Integrated Bayesian Credibility Premium Workshop

From Conjugate Priors to a Predictive Pure Premium

A Bayesian premium calculation can combine two distinct sources of uncertainty: how often claims occur and how large those claims are. The calculation is most useful when the data are not equally exposed. A risk observed for half a year should not have the same influence on an estimated claim rate as a risk observed for several years.

This tutorial uses a Poisson model for claim counts and an exponential model for individual claim amounts. Independent Gamma priors make the updates tractable. The goal is a posterior predictive pure premium: the expected aggregate claim amount for a future exposure, conditional on the observed data. It includes neither expenses nor profit or risk margins.

Model: For risk $j$, let $e_j$ be its exposure, $N_j$ its claim count, and $X_{jk}$ its $k$th claim amount. Conditional on the frequency rate $\lambda$, assume $N_j\sim\operatorname{Poisson}(e_j\lambda)$. Conditional on the severity rate $\theta$, assume claim amounts are independent with $X_{jk}\sim\operatorname{Exponential}(\theta)$, so $E[X_{jk}\mid\theta]=1/\theta$. The rates $\lambda$ and $\theta$ are independent, with priors $\lambda\sim\operatorname{Gamma}(\alpha,\beta)$ and $\theta\sim\operatorname{Gamma}(a,b)$, using the shape-rate parameterization. Thus a $\operatorname{Gamma}(c,r)$ variable has density proportional to $x^{c-1}e^{-rx}$, mean $c/r$, and rate parameter $r$. Assume exposures are measured in a consistent unit, all claims are fully observed, and the frequency and severity model assumptions hold.

Under this parameterization, $\lambda$ is measured in claims per exposure unit and $\theta$ in inverse dollars if claim amounts are in dollars. The Gamma rate parameters $\beta$ and $b$ therefore have units of exposure and dollars, respectively. Keeping track of whether the second Gamma parameter is a rate rather than a scale is essential.

Eliciting the Priors

For the frequency prior, $\alpha/\beta$ is the prior mean claim rate. A common elicitation treats $\beta$ as a prior exposure strength: if the desired prior mean is $m_\lambda$ and the strength is $k$ exposure units, set $\beta=k$ and $\alpha=m_\lambda k$. This construction gives a transparent pseudo-exposure interpretation, although it does not make the prior actual observed data.

For severity, if $\theta\sim\operatorname{Gamma}(a,b)$, then, provided $a>1$,

E[X]=E\!\left[\frac{1}{\theta}\right]=\frac{b}{a-1}.

Thus a desired prior mean severity $m_X$ and a chosen shape $a>1$ imply $b=m_X(a-1)$. The mean alone does not determine both Gamma parameters: the shape must also be chosen to represent the intended prior concentration. This inverse-rate expectation differs from the familiar Gamma mean $a/b$, which is the mean of $\theta$, not of the claim amount.

Worked Example: Eliciting Frequency and Severity Priors

A hypothetical portfolio actuary wants a prior mean frequency of 1.5 claims per policy-year, with prior exposure strength 2 policy-years. For claim severity, the desired prior mean is $2,000, and the selected Gamma shape for the exponential rate is 4. Determine both priors and verify their implied means.

Frequency prior. Set $\beta=2$ policy-years and calculate $\alpha=1.5(2)=3$. Therefore $\lambda\sim\operatorname{Gamma}(3,2)$, with $E[\lambda]=3/2=1.5$ claims per policy-year.

Severity prior. With $a=4$, solve $2000=b/(4-1)$, giving $b=2000(3)=6000$. Thus $\theta\sim\operatorname{Gamma}(4,6000)$, and its implied mean claim amount is $6000/(4-1)=\$2{,}000$.

Check and method. Substitution recovers both requested prior means. The frequency parameters encode two prior exposure units and three prior claims in the mean-rate sense. The severity shape was an additional modeling choice; it was not deduced from the mean alone. A common mistake is to set $b=am_X=8000$, using the formula for $E[\theta]$ as if it were the formula for $E[X]$. That would imply $E[X]=8000/3$, not $2,000.

