I'm currently looking into exposure curves for the excess of loss reinsurance treaty. The idea is, that for a single risk (claim) $X$, we can decompose it into $X=D+R$, a deductible part $D$ and reinsurance part $R$. Then, say we want to reinsure losses that exceed some threshold $M$, and $X$ follows some distribution function $F$, $$ E[R] = \int_M^\infty (1-F(x)) dx, \quad E[D] = \int_0^M (1-F(x)) dx. $$ Next, say our risk follows a Pareto distribution over some other threshold $0<u<M$, so that $$ F(x) = 1-Cx^{-\alpha}, \quad x\geq u. $$ A quite important quantity in the calculation of a pure premium in the excess of loss contract, we consider the exposure curve, given by $$ r(M) = \frac{1}{E[X]} \int_0^M (1-F(x)) dx $$ and thus, in our case, we would calculate $$ E[X] = \int_u^\infty x dF(x) = \frac{\alpha Cu^{-\alpha+1}}{\alpha-1} $$ for $\alpha>1$ and $$ \int_u^M Cx^{-\alpha} dx = \frac{C}{\alpha-1}(u^{-\alpha+1} - M^{-\alpha+1}). $$ Hence, the exposure curve in our case is given by $$ r(M) = \frac{1}{\alpha}(1-(\frac{M}{u})^{-\alpha+1}). $$ However, for $M\rightarrow\infty$ this quantity converges to $1/\alpha$, which I find quite odd. In my understanding the exposure curve is in some sense the equilibrium distribution function of $X$ and thus should be 1 as $M\rightarrow\infty$. Am I doing something wrong here, or what am I missing.
Thank you in advance.