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精算代写-MATH352001
时间：2021-04-22
Module Code: MATH352001
Module Title: Actuarial Mathematics 2 c©UNIVERSITY OF LEEDS
School of Mathematics Semester Two 201920
Calculator instructions:
• You are allowed to use a calculator.
Exam information:
• There are 10 pages to this exam.
• Answer all questions.
• The numbers in brackets indicate the marks available for each question.
• Actuarial tables are included from page 7.
• You must submit your solutions through Minerva (Submit My Work)
Page 1 of 10 Turn the page over
Module Code: MATH352001
1. (a) Suppose we have two independent lives aged 40 and 50 respectively. Using the
actuarial tables provided, calculate the following assuming that A40:50 = 0.13,
A60:70 = 0.35 and i =5% p.a. Clearly state any approximations you make.
(i) 20q40:50
(ii) A
(12)
40:50:20
[6 marks]
(b) Consider two independent lives initially aged x and y respectively. Explain the
meaning of nq
1
xy and nq
2
xy, and prove that
nq
1
xy + nq
2
xy = nqx.
[4 marks]
(c) Suppose that µx denotes the force of mortality for a life aged x, and δ denotes the
force of interest.
(i) Show that:
a¯x =
∫ ∞
0
e−δttpx dt ≈ a¨x − 1
2
− 1
12
(δ + µx).
Hint: Use the Woolhouse formula (stated below) with g(t) = e−δttpx and
h = 1. ∫ ∞
0
g(t) dt = h
∞∑
k=0
g(kh)− h
2
g(0) +
h2
12
g′(0)− · · ·
(ii) Use the Woolhouse formula to show that:
a¨(2)x − a¨(4)x ≈
1
8
+
1
64
(δ + µx).
[5 marks]
(d) Consider a whole life assurance issued to a select life aged [30]. Premiums are
paid annually in advance and a death benefit B = £100, 000 is paid at the end of
year of death. You may assume an interest rate of 5% p.a. applies throughout the
period. The select period has a duration of two years.
Policy expenses are incurred annually as detailed below:
- Initial expenses are E1 =£500.
- Renewal expenses are i1 = 2% of the annual gross premium. Renewal expenses
are only due at the start of the second and subsequent policy years.
(i) Write down the future gross loss random variable LGt at t = 0. [3 marks]
(ii) Determine the minimum future lifetime for the policyholder in order that the
insurance company makes a profit on the policy. [7 marks]
[ Total 25 marks]
Page 2 of 10 Turn the page over
Module Code: MATH352001
2. Consider a 45-year term insurance issued to a select life aged [35]. Premiums are paid
annually in advance and benefits due are paid at the end of year of death. You may
assume an interest rate of 5% p.a. applies throughout the period. The select period has
a duration of two years.
The death benefits for the policy are as set out below:
- If death occurs within the first 10 years of the policy (i.e. at time t < 10), then a
benefit of B1 = £10, 000 is paid.
- If death occurs at time t ≥ 10 and within the 45-year term, then a benefit of
B2 = £300, 000 is paid.
Policy expenses are incurred annually as detailed below:
- Initial expenses are E1 =£200 and i1 = 5% of the annual gross premium at the
start of the first policy year.
- Renewal expenses are E2 =£50 and i2 = 1% of the annual gross premium. Renewal
expenses are only due at the start of the second and subsequent policy years.
- A terminal expense of 2% of the benefits is due at the time of payment of those
benefits.
(a) (i) Write down the future gross loss random variable LGt for t = 0.
(ii) Write down recursive relationships between the gross policy values tV
G for
each year of the policy.
Hint: You may wish to group similar years into a single relationship. [9 marks]
(b) Use the equivalence principle to calculate the gross premium G and compute the
gross policy value at time t = 9. [12 marks]
(c) Use the gross policy value calculated in part (b) to calculate the death strain at
risk at time t = 9. [2 marks]
Now suppose that the insurer issued 2,000 identical policies to independent select lives
aged [35].
(d) Calculate the mortality profit to the insurer on this portfolio arising during the ninth
year, knowing that there was 1 death during the ninth year of the policy and there
had been only 1 death up to time t = 8. [2 marks]
[ Total 25 marks]
Page 3 of 10 Turn the page over
Module Code: MATH352001
3. (a) You are given the following population data.
Age Leeds Population Leeds Deaths UK Population UK Population Deaths
34 35, 000 150 1, 800, 000 5, 500
35 42, 000 182 1, 600, 000 3, 100
36 40, 000 120 980, 000 2, 300
(i) Calculate the crude mortality rate for Leeds.
(ii) Calculate the directly standardised mortality rate for Leeds.
(iii) Calculate the area compatibility factor for Leeds.
[7 marks]
(b) Assume the following 4 state Markov model for a life initially aged x. The possible
transitions and the corresponding intensities are shown in the graph below:
State S: Sick - State D: Dead
- State C: Critical Sick
?
