Experience Rating with Pure Premium

Why Pure-Premium Experience Rating Uses Exposure

A pure premium is the expected loss cost per exposure unit, before expenses, profit provisions, or other loads. In experience rating, an insurer compares a group’s observed aggregate losses per exposure unit with a manual pure premium, which serves as the benchmark. When experience is not fully credible, the indicated pure premium gives only partial weight to the group’s observed result.

The key quantity is not the aggregate loss total by itself. It is the experience pure premium: aggregate losses divided by the corresponding exposure. The exposure must use the same basis as the manual pure premium and the full-credibility standard. For example, a manual rate stated per earned vehicle-year must be compared with losses divided by earned vehicle-years, not by the number of policies or claims.

Definition: Let $L$ be the aggregate losses for a group over a specified experience period, and let $e$ be its exposure on the rating basis. The experience pure premium is $\widehat{P}=L/e$. If $P_M$ is the manual pure premium and $Z$ is the credibility assigned to the experience, the indicated pure premium is $P_I=Z\widehat{P}+(1-Z)P_M$, where $0\leq Z\leq 1$.

This is a limited-fluctuation approach: the credibility weight is chosen to reflect the amount of exposure relative to a full-credibility standard. It is a rating procedure, not a claim that the observed result is the group’s true expected loss cost. The manual benchmark remains in the calculation whenever $Z<1$.

Full and Partial Credibility for Pure Premium

A full-credibility standard specifies how much exposure is needed for the experience pure premium to meet a chosen precision requirement with a chosen probability. For a standard based on aggregate losses, a common model assumes that claim count is Poisson and severities are independent and identically distributed, independent of claim count. Let $N\sim\operatorname{Poisson}(e\lambda)$, where $\lambda$ is the expected claim frequency per exposure unit, and let claim severity be $Y$. Aggregate loss is $L=\sum_{j=1}^{N}Y_j$.

Under these assumptions, the mean and variance of aggregate loss are $E[L]=e\lambda E[Y]$ and $\operatorname{Var}(L)=e\lambda E[Y^2]$. Thus, the expected pure premium is $P=\lambda E[Y]$, and the variance of the experience pure premium is $\lambda E[Y^2]/e$. This variance includes both claim-count and severity variation. It is not generally correct to use only the variance of claim severity.

Full-credibility standard: Suppose the requirement is that the experience pure premium be within $k$ of its expected value, as a proportion of that value, with probability at least $p$. Under a normal approximation, the required exposure is $$ e_0=\frac{z_{(1+p)/2}^{\,2}\lambda E[Y^2]}{k^2\{\lambda E[Y]\}^2} =\frac{z_{(1+p)/2}^{\,2}E[Y^2]}{k^2\lambda\{E[Y]\}^2}, $$ where $z_{(1+p)/2}$ is the standard normal quantile. Use this formula only when its stated model and normal-approximation basis are appropriate; if a problem supplies a full-credibility standard, use that standard as given.

The limited-fluctuation square-root rule assigns partial credibility in proportion to the square root of exposure relative to the full-credibility exposure. It caps the weight at one: exposure beyond the standard does not create credibility greater than full credibility.

Z=\min\left(1,\sqrt{\frac{e}{e_0}}\right)

Why the Square-Root Rule Appears

The square-root rule follows from the way random error decreases as exposure increases. In the compound Poisson model, the variance of the experience pure premium is proportional to $1/e$. Its standard deviation is therefore proportional to $1/\sqrt{e}$. If $e_0$ is the exposure that meets the chosen full-credibility precision, then the relative standard deviation at exposure $e$ is larger by the factor $\sqrt{e_0/e}$. A weight of $\sqrt{e/e_0}$ gives the standard limited-fluctuation scaling; the cap prevents a weight above one.

The rule is a prescribed credibility convention, rather than a claim that blending automatically makes the result unbiased for every group. Its practical role is to reduce the influence of volatile experience when exposure is small while allowing the experience to dominate once it meets the standard.

Calculate the experience pure premium.
Divide aggregate losses by exposure measured on the rating basis.

Determine the credibility weight.
Use the given full-credibility exposure, or calculate it from the stated model and standard. Then apply the square-root rule and cap at one.

Blend with the manual benchmark.
Multiply experience by $Z$, multiply the manual pure premium by $1-Z$, and add the two weighted amounts.

Check the result.
The indicated pure premium must lie between the experience and manual pure premiums when $0\leq Z\leq1$.

Worked Pure-Premium Rating Calculations

Worked Example: Blending Experience Below Full Credibility

A hypothetical book has aggregate losses of $57,728 over 225.5 earned vehicle-years. Its manual pure premium is $240 per earned vehicle-year, and the applicable full-credibility standard is 902 earned vehicle-years. Calculate the indicated pure premium. Report the final amount to the cent.

Method. First calculate the experience pure premium, then use the square-root rule to find the experience weight and blend it with the manual benchmark.

The experience pure premium is

\widehat{P}=\frac{\$57{,}728}{225.5}=\$256.00.

The exposure is below the full-credibility standard, so

Z=\sqrt{\frac{225.5}{902}}=\sqrt{0.25}=0.5.

The manual weight is $1-Z=0.5$. Therefore,

P_I=0.5(\$256.00)+0.5(\$240.00) =\$128.00+\$120.00 =\$248.00.

