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Unformatted text preview: ﬂit; 33‘) whme Hm 1 You are given the following extract from a select life table with 4year 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 fer1. .«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/lnviQlitv 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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 Spring '09
 david

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