Exercises_-_Chapter_7_-_Solutions

Exercises_-_Chapter_7_-_Solutions - ACTSCI 331 - Life...

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Unformatted text preview: ACTSCI 331 - Life Contingencies I Solutions (Chapter 7) 1. (a) The bene&t premium is 2000 20 P & A 1 35: 30 j which is obtained by 2000 20 P & A 1 35: 30 j = 2000 A 1 35: 30 j a 35: 20 j = 2000 (0 : 06873) 11 : 39534 = 12 : 063 (b) The prospective loss at time 10 is de&ned as 10 L = 8 > > > > > > < > > > > > > : 2000 v T & 10 & 2000 20 P & A 1 35: 30 j a T & 10 j , 10 < T < 20 2000 v T & 10 & 2000 20 P & A 1 35: 30 j a 10 j , 20 < T < 30 & 2000 20 P & A 1 35: 30 j a 10 j , T > 30 , while the prospective loss at time 25 is de&ned as 25 L = 2000 v T & 25 , 25 < T < 30 , T > 30 . (c) To obtain the values of 10 L at T = 12 : 3 , we simply replace T by 12.3 in the de&nition of 10 L by choosing the expression in the appropriate domain of de&nition. It follows that ( 10 L j T = 12 : 3) = 2000 v 12 : 3 & 10 & 2000 20 P & A 1 35: 30 j a 12 : 3 & 10 j = 2000 v 2 : 3 & 12 : 063 a 2 : 3 j = 2000 e & : 06(2 : 3) & 12 : 063 & 1 & e & : 06(2 : 3) : 06 = 1716 : 3 Identical technique for 25 L at T = 27 : 4 ( 25 L j T = 27 : 4) = 2000 v 27 : 4 & 25 = 2000 v 2 : 4 = 2000 e & : 06(2 : 4) = 1731 : 8 (d) The bene&t reserve 10 V is given by 10 V = 2000 A 1 45: 20 j & 2000 20 P & A 1 35: 30 j a 45: 10 j = 2000 (0 : 09710) & 12 : 063 (7 : 35175) = 105 : 52 1 while 25 V = 2000 A 1 60: 5 j = 2000 (0 : 07952) = 159 : 04 (e) From the de&nition of 10 L , one identi&es that 10 L is a strictly decreasing function of the time until death T for which ( 10 L j T = 10) = 2000 . Thus, Pr ( 10 L & 1800 j T > 10) = Pr ( T > t & j T > 10) , (1) for which t & is such that 10 L is taken the value 1800 , i.e. ( 10 L j T = t & ) = 1800 . To do so, we shall identify in which brackets of the de&nition of 10 L the value t & belongs ( (10 ; 20) , (20 ; 30) or (30 ; 1 ) ). Note that ( 10 L j T = 10) = 2000 and ( 10 L j T = 20) = 2000 v 10 2000 20 P & A 1 35: 30 j a 10 j = 2000 e : 06(10) (12 : 063) & 1 e : 06(10) : 06 = 1006 : 9 . Given that 10 L is a decreasing function of T , it follows that t & 2 (10 ; 20) . Then, t & is the unique solution of 2000 v t & 10 2000 20 P & A 1 35: 30 j a t & 10 j = 1800 , which is equivalent to 2000 v t & 10 12 : 063 & 1 v t & 10 : 06 = 1800 & 2000 + 12 : 063 : 06 v t & 10 = 1800 + 12 : 063 : 06 . It follows that v t & 10 = 1800 + 12 : 063 : 06 2000 + 12 : 063 : 06 , or t & = 10 1 : 06 ln & 1800 + 12 : 063 : 06 2000 + 12 : 063 : 06 = 11 : 5877 . Substituting back t & in (1), one &nds Pr ( 10 L & 1800 j T > 10) = Pr ( T > 11 : 5877 j T > 10) = 1 : 5877 p 45 = exp : 0005246 1 : 1 45 1 : 1 1 : 5877 1 = 0 : 99377 , 2 2. (a) The bene&t premium 4000 P & 20 j A 45 is calculated as follows: 4000 P & 20 j A 45 = 4000 20 j A 45 a 45: 20 j = 4000 (0 : 11630) 11 : 01577 = 42 : 23 (b) The prospective loss at time 10 is de&ned as 10 L = ( & 4000 P & 20 j A 45 a T & 10 j , 10 < T < 20 4000 v T & 10 & 4000 P & 20 j A 45 a 10 j , T > 20 , while the prospective loss at time 25 is de&ned as...
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This note was uploaded on 10/30/2009 for the course ACTSC 331 taught by Professor David during the Fall '09 term at Waterloo.

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Exercises_-_Chapter_7_-_Solutions - ACTSCI 331 - Life...

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