= Solution
Write $A=\alpha+\beta$ and keep $p_0=\alpha/A$ fixed. Then $\alpha=p_0A$, $\beta=(1-p_0)A$, while the fixed observed years and exposures leave $M$ unchanged. The <credibility factor> is
$$
Z=\frac{M}{M+A},\qquad \frac{dZ}{dA}=-\frac{M}{(M+A)^2}<0.
$$
As prior concentration $A$ increases without bound,
$$
\boxed{Z\longrightarrow0,\qquad
\widehat\mu_{k+1}\longrightarrow m_{k+1}p_0.}
$$
The beta prior <variance> is $p_0(1-p_0)/(A+1)$, which tends to zero while its mean stays fixed. Increasing $A$ therefore makes the prior increasingly concentrated and reduces the influence of the same finite data. This explains the limiting <Bayesian credibility> estimate, rather than merely treating $\alpha,\beta$ as two larger arbitrary parameters.
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