1. Collision asked about Daykin p. 199 where high loading implies high retention. All of Daykin's analyses are "all else equal" and so the question is what should be done for a given policy if the premium is increased (so it is no riskier, the company just collects more). Then it makes sense to retain more of the risk.
2. Michael Sce asked about the calculations on p. 370 of Bowers. For such problems, col. (2) is always a 1 followed by zeros (probabilities for S when there are no claims); col. (3) is always p(x), the probabilities for X, which are also probabilities for S when there is one claim; col. (4) are probabilities for X1+X2 calculated by convolution (see pages 35 and 36 for more on convolutions of discrete distribution; col. (5) is the distribution of X1+X2+X3 found by doing the convolution of cols (3) and (4). In similar problems, where more than three claims might be possible, further columns are needed. Each is the convolution of the previous column with column (3).
Across the bottom of the table of the probabilities of N, the number of claims. The second to last column gives probabilities for S, aggregate claims. Each entry is the cross-product (dot-product) of colums (2)-(5) in that row with the probabilities at the bottom of the table.
The final column is the cumulative sums of the preceding column.
3. Collision asked about formula (1.2.5) in Daykin, wondering if Wnew is missing. That appears to be the case.
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Stuart Klugman, FSA, PhD
Principal Financial Group Professor of Actuarial Science
Drake University
2507 University Avenue
Des Moines, IA 50311 USA
ph: 515-271-4097
e-mail: Stuart.Klugman@drake.edu
Drake Act. Sci. web site: www.drake.edu/cbpa/acts