Consider a multi-state model for the three lives (x), (y), (z) where we have eight states described below, where the stipulated lives are those still living,
State 0: all; State 1: (x), (y) only; State 2: (x), (z) only; State 3: (y), (z) only; State 4: (x) only; State 5: (y) only; State 6: (z) only; State 7: None.
You are given that μ01 = 0.02, μ02 = 0.01, μ03 = 0.04, μ34 = 0.02, μ35 = 0.06, μ36 = 0.05, μ67 = 0.10 and the force of interest is a constant 0.05.
(a) Find the actuarial present value of an insurance policy on the three lives which pays 1 at the moment of death of (z) provided that (x) dies first and (y) dies second.
(b) Find the probability that the lives die in the order (x), (y), (z).