Kindly answer the following questions appropriately.

An office uses the multiple state sickness model to calculate premiums for a three-year combined sickness and endowment assurance policy issued to a healthy policyholder aged 57 at inception. Premiums are payable annually in advance for 3 years and are waived during periods of sickness. At the end of each of the first 2 years a benefit of $10,000 is payable if the life is then sick. A sum assured of f15,000 is payable at the end of the year of death if this occurs during the term of the policy or at the end of the three years if the life is alive and has never claimed any sickness benefit. The benefit payable at the end of year 3 is $10,000 if the life has previously claimed sickness benefit. The transition probabilities are as follows (for / = 0, 1 ,2): P(dead at / + 1 | healthy at /) = 0.02 P( dead at / + 1 | sick at /) = 0.05 P( sick at / + 1 | healthy at () =0.10 P( sick at / + 1 | sick at () = 0.09 (i) Calculate the probabilities of being in each state at times / = 0, 1, 2, 3. [3] (ii) Calculate the profit margin for the contract by carrying out a profit test using the following assumptions: Annual premium: E5,300 Risk discount rate: 10% pa Interest on cash flows and reserves: 8% pa Initial expenses: E200 incurred on payment of the first premium Renewal expenses: E40 at times / = 1, 2 whether healthy or sick Claim expense: $30 at the date of claim of any benefit Reserves: oV= =0 V = 5,000r if the policyholder is healthy at time ? (t = 1, 2) ,1 = 4,000 if the policyholder is sick at time / (/ = 1, 2) [16] [Total 19]E200 is invested for 12 years. In any year the yield on the investment will be 3% with probability 0.25, 5% with probability 0.6 and 6% with probability 0.15 and is independent of the yield in any other year. (i) Calculate the mean accumulation at the end of 12 years. [2] (ii) Calculate the standard deviation of the accumulation at the end of 12 years. [5] [Total 7]