By by Bjørn Sundt.
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Additional resources for An introduction to non-life insurance mathematics
X .. n g. )2 under the con,_ 1 J= 1 t) tJ straint. N n E. IE. 1g .. =l. 33). 6. We have N independent ceded portfolios that have been observed for n years. Let Xij be the loss ratio and Pij the risk volume of portfolio i in year j. We assume that Xil'Xi ,... \; Var[X .. Ie·)=~ 'J ' ij Ev(ei) = cp. ,.. ,H) * =~E. 1 N E. n P. (X . ~X . )2 1 1 1n ).. = }:2 J= z= "\n-~, P N - E. ,. ~- ~J ~J -2 N . _ 1 P. (X. -X) -(N-1)tp) ~- ~· E[m(E>i,H)jH) = v(H); Var(X .. ; Ev(8i,H) = tp; E(Xi)E>i,H) = m(8i,H); ~· ~· are unbiased estimators of 1fJ and>-..
Comment on this result. meter ()is the value of a random variable e. The insurance company wants to reinsure the portfolio with an excess of loss reinsurance for the layer m xs l. The reinsurance company calculates the premium using the variance principle. 6 We consider an insurance policy Let N. be the number of claims in year j and J Y.. the amount of the ith of these claims (i=l, ... ,N ; j=1,2, ... ). Let ~1 1 -62- be _the aggregate claim amount of year j. We assume that N ,N ,... are conditional1 2 ly mdependent and identically distributed given a random variable e, and that the -63- This exercise is based on Sundt (1991a).
Consider in particular the limiting cases where egoes to zero or infinity. Comment. 7. 18 Var v(H) -70- 7. Bonus systems -71m(e) = E(XI e). It is assumed that EX and Var X are finite. 7A. Credibility rating of individual policies is one example of experience rating. In this chapter we shall study another example, the bonus systems used in motor insurance. We shall assume that there is a finite number K of bonus classes numbered from 1 to K. All policies are placed in the same initial class k in the first insurance year.