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

equation

The recursive formula is then used with img. Suppose that a deductible of d, limit of u, and coinsurance of img are to be applied. If the modifications are to be applied before the discretization, then

equation

where img and Z is the modified distribution. This method does not require that the limits be multiples of h, but does require that img 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 img and also change the numerator of img to img.

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

equation

For the modified distribution,

equation

To incorporate truncation from above, change the denominators to

equation

and subtract img from the numerators of each of img and img.

E.3 Undiscretization of a Discretized Distribution

Assume that we have img, the true probability that the random variable is zero. Let img, where img 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 img. They are all based on the assumption that S has a uniform distribution over the interval from img to img for integral j. The first interval is from 0 to img, and the probability img is assumed to be uniformly distributed over it. Let img be the random variable with this approximate mixed distribution. (It is continuous, except for discrete probability img at zero.) The approximate distribution function can be found by interpolation as follows. First, let

equation

Then, for x in the interval img to img,

equation

Because the first interval is only half as wide, the formula for img is

equation

It is also possible to express these formulas in terms of the discrete probabilities:

equation

With regard to the limited expected value, expressions for the first and kth LEVs are

equation

and, for img,

equation

while for img,

equation
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