Actuarial Modelling of Claim Counts Risk Classification, Credibility ...

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178 Actuarial Modelling of Claim Counts The convergence is now much slower: ⎛ P 5 01 = ⎜ ⎝ 0791523 0081985 0088694 0018169 0015456 0004173 0788490 0066822 0106890 0017259 0014546 0005992 0788490 0063789 0073531 0053651 0013637 0006902 0606531 0245749 0070498 0020292 0050028 0006902 0606531 0063789 0252457 0017259 0034865 0025098 0606531 0063789 0070498 0199219 0031833 0028131 ⎞ ⎟ ⎠ ⎛ P 10 01 = ⎜ ⎝ 0784013 0081747 0090966 0022022 0016217 0005037 0784003 0081480 0090248 0023009 0016178 0005081 0777382 0088092 0089871 0022071 0017497 0005087 0776278 0079263 0099795 0021694 0016890 0006080 0776278 0078160 0090966 0031618 0016623 0006356 0743169 0111269 0089862 0026100 0023236 0006365 ⎞ ⎟ ⎠ ⎛ P 20 01 = ⎜ ⎝ which slowly converges to ⎛ 01 = ⎜ ⎝ 0782907 0082338 0090996 0022276 0016387 0005096 0782903 0082332 0091006 0022275 0016387 0005097 0782902 0082326 0090993 0022295 0016386 0005098 0782803 0082424 0090984 0022285 0016406 0005098 0782776 0082352 0091082 0022278 0016403 0005108 0782774 0082327 0091011 0022376 0016399 0005113 0782901 0082338 0090998 0022278 0016387 0005097 0782901 0082338 0090998 0022278 0016387 0005097 0782901 0082338 0090998 0022278 0016387 0005097 0782901 0082338 0090998 0022278 0016387 0005097 0782901 0082338 0090998 0022278 0016387 0005097 0782901 0082338 0090998 0022278 0016387 0005097 In this case, the system is not stable after 20 years. Let us consider the trajectory of a policyholder with expected claim frequency accross the levels of the bonus-malus scale. We define the stationary distribution = 0 1 s T as follows: l is the stationary probability for a policyholder with mean frequency to be in level l i.e. l2 = lim n→+ pn l 1 l 2 The term l is the limit value of the probability that the policyholder is in level l, when the number of periods tends to +. It is also the fraction of the time a policyholder with claim frequency spends in level l, once the steady state has been reached. Note that ⎞ ⎟ ⎠ ⎞ ⎟ ⎠
Bonus-Malus Scales 179 does not depend on the starting class. This means that the nth power P n of the one-step transition matrix P converges to a matrix with all the same rows T , that is ⎛ lim n→+ Pn = = ⎜ ⎝ T T T exactly as we saw in the introductory examples. Let us now explain how to compute the l s. Taking the limit in both sides of (4.5) for n →+, we see that the vector is the unique probabilistic solution to the system of linear equations j = ⎞ ⎟ ⎠ s∑ l p lj j ∈ 0s (4.7) l=0 In matrix notation, (4.7) can be written as { T = T P T e = 1 (4.8) where e is a column vector of 1s. This means that is the left eigenvector u 0 of P encountered above. Thus we see that if the initial distribution in the scale is , then the probability distribution remains equal to . 4.4.2 Rolski–Schmidli–Schmidt–Teugels Formula Let E be the s +1×s +1 matrix all of whose entries are 1, i.e. consisting of s +1 column vectors e. Then, the following result provides a direct method to get . Property 4.2 Assume that the stochastic matrix P is regular. Then the matrix I −P+ E is invertible and the solution of (4.8) is given by T = e T I − P + E −1 (4.9) Proof Let us first check that I − P + E is invertible. We must show that I − P + Ex = 0 ⇒ x = 0 From (4.8), we have T I − P = 0. Thus, I − P + Ex = 0 ⇒ 0 = T I − P + Ex = 0 + T Ex ⇔ T Ex = 0
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Contents Foreword Preface Notation
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Contents vii 2.4 Overdispersion 79
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Contents ix 4.2.3 Trajectory 170 4.
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Contents xi 7.4 Numerical Illustrat
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Part I Modelling Claim Counts Actua
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3 Credibility Models for Claim Coun
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Credibility Models for Claim Counts
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Credibility Models for Claim Counts
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8 Transient Maximum Accuracy Criter
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Transient Maximum Accuracy Criterio
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Bibliography Albrecht, P. (1983a).
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Bibliography 347 Daengdej, J., Luko
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Bibliography 349 Greenwood, M., & Y
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Bibliography 351 Norberg, R. (1976)
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Bibliography 353 Walhin, J.-F., & P
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