A no-claims discount (NCD) system for motor insurance is to be modelled using a Markov chain with constant transition probabilities, as shown in the diagram: 80% 85% 90% 90% 1 2 3+ 4 15% 10% 20% 80% 10% 20% 3- State 1 No discount State 2 15% discount State 3- State 3+ State 4 30% discount (at least one claim in previous year) 30% discount (no claims in previous year) 40% discount (i) Set up the one-step transition probability matrix for this process in R and check that your matrix is correct. A new policyholder is currently on the 0% discount level. [4] (ii) Calculate the probabilities for the discount rate this policyholder will be receiving after five years. [4] (iii) Calculate the probabilities for the discount rate for this policyholder after 20 years and comment on your answers. [6]