Pure Premiums from Independent Frequency and Severity Models

From Posterior Estimates to a Pure Premium

A pure premium is the expected claim cost for a specified amount of exposure, before expenses, profit provisions, or risk margins. When both frequency and severity are uncertain, the calculation must account for the data and prior information about each part. The central idea is to calculate the posterior expected number of future claims and the posterior expected cost per claim, then combine them under clearly stated assumptions.

This tutorial considers models in which the frequency parameter and severity parameter have independent priors, and frequency data and severity data contribute separate likelihoods. Under those conditions, the two parameters remain independent given the data. This lets us express the posterior expected aggregate claim cost as a product. If the parameters are dependent, or if the observation process links claim counts and observed severities, that product generally needs further justification.

Definition: For a future exposure amount $e$, let $N_e$ be the number of claims and let $X_1,\ldots,X_{N_e}$ be their severities. The posterior expected pure premium is $E[\sum_{j=1}^{N_e}X_j\mid\text{data}]$. It is an expected aggregate claim cost for the stated future exposure, conditional on the observed data and the specified model.

The frequency model describes how many claims occur; the severity model describes the size of a claim, conditional on a claim occurring. Write $\theta$ for a Poisson frequency rate and $\lambda$ for an exponential severity rate. These are different parameters: $\theta$ is measured in claims per exposure unit, while $\lambda$ is measured in inverse dollars when severity is in dollars.

Why the Product Formula Holds

Suppose that, conditional on the frequency parameter, the expected future claim count is $m(\theta)=E[N_e\mid\theta]$. Suppose also that, conditional on the severity parameter, each claim has mean severity $s(\lambda)=E[X_j\mid\lambda]$, and claim severities are independent of the count given the model parameters. Let $D$ denote all observed frequency and severity data.

Condition first on the parameters and the future count. The expected aggregate cost is $N_e s(\lambda)$. Taking the expectation over the future count and then over the posterior distribution of the parameters gives

E\!\left[\sum_{j=1}^{N_e}X_j\mid D\right] =E[m(\theta)s(\lambda)\mid D].

If the posterior distributions of $\theta$ and $\lambda$ are independent given $D$, the expectation of this product factors. Thus

E\!\left[\sum_{j=1}^{N_e}X_j\mid D\right] =E[m(\theta)\mid D]E[s(\lambda)\mid D].

This is the required justification: each factor is a posterior expectation, not necessarily a parameter estimate substituted into a formula. Posterior independence follows, for example, when the prior factors into independent frequency and severity priors and the likelihood factors into a frequency part and a severity part.

Conjugate Updates Used in the Calculation

For a Poisson frequency model, suppose the count over exposure $t$ is Poisson with mean $t\theta$, and the prior is $\theta\sim\operatorname{Gamma}(\alpha,\beta)$, using shape $\alpha$ and rate $\beta$. After observing $n$ claims over $t$ exposure units, the posterior is $\operatorname{Gamma}(\alpha+n,\beta+t)$. A future exposure of $e$ has posterior expected count

E[N_e\mid D]=e\frac{\alpha+n}{\beta+t}.

For the overdispersed negative-binomial model, take a fixed shape $r$ per exposure unit and a shared probability parameter $q$. Conditional on $q$, the count over exposure $t$ has probability mass proportional to $(1-q)^{rt}q^n$, and its mean is $rtq/(1-q)$. If $q\sim\operatorname{Beta}(a,b)$, observing $n$ claims over exposure $t$ gives $q\mid D\sim\operatorname{Beta}(a+n,b+rt)$. For future exposure $e$, provided $b+rt>1$, the posterior expected count is

E[N_e\mid D]=re\,E\!\left[\frac{q}{1-q}\middle|D\right] =re\frac{a+n}{b+rt-1}.

The last equality uses the Beta expectation $E[q/(1-q)]=a'/(b'-1)$ for $q\sim\operatorname{Beta}(a',b')$ with $b'>1$. It is not generally equal to $E[q]/(1-E[q])$.