6
State H: Healthy
?
Q
Q
Q
Q
Q
Q
QQs
3
µHCx+t
µSDx+t
µHSx+tµ
SH
x+t µ
CD
x+t
µHDx+t
µSCx+t
(i) By considering t+hp
HH
x , for small h > 0, derive the following expression,
tp
HH
x = e
− ∫ t0 (µHCx+s+µHSx+s+µHDx+s)ds,
where µijx+s denotes the transition rate from state i to state j at time s.
[5 marks]
(ii) Assuming that all transition intensities are constant and that
µHSx+t = 4µ
HC
x+t
µHDx+t = 3
(
µHCx+t
)2
3p
HH
x = e
−0.012,
calculate µHSx+t, µ
HC
x+t and µ
HD
x+t. [5 marks]
(iii) Consider the above 4 state Markov model for a whole life insurance policy
issued to a life aged x with the following features:
- A benefit at a rate b1 per annum is payable continuously while the life is
in state S.
- A benefit at a rate b2 per annum is payable continuously while the life is
in state C.
Page 4 of 10 Turn the page over
Module Code: MATH352001
- There is a death benefit B1 on transition from H to D, a death benefit
B2 on transition from S to D and a death benefit B3 on transition from
C to D.
- Premiums are paid continuously at a rate P¯ per annum while the life is in
state H.
Derive the Thiele’s differential equations for the policy values tV
(H) and tV
(C),
by considering what happens over a short period of time from t to t+h. Assume
the force of interest is δ > 0.
Hint: You may wish to draw a graph to reflect the cashflows during [t, t+ h].
[8 marks]
[ Total 25 marks]
Page 5 of 10 Turn the page over
Module Code: MATH352001
4. (a) An insurance company issues a special three-year term insurance policy available
to lives aged 65. Premiums of £1,000 are paid annually in advance. The death
benefit of £130,000 is paid at the end of year of death. In addition to the death
benefit, the insurer provides a surrender benefit which is payable at the end of
year of surrender, and the amount is equal to 60% of the total premiums (ignore
interest) paid prior to surrender. Policy expenses are as follows:
- Initial expenses of £150 and 3% of the premium are due at the same time as
the first premium.
- Renewal expenses of 1.5% of the premium are due at the same time as subse-
quent premiums.
- Terminal expenses due on payment of death or surrender are £30.
The rate of interest on the cashflow account is 2% p.a.
The insurer uses the following multiple decrement table as the basis for calcu-
lations.
Age(x) (aq)dx (aq)
s
x (ap)x
65 0.0060 0.0300 0.964
66 0.0065 0.0200 0.9735
67 0.0070 0.0200 0.973
Calculate the profit margin for the policy described above using a risk discount rate
of 5% p.a. [10 marks]
(b) The insurer wants to introduce a survival benefit of £200 payable at the end of
the three-year term of the contract. No terminal expense occurs in that case.
Calculate the profit margin for the new policy assuming that the risk discount rate
and premiums are left unchanged. [5 marks]
(c) It has been some time since the above policy was designed and the insurer believes
that the multiple decrement table above is no longer appropriate.
Assuming that the underlying rates of decrement are constant in each year of
age, calculate an updated table based on a decrease in the rates µdx of 10% in each
year of age and an increase in the rates µsx of 5% in each year of age. You may
assume independence of decrements. [8 marks]
(d) Calculate the probability that the policyholder will get the survival benefit of £200
under the updated multiple decrement table.
[2 marks]
[ Total 25 marks]
End of Questions.
Page 6 of 10 Turn the page over
Module Code: MATH352001
Actuarial Tables for University of Leeds Examinations 2019/20
i = 5% p.a.
