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

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176 Actuarial Modelling of Claim Counts 4.3.4 Ergodicity and Regular Transition Matrix A Markov chain with transition matrix P is said to be ergodic if P is regular, that is, if there exists some n 0 ≥ 1 such that all entries of P n 0 are strictly positive. This condition means that it is possible, with a strictly positive probability, to go from one level i to another level j in a finite number of transitions or, in other words, that all states of the Markov chain are accessible from any initial state in a finite number of steps. All bonus-malus scales in practical use have a ‘best’ level, with the property that a policy in that level remains in the same level after a claim-free period. In our framework, the best level is level 0 and, for any level, it is possible to reach the superbonus level 0 after a sufficiently large number of claim-free years, resulting in p n l0 > 0 for all sufficiently large n. In the following, we restrict our attention to such non-periodic bonus rules. The transition matrix P associated with such a bonus-malus scale is regular, i.e. there exists some integer n 0 ≥ 1 such that all entries of the n 0 th power P n 0 of the one-step transition matrix are strictly positive. 4.4 Long-Term Behaviour of Bonus-Malus Systems 4.4.1 Stationary Distribution A natural question that arises concerns the long term behaviour of a bonus-malus system. Intuitively, we expect that the system will stabilize in the long run. Since the annual claim numbers have been assumed to be independent and identically distributed, each policyholder will ultimately stabilize around an equilibrium level correponding to the expected annual claim frequency , and will gravitate around this level. To formalize this intuitive idea, let us compute the powers of the transition matrix P for = 01 inthe−1/top and −1/ + 2 bonus-malus scales. This is done in the following examples. Example 4.7 (−1/Top Scale) ⎛ P01 = ⎜ ⎝ we get ⎛ P 2 01 = ⎜ ⎝ Starting from 0904837 0 0 0 0 0095163 0904837 0 0 0 0 0095163 0 0904837 0 0 0 0095163 0 0 0904837 0 0 0095163 0 0 0 0904837 0 0095163 0 0 0 0 0904837 0095163 0818731 0 0 0 0086107 0095163 0818731 0 0 0 0086107 0095163 0818731 0 0 0 0086107 0095163 0 0818731 0 0 0086107 0095163 0 0 0818731 0 0086107 0095163 0 0 0 0818731 0086107 0095163 ⎞ ⎟ ⎠ ⎞ ⎟ ⎠
Bonus-Malus Scales 177 ⎛ P 3 01 = ⎜ ⎝ 0740818 0 0 0077913 0086107 0095163 0740818 0 0 0077913 0086107 0095163 0740818 0 0 0077913 0086107 0095163 0740818 0 0 0077913 0086107 0095163 0 0740818 0 0077913 0086107 0095163 0 0 0740818 0077913 0086107 0095163 ⎞ ⎟ ⎠ and ⎛ P 4 01 = ⎜ ⎝ ⎛ P 5 01 = ⎜ ⎝ 067032 000000 0070498 0077913 0086107 0095163 067032 000000 0070498 0077913 0086107 0095163 067032 000000 0070498 0077913 0086107 0095163 067032 000000 0070498 0077913 0086107 0095163 067032 000000 0070498 0077913 0086107 0095163 000000 067032 0070498 0077913 0086107 0095163 0606531 0063789 0070498 0077913 0086107 0095163 0606531 0063789 0070498 0077913 0086107 0095163 0606531 0063789 0070498 0077913 0086107 0095163 0606531 0063789 0070498 0077913 0086107 0095163 0606531 0063789 0070498 0077913 0086107 0095163 0606531 0063789 0070498 0077913 0086107 0095163 where all the rows are identical. Of course, ⎞ ⎟ ⎠ ⎞ ⎟ ⎠ P k 01 = P 5 01 for any integer k ≥ 6 This means that, whatever the initial distribution, p k 01 = 0606531 0063789 0070498 0077913 0086107 0095163 T for any k ≥ 5. The proportion of policyholders occupying each of the levels of the −1/top scale thus remains unchanged after 5 years. Example 4.8 (−1/+2 Scale) ⎛ P01 = ⎜ ⎝ In this case, 0904837 0 0090484 0 0004524 0000155 0904837 0 0 0090484 0 0004679 0 0904837 0 0 0090484 0004679 0 0 0904837 0 0 0095163 0 0 0 0904837 0 0095163 0 0 0 0 0904837 0095163 ⎞ ⎟ ⎠
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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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8 Transient Maximum Accuracy Criter
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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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