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A2Sol - ACTSC 232 Spring 2010 Assignment 2 SOLUTIONS Due...

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ACTSC 232 - Spring 2010 - Assignment 2 SOLUTIONS Due: Friday, June 25 at 11:30 am 1. (a) Describe in words the insurances with the present values given. i. Z 1 = braceleftbigg 20000 v T x , T x 10 10000 v T x , T x > 10 This benefit pays 20000 immediately on death if death occurs within 10 years, but the benefit reduces to 10000 (still paid immediately on death) if the life survives past 10 years. It is a whole life insurance but with a change in sum insured at time 10. ii. Z 2 = 0 , T x 15 10000 v T x , 15 < T x 25 10000 v 25 , T x > 25 This benefit is a 15-year deferred 10-year endowment insurance with sum insured 10000. It pays nothing if death occurs within 15 years, 10 imme- diately on death in the following 10 years (from time 15 to time 25) or on survival to 25 years. (b) Write down in integral form the formula for the expected value for Z 1 and Z 2 . i. Z 1 = 20000 integraltext 10 0 e - δt t p x μ x + t dt + 10000 integraltext 10 e - δt t p x μ x + t dt = 20000 integraltext 10 0 e - δt t p x μ x + t dt + 10000 v 10 10 p x integraltext 0 e - δt t p x +10 μ x +10+ t dt OR 20000 integraltext 0 e - δt t p x μ x + t dt - 10000 v 10 10 p x integraltext 0 e - δt t p x +10 μ x +10+ t dt OR 10000 integraltext 10 0 e - δt t p x μ x + t dt + 10000 integraltext 0 e - δt t p x μ x + t dt ii. Z 2 = 10000 integraltext 25 15 e - δt t p x μ x + t dt + 10000 v 25 25 p x = 10000 v 15 15 p x integraltext 10 0 e - δt t p x +15 μ x +15+ t dt + 10000 v 25 25 p x (c) Derive expressions in terms of standard actuarial functions for the expected values in (b). i. Z 1 = 20000 ¯ A 1 x : 10 + 10000 10 | ¯ A x = 20000 ¯ A 1 x : 15 + 10000 10 E x ¯ A x +10 OR 20000 ¯ A x - 10000 10 E x ¯ A x +10 OR 10000 ¯ A 1 x : 10 + 10000 ¯ A x ii. Z 2 = 10000 15 | ¯ A x : 10 = 10000 15 E x ¯ A x +15: 10 (d) Derive an expression in terms of standard actuarial functions for the covariance of

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A2Sol - ACTSC 232 Spring 2010 Assignment 2 SOLUTIONS Due...

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