w2004Final - flit; 33‘) whme Hm 1 You are given the...

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Unformatted text preview: flit; 33‘) whme Hm 1 You are given the following extract from a select life table with 4-year select period. A 4.. select individual aged 41 purchased a fully discrete 3—year term insurance with a sum ‘ insured Of $200,000, With premium P 1n per unit sum insured payable annually. ['41 :3 mJ. my bl} l3] 3in l[ac]+1 lIa=l+2 l[n+3 l:c+4 [40] 100 000 99 899 99 724 99 529 99 288 44 [41] 99 802 99 689 99 502 99 283 99 033 '45 [42] 99 597 99 471 99 628 99 030 98 752 46 Use an effective rate of interest of 6% per year. ( a) Show that the benefit premium is P = 323.58 [2 marks] (b) Calculate the mean and standard deviation of the present value of loss after one year, 1L, conditional on survival 1 year. I [4' marks] (c) Calculate the sum insured for a three year endowment insurance for a select life age 41, with the same premium as the term insurance, P = 323.58. [2 marks] (d) Calculate the mean and standard deviation of the present value of loss after one year, 1L, conditional on survival 1 year, for the endowment insurance. [2 marks] (e) Comment on the difierences between the term insurance and the endowment insur- ance. [2 marks] [Total 12 marks] '2. fully discrete whole hf“ assets“ 0* ""3" r d1 50. The sum assured payable on death in the first 2 years is equal to $1000 plus the end year reserve in the year of death (that is, the reserve that would have been required if the life had survived). The benefit payable at the end of the year of death in any subsequent year is $20,000. The annual premium P is calculated using the equivalence principle. The insurer calculates “remiums and reserves using the Illustrative Life Table mortality at 6% per year interest. er See below for an excerpt from this table. (a) i) Write down the equations for the recursive relationship between successive re— serves for the reserves in the first two years of the contract, and simplify as far as possible. [3 marks] (ii) Write down the prospective reserve at time 2, 2V, say, in terms of the premium - P and standard actuarial tunetiensrw » r [2 marks]n Mm (iii) Using ( i) and ( ii) above, or otherwise, show that the annual premium is $360 (to the nearest $10) and calculate the reserve at the end of the second year, 2V. [5 marks] (b) Estimate the reserve that the insurer should hold for the contract after 2% years using three different approximations. State any assumptions. [6 marks] [Total 16 marks] uxcerpt from the Illustrative Life Table, with annuity and insurance functions at 6% 2 £2: a}. 40 Am 50 89 509.00 529.8844 0.0059199 13.26683 0.24905 51 88979.11 571.4316 0.0064221 13.08027 0.25961 2 88407.68 616.4165 0.0069724 12.88785 0.27050 3 87701.26 665.0646 0.0075755 12.68960 0.28172 =4 87126.20 717.6041 0.0082364 12.48556 0.29327 55 85634.33 835.2636 0.0089605 12.27581 0.30514 DD 3. You are given that 10pm = 0.80 and that 1pr = 0.75. ( a) Assuming that TI and Ty are independent, calculate: (i) The probability that (Ky) will survive 10 years. (ii) The probability that exactly one life will survive 10 years. (iii) The probability that at least one life will die within 10 years. (ix?) [6 marks] (b) Now assume that T2 and Ty follow the common shock model, with common shock parameter A = 0.002. The individual survival probabilities are still assumed to be mpz = 0.80 and 101% = 0.75. Recalculate the probabilities in (a)(i), (ii), (iii) and (iv). [9 marks] [Total 15 marks] [FR [1?) i” ‘" n n: .— n - f —. .—--—. w. v. ' :2 fer-1-. .«I Let atrial “min kit-’38 LO Esme endOWmeiit and centingent assurance 0) m "yawn .: n n .— . -..:.-..,.‘. ll) (I) the net single premium for an insurance of 1 payable exactly 11 years after the death of (as) provided (y) is still alive at the payment date. [Total 5 marks] « . A husband and Wife, 65 and 60 respectively, purchas e an insurance policy, under which the benefits payable on first death are a lump sum of $10000, payable immediately on death, plus an annuity of $5000 per year payable continuously throughout the lifetime of the surviving spouse. A benefit of $1000 is paid immediately on the second death. Premiums are payable continuously until the first death. You are given that A60 = 0.353789, A6; = 0.473229 and that flew = 0.512589 at 4% per year effective rate of interest. The lives are assumed to be independent. (a) Calculate the expected present value of the lump sum death benefits, at 4% per year interest. 7 ~ [3 marks] (b) Calculate the expected present value of the reversionary annuity benefit, at 4% per year interest. [5 marks] (c) Calculate the annual rate of benefit premium, at 4% per year interest. [3 marks] (d) Ten years after the contract is issued the insurer is calculating the reserve. (i) Write down an expression for the reserve at that time assuming that both lives are still surviving. (ii) Write down an expression for the reserve assuming that the husband has died but the wife is still alive. (i) Write down the Thiele difierential equation for the reserve assuming (a)- both lives are still alive and only the wife is [6 marks] [Total 17 marks] an-” M My,“ ex, NV,” as man, n,,,, 9} -1. l‘i/lnvi-Qlitv r5110 tn 9 n; , . Lvoum.-dl) “My JV 5.1 ‘1 tain d‘i rm“- ; mo» ‘elled hv tle {chewing four state model in which the transition intensities are all assumed to be constant, with values: 7/ = 0.004; n = 0.05, and pm) = 0.006; 0(1) = 0.01, and M2) = 0. (a) Derive the Kolmogorov forward equations for $392k for k = 1, 2, 3 for this model. State any assumptions required. l8 (b) Show that W21 = 0.04 (6—0.01t _ e—onst) [4 marks] (c) Using a force of interest 6 = .04, calculate: /*\ w . (i) the expected present value of an annuity of 1 per year payable continuously is: \..J \ 4.1. 4... -..: 4. I ' A " present vasue of a “ayment of 10,000 payable {and while a life is in state 0, given that the life is in state 0 at time 0. [3 marks] but: alien's.“ mmedza 31v 911 death, if the life dies immediately from the able state, given that the life is in state 0 at time 0. [3 marks] the actuarial present value of an annuity of 10 000 per year payable continuously [-3 marks] 1 while a life is in state 1 given that the life is in state 0 at time 0. 'l i [Total 2}. mar ’63 60 61 62 lgr) d9) (if) a?) 10 000 350 150 25 9 475 360 125 45 8 945 380 110 70 Calculate 3%? [1 mark] C*~lculate 2225) 1 mark] Calculate the expected present value of a benefit of $10 000 payable at the end of the year of exit, if a life age 60 leaves by decrement 3 before age 63. Use a rate of interest of 5% per year. [3 marks] (d) Calculate the expected present Value of an annuity of $1 000 per year‘payable at the ' start of each of the next 3 years if a life currently age 60 remains in service. Use a rate of interest of 5% per year. [3 marks] (e) By calculating the value to 5 decimal places, show that qul) = 0.0429,assuming each decrement is unifome distributed over each year of age in the double decrement table. [2 marks] (f) Calculate the revised service table for age 62 if qul) is increased to 0.1, with the [4 marks] [Total 14 marks] other independent rates remaining unchanged. ...
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w2004Final - flit; 33‘) whme Hm 1 You are given the...

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