Find the actuarial present value of an insurance policy on

Question

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).