x l[x] l[x]+1 lx+2 A[x] A[x]+1 Ax+2 (IA)x+2 x+ 2
28 99, 781.36 99, 756.17 99, 727.29 0.07032 0.07357 0.07698 3.652248809 30
29 99, 751.69 99, 725.70 99, 695.83 0.07356 0.07697 0.08054 3.755213856 31
30 99, 721.06 99, 694.18 99, 663.20 0.07696 0.08053 0.08427 3.859668736 32
31 99, 689.36 99, 661.48 99, 629.26 0.08051 0.08425 0.08817 3.965520005 33
32 99, 656.47 99, 627.47 99, 593.83 0.08424 0.08816 0.09226 4.072663183 34
33 99, 622.23 99, 591.96 99, 556.75 0.08814 0.09224 0.09653 4.180982148 35
34 99, 586.47 99, 554.78 99, 517.80 0.09223 0.09652 0.10101 4.290348521 36
35 99, 549.01 99, 515.73 99, 476.75 0.09650 0.10099 0.10569 4.400621053 37
36 99, 509.64 99, 474.56 99, 433.34 0.10097 0.10567 0.11059 4.511645019 38
37 99, 468.12 99, 431.02 99, 387.29 0.10565 0.11057 0.11571 4.623251623 39
38 99, 424.18 99, 384.82 99, 338.26 0.11055 0.11569 0.12106 4.735257429 40
39 99, 377.52 99, 335.62 99, 285.88 0.11567 0.12104 0.12665 4.847463812 41
40 99, 327.82 99, 283.06 99, 229.76 0.12101 0.12663 0.13249 4.959656446 42
41 99, 274.69 99, 226.72 99, 169.41 0.12660 0.13247 0.13859 5.071604847 43
42 99, 217.72 99, 166.14 99, 104.33 0.13244 0.13857 0.14496 5.183061963 44
43 99, 156.42 99, 100.80 99, 033.94 0.13854 0.14493 0.15161 5.293763836 45
44 99, 090.27 99, 030.10 98, 957.57 0.14491 0.15158 0.15854 5.403429353 46
45 99, 018.67 98, 953.40 98, 874.50 0.15155 0.15851 0.16577 5.511760086 47
46 98, 940.96 98, 869.96 98, 783.91 0.15847 0.16573 0.17330 5.618440252 48
47 98, 856.38 98, 778.94 98, 684.88 0.16569 0.17326 0.18114 5.723136801 49
48 98, 764.09 98, 679.44 98, 576.37 0.17322 0.18110 0.18931 5.825499653 50
49 98, 663.15 98, 570.40 98, 457.24 0.18106 0.18926 0.19780 5.925162103 51
50 98, 552.51 98, 450.67 98, 326.19 0.18921 0.19775 0.20664 6.02174142 52
51 98, 430.98 98, 318.95 98, 181.77 0.19770 0.20658 0.21582 6.114839644 53
52 98, 297.24 98, 173.79 98, 022.38 0.20653 0.21576 0.22535 6.20404463 54
53 98, 149.81 98, 013.56 97, 846.20 0.21570 0.22529 0.23524 6.28893133 55
54 97, 987.03 97, 836.44 97, 651.21 0.22522 0.23517 0.24550 6.369063361 56
55 97, 807.07 97, 640.40 97, 435.17 0.23510 0.24542 0.25613 6.443994859 57
56 97, 607.84 97, 423.18 97, 195.56 0.24534 0.25605 0.26714 6.513272655 58
57 97, 387.05 97, 182.25 96, 929.59 0.25596 0.26704 0.27852 6.576438778 59
58 97, 142.13 96, 914.80 96, 634.14 0.26695 0.27842 0.29028 6.633033303 60
59 96, 870.22 96, 617.70 96, 305.75 0.27831 0.29017 0.30243 6.682597561 61
60 96, 568.13 96, 287.48 95, 940.60 0.29005 0.30230 0.31495 6.72467771 62
61 96, 232.34 95, 920.27 95, 534.43 0.30217 0.31481 0.32785 6.758828664 63
62 95, 858.91 95, 511.80 95, 082.53 0.31467 0.32770 0.34113 6.784618387 64
63 95, 443.51 95, 057.36 94, 579.73 0.32755 0.34097 0.35477 6.801632525 65
64 94, 981.34 94, 551.72 94, 020.33 0.34080 0.35459 0.36878 6.809479354 66
65 94, 467.11 93, 989.16 93, 398.05 0.35441 0.36858 0.38313 6.807795016 67
66 93, 895.00 93, 363.38 92, 706.06 0.36838 0.38292 0.39783 6.796248988 68
67 93, 258.63 92, 667.50 91, 936.88 0.38270 0.39760 0.41285 6.774549735 69
Page 7 of 10 Turn the page over
Module Code: MATH352001
x l[x] l[x]+1 lx+2 A[x] A[x]+1 Ax+2 (IA)x+2 x+ 2
68 92, 551.02 91, 894.03 91, 082.43 0.39736 0.41260 0.42818 6.742450463 70
69 91, 764.58 91, 034.84 90, 133.96 0.41234 0.42790 0.44379 6.699754898 71
70 90, 891.07 90, 081.15 89, 082.09 0.42762 0.44349 0.45968 6.646322984 72
71 89, 921.62 89, 023.56 87, 916.84 0.44319 0.45935 0.47580 6.582076391 73
72 88, 846.72 87, 852.03 86, 627.64 0.45902 0.47545 0.49215 6.507003704 74
73 87, 656.25 86, 555.99 85, 203.46 0.47509 0.49177 0.50868 6.421165157 75
