Assignment 3 soln

Assignment 3 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 3 1. The present value of payments with the associated probabilities are Curtate future lifetime Probability Present value of payment 0 1 0.8 0.2 x q = −= 1 1 ( ) 1 0.8 1 0.75 0.2 xx pq + =−= 120 . 9 2 . 8 + ×= 2 1 0.8 0.75 0.6 pp + = 2 1 2 0.9 3 0.9 5.23 +× +× = Then the expected PV of the payments 1 0.2 2.8 0.2 5.23 0.6 3.898 K = ×+×+ . Hence the probability that the PV of the payments actually made will exceed K is Pr( PV > K ) = Pr(curtate future lifetime 2) = 0.6. 2. It is easy to show that 30 30 1 150, 150 qq == and 23 0 48 50 p = . Hence we have () 30 30 2 30 1 12 3 11 2 1 1 48 1 1 48 1 1 1.05 1 1.05 1.05 1 1.952381 2.859410 50 50 50 50 50 50 2.804082, EY a q a p −− =⋅ +⋅ = ⋅ ++ ++ + = ⋅ + = ±± 2 22 30 30 2 30 1 3 114 8 1 1.952381 2.859410 50 50 50 7.945415, a q a q a p = + + = () () () { } 2 2 2 Var 7.945415 2.804082 0.082539. YE Y E Y =− = = 3. ( ) ( ) ( ) 44 4 1 17.287 1 17.287 0.057847 ad d = = . By the relation, ( ) 4 4 4 4 0.058696 and 0.06. 14 i di i i =⇒ = = + Hence we have 4 4 4 4 4 0.06 0.1025 0.104778, 0.058696 1 1 0.104778 15.4757. 0.057847 x x i AA i A a d = =
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2 4. (a) The random variable Z is 1 1,000 1,000 1,000 5,000 1,000 5,000 5,000 .
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Assignment 3 soln - FIN 3220A Actuarial Models I First Term...

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