 Solution

For an exponential distribution with mean $\mu$, the claim moment-generating function is $M_I(r)=(1-\mu r{)}^{-1}$ for $r<1/\mu$. The calculated adjustment coefficient must solveSubtract one, divide by the nonzero factor $\mu R_I$, and obtain $(1-\mu R_I{)}^{-1}=1+\rho$. ThusIt belongs to $(0,1/\mu )$ for every $\rho >0$, as required by the transform domain. The Poisson intensity cancels because the premium rate is specified using the same relative loading and mean claim cost.