act348-1010-solution - The University of Toronto ACT348...

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Unformatted text preview: The University of Toronto ACT348 Intermediate Life Contingencies Midterm Test Duration: One and A Half Hours You must show your steps or no point will be awarded N mew ID # 1. A whole life insurance of 1 is issued to (30) on a fully continuous basis. The force of mortality is constant at [13005) = 2%. The force of interest is 6 == 6%. a) 5 points Determine the variance of the loss-at-issue. \/M(L) : (H 7T 1A); Alf) ~ 3- M. P:—:X—: AW” :j/tx:otoz _ ‘ Z Z/D‘X : fl 1’ ri—L/ : 1;] 44326 0_01+0rll b) 5 points Determine the benefit reserve 5V(Zgo). , \ ' ' S‘an t/w imW‘tOtl/it} ‘yé @X/pon-MJUML, Kflt : AX, ‘7’ “’ ~ V. a: o'— o a “ 5V4 AM) : A775 ” F3” a” A; P}. 30 2 2. For a special fully discrete Whole life insurance of 1000 issued on the life of (75), increasing premiums, 7r,“ are payable at time k, for k = 0,1, - - - . You are given: I (i) The annual effective rate of interest 2' = 0.05; (ii) Mortality follows DeMoivre’s Law with w = 105; (iii) 77k = (1 + i)k7ro; and (iv) Premiums are calculated in accordance with the percentile premium principle With 0 = 0.25, i.e. Pr(L > 0) S c, Where is the loss-at—issue random variable. a) 5 points Calculate 11'0. ’ - ~ ' e, ‘ Qfimfl SWIM” l/\ 75 CWJCMQ fulfil/V WC ' 3 » b) 5 points Now assume that me = (1.03)k7r0 and premiums are calculated in accordance with the equivalent premium principle. Recalculate 7T0. 1179 (A) (Um) L: [0072 (my) Shim} :(000 A7: —- 0 £75 :0 Wile/“9 A73“ I if“ @3972; @ L :00) a7: : i’A7y . —/ I - A ~ @ b Lao/fit I ' / l'ifii’fv ‘ A7: 55 437271 — 7;; i :o 512% i 073, : QILIEEJ—L— .: /Z 132. $1 1:; : {000/47i’ {-900 xoi’I?/}Z _ ‘ a ’ 2 274544 ‘ 4 3. 5 points For a. 10-year endowment insurance of 1 issued to (35) on a fully discrete basis, you are given: (1) the benefit premium, Pasz-l—dl, is determined using the equivalent premium principle; (ii) the force of mortality Mm) = 0.05, for a; _>_ 35; and (iii) the force of interest 6 = 0.06. ' ' - 5 Determine the benefit reserve 3V35m . the MnW/l @fifiofiv/z filmy/adj Yam-Q 1: i5 ‘ \\ A55:Toj “ 3V552ra 7- IAV'EQLfl F F351W :A'38’Lfl' HUI-ta ‘\ ~mobh £3515]: P3), 7 0.0612 4%ka : i 6 Q/ [03,0 -37 you) 9 ~ 11/2 v 0 _ 2 Q 0 ; l e ‘a : AlfiOlZZ th [— e—Vll q r A2751!“ I l’AZ/Ué'iffl : l' M AQZM/Z’ l 05.13; . 1: (j 62/72é 6 4-042 (138:?! ’ w @ Mag 5 -ooe/z_o;o;k :3: (i 6 [1:0 2%: Q'O‘Hk T: 9 hm 5 4. 5 points A fully continuous whole life insurance of l issued to (:13) has a net single premium E; = 0.4 it i = 0.06, and Var(L) = 0.25 where L is the loss-at—issue associated With the annual benefit premium P(A$). The issuer charges a gross annual premium of 0.05, payable continuously, and this results in a new loss—at—issue L’. Calculate Var(L’ 5. Solving the following questions. a) 5 points You are given: Pac = 0.01212, 15Pm = 0.02508, P L = 0.06942, 10% = 0.1143. Calculate 3:101 g. - \\ I - ' _ I” \/ X ’ (0V0 ' [ (FPX F 5X: W z/MLE‘W eaofz-fl] ~ I ’f I /7><—~fv7 :(0.0Lf%/0‘0/12/2 '_ \/____q_ ' _ 0.01274” i£\/)( :(VVX “P Ddgvé7 : 04(4ng 0.(%7 : (230/ b) 5 points You are given: tV(:4_.m) = 0.1, fiCZl-z) = 0.01005, 6 = 0.03. Calculate 53.”. .: 1., {no/oerflms) Aw —’ Ival am : ————/ :22\LL7 0 04003“ 7 6. 5 points You are given: 15035] = 044860325053] = 0.21040 and 6 = 0.06. A number of 15-year 1,000 endowment insurance policies are issued with issue age 50. Each policy has annual premiums of 55.00 and is fully continuous. Find the smallest number of independent policies for which the probability, using the normal approximation of a positive aggregate loss-at—issue is less than 0.05. The 95th percentile of the standard normal distribution is 1.645. Fm each Foliage, V2”? boss-atr'f/Cme Aw; meow fit: [fiflo/Zlgozifi '/ F 265771-57 :[evv A50”? ,5 ”' r 55 :(‘7””‘°55e) AM- " 2 ma fixawfl Vj/M] = ~5Mn/ wwle oft—(fwvt [ A, A; - L—nM ~fi/vt e_ j, no" > (T ) J00 . HM ‘ __ , - 0‘ K ‘ '/_ (UL—,[MU ~> ‘ln:/tbu $25 307/) n- ...
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This note was uploaded on 06/12/2011 for the course ACTUARIAL 348 taught by Professor Lin during the Fall '10 term at University of Toronto.

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act348-1010-solution - The University of Toronto ACT348...

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