A transition matrix for a four-state homogeneous Markov chain is given by
At time 0, the process is in state 0. A contract provides for payments at the end of the year of the first transition to state 3 from any other state, provide this occurs within 4 years. The amount of the payment is 100 if the transition is from state 0, 200 if the transition is from state 1, or 300 if the transition is from state 2. Level net annual premiums are paid beginning at time 0, and continuing until the time of transition to state 3. The interest rate is a constant 5%.
(a) Find the annual premium.
(b) Find the reserve at time 2, assuming the process is at that time in (i) state 0, (ii) state 1, (iii) state 2.