Tutorial-7-Problems

THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT3801/3909 Advanced Life Contingencies Tutorial 7 Problems 1. Consider a policy with a term of n years issued to a life aged ( x ) . Premiums are payable continuously throughout the term at rate P per year while the life is healthy, an annuity benefit is payable continuously at rate B per year while the life is sick, and a lump sum, S , is payable immediately on death within the term. Recovery from sick to healthy is possible and the disability income insurance model is appropriate. We assume that the premium, the benefits and the force of interest, δ per year, are constants rather than functions of time. (a) Show that, for 0 < t < n , t V (0) = B ¯ a 01 x + t : n - t | + S ¯ A 02 x + t : n - t | - P ¯ a 00 x + t : n - t | and derive a similar expression for t V (1) . (b) Show that, for 0 < t < n , d dt t V (0) = δ t V (0) + P - μ 01 x + t ( t V (1) - t V (0) ) - μ 02 x + t ( S - t V (0) ) and d dt t V (1) = δ t V (1) - B - μ 10 x + t ( t V (0) - t V (1) ) - μ 12 x + t ( S - t V (1) ) 2. The mortality of ( x ) and ( y ) follows a common shock model with the following states: State 0 - both alive State 1 - only ( x ) alive State 2 - only ( y ) alive State 3 - both dead You are given: • μ x + t = μ 02 x + t : y + t + μ 03 x + t : y + t = μ 13 x + t : y + t = g , a constant, for 0 ≤ t < 5 ; • μ y + t = μ 01 x + t : y + t + μ 03 x + t : y + t = μ 23 x + t : y + t = h , a constant, for 0 ≤ t < 5 ; • p x + t = 0 . 96 , for 0 ≤ t ≤ 4 ; • p y + t = 0 . 97 , for 0 ≤ t ≤ 4 ; 1