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Exercises_-_Chapter_7_-_Solutions

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

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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 0 ° 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 = ² 2000 v T ° 25 , 25 < T < 30 0 , 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 ° 0 : 06(2 : 3) ° 12 : 063 ° 1 ° e ° 0 : 06(2 : 3) 0 : 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 ° 0 : 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
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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 ° 0 : 06(10) ° (12 : 063) ° 1 ° e ° 0 : 06(10) 0 : 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 0 : 06 ± = 1800 ° 2000 + 12 : 063 0 : 06 ± v t ° ° 10 = 1800 + 12 : 063 0 : 06 . It follows that v t ° ° 10 = 1800 + 12 : 063 0 : 06 2000 + 12 : 063 0 : 06 , or t ± = 10 ° 1 0 : 06 ln ° 1800 + 12 : 063 0 : 06 2000 + 12 : 063 0 : 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 ³ ° 0 : 0005246 ² 1 : 1 45 ´ 1 : 1 1 : 5877 ° 1 µ¶ = 0 : 99377 , 2
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2. (a) The bene°t premium 4000 P ´ 20 j A 45 µ is calculated as follows: 4000 P ´ 20 j A 45 µ = 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 = ( 0 ° 4000 P ´ 20 j A 45 µ a T ° 10 j , 10 < T < 20 4000 v T ° 10 ° 4000 P ´ 20 j A 45 µ a 10 j , T > 20 , while the prospective loss at time 25 is de°ned as 25 L = ³ 4000 v T ° 25 , T > 25 . (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) = 0 ° 4000 P ´ 20 j A 45 µ a 12 : 3 ° 10 j = ° 4000 P ´ 20 j A 45 µ a 2 : 3 j = ° 42 : 23 ° 1 ° e ° 0 : 06(2 : 3) 0 : 06 ± = ° 90 : 725 Identical technique for 25 L at T = 27 : 4 ( 25 L j T = 27 : 4) = 4000 v 27 : 4 ° 25 = 4000 v 2 : 4 = 4000 e ° 0 : 06(2 : 4) = 3463 : 6 (d) The bene°t reserve 10 V is given by 10 V = 4000 ´ 10 j A 55 µ ° 4000 P ´ 20 j A 45 µ a 55: 10 j = 4000 (0 : 22524) ° 42 : 23 (7 : 09581) = 601 : 3 while 25 V = 4000 A 70 = 4000 (0 : 56246) = 2249 : 8 (e) From the de°nition of 10 L , one notices that ( 10 L j T = 10) = 0 . Also, 10 L is a strictly decreasing function in (10 ; 20) . At T = 20 , 10 L experiences a jump ´ 10 L j T = 20 ° µ = 0 ° 4000 P ´ 20 j A 45 µ a 10 j ´ 10 L j T = 20 + µ = 4000 v 10 ° 4000 P ´ 20 j A 45 µ a 10
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