Updating with Unequal Exposures

Let total exposure be $E=\sum_j e_j$, total claims be $N=\sum_j N_j$, and total observed claim amount be $T=\sum_j\sum_k X_{jk}$. Given $\lambda$, the count likelihood is proportional to $\lambda^N e^{-\lambda E}$. Given $\theta$, the exponential severity likelihood is proportional to $\theta^N e^{-\theta T}$. Multiplying each likelihood by its Gamma prior gives the posterior distributions:

\lambda\mid\text{data}\sim\operatorname{Gamma}(\alpha+N,\beta+E), \qquad \theta\mid\text{data}\sim\operatorname{Gamma}(a+N,b+T).

The frequency update uses all exposure, not the number of risks. The severity update uses all claim amounts and the number of claims. If claim amounts are missing or censored, the severity likelihood above is not the appropriate complete-data likelihood without adjustment. The stated update also presumes the observed claims are represented by the exponential severity model.

Worked Example: Updating from Four Unequal Exposures

Use the priors from the preceding example. A hypothetical portfolio has the following fully observed experience; amounts are in dollars.

Risk	Exposure (policy-years)	Claims	Observed claim amounts
A	0.5	0	None
B	1.0	1	$1,800
C	1.5	2	$2,200; $3,200
D	1.0	3	$2,400; $3,600; $4,800

Step 1: Add exposure and claims. Total exposure is $E=0.5+1.0+1.5+1.0=4.0$ policy-years. Total claims are $N=0+1+2+3=6$.

Step 2: Update frequency. The posterior is $\lambda\mid\text{data}\sim\operatorname{Gamma}(3+6,2+4)=\operatorname{Gamma}(9,6)$. Its mean is $9/6=1.5$ claims per policy-year.

Step 3: Add claim amounts and update severity. The total is $T=1{,}800+2{,}200+3{,}200+2{,}400+3{,}600+4{,}800=\$18{,}000$. Therefore $\theta\mid\text{data}\sim\operatorname{Gamma}(4+6,6{,}000+18{,}000)=\operatorname{Gamma}(10,24{,}000)$. The posterior expected severity is $E[1/\theta\mid\text{data}]=24{,}000/(10-1)=\$2{,}666.666\ldots$.

Check and method. The observed portfolio rate is also $6/4=1.5$ claims per policy-year; here it equals the posterior mean because the prior mean rate is 1.5. The observed mean claim amount is $18{,}000/6=\$3{,}000$, while the posterior expected amount is pulled toward the $2,000 prior mean. The method is conjugate updating from sufficient totals. Do not average the four risk-level rates without exposure weights: their rates are 0, 1, $2/1.5=1.333\ldots$, and 3, whose unweighted average is $1.333\ldots$, not the correct $6/4=1.5$.

Deriving the Posterior Predictive Pure Premium

Let a future risk have exposure $e_*$, and let $S_*$ denote its aggregate claim amount. Conditional on $\lambda$ and $\theta$, the expected number of claims is $e_*\lambda$, and each claim has mean amount $1/\theta$. The expected aggregate amount is therefore $e_*\lambda/\theta$. Taking the posterior expectation gives the predictive premium.

E[S_*\mid\text{data}] =e_*E\!\left[\frac{\lambda}{\theta}\,\middle|\,\text{data}\right] =e_*E[\lambda\mid\text{data}]E[1/\theta\mid\text{data}].

The last equality holds because the priors are independent and the likelihood factors into a term involving $\lambda$ and a term involving $\theta$; consequently their posterior distributions remain independent. The expectations exist when the posterior severity shape $a+N$ exceeds 1. In terms of the posterior parameters, the result is

E[S_*\mid\text{data}] =e_*\left(\frac{\alpha+N}{\beta+E}\right) \left(\frac{b+T}{a+N-1}\right).

This is a posterior predictive expected loss, not a predictive interval or a risk-loaded premium. It combines the expected frequency and expected severity, rather than treating the observed aggregate amount per exposure as a guaranteed future result.

Worked Example: Pricing a New Exposure and Testing Prior Sensitivity

For the portfolio experience above, calculate the pure premium for a future risk with exposure 0.75 policy-years. Then repeat the calculation under an alternative prior: $\lambda\sim\operatorname{Gamma}(2,2)$ and $\theta\sim\operatorname{Gamma}(4,7{,}500)$. The observed experience is unchanged.

