text{ dollars}^2.

The total variance of payment per claim is \(680{,}000+40{,}000=720{,}000\) dollars squared. Here process variance dominates severity variation across risk means. For frequency, the components are 2 and 1. These contrasts describe the assumed models; the raw frequency and severity magnitudes are not directly comparable.

Comparing Components Across a Common Risk Mix

A useful comparison holds the population of risks explicit and calculates both responses over that same mix. The relative sizes of EPV and VHM within each response can indicate whether the assumed variability is mainly within risks or among their expected outcomes. A ratio such as VHM/EPV is dimensionless for one response, but it is still response-specific: it does not put frequency and severity on a common dollar scale.

Worked Example: Comparing Both Responses Across Three Risk Types

Suppose three hypothetical risk types are equally represented. For annual frequency, their conditional models are Poisson with rates \(\lambda=1,2,3\) claims per policy-year. For severity, their conditional payment distributions are Normal with means 500, 1,000, and 1,500 dollars, respectively, and variance \(300^2=90{,}000\) dollars squared for each type. There is no interest or mortality assumption.

Frequency EPV. The Poisson conditional variance equals its rate. Averaging the three rates gives

\operatorname{EPV}_N=\frac{1+2+3}{3}=2.

Frequency VHM. The mean annual rate across types is 2. The deviations are \(-1,0,1\), so the variance across the equally weighted types is

\operatorname{VHM}_N =\frac{(-1)^2+0^2+1^2}{3} =\frac{2}{3} \approx 0.667.

The annual count variance is \(2+2/3=8/3\), or approximately \(2.667\) counts squared per policy-year. The relative component ratio is \(\operatorname{VHM}_N/\operatorname{EPV}_N=(2/3)/2=1/3\).

Severity EPV. Each type has conditional severity variance 90,000 dollars squared, so \(\operatorname{EPV}_X=90{,}000\) dollars squared per claim.

Severity VHM. The mean across the three type means is 1,000 dollars. Their deviations are \(-500,0,500\), giving

\operatorname{VHM}_X =\frac{(-500)^2+0^2+500^2}{3} =\frac{500{,}000}{3} =166{,}666.67\text{ dollars}^2.

Total severity variance is \(90{,}000+166{,}666.67=256{,}666.67\) dollars squared per claim, and \(\operatorname{VHM}_X/\operatorname{EPV}_X=166{,}666.67/90{,}000\approx1.852\). Relative to process variance, these assumptions have more variation among severity means than within a risk, while the frequency ratio is \(1/3\). This is a meaningful within-response comparison, not evidence that severity variance is “larger” than frequency variance in an absolute, unit-free sense.

Common Mistakes and Exam Traps

A frequent error is swapping EPV and VHM because both are variances. In the Normal severity model from the first example, the conditional variance is 40,000, while variation in the conditional mean is 90,000. The wrong assignment would be \(\operatorname{EPV}_X=90{,}000\) and \(\operatorname{VHM}_X=40{,}000\). That reverses the meanings: EPV averages variance conditional on the risk parameter, whereas VHM takes the variance of the conditional mean.

Another trap is using the wrong observational unit. A severity distribution describes payment per claim, not payment per policy-year. If the question instead defines \(Y\) as annual aggregate loss, the frequency and severity randomness jointly affect its components; the per-claim severity EPV cannot simply be relabeled as annual-loss EPV.

Before comparing results, check these points:

Name the response. Is the observation a count, a payment per claim, or an aggregate amount per policy period?
Condition first. Find the conditional mean and variance from the stated model before averaging or taking a variance across risks.
Use consistent units. A frequency component is in squared counts for its specified exposure unit; a severity component is in squared dollars per claim.
Separate absolute from relative comparisons. Compare EPV with VHM within the same response, or use a clearly defined dimensionless ratio. Do not compare unlike raw units.

Key takeaway: EPV measures average conditional process variation; VHM measures variation across risk-specific expected outcomes. Both depend on the response and observation unit. Frequency and severity components can be compared structurally, but their raw magnitudes have different units and interpretations.

Check Your Understanding

For each problem, identify the observation unit before calculating or interpreting the components.

A hypothetical annual claim count satisfies \(N\mid\lambda\sim\operatorname{Poisson}(\lambda)\), with \(\lambda\) equally likely to be 1 or 3. Find frequency EPV and VHM. (Answer: EPV \(=2\) and VHM \(=1\), in squared annual-count units.)
For claim severity, \(X\mid\mu\sim N(\mu,25{,}000)\), and across risks \(\mu\sim N(700,40{,}000)\). Find severity EPV and VHM. (Answer: EPV \(=25{,}000\) dollars squared per claim; VHM \(=40{,}000\) dollars squared per claim.)
For a hypothetical portfolio, \(p\sim\operatorname{Beta}(1,4)\) and one-period claim count \(N\mid p\sim\operatorname{Bernoulli}(p)\). Find both frequency components. (Answer: EPV \(=4/(5\cdot6)=0.1333\) and VHM \(=4/(25\cdot6)=0.0267\), in squared counts per period.)
Risk types are equally likely to have exponential claim severity with conditional means 400 and 800 dollars. Find EPV and VHM. (Answer: EPV \(=(400^2+800^2)/2=400{,}000\) dollars squared per claim; VHM \(=40{,}000\) dollars squared per claim.)
A portfolio has frequency VHM \(=0.5\) and EPV \(=2.0\), while severity VHM \(=120{,}000\) and EPV \(=80{,}000\) dollars squared per claim. Calculate each response’s VHM/EPV ratio and state what can be compared. (Answer: frequency ratio \(=0.25\); severity ratio \(=1.5\). These within-response ratios can be compared as relative component balances, but the raw frequency and severity variances cannot be compared directly.)

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