Check. The result, $248.00, lies halfway between $240.00 and $256.00, as expected for equal weights. The common mistake to avoid is applying the experience pure premium at full weight simply because it has already been calculated; the exposure is only one-quarter of the standard, so its credibility is one-half.

Worked Example: Calculating a Full-Credibility Exposure

For a hypothetical portfolio, claim count per exposure unit follows a Poisson model with $\lambda=0.5$. Each claim has a fixed severity of $1,000, so $E[Y]=\$1{,}000$ and $E[Y^2]=\$1{,}000{,}000$. Find the exposure needed for a 90% probability that the experience pure premium is within 20% of its expected value, using the normal approximation. If the portfolio has 140 exposure units and aggregate losses of $84,000, calculate the indicated pure premium using a manual pure premium of $500. Report the full-credibility exposure to one decimal and the indicated pure premium to the cent.

Method. Apply the compound Poisson full-credibility formula, then compare observed exposure with the required exposure. For a central 90% probability, use $z_{0.95}=1.645$.

The expected pure premium is $\lambda E[Y]=0.5(\$1{,}000)=\$500$. Substitute into the full-credibility formula:

e_0= \frac{1.645^2(0.5)(1{,}000{,}000)} {0.20^2(\$500)^2}.

The numerator is $2.706025(500{,}000)=1{,}353{,}012.5$. The denominator is $0.04(250{,}000)=10{,}000$. Hence

e_0=\frac{1{,}353{,}012.5}{10{,}000}=135.30125\approx135.3.

Since 140 exposure units exceed 135.3, the capped credibility weight is $Z=1$. The observed pure premium is

\widehat{P}=\frac{\$84{,}000}{140}=\$600.

The indicated pure premium is therefore $1(\$600)+0(\$500)=\$600.00$. Check. The experience is fully credible under the stated standard, so the manual benchmark receives zero weight. The common mistake to avoid is using the standard normal quantile for a one-sided 90% interval; the requirement is central, so the quantile is $z_{0.95}$, not $z_{0.90}$.

Worked Example: Keeping Exposure and Loss Basis Consistent

A hypothetical group incurred $49,500 in covered losses over 180 earned policy-years. The manual pure premium is $300 per earned policy-year, and the full-credibility standard is 720 earned policy-years. Calculate the indicated pure premium. Report the final amount to the cent.

Method. Use the same earned-policy-year denominator for the observed pure premium and the credibility calculation.

The observed pure premium is

\widehat{P}=\frac{\$49{,}500}{180}=\$275.

The credibility weight is

Z=\sqrt{\frac{180}{720}}=\sqrt{0.25}=0.5.

Thus the indicated pure premium is

P_I=0.5(\$275)+0.5(\$300) =\$137.50+\$150.00 =\$287.50.

Check. The indication lies between $275 and $300, and is closer to neither because the weights are equal. The common mistake to avoid is substituting claim count for exposure: the full standard and the loss total are both stated on an earned-policy-year basis, so that is the denominator required for the pure premium.

Common Exam Traps

A frequent error is to treat the credibility weight as an adjustment to the experience rate instead of a blend with a benchmark. For the first worked example, the incorrect calculation $0.5(\$256)=\$128$ is not an indicated pure premium: it has discarded the manual component. The correction is $0.5(\$256)+0.5(\$240)=\$248$. A credibility weight allocates weight between two rates; it does not scale the experience rate by itself.

Other checks help prevent avoidable errors:

Use the stated exposure basis. If losses are divided by vehicle-years, compare the result with a manual rate per vehicle-year.
Use aggregate losses, not average severity alone. The experience pure premium reflects the combined effect of frequency and severity through total losses per exposure.
Cap credibility at one. If $e>e_0$, do not use a weight greater than one.
Keep the manual benchmark distinct from experience. It is the comparison rate in the blend, not an additional loss amount to add to the observed losses.
Check the direction of the result. For a valid weight between zero and one, the indication must be between the manual and experience pure premiums.

Pure-premium experience rating is therefore a short sequence of linked calculations: establish the loss cost per exposure, measure exposure against a full-credibility standard, and blend the result with the manual benchmark. This procedure is aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, calculate the requested credibility result and check that the indicated pure premium is consistent with its two inputs.

A hypothetical class has $22,500 in aggregate losses over 100 exposure units. The manual pure premium is $180, and the full-credibility standard is 400 exposure units. Find the credibility weight and indicated pure premium. (Answer: $Z=0.5$; indicated pure premium = $202.50.)
Claim count per exposure unit is Poisson with $\lambda=0.4$, and each claim has fixed severity of $500. Find the exposure for 90% central probability that the experience pure premium is within 25% of its expected value, using $z_{0.95}=1.645$ and the normal approximation. (Answer: $e_0=108.241$, or 108.2 exposure units to one decimal.)
A hypothetical group has $99,000 in losses over 360 exposure units. Its manual pure premium is $260, and its full-credibility standard is 640 units. Find the indicated pure premium. (Answer: $Z=0.75$; indicated pure premium = $271.25.)
A group has $306,000 in aggregate losses over 1,200 exposure units. The manual pure premium is $230, and the full-credibility standard is 900 units. Find the capped credibility weight and indicated pure premium. (Answer: $Z=1$; indicated pure premium = $255.00.)
A hypothetical portfolio has $36,250 in aggregate losses over 125 exposure units. Its manual pure premium is $250, and its full-credibility standard is 500 units. Find the credibility weight and indicated pure premium. (Answer: $Z=0.5$; indicated pure premium = $270.00.)

What You'll Learn