A no claims discount system operates with three levels of discount, 0%, 15% and 40%. If a policyholder makes no claim during the year he moves up a level of discount (or remains at the maximum level). If he makes one claim during the year he moves down one level of discount (or remains at the minimum level) and if he makes two or more claims he moves down to, or remains at, the minimum level. The probability for each policyholder of making two or more claims in a year is 25% of the probability of making only one claim. The long-term probability of being at the 15% level is the same as the long-term probability of being at the 40% level. (i) Derive the probability of a policyholder making only one claim in a given year. [4] (ii) Determine the probability that a policyholder at the 0% level this year will be at the 40% level after three years. [2] (iii) Estimate the probability that a policyholder at the 0% level this year will be at the 40% level after 20 years, without calculating the associated transition matrix. [3] [Total 9]