Loss Models: From Data to Decisions, 4th Edition

E.1 The method of rounding

This method has two features: All probabilities are positive and the probabilities add to 1. Let h be the span and let Y be the discretized version of X. If there are no modifications, then

The recursive formula is then used with f_X(j) = f_j. Suppose a deductible of d, limit of u, and coinsurance of α are to be applied. If the modifications are to be applied before the discretization, then

where g_j = Pr(Z = jαh) and Z is the modified distribution. This method does not require that the limits be multiples of h but does require that u − d be a multiple of h. This method gives the probabilities of payments per payment.

Finally, if there is truncation from above at u, change all denominators to F_X(u) − F_X(d) and also change the numerator of g_{(u-d) / h} to F_X(u) − F_X(u − h/2).