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Assignment 2 soln

# Assignment 2 soln - FIN 3220A Actuarial Models I First Term...

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1 FIN 3220A Actuarial Models I First Term 2005-2006 Solutions to Assignment 2 1. Let Z be the random variable of the present value of benefit. Since i = 0, we have 1 30 0 1 30 1 2 30 with probability 0.1 10 with probability 0.9 0.6 0.54 0 with probability 0.9 0.4 0.36 b q Z b q p = = = = = = Hence the variance of Z is given by ( ) ( ) ( ) { } ( ) ( ) ( ) { } ( ) ( )( ) { } 2 2 2 2 2 1 1 1 1 Var 0.1 10 0.54 0.1 10 0.54 . Z E Z E Z b b b b = = + + Differentiate Var( Z ) with respect to b 1 and set it to zero, we have ( ) ( ) ( )( ) ( ) ( )( ) { } ( ) 1 1 1 1 1 1 1 Var 2 0.1 2 10 0.54 2 0.1 10 0.54 0.44 0 0.8928 6.048 or 6.7742. d Z b b b b d b b b = + = = = 2. Let Z be the random variable of present value of benefit. We have ( ) ( ) ( ) ( ) ( ) ( ) { } ( ) ( ) 1 1 : : 2 2 2 2 1 2 1 : : 2 2 2 2 2 2 1 2 1 1 1 : : : : 1,000 , 1,000 , Var 1,000 1,000 . x n x n x n x n x n x n x n x n E Z A A E Z A A Z E Z E Z A A A A = π + = π ⋅ + = = π ⋅ + − π + Expanding out the squared term and we have ( ) ( ) 1 1 : : 2 1,000 . x n x n K A A = − π 3. The net single premium is given by ( ) ( ) ( ) 100 0.10 0 100 0.10 0 100 100 0.10 0.10 2 0 0 10 10 10 50 50 50 50 5000 50 5000 0.10 0.10 100 1 0.01 0 0.1 0.01 100 1 11 0.999501.

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