74 86, 339.55 85, 124.37 83, 632.89 0.49138 0.50826 0.52536 6.324696777 76
75 84, 885.49 83, 545.75 81, 904.34 0.50785 0.52492 0.54217 6.21781377 77
76 83, 282.61 81, 808.54 80, 006.23 0.52447 0.54169 0.55906 6.100813016 78
77 81, 519.30 79, 901.17 77, 927.35 0.54121 0.55854 0.57599 5.974074522 79
78 79, 584.04 77, 812.44 75, 657.16 0.55802 0.57544 0.59293 5.838061685 80
79 77, 465.70 75, 531.88 73, 186.31 0.57488 0.59234 0.60984 5.693320261 81
80 75, 153.97 73, 050.22 70, 507.19 0.59175 0.60920 0.62666 5.540475924 82
81 72, 639.81 70, 359.94 67, 614.60 0.60857 0.62598 0.64336 5.380230363 83
82 69, 916.06 67, 455.99 64, 506.50 0.62532 0.64264 0.65990 5.213355855 84
83 66, 978.12 64, 336.53 61, 184.88 0.64193 0.65913 0.67622 5.040688357 85
84 63, 824.72 61, 003.79 57, 656.68 0.65838 0.67540 0.69229 4.86311913 86
85 60, 458.83 57, 464.98 53, 934.73 0.67462 0.69142 0.70806 4.681585034 87
86 56, 888.53 53, 733.29 50, 038.65 0.69060 0.70714 0.72349 4.49705762 88
87 53, 128.02 49, 828.70 45, 995.64 0.70629 0.72252 0.73853 4.310531237 89
88 49, 198.45 45, 778.84 41, 841.05 0.72163 0.73752 0.75317 4.123010406 90
89 45, 128.71 41, 619.51 37, 618.56 0.73660 0.75211 0.76735 3.935496741 91
90 40, 955.90 37, 394.80 33, 379.88 0.75115 0.76624 0.78104 3.748975755 92
91 36, 725.43 33, 156.84 29, 183.78 0.76526 0.77989 0.79423 3.564403889 93
92 32, 490.61 28, 964.73 25, 094.33 0.77889 0.79303 0.80688 3.382696112 94
93 28, 311.50 24, 882.75 21, 178.30 0.79201 0.80564 0.81897 3.204714446 95
94 24, 252.97 20, 977.73 17, 501.76 0.80460 0.81769 0.83049 3.031257728 96
95 20, 381.98 17, 315.59 14, 125.89 0.81665 0.82918 0.84143 2.863052897 97
96 16, 763.92 13, 957.13 11, 102.53 0.82814 0.84009 0.85177 2.70074804 98
97 13, 458.37 10, 953.56 8, 469.73 0.83905 0.85041 0.86153 2.54490735 99
98 10, 514.62 8, 342.12 6, 248.17 0.84938 0.86014 0.87068 2.396008116 100
99 7, 967.35 6, 142.47 4, 438.80 0.85913 0.86928 0.87925 2.254439739 101
100 5, 833.20 4, 354.49 3, 022.58 0.86830 0.87784 0.88724 2.120504722 102
101 4, 108.86 2, 958.13 1, 962.49 0.87690 0.88582 0.89465 1.994421504 103
102 2, 771.25 1, 915.52 1, 207.79 0.88492 0.89323 0.90150 1.87632892 104
103 1, 780.05 1, 175.35 699.91 0.89239 0.90010 0.90781 1.766292023 105
104 1, 082.34 678.82 379.08 0.89931 0.90643 0.91360 1.664308939 106
105 618.75 366.26 190.28 0.90571 0.91224 0.91888 1.570318406 107
106 330.04 183.07 87.69 0.91159 0.91756 0.92367 1.484207571 108
107 162.85 83.97 36.71 0.91698 0.92240 0.92800 1.405819641 109
108 73.61 34.96 13.80 0.92189 0.92678 0.93188 1.334960925 110
109 30.16 13.06 4.59 0.92634 0.93072 0.93534 1.271406814 111
110 11.06 4.32 1.33 0.93036 0.93426 0.93840 1.214906225 112
111 3.58 1.24 0.33 0.93396 0.93739 0.94107 1.165184082 113
112 1.01 0.31 0.07 0.93716 0.94016 0.94339 1.121941457 114
113 0.24 0.06 0.01 0.93998 0.94257 0.94537 1.08485318 115
114 0.05 0.01 0.00 0.94244 0.94464 0.94703 1.053563039 116
115 0.01 0.00 0.00 0.94455 0.94641 0.94841 1.0276772 117
Page 8 of 10 Turn the page over
Module Code: MATH352001
x a¨[x] a¨[x]+1 a¨x+2 (Ia¨)[x] (Ia¨)[x]+1 (Ia¨)x+2 x+ 2
30 19.38449 19.30892 19.23034 330.37389 326.62691 322.78419 32
31 19.30979 19.23066 19.14838 326.64260 322.78973 318.84013 33
32 19.23156 19.14871 19.06258 322.80588 318.84584 314.78826 34
33 19.14965 19.06292 18.97277 318.86251 314.79415 310.62762 35
34 19.06390 18.97313 18.87880 314.81141 310.63373 306.35745 36
35 18.97415 18.87917 18.78049 310.65164 306.36380 301.97715 37
36 18.88024 18.78088 18.67766 306.38244 301.98377 297.48630 38
37 18.78201 18.67807 18.57014 302.00322 297.49322 292.88473 39