Baseline posterior premium. From the update, $E[\lambda\mid\text{data}]=9/6=1.5$ and $E[1/\theta\mid\text{data}]=24{,}000/9=\$2{,}666.666\ldots$. The pure premium per policy-year is $1.5(2{,}666.666\ldots)=\$4{,}000$. For 0.75 policy-years, the predictive expected aggregate loss is $0.75(4{,}000)=\$3{,}000.00$.

Alternative frequency prior. Its posterior is $\operatorname{Gamma}(2+6,2+4)=\operatorname{Gamma}(8,6)$, with mean $8/6=4/3$ claims per policy-year.

Alternative severity prior. Its prior mean is $7{,}500/(4-1)=\$2{,}500$. After observing six claims totaling $18,000, the posterior is $\operatorname{Gamma}(4+6,7{,}500+18{,}000)=\operatorname{Gamma}(10,25{,}500)$. Thus the posterior expected severity is $25{,}500/9=\$2{,}833.333\ldots$.

Alternative premium and comparison. The pure premium per policy-year is $(4/3)(25{,}500/9)=\$3{,}777.777\ldots$, or $3,777.78 to the nearest cent. For 0.75 policy-years, it is $0.75(3{,}777.777\ldots)=\$2{,}833.333\ldots$, or $2,833.33. Relative to the baseline $3,000.00, this is lower by $166.67, or about 5.56%.

Check and method. The changed prior lowers the frequency mean and raises the severity mean, so the two effects partly offset. The computed premium is lower overall. The multiplication is $ (4/3)(25{,}500/9)=34{,}000/9$, approximately $3,777.78 per policy-year; multiplying by 0.75 gives $2,833.33. A common trap is to multiply incorrectly or apply the future exposure twice. The product of the two posterior means is already a per-unit-exposure premium; multiply it by $e_*=0.75$ exactly once.

What the Calculation Depends On

This workshop illustrates why an integrated calculation should keep its sufficient statistics and units visible. Frequency is informed by total claims and total exposure; severity is informed by the total number of observed claims and their total amount. The posterior predictive premium then combines the two posterior expectations and scales by the future exposure.

Prior sensitivity is not a defect in Bayesian analysis. It is a consequence of combining finite data with prior information. It is especially relevant when the experience is limited, when a selected prior strength is substantial relative to observed exposure, or when a severity prior has a mean materially different from the observed average. A useful review is to calculate the result under plausible alternative prior means or strengths and explain which component drives the change.

Key takeaway: With independent Gamma priors on a Poisson frequency rate and an exponential severity rate, unequal exposures update through the totals $N$, $E$, and $T$. The posterior predictive pure premium for exposure $e_*$ is $e_*E[\lambda\mid\text{data}]E[1/\theta\mid\text{data}]$, provided the posterior severity shape exceeds 1.

Check Your Understanding

For each problem, state the relevant posterior or premium and check that the parameterization and exposure units are consistent.

A prior mean frequency is 0.8 claims per exposure unit, with prior exposure strength 5 units. What Gamma shape-rate prior follows from the pseudo-exposure elicitation? (Answer: $\operatorname{Gamma}(4,5)$.)
An exponential severity rate has a Gamma prior shape of 3, and the desired prior mean claim amount is $1,500. Find the Gamma rate parameter. (Answer: 3,000 dollars.)
The frequency prior is $\operatorname{Gamma}(2,1)$. A portfolio has 4 claims over 2 exposure units. Find the posterior distribution and posterior mean frequency. (Answer: $\operatorname{Gamma}(6,3)$, with mean 2 claims per exposure unit.)
The severity-rate prior is $\operatorname{Gamma}(3,4{,}000)$. There are 2 fully observed claims totaling $10,000. Find the posterior expected claim amount. (Answer: posterior $\operatorname{Gamma}(5,14{,}000)$ for the rate; expected severity $3,500.)
Independent priors are $\lambda\sim\operatorname{Gamma}(2,2)$ and $\theta\sim\operatorname{Gamma}(3,3{,}000)$. There are 3 claims over 2 exposure units, totaling $9,000. Find the posterior predictive pure premium per exposure unit. (Answer: posterior means $E[\lambda\mid\text{data}]=5/4=1.25$ and $E[1/\theta\mid\text{data}]=12{,}000/5=\$2{,}400$, giving $3,000 per exposure unit.)

This integrated conjugate calculation is aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus.

What You'll Learn