Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-23T15:13:34.020Z Has data issue: false hasContentIssue false

On The Approximation of the Total Amount of Claims

Published online by Cambridge University Press:  29 August 2014

Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Several “short cut” methods exist to approximate the total amount of claims ( = χ) of an insurance collective. The classical one is the normal approximation

where and σx are the mean value and standard deviation of x. Φ is the normal distribution function.

It is well-known that the normal approximation gives acceptable accuracy only when the volume of risk business is fairly large and the distribution of the amounts of the individual claims is not “too dangerous”, i.e. not too heterogeneous (cf. fig. 2).

One way to improve the normal approximation is the so called NP-method, which provides for the standardized variable a correction Δz

where

is the skewness of the distribution F(χ). Another variant (NP3) of the NP-method also makes use of the moment μ4, but, in the following, we limit our discussion mainly to the variant (2) (= NP2).

If Δz is small, a simpler formula

is available (cf. fig. 2).

Another approximation was introduced by Bohman and Esscher (1963). It is based on the incomplete gamma function

where

Experiments have been made with both formulae (2) and (4); they have been applied to various F functions, from which the exact (or at least controlled) values are otherwise known. It has been proved that the accuracy is satisfactory provided that the distribution F is not very “dangerous”.

Type
Research Article
Copyright
Copyright © International Actuarial Association 1977

References

Beard, R., Pentikäinen, T. and Pesonen, E., (1969, a new edition 1977): Risk Theory (Methuen & Co, London)CrossRefGoogle Scholar
Bohman, H. and Esscher, F., (19631964): “Studies in risk theory with numerical illustrations” (Skand. Aktuarietidskrift).Google Scholar
Hovinen, E., (1964): “A method to compute convolution”, (Congress of Actuaries).Google Scholar
Kauppi, L. and Ojantakanen, P., (1969): “Approximations of the generalised Poisson function” (Astin Bulletin).CrossRefGoogle Scholar
Pesonen, E., (1969): “NP-approximation of risk processes” (Skand. Aktuarietidskrift).CrossRefGoogle Scholar
Seal, H., (1977): “Approximations to risk theory's F(x, t) by means of the gamma distribution” (Astin Bulletin).Google Scholar