PQ-3-Answer-331-2011-F

PQ-3-Answer-331-2011-F - Answers to Practice Questions 3...

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Unformatted text preview: Answers to Practice Questions 3 – ACTSC 331, Fall 2011 1. (a) b = 5225.32. (b) The benefit reserve is 15 V = 1287.93. (c) V ar(15 L|T50 > 15) = 3.512 × 106 . ¯ 2. (a) π a40:20 = 1500 A1 30 + 1000 30 E40 . ¯ 40: (b) Given T40 > 25, = 1500 v T40 −25 , 5 25 L = 1000 v , if T40 ≤ 30, if T40 > 30. 25 L ¯ = 1500 A1 5 + 1000 5 E 65 . 65: 1 ¯ (d) The formula is 25 V = 25 E 40 (π a40:20 − 1500 A1 25 ). ¯ 40: (c) The formula is 25 V ¯ ¯ ¯ (e) Hint: Use (a), A1 30| = A1 25 + A1 5 40: 40: 65: (f) i. π = 40.91. ii. The benefit reserve is iii. The benefit reserve is 25 E 40 and 30 E 40 = 25 E 40 × 5 E 65 . 25 V = 887.00. 10 V = 318.60. d d 3. (a) For 0 < t < 10, dt t V = (δ + µ) t V + π − µb, and for 10 < t < 20, dt t V = δ t V + π. The boundary conditions: 0 V = 0, 20 V = 60000, and limt→10− t V = limt→10+ t V . (b) π = 1581.68. (c) The explicit solution is 19.90 V = 59483.40. (d) The approximate solution, by Euler’s method, is 19.90 V = 59484.20. 4. (a) t V = q60+t v + p60+t t+1 V v for t = 1, ..., 19. (b) t V = h q60+t v h + h p60+t t+h V v h for t = h, 2h, ..., 20 − h. (c) In this case, we have that x = 60 and for 0 < t < 20, δt = δ , πt = 0, αt = 0, d bt = 1, βt = 0. Thus, Thiele’s differential equation for t V is dt t V = (µ60+t + δ ) t V − µ60+t , 0 < t < 20. ¯ ¯ (d) i. The retrospective formula is t V = (A1 − A1 ) 1 . 60: t 60: 20 t E60 ¯ ii. The prospective formula is t V = A1 t: 20−t . 60+ ¯1 ¯1 + t E60 A1 ¯ iii. Since A60: 20 = A60: t , the two formulas in (i) and (ii) are equal. 60+t: 20−t (e) We have ¯ A1 t: 20−t = 60+ 20−t 0 e−δs µ60+t+s s p60+t ds = Thus, by the differential rule da dt t 20 t e−δ(x−t) µ60+x e− f (t, x)dx = −f (t, t) + a t µ60+y dy d f (t, x) dt d ¯1 ¯ A = −µ60+t + (µ60+t + δ ) A1 t: 20−t , 60+ dt 60+t: 20−t ¯ which means t V = A1 t: 20−t is the solution to the DE in (c). 60+ 1 x t dx. dx, we have ...
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This note was uploaded on 01/04/2012 for the course ACTSC 331 taught by Professor David during the Fall '09 term at Waterloo.

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