38 18.67927 18.57058 18.45776 297.51358 292.89198 288.17248 40
39 18.57184 18.45822 18.34031 292.91336 288.18010 283.34985 41
40 18.45956 18.34081 18.21763 288.20260 283.35788 278.41741 42
41 18.34224 18.21815 18.08951 283.38162 278.42589 273.37601 43
42 18.21969 18.09007 17.95577 278.45100 273.38499 268.22685 44
43 18.09172 17.95637 17.81621 273.41163 268.23638 262.97143 45
44 17.95814 17.81686 17.67065 268.26469 262.98157 257.61163 46
45 17.81876 17.67135 17.51889 263.01173 257.62244 252.14969 47
46 17.67340 17.51965 17.36074 257.65462 252.16123 246.58827 48
47 17.52187 17.36156 17.19602 252.19564 246.60062 240.93045 49
48 17.36397 17.19691 17.02453 246.63745 240.94367 235.17974 50
49 17.19952 17.02551 16.84612 240.98317 235.19392 229.34013 51
50 17.02835 16.84718 16.66060 235.23632 229.35536 223.41609 52
51 16.85028 16.66175 16.46782 229.40091 223.43245 217.41258 53
52 16.66514 16.46908 16.26762 223.48144 217.43017 211.33509 54
53 16.47277 16.26899 16.05987 217.48289 211.35402 205.18964 55
54 16.27303 16.06137 15.84443 211.41077 205.21001 198.98279 56
55 16.06579 15.84608 15.62122 205.27112 199.00472 192.72165 57
56 15.85091 15.62302 15.39012 199.07053 192.74525 186.41388 58
57 15.62831 15.39210 15.15109 192.81612 186.43928 180.06769 59
58 15.39789 15.15325 14.90407 186.51557 180.09501 173.69186 60
59 15.15960 14.90644 14.64906 180.17711 173.72124 167.29568 61
60 14.91340 14.65165 14.38606 173.80954 167.32725 160.88898 62
61 14.65927 14.38890 14.11512 167.42215 160.92287 154.48209 63
62 14.39724 14.11822 13.83632 161.02479 154.51844 148.08579 64
63 14.12736 13.83972 13.54979 154.62778 148.12474 141.71131 65
64 13.84972 13.55351 13.25568 148.24192 141.75300 135.37026 66
65 13.56444 13.25975 12.95420 141.87843 135.41483 129.07459 67
66 13.27169 12.95864 12.64561 135.54890 129.12216 122.83651 68
67 12.97168 12.65045 12.33019 129.26527 122.88721 116.66844 69
68 12.66467 12.33547 12.00830 123.03972 116.72238 110.58291 70
69 12.35097 12.01406 11.68035 116.88464 110.64020 104.59250 71
70 12.03093 11.68661 11.34678 110.81252 104.65322 98.70970 72
Page 9 of 10 Turn the page over
Module Code: MATH352001
x a¨[x] a¨[x]+1 a¨x+2 (Ia¨)[x] (Ia¨)[x]+1 (Ia¨)x+2 x+ 2
71 11.70495 11.35359 11.00812 104.83588 98.77394 92.94687 73
72 11.37350 11.01550 10.66491 98.96716 93.01470 87.31608 74
73 11.03709 10.67291 10.31778 93.21865 87.38753 81.82901 75
74 10.69629 10.32644 9.96740 87.60234 81.90411 76.49687 76
75 10.35171 9.97676 9.61449 82.12983 76.57561 71.33021 77
76 10.00402 9.62458 9.25981 76.81221 71.41257 66.33889 78
77 9.65395 9.27067 8.90416 71.65994 66.42480 61.53188 79
78 9.30225 8.91584 8.54841 66.68276 61.62127 56.91722 80
79 8.94973 8.56093 8.19341 61.88954 57.00997 52.50189 81
80 8.59722 8.20681 7.84008 57.28817 52.59784 48.29171 82
81 8.24560 7.85440 7.48934 52.88550 48.39069 44.29128 83
82 7.89575 7.50460 7.14211 48.68723 44.39307 40.50392 84
83 7.54859 7.15834 6.79934 44.69784 40.60828 36.93161 85
84 7.20502 6.81655 6.46193 40.92050 37.03827 33.57497 86
85 6.86596 6.48015 6.13079 37.35707 33.68364 30.43327 87
86 6.53231 6.15002 5.80679 34.00808 30.54362 27.50445 88
87 6.20493 5.82705 5.49077 30.87267 27.61612 24.78512 89
88 5.88469 5.51204 5.18352 27.94869 24.89776 22.27067 90
89 5.57237 5.20579 4.88575 25.23268 22.38390 19.95536 91
90 5.26873 4.90900 4.59814 22.71998 20.06878 17.83235 92
91 4.97447 4.62234 4.32126 20.40476 17.94558 15.89391 93
92 4.69020 4.34637 4.05562 18.28021 16.00655 14.13150 94
93 4.41647 4.08161 3.80166 16.33856 14.24315 12.53592 95
94 4.15375 3.82845 3.55971 14.57128 12.64622 11.09749 96
95 3.90242 3.58723 3.33001 12.96923 11.20605 9.80615 97
