Stats and Probability
The no claims discount (NCD) system operated by an insurance company for their annual motor insurance business has four levels of discount: The probability of an accident Level 1: 0% 0.30 Level 2: 15% 0.25 Level 3: 25% 0.20 Level 4: 45% 0.10 If a policyholder does not make a claim under the policy in a particular year then he or she will go up one level (or stay at level 4), whereas if any claims are made he or she will go down by two levels (or remain at, or move to, level 1). The full premium payable at the 0% discount level is 900. Losses are assumed to follow a lognormal distribution with mean 1,188 and standard deviation 495. However, policyholders claim only if the loss is greater than the total additional premiums that would have to be paid over the next three years, assuming that no further accidents occur. () Calculate the smallest loss for which a claim will be made for each of the four states in the NCD system. [3 marks) (1) Determine the transition matrix for this NCD system. [10 marks) (iii) Calculate the proportion of policyholders at each discount level when the system reaches a stable state. [3 marks] (iv) Determine the average premium paid once the system reaches a stable state. [2 marks] (v) Describe the limitations of simple NCD systems such as this one. [3 marks] The no claims discount (NCD) system operated by an insurance company for their annual motor insurance business has four levels of discount: The probability of an accident Level 1: 0% 0.30 Level 2: 15% 0.25 Level 3: 25% 0.20 Level 4: 45% 0.10 If a policyholder does not make a claim under the policy in a particular year then he or she will go up one level (or stay at level 4), whereas if any claims are made he or she will go down by two levels (or remain at, or move to, level 1). The full premium payable at the 0% discount level is 900. Losses are assumed to follow a lognormal distribution with mean 1,188 and standard deviation 495. However, policyholders claim only if the loss is greater than the total additional premiums that would have to be paid over the next three years, assuming that no further accidents occur. () Calculate the smallest loss for which a claim will be made for each of the four states in the NCD system. [3 marks) (1) Determine the transition matrix for this NCD system. [10 marks) (iii) Calculate the proportion of policyholders at each discount level when the system reaches a stable state. [3 marks] (iv) Determine the average premium paid once the system reaches a stable state. [2 marks] (v) Describe the limitations of simple NCD systems such as this one. [3 marks]
