Question

1. A motor insurer operates a no-claims discount system that has five levels. The percentage of the basic premium paid by the insured in each level is as follows: level % premium charged 5 100 4 90 3 80 2 70 1 60 Insured motorists move between levels depending on the number of claims in the previous year. For each policy holder, the number of claims per year follows a poisson distribution with mean 0.25. For those in levels 2, 3, 4 and 5 at the start of the previous year; i) If no claims are made during the previous year, the insured moves down one level (example, from level 4 to level 3). ii) If one claim is made during the previous year, the insured moves up one level (except those in level 5 at the start of the previous year who will remain in level 5). iii) If two claims are made during the previous year, the insured moves up two levels (except those in level 5 at the start of the previous year who will remain in level 5 and those in level 4, who will move to level 5). iv) If three or more claims are made during the previous year, the insured moves to level 5. For those in level 1 at the start of the previous year a no claims discount protection policy applies whereby they remain in level 1 if they make one claim. If they make two claims, they move to level 5. If they make no claims, they remain in level 1. a) Determine the transition matrix for the no-claims discount system (Assuming that all motorists activate their policies). b) A policy holder is in level 3 for the first year of the policy. Assuming that the policy is maintained, calculate the probability that at the start of the third year the policy holder will be; (i) In level 1 (ii) In level 3 c) (i) state conditions under which the probability of being in a particular state after n years converges as n ? ? to some limit which is independent of the initial state. (ii) verify that the conditions are satisfied in this instance. (iii) Determine the ultimate probability that the insured will be in level 1. d) The insurer suspects that the model used for its calculation may be too simplistic. Given annual data listing numbers of claims per policy, broken down by discount level, state which test would be most appropriate to test the assumption the distribution of the number of claims per policy per year is poisson with mean 0.25.

1. A motor insurer operates a no-claims discount system that has five levels. The percentage of the basic premium paid by the insured in each level is as follows:
level % premium charged
5 100
4 90
3 80
2 70
1 60

Insured motorists move between levels depending on the number of claims in the previous year. For each policy holder, the number of claims per year follows a poisson distribution with mean 0.25.
For those in levels 2, 3, 4 and 5 at the start of the previous year;
i) If no claims are made during the previous year, the insured moves down one level (example, from level 4 to level 3).
ii) If one claim is made during the previous year, the insured moves up one level (except those in level 5 at the start of the previous year who will remain in level 5).
iii) If two claims are made during the previous year, the insured moves up two levels (except those in level 5 at the start of the previous year who will remain in level 5 and those in level 4, who will move to level 5).
iv) If three or more claims are made during the previous year, the insured moves to level 5.
For those in level 1 at the start of the previous year a no claims discount protection policy applies whereby they remain in level 1 if they make one claim. If they make two claims, they move to level 5. If they make no claims, they remain in level 1.

a) Determine the transition matrix for the no-claims discount system (Assuming that all motorists activate their policies).
b) A policy holder is in level 3 for the first year of the policy. Assuming that the policy is maintained, calculate the probability that at the start of the third year the policy holder will be;
(i) In level 1 (ii) In level 3
c) (i) state conditions under which the probability of being in a particular state after n years converges as n ? ? to some limit which is independent of the initial state.
(ii) verify that the conditions are satisfied in this instance.
(iii) Determine the ultimate probability that the insured will be in level 1.
d) The insurer suspects that the model used for its calculation may be too simplistic. Given annual data listing numbers of claims per policy, broken down by discount level, state which test would be most appropriate to test the assumption the distribution of the number of claims per policy per year is poisson with mean 0.25.

1. A motor insurer operates a no-claims discount system that has five levels. The percentage of the basic premium paid by the insured in each level is as follows:
level % premium charged
5 100
4 90
3 80
2 70
1 60

Insured motorists move between levels depending on the number of claims in the previous year. For each policy holder, the number of claims per year follows a poisson distribution with mean 0.25.
For those in levels 2, 3, 4 and 5 at the start of the previous year;
i) If no claims are made during the previous year, the insured moves down one level (example, from level 4 to level 3).
ii) If one claim is made during the previous year, the insured moves up one level (except those in level 5 at the start of the previous year who will remain in level 5).
iii) If two claims are made during the previous year, the insured moves up two levels (except those in level 5 at the start of the previous year who will remain in level 5 and those in level 4, who will move to level 5).
iv) If three or more claims are made during the previous year, the insured moves to level 5.
For those in level 1 at the start of the previous year a no claims discount protection policy applies whereby they remain in level 1 if they make one claim. If they make two claims, they move to level 5. If they make no claims, they remain in level 1.

a) Determine the transition matrix for the no-claims discount system (Assuming that all motorists activate their policies).
b) A policy holder is in level 3 for the first year of the policy. Assuming that the policy is maintained, calculate the probability that at the start of the third year the policy holder will be;
(i) In level 1 (ii) In level 3
c) (i) state conditions under which the probability of being in a particular state after n years converges as n ? ? to some limit which is independent of the initial state.
(ii) verify that the conditions are satisfied in this instance.
(iii) Determine the ultimate probability that the insured will be in level 1.
d) The insurer suspects that the model used for its calculation may be too simplistic. Given annual data listing numbers of claims per policy, broken down by discount level, state which test would be most appropriate to test the assumption the distribution of the number of claims per policy per year is poisson with mean 0.25.

Added by Jennifer G.

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