96 3.66279 3.35819 3.11273 11.52274 9.91263 8.65166 98
97 3.43504 3.14146 2.90794 10.22185 8.75572 7.62373 99
98 3.21930 2.93713 2.71563 9.05637 7.72505 6.71212 100
99 3.01561 2.74516 2.53572 8.01611 6.81042 5.90684 101
100 2.82392 2.56545 2.36804 7.09095 6.00184 5.19818 102
101 2.64410 2.39784 2.21237 6.27099 5.28965 4.57686 103
102 2.47597 2.24210 2.06843 5.54664 4.66457 4.03408 104
103 2.31927 2.09792 1.93589 4.90874 4.11782 3.56158 105
104 2.17370 1.96497 1.81439 4.34862 3.64118 3.15168 106
105 2.03893 1.84289 1.70352 3.85814 3.22697 2.79731 107
106 1.91458 1.73125 1.60288 3.42975 2.86816 2.49204 108
107 1.80025 1.62965 1.51201 3.05652 2.55830 2.23002 109
108 1.69554 1.53764 1.43049 2.73213 2.29158 2.00601 110
109 1.60004 1.45478 1.35785 2.45087 2.06278 1.81535 111
110 1.51334 1.38063 1.29366 2.20762 1.86724 1.65391 112
111 1.43504 1.31475 1.23747 1.99785 1.70086 1.51803 113
112 1.36476 1.25672 1.18882 1.81756 1.56002 1.40453 114
113 1.30212 1.20611 1.14726 1.66322 1.44157 1.31061 115
114 1.24676 1.16250 1.11232 1.53178 1.34275 1.23380 116
115 1.19833 1.12546 1.08348 1.42056 1.26116 1.17190 117
Page 10 of 10 End of Paper.
学霸联盟
Module Title: Actuarial Mathematics 2 c©UNIVERSITY OF LEEDS
School of Mathematics Semester Two 201920
Calculator instructions:
• You are allowed to use a calculator.
Exam information:
• There are 10 pages to this exam.
• Answer all questions.
• The numbers in brackets indicate the marks available for each question.
• Actuarial tables are included from page 7.
• You must submit your solutions through Minerva (Submit My Work)
Page 1 of 10 Turn the page over
Module Code: MATH352001
1. (a) Suppose we have two independent lives aged 40 and 50 respectively. Using the
actuarial tables provided, calculate the following assuming that A40:50 = 0.13,
A60:70 = 0.35 and i =5% p.a. Clearly state any approximations you make.
(i) 20q40:50
(ii) A
(12)
40:50:20
[6 marks]
(b) Consider two independent lives initially aged x and y respectively. Explain the
meaning of nq
1
xy and nq
2
xy, and prove that
nq
1
xy + nq
2
xy = nqx.
[4 marks]
(c) Suppose that µx denotes the force of mortality for a life aged x, and δ denotes the
force of interest.
(i) Show that:
a¯x =
∫ ∞
0
e−δttpx dt ≈ a¨x − 1
2
− 1
12
(δ + µx).
Hint: Use the Woolhouse formula (stated below) with g(t) = e−δttpx and
h = 1. ∫ ∞
0
g(t) dt = h
∞∑
k=0
g(kh)− h
2
g(0) +
h2
12
g′(0)− · · ·
(ii) Use the Woolhouse formula to show that:
a¨(2)x − a¨(4)x ≈
1
8
+
1
64
(δ + µx).
[5 marks]
(d) Consider a whole life assurance issued to a select life aged [30]. Premiums are
paid annually in advance and a death benefit B = £100, 000 is paid at the end of
year of death. You may assume an interest rate of 5% p.a. applies throughout the
period. The select period has a duration of two years.
Policy expenses are incurred annually as detailed below:
- Initial expenses are E1 =£500.
- Renewal expenses are i1 = 2% of the annual gross premium. Renewal expenses
are only due at the start of the second and subsequent policy years.
(i) Write down the future gross loss random variable LGt at t = 0. [3 marks]
(ii) Determine the minimum future lifetime for the policyholder in order that the
insurance company makes a profit on the policy. [7 marks]
[ Total 25 marks]
Page 2 of 10 Turn the page over
Module Code: MATH352001
2. Consider a 45-year term insurance issued to a select life aged [35]. Premiums are paid
annually in advance and benefits due are paid at the end of year of death. You may
assume an interest rate of 5% p.a. applies throughout the period. The select period has
a duration of two years.
