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Claims claims inflation changes the monetary size of a claim over calendar time. With a fixed annual rate $r$, a time-zero size $c_0$ becomes the displayed end-of-year amount. Historical claim amounts must be converted to a common price year before their empirical monetary averages are combined. When the per-claim amounts are known, dividing total amount by the year's per-claim amount recovers the claim count.

Let $Y=(\log X-\mu )/\sigma$ and write $\varphi$ for the standard normal density, with $\sigma >0$. The change of variables converts the density integral intoComplete the square using $e^{k\sigma y}\varphi (y)=e^{k^2{\sigma }^2/2}\varphi (y-k\sigma )$. Integration gives the truncated lognormal momentHere $\Phi$ is the standard normal distribution function; take $\log 0=-\infty$ and $\log \infty =\infty$. The identity holds for every real $k$, including the full-interval limits, because the shifted Gaussian integral remains finite.

The limited excess of loss reinsurance payment is $L(X)=\min \{(X-M{)}_{+},A\}$. It is zero below $M$, equals $X-M$ for $M<X<M+A$, and equals $A$ above $M+A$. Since the lognormal distribution has a continuous density,Put $d_j(t)=(\log t-\mu -j{\sigma }^2)/\sigma$, $j=0,1$, and $\overline{\Phi }=1-\Phi$. Applying the moment identity with $k=1$ and $k=0$ gives the conditional lognormal layer recoveryThe denominator is $\mathbb{P}(X>M)$, and payments outside that event are zero. The three numerator terms are respectively the truncated first moment, the attachment subtracted from partial payments, and the capped payments on claims above the layer. This proves the conditional expectation including the limit on each reinsurer payment.

For claims inflation by $h=1.1$, set $X_h=hX$. Its log-mean is ${\mu }_h=\mu +\log h$ and its log-standard-deviation stays $\sigma$. The attachment and payment limit stay fixed. Therefore the requested new conditional mean isExplicitly, in the preceding fraction replace $d_j(t)$ by $d_j^{(h)}(t)=(\log t-\mu -\log 1.1-j{\sigma }^2)/\sigma$ and replace $e^{\mu +{\sigma }^2/2}$ by $1.1e^{\mu +{\sigma }^2/2}$. Equivalently, claims inflation of a fixed reinsurance layer givesbecause $L_{M,A}(hX)=h{L}_{M/h,A/h}(X)$ and the new conditioning event is $X>M/h$. Merely multiplying the original conditional mean by $1.1$ would omit both of these threshold changes.

Use the shape-rate convention for the gamma distribution $\theta \sim \mathrm{Gamma}(\alpha ,\beta )$, with prior distribution probability density function proportional to ${\theta }^{\alpha -1}{e}^{-\beta \theta }$. The observations are independent conditional on the common parameter $\theta$. Unconditionally a common nondegenerate prior distribution induces covariance $\alpha /{\beta }^2$ between distinct observations, so the Bayesian inference model requires conditional independence.

For unit-exposure counts, the likelihood is proportional to ${\theta }^{{\sum }_i{x}_i}{e}^{-n\theta }$. Multiplying by the prior distribution givesThe future count has conditional expectation $\theta$, so posterior distribution averaging givesThis is an exact Bayesian credibility estimate, with credibility factor $Z=n/(n+\beta )$ and collective mean $\alpha /\beta$.

For the insurance exposure and claims inflation calculation put $c_j=c(1+r{)}^j$, assuming $c>0$, $1+r>0$ and positive policy counts $n_j$. Let $K_j$ be the total claim count in year $j$. Conditional independence of the policy counts gives $K_j\mid \theta \sim \mathrm{Pois}(n_j\theta )$, and the observed average amount satisfiesFor possible observations, the recovered $k_j$ are nonnegative integers. If they are not, the data have zero likelihood under the model and no conditional posterior distribution is defined for those impossible values.

Write $W={\sum }_{j=1}^n{n}_j$ and $K={\sum }_{j=1}^n{k}_j$. Since $Y_j$ and $K_j$ determine each other, conditioning on the observed average amounts is equivalent to conditioning on these counts. The independent yearly count likelihood isThe Poisson-gamma conjugacy with unequal exposures therefore givesFor the future year, $\mathbb{E}[Y_{n+1}\mid \theta ]=c_{n+1}\theta$; the number of future policies cancels because $Y_{n+1}$ is a per-policy average. HencewithThe collective mean $m$ is the prior distribution expected claim amount per policy in the future year's monetary units. The experience mean $m(\mathbf{y})$ first inflates each past per-policy amount to that same future year and then averages with policy-exposure weights. Equivalently, it estimates annual claim frequency by total observed claims divided by total past insurance exposure and converts that frequency to the future claim cost. This is inflation-adjusted Poisson-gamma credibility.

For fixed prior distribution parameters, $Z$ increases strictly with total past insurance exposure: $dZ/dW=\beta /(W+\beta {)}^2>0$, and $Z\to 1$ as $W\to \infty$. The prior distribution weight decreases to zero; $\beta$ acts as equivalent prior distribution insurance exposure. With no past insurance exposure the appropriate weight is zero and the prediction is $m$, without defining an empirical experience mean.