By Robert Eric Beard O.B.E., F.I.A., F.I.M.A., Professor, Teivo Pentikäinen Phil. Dr, Professor h.c., Erkki Pesonen Phil. Dr (auth.)

The thought of threat already has its traditions. A evaluation of its classical effects is contained in Bohlmann (1909). This classical thought was once linked to existence coverage arithmetic, and dealt ordinarily with deviations that have been anticipated to be produced via random fluctua tions in person regulations. in response to this thought, those deviations are discounted to a couple preliminary immediate; the sq. root of the sum of the squares of the capital values calculated during this approach then provides a degree for the soundness of the portfolio. A thought constituted during this demeanour isn't, in spite of the fact that, very applicable for useful reasons. in truth that it doesn't supply a solution to such questions as, for instance, inside what limits a company's possible achieve or loss will lie in the course of assorted sessions. additional, non-life assurance, to which danger idea has, in reality, its so much worthwhile purposes, used to be regularly outdoors the sphere of curiosity of the chance theorists. hence it's relatively comprehensible that this thought didn't obtain a great deal cognizance and that its functions to sensible difficulties of coverage job remained relatively unimportant. a brand new section of improvement all started following the reviews of Filip Lundberg (1909, 1919), which, because of H. Cramer (1926), e.O.

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1. 1 Comparison of the Wilson-Hilferty approximated values with exact Poisson values. For k ~ n values of F are given and for k > n. values of 1- F. k Exact Approx. 00354 1 4 7 10 50 25 10 n= Exact Approx. k Exact Approx. 00203 (e) Gamma formula The dJ. 7. 1 A friendly society has 1000 members. In the event of death a fixed sum S = £1000 is paid. 01, the premium p;. 1 is a safety loading and k is the number of deaths. The actuarial status of the society is examined every year. How large a security reserve U0 should the society have to be sure, at a 99% probability level, that the balance does not show any deficit?

1 Introduction (a) Definition of the problem As already mentioned, the simplest problem is considered first. e. a function p(k; t) which gives the probability that the number of claims k t in time t is equal to k. In terms of conventional symbols p(k; t) = prob {kt = k}. 1 ) In the following analysis the collective concerned can be the whole portfolio of an insurer or some special part thereof. (b) The individual and collective approaches The problem can be solved in a number of different ways.

It can be seen that the Poisson dJ. f. as n increases. 7). 1. 1 Comparison of the Wilson-Hilferty approximated values with exact Poisson values. For k ~ n values of F are given and for k > n. values of 1- F. k Exact Approx. 00354 1 4 7 10 50 25 10 n= Exact Approx. k Exact Approx. 00203 (e) Gamma formula The dJ. 7. 1 A friendly society has 1000 members. In the event of death a fixed sum S = £1000 is paid. 01, the premium p;. 1 is a safety loading and k is the number of deaths. The actuarial status of the society is examined every year.