For exponential severities, let $X_j\mid\lambda\sim\operatorname{Exponential}(\lambda)$, where $\lambda$ is the rate, so the mean severity is $1/\lambda$. Use the Gamma prior $\lambda\sim\operatorname{Gamma}(c,d)$ in shape-rate form. Given $m$ observed claim amounts with total $T$, the posterior is $\operatorname{Gamma}(c+m,d+T)$. When $c+m>1$,

E[X_{\mathrm{new}}\mid D] =E[1/\lambda\mid D] =\frac{d+T}{c+m-1}.

The posterior expectation of the reciprocal rate is needed because the claim mean is $1/\lambda$. Substituting the posterior mean of $\lambda$ into $1/\lambda$ would not give this posterior expected severity.

Worked Applications

Worked Example: Poisson-Gamma Frequency with Exponential-Gamma Severity

A hypothetical portfolio has a Poisson claim rate $\theta$ per exposure unit, with prior $\theta\sim\operatorname{Gamma}(2,3)$ in shape-rate form. During 2 exposure units, 4 claims are observed. Claim severities are exponential with rate $\lambda$, whose prior is $\operatorname{Gamma}(3,200)$. The 4 observed severities total $2,000. The frequency and severity parameters have independent priors. Find the posterior expected pure premium for 1.5 future exposure units.

Method and frequency update. The Poisson-Gamma posterior is $\operatorname{Gamma}(\alpha+n,\beta+t)$. Therefore

\theta\mid D\sim\operatorname{Gamma}(2+4,3+2) =\operatorname{Gamma}(6,5).

Its posterior mean is $6/5=1.2$ claims per exposure unit. For 1.5 future units, the posterior expected count is $1.5(1.2)=1.8$ claims.

Severity update. The severity posterior is

\lambda\mid D\sim\operatorname{Gamma}(3+4,200+2{,}000) =\operatorname{Gamma}(7,2{,}200).

The posterior expected claim size is $2{,}200/(7-1)=\$366.666\ldots$. The prior rate parameter 200 and observed total are both in dollars in this shape-rate specification.

Combine. Independent priors and separate count and severity likelihoods give independent posterior parameters. Hence the premium is

1.8\times\frac{\$2{,}200}{6}=\$660.00.

Check. The expected count is for 1.5 exposure units, while the expected severity is per claim. Their product is an aggregate cost for the requested future exposure. The common mistake of using the observed count, 4, instead of the future posterior expected count would not answer the question.

Worked Example: Beta-Negative-Binomial Frequency with an Exponential Severity Model

A hypothetical coverage has negative-binomial claim counts with fixed shape $r=2$ per exposure unit and $q\sim\operatorname{Beta}(2,5)$. The observed count is 3 over 1.5 exposure units. Claim severities are exponential with rate $\lambda\sim\operatorname{Gamma}(2,100)$, independently of $q$. The three observed claims total $900. Calculate the posterior expected pure premium for 0.5 future exposure unit.

Method and frequency update. For exposure $t=1.5$, the posterior Beta parameters are $a+n=2+3=5$ and $b+rt=5+(2)(1.5)=8$. Thus $q\mid D\sim\operatorname{Beta}(5,8)$. The expected future count is

re\frac{a+n}{b+rt-1} =(2)(0.5)\frac{5}{8-1} =\frac{5}{7}.

The condition on the posterior second Beta parameter holds because $8>1$.

Severity update. There are 3 observed claim amounts, with total $T=\$900$, so

\lambda\mid D\sim\operatorname{Gamma}(2+3,100+900) =\operatorname{Gamma}(5,1{,}000).

The posterior expected severity is $1{,}000/(5-1)=\$250$.

Combine and check. The expected pure premium is

\frac{5}{7}\times\$250 =\frac{\$1{,}250}{7} =\$178.57.

The final amount is rounded to the nearest cent. The frequency factor is an expected claim count for half an exposure unit, not a probability of at least one claim; the model permits more than one claim.

Worked Example: Binomial-Beta Frequency with Exponential-Gamma Severity

A hypothetical policy can have at most one claim in each exposure unit. Its claim probability $p$ has prior $\operatorname{Beta}(2,6)$. In 20 observed exposure units there are 3 claims. Their severities are exponential with rate $\lambda\sim\operatorname{Gamma}(4,300)$, independently of $p$, and the three amounts total $1,800. Find the expected pure premium for one future exposure unit.

Method and frequency update. For Bernoulli claim indicators, the Beta posterior adds claims to its first parameter and claim-free units to its second. There are $20-3=17$ claim-free units, giving

p\mid D\sim\operatorname{Beta}(2+3,6+17) =\operatorname{Beta}(5,23).

The posterior expected claim count in one exposure unit is the posterior expected claim probability, $5/(5+23)=5/28$.

Severity update. The posterior rate distribution is

\lambda\mid D\sim\operatorname{Gamma}(4+3,300+1{,}800) =\operatorname{Gamma}(7,2{,}100).

Thus the posterior expected severity is $2{,}100/(7-1)=\$350$.

Combine and check. The premium is

\frac{5}{28}\times\$350=\$62.50.

This result is an expected cost per exposure unit. It is not the expected size conditional on a claim, which is $350, nor the probability of a claim, which is $5/28$. The product combines both components.

Common Exam Trap: Averaging a Nonlinear Function

For the Beta-negative-binomial model, the expected count depends on $q/(1-q)$, not directly on $q$. In the second example, the posterior is $\operatorname{Beta}(5,8)$. Its mean is $5/13$. A plug-in calculation would give $2(0.5)(5/13)/(1-5/13)=5/8=0.625$, which is not the posterior expected count. The correct calculation is

2(0.5)E\!\left[\frac{q}{1-q}\middle|D\right] =2(0.5)\frac{5}{8-1} =\frac{5}{7}.

Using the incorrect count with the correct $250 expected severity would produce $156.25 instead of $178.57. The error is replacing the expectation of a nonlinear function by the function of an expectation.

Other checks help prevent related mistakes:

For a Poisson-Gamma model, add observed exposure to the prior Gamma rate, not to its shape.
For exponential severity with a Gamma prior on the rate, add the number of claims to the prior shape and the sum of claim amounts to the prior rate parameter.
Use the posterior expected severity $(d+T)/(c+m-1)$, not the reciprocal of the posterior mean rate.
Match the frequency expectation to the requested future exposure. A per-unit rate is not itself the expected count over every future period.
Check finiteness conditions: the posterior Gamma shape for the severity rate must exceed 1, and the posterior second Beta parameter in the negative-binomial model must exceed 1.

Key takeaway: Under independent frequency and severity models with independent posterior parameters, calculate the posterior expected future count for the stated exposure and the posterior expected severity per claim, then multiply. State the parameterization and verify the required expectations exist before reporting a premium.

Check Your Understanding

For each hypothetical model, calculate the posterior expected pure premium for the stated future exposure. Gamma distributions use shape-rate parameterization; monetary amounts are in dollars.

Poisson frequency has $\theta\sim\operatorname{Gamma}(1,2)$; 3 claims are observed over 3 exposure units. Exponential severities have $\lambda\sim\operatorname{Gamma}(3,100)$, and the 3 observed claims total $900. Find the premium for 2 future exposure units. (Answer: $320.00)
A policy has at most one claim per exposure unit, with $p\sim\operatorname{Beta}(1,4)$. There are 2 claims in 10 observed units. Exponential severity has $\lambda\sim\operatorname{Gamma}(2,100)$, and the 2 claim amounts total $500. Find the premium for one future unit. (Answer: $40.00)
Negative-binomial frequency has $r=1.5$ and $q\sim\operatorname{Beta}(3,4)$. There are 2 claims over 2 observed units. Exponential severity has $\lambda\sim\operatorname{Gamma}(3,200)$, and the 2 claims total $800. Find the premium for one future unit. (Answer: $312.50)
Poisson frequency has $\theta\sim\operatorname{Gamma}(2,4)$; 2 claims are observed over 1 exposure unit. Exponential severity has $\lambda\sim\operatorname{Gamma}(4,300)$, and the 2 claims total $900. Find the premium for 0.75 future exposure unit. (Answer: $144.00)

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