T1-331F09

T1-331F09 - 3 Let 15 L be the prospective loss function at...

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Tutorial 1 – ACTSC 331, Fall 2009 September 30, Wednesday, 3:30-4:20, MC 4021 Problem 1 Let t L be the propective loss function at time t for a fully continuous whole life insurance of 1 on ( x ). Prove that Var[ t L | T ( x ) > t ] = " 1 + ¯ P ( ¯ A x ) δ # 2 [ 2 ¯ A x + t - ( ¯ A x + t ) 2 ] = 2 ¯ A x + t - ( ¯ A x + t ) 2 ( δ ¯ a x ) 2 . Problem 2 You are given that μ 35 ( t ) = 0 . 03 for t 0 and δ = 0 . 05. Calculate 5 ¯ V ( ¯ A 35: 25 ). Problem 3 Assume mortality follows De Moivre’s law with l x = 100 - x and i = 6%. Calculate ( a ) ¯ P ( ¯ A 35: 20 ), ( b ) 10 ¯ V ( ¯ A 35: 20 ), and ( c ) 20 ¯ V ( ¯ A 35: 20 ). Problem 4 In a whole life insurance on (50), premiums are charged continuously at an annual rate of 200. A death beneﬁt of b will be paid at the time of death. Assume that mortality follows De Moivre’s law with l x = 90 - x and the rate of interest is 5%. 1. Calculate b by the equivalence principle. 2. Calculate the beneﬁt reserve at the end of the 15th policy year.
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Unformatted text preview: 3. Let 15 L be the prospective loss function at the end of the 15th policy year, given that (50) is still alive then. Calculate V ar ( 15 L | T (50) > 15). Problem 5 Consider a fully continuous 20-Payment years, whole life insurance on (35). A death beneﬁt of 1000 will be paid at the moment of death. Premiums are paid continuously at an annual rate of 1000 20 ¯ P ( ¯ A 35 ). You are given a constant force of mortality μ = 0 . 015 and the force of interest δ = 0 . 035 . Let t L denote the present value at time t of the future loss after time t , given T (35) > t . 1. Calculate E[ 30 L | T (35) > 30]. 2. Calculate Var[ 30 L | T (35) > 30]. 1...
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