LDF Curve-Fitting and Stochastic Reserving: A Maximum Likelihood ...

Unfortunately, for the Logtogistic and Weibull growth curves, the moment generatingfunction is intractable and so does not simplify our calculation. For practical purposeswe will use the incremental approximation instead.y-xR~ ~ ~,YyLTA,.v ..... (G(~ +k)-G(~ +k- 1))AY k-[The variance can then be calculated for the discounted reserve in two pieces: the processvariance and the parameter variance.Process VarianceThe process variance component is actually trivial to calculate. We already know that thevariance of the full value reserve is estimated by multiplying by the scale factor a 2. Wethen need to recall that the variance for some random variable times a constant is givenby Var(v ~ .R) = v 2k .Var(R).The process variance of the discounted reserve is therefore:yxVar(Rd) = crZ. y~yULTA,..v2k-l.(G(x +k)-G(x +k-l))AY k~lParameter VarianceThe parameter variance again makes use of the covariance matrix of the modelparameters Z. The formula is then given below.Var(E[Ra] ) = (ORa)'.Z.(c3R,)where= L ORa ORa 0-~) fortheCapeCodmethod~Ra \~ELR' ~' -aR~orlfor the LDF method90