Appendix E
Discretization of the Severity Distribution
There are two relatively simple ways to discretize the severity distribution. One is the method of rounding and the other is a mean-preserving method.
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 
. Suppose that a deductible of d, limit of u, and coinsurance of 
are to be applied. If the modifications are to be applied before the discretization, then
where 
and Z is the modified distribution. This method does not require that the limits be multiples of h, but does require that 
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 
and also change the numerator of 
to 
.
E.2 Mean Preserving
This method ensures that the discretized distribution has the same mean as the original severity distribution. With no modifications, the discretization is
For the modified distribution,
To incorporate truncation from above, change the denominators to
and subtract 
from the numerators of each of 
and 
.
E.3 Undiscretization of a Discretized Distribution
Assume that we have 
, the true probability that the random variable is zero. Let 
, where 
is a discretized distribution and h is the span. The following are approximations for the cdf and limited expected value of S, the true distribution that was discretized as 
. They are all based on the assumption that S has a uniform distribution over the interval from 
to 
for integral j. The first interval is from 0 to 
, and the probability 
is assumed to be uniformly distributed over it. Let 
be the random variable with this approximate mixed distribution. (It is continuous, except for discrete probability 
at zero.) The approximate distribution function can be found by interpolation as follows. First, let
Then, for x in the interval 
to 
,
Because the first interval is only half as wide, the formula for 
is
It is also possible to express these formulas in terms of the discrete probabilities:
With regard to the limited expected value, expressions for the first and kth LEVs are
and, for 
,
while for 
,
