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 13 0 0 0 1 23 0 with probability 0.1 10 with probability 0.9 0.6 0.54 0 with probability 0.9 0.4 0.36 bq Zb q p = =− = = =⋅= Hence the variance of Z is given by () () () { } () ( ) ( ) { } ( ) ( ) { } 2 2 2 22 11 1 1 Var 0.1 10 0.54 0.1 10 0.54 . ZE 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 Var 2 0.1 2 10 0.54 2 0.1 10 0.54 0.44 0 0.8928 6.048 or 6.7742. dZ bb b b db + = ⇒= = 2. Let Z be the random variable of present value of benefit. We have :: 2 2 1 2 1 2 2 2 2 1 2 1 1 1 : : 1,000 , 1,000 , Var 1,000 1,000 . xn EZ A A A A Z E Z A A A A + =π ⋅ + = = π ⋅+ ⋅− π + Expanding out the squared term and we have ( ) ( ) 21 , 0 0 0 . KA 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 0.999501. t x t T t tt AE e ef t d t t ed t te e ee e −δ −− = = = ⎩⎭ ⎛⎞ =+ + ⎜⎟ ⎝⎠ =
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This note was uploaded on 05/21/2011 for the course FIN 3220 taught by Professor Cswong during the Spring '05 term at CUHK.

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

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