The death benefits for the policy are as set out below:
- If death occurs within the first 10 years of the policy (i.e. at time t < 10), then a
benefit of B1 = £10, 000 is paid.
- If death occurs at time t ≥ 10 and within the 45-year term, then a benefit of
B2 = £300, 000 is paid.
Policy expenses are incurred annually as detailed below:
- Initial expenses are E1 =£200 and i1 = 5% of the annual gross premium at the
start of the first policy year.
- Renewal expenses are E2 =£50 and i2 = 1% of the annual gross premium. Renewal
expenses are only due at the start of the second and subsequent policy years.
- A terminal expense of 2% of the benefits is due at the time of payment of those
benefits.
(a) (i) Write down the future gross loss random variable LGt for t = 0.
(ii) Write down recursive relationships between the gross policy values tV
G for
each year of the policy.
Hint: You may wish to group similar years into a single relationship. [9 marks]
(b) Use the equivalence principle to calculate the gross premium G and compute the
gross policy value at time t = 9. [12 marks]
(c) Use the gross policy value calculated in part (b) to calculate the death strain at
risk at time t = 9. [2 marks]
Now suppose that the insurer issued 2,000 identical policies to independent select lives
aged [35].
(d) Calculate the mortality profit to the insurer on this portfolio arising during the ninth
year, knowing that there was 1 death during the ninth year of the policy and there
had been only 1 death up to time t = 8. [2 marks]
[ Total 25 marks]
Page 3 of 10 Turn the page over
Module Code: MATH352001
3. (a) You are given the following population data.
Age Leeds Population Leeds Deaths UK Population UK Population Deaths
34 35, 000 150 1, 800, 000 5, 500
35 42, 000 182 1, 600, 000 3, 100
36 40, 000 120 980, 000 2, 300
(i) Calculate the crude mortality rate for Leeds.
(ii) Calculate the directly standardised mortality rate for Leeds.
(iii) Calculate the area compatibility factor for Leeds.
[7 marks]
(b) Assume the following 4 state Markov model for a life initially aged x. The possible
transitions and the corresponding intensities are shown in the graph below:
State S: Sick - State D: Dead
- State C: Critical Sick
?
6
State H: Healthy
?
Q
Q
Q
Q
Q
Q
QQs
3
µHCx+t
µSDx+t
µHSx+tµ
SH
x+t µ
CD
x+t
µHDx+t
µSCx+t
(i) By considering t+hp
HH
x , for small h > 0, derive the following expression,
tp
HH
x = e
− ∫ t0 (µHCx+s+µHSx+s+µHDx+s)ds,
where µijx+s denotes the transition rate from state i to state j at time s.
[5 marks]
(ii) Assuming that all transition intensities are constant and that
µHSx+t = 4µ
HC
x+t
µHDx+t = 3
(
µHCx+t
)2
3p
HH
x = e
−0.012,
calculate µHSx+t, µ
HC
x+t and µ
HD
x+t. [5 marks]
(iii) Consider the above 4 state Markov model for a whole life insurance policy
issued to a life aged x with the following features:
- A benefit at a rate b1 per annum is payable continuously while the life is
in state S.
- A benefit at a rate b2 per annum is payable continuously while the life is
in state C.
Page 4 of 10 Turn the page over
Module Code: MATH352001
- There is a death benefit B1 on transition from H to D, a death benefit
B2 on transition from S to D and a death benefit B3 on transition from
C to D.
- Premiums are paid continuously at a rate P¯ per annum while the life is in
state H.
Derive the Thiele’s differential equations for the policy values tV
(H) and tV
(C),
by considering what happens over a short period of time from t to t+h. Assume
the force of interest is δ > 0.
Hint: You may wish to draw a graph to reflect the cashflows during [t, t+ h].
[8 marks]
[ Total 25 marks]
Page 5 of 10 Turn the page over
Module Code: MATH352001
4. (a) An insurance company issues a special three-year term insurance policy available
to lives aged 65. Premiums of £1,000 are paid annually in advance. The death
benefit of £130,000 is paid at the end of year of death. In addition to the death
benefit, the insurer provides a surrender benefit which is payable at the end of
year of surrender, and the amount is equal to 60% of the total premiums (ignore
interest) paid prior to surrender. Policy expenses are as follows:
- Initial expenses of £150 and 3% of the premium are due at the same time as
the first premium.
- Renewal expenses of 1.5% of the premium are due at the same time as subse-
quent premiums.
- Terminal expenses due on payment of death or surrender are £30.
The rate of interest on the cashflow account is 2% p.a.
The insurer uses the following multiple decrement table as the basis for calcu-
lations.
Age(x) (aq)dx (aq)
s
x (ap)x
65 0.0060 0.0300 0.964
66 0.0065 0.0200 0.9735
67 0.0070 0.0200 0.973
Calculate the profit margin for the policy described above using a risk discount rate
of 5% p.a. [10 marks]
(b) The insurer wants to introduce a survival benefit of £200 payable at the end of
the three-year term of the contract. No terminal expense occurs in that case.
Calculate the profit margin for the new policy assuming that the risk discount rate
and premiums are left unchanged. [5 marks]
(c) It has been some time since the above policy was designed and the insurer believes
that the multiple decrement table above is no longer appropriate.
Assuming that the underlying rates of decrement are constant in each year of
age, calculate an updated table based on a decrease in the rates µdx of 10% in each
year of age and an increase in the rates µsx of 5% in each year of age. You may
assume independence of decrements. [8 marks]
(d) Calculate the probability that the policyholder will get the survival benefit of £200
under the updated multiple decrement table.
[2 marks]
[ Total 25 marks]
End of Questions.
Page 6 of 10 Turn the page over
Module Code: MATH352001
Actuarial Tables for University of Leeds Examinations 2019/20
i = 5% p.a.
x l[x] l[x]+1 lx+2 A[x] A[x]+1 Ax+2 (IA)x+2 x+ 2
28 99, 781.36 99, 756.17 99, 727.29 0.07032 0.07357 0.07698 3.652248809 30
29 99, 751.69 99, 725.70 99, 695.83 0.07356 0.07697 0.08054 3.755213856 31
30 99, 721.06 99, 694.18 99, 663.20 0.07696 0.08053 0.08427 3.859668736 32
31 99, 689.36 99, 661.48 99, 629.26 0.08051 0.08425 0.08817 3.965520005 33
32 99, 656.47 99, 627.47 99, 593.83 0.08424 0.08816 0.09226 4.072663183 34
33 99, 622.23 99, 591.96 99, 556.75 0.08814 0.09224 0.09653 4.180982148 35
34 99, 586.47 99, 554.78 99, 517.80 0.09223 0.09652 0.10101 4.290348521 36
35 99, 549.01 99, 515.73 99, 476.75 0.09650 0.10099 0.10569 4.400621053 37
36 99, 509.64 99, 474.56 99, 433.34 0.10097 0.10567 0.11059 4.511645019 38
37 99, 468.12 99, 431.02 99, 387.29 0.10565 0.11057 0.11571 4.623251623 39
38 99, 424.18 99, 384.82 99, 338.26 0.11055 0.11569 0.12106 4.735257429 40
39 99, 377.52 99, 335.62 99, 285.88 0.11567 0.12104 0.12665 4.847463812 41
40 99, 327.82 99, 283.06 99, 229.76 0.12101 0.12663 0.13249 4.959656446 42
41 99, 274.69 99, 226.72 99, 169.41 0.12660 0.13247 0.13859 5.071604847 43
42 99, 217.72 99, 166.14 99, 104.33 0.13244 0.13857 0.14496 5.183061963 44
43 99, 156.42 99, 100.80 99, 033.94 0.13854 0.14493 0.15161 5.293763836 45
44 99, 090.27 99, 030.10 98, 957.57 0.14491 0.15158 0.15854 5.403429353 46
45 99, 018.67 98, 953.40 98, 874.50 0.15155 0.15851 0.16577 5.511760086 47
46 98, 940.96 98, 869.96 98, 783.91 0.15847 0.16573 0.17330 5.618440252 48
47 98, 856.38 98, 778.94 98, 684.88 0.16569 0.17326 0.18114 5.723136801 49
48 98, 764.09 98, 679.44 98, 576.37 0.17322 0.18110 0.18931 5.825499653 50
49 98, 663.15 98, 570.40 98, 457.24 0.18106 0.18926 0.19780 5.925162103 51
50 98, 552.51 98, 450.67 98, 326.19 0.18921 0.19775 0.20664 6.02174142 52
51 98, 430.98 98, 318.95 98, 181.77 0.19770 0.20658 0.21582 6.114839644 53
52 98, 297.24 98, 173.79 98, 022.38 0.20653 0.21576 0.22535 6.20404463 54
53 98, 149.81 98, 013.56 97, 846.20 0.21570 0.22529 0.23524 6.28893133 55
54 97, 987.03 97, 836.44 97, 651.21 0.22522 0.23517 0.24550 6.369063361 56
55 97, 807.07 97, 640.40 97, 435.17 0.23510 0.24542 0.25613 6.443994859 57
56 97, 607.84 97, 423.18 97, 195.56 0.24534 0.25605 0.26714 6.513272655 58
57 97, 387.05 97, 182.25 96, 929.59 0.25596 0.26704 0.27852 6.576438778 59
58 97, 142.13 96, 914.