# Insurance risk and ruin by Dickson D.C.M.

By Dickson D.C.M.

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However, for most practical purposes, numerical stability is not an issue. 1 X i . Recursively calculate Pr(S4 = r ) for r = 1, 2, 3 and 4. 0256. Now note that as f j = 0 for j = 4, 5, 6, . . 1905. 7 Notes and references Further details of the distributions discussed in this chapter, including a discussion of how to fit parameters to these distributions, can be found in Hogg and Klugman (1984). See also Klugman et al. (1998). 3 was derived by De Pril (1985), and a very elegant proof of the result can be found in his paper.

Consistency. This property requires that if Y = X + c where c > 0, then we should have Y = X + c. Thus, if the distribution of Y is the distribution of X shifted by c units, then the premium for risk Y should be that for risk X increased by c. No ripoff. This property requires that if there is a (finite) maximum claim amount for the risk, say xm , then we should have X ≤ x m . If this condition is not satisfied, then there is no incentive for an individual to effect insurance. 1 The pure premium principle The pure premium principle sets X = E [X ] .

Jensen’s inequality states that if u is a concave function, then E [u(X )] ≤ u (E [X ]) provided that these quantities exist. 3) 30 Utility theory We now prove Jensen’s inequality on the assumption that there is a Taylor series expansion of u about the point a. Thus, writing the Taylor series expansion with a remainder term as u(x) = u(a) + u (a)(x − a) + u (z) (x − z)2 2 where z lies between a and x, and noting that u (z) < 0, we have u(x) ≤ u(a) + u (a)(x − a). 3) by taking expected values. We can use Jensen’s inequality to obtain results relating to appropriate premium levels for insurance cover, from the viewpoint of both an individual and an insurer.