De Pril recursive method

See also: Aggregate distributions introduction, Aggregate modeling in ModelRisk, Vose Aggregate DePril, Insurance and finance risk analysis modeling introduction,

De Pril's recursive method is often used in insurance risk analysis modeling.

For a portfolio of n independent life insurance policies, each policy y has a particular probability of a claim py in some period (usually a year) and benefit By. There are various methods for calculating the aggregate payout. Dickson (2005) is an excellent (and very readable) review of these methods and other areas of insurance risk and ruin.

The de Pril method is an exact method for determining the aggregate payout distribution. The compound Poisson approximation (discussed here) is a faster method that will usually work too.

De Pril (1986) offers an exact calculation of the aggregate distribution under the assumptions that:

  • The benefits are fixed values rather than random variables and take integer multiples of some convenient base (e.g. $1000) with a maximum value M*base, i.e. Bi = {1...M}*base; and

  • The probability of claims can similarly be grouped into a set of J values (i.e. into tranches of mortality rates) pj = {p1...pJ}.

Let nij be the number of policies with benefit i and probability of claim pj. Then De Pril's paper demonstrates that P(Y), the probability that the aggregate payout will be Y, is equal to y*base is given by the recursive formula:

    for y = 1,2,3,...

and

            

where

            

The formula has the benefit of being exact, but it is very computationally intensive. However, the number of computations can usually be significantly reduced if one accepts ignoring small aggregate costs to the insurer. Let K be a positive integer. Then the recursive formulae above are modified as follows:

 

Dickson (2005) recommends using a value of 4 for K. The de Pril method can be seen as the counterpart to the Panjer's recursive method for the collective model.

The aggregate De Pril method is implemented in ModelRisk with the Aggregate DePril window.

Read on: Aggregate modeling - compound Poisson approximation
 

 

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