STAT3801_3909 AS4

14 / 15 THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT3801/STAT3909 Advanced Life Contingencies Assignment 4 Due: May 11, 2015 (Monday) 1. For a last-survivor whole life insurance of 1000 on ( x ) and ( y ): (a) The death benefit is payable at the moment of the second death. (b) The independent random variables T * x , T * y and Z are the components of a common shock model. (c) T * x has an exponential distribution with a failure rate of 0.03, t ≥ 0. (d) T * y has an exponential distribution with a failure rate of 0.05, t ≥ 0. (e) Z , the common shock random variable, has an exponential distribution with a failure rate of 0.02, t ≥ 0. (f) δ = 0 . 06. Calculate the actuarial present value of this insurance. 2. The mortality of ( x ) and ( y ) follows a common shock model with components T * x , T * y and Z . (a) T * x , T * y and Z are independent and have exponential distributions with respec- tive forces μ 1 , μ 2 and λ . (b) The probability that ( x ) survives 1 year is 0.96. (c) The probability that ( y ) survives 1 year is 0.97. (d) λ = 0 . 01. Calculate the probability that both ( x ) and ( y ) survive 5 years. 3. Consider a risk free investment that pays $ 1500 one year from now, $ 2000 two years from now and $ 1000 three years from now. Let y t be the t -year spot interest rate. Calculate the present value of this investment by using the following interest rates. (a) A constant effective interest rate of 8% per annum for all durations. (b) y 1 = 0 . 04, y 2 = 0 . 08 and y 3 = 0 . 12. (c) y 1 = 0 . 12, y 2 = 0 . 08 and y 3 = 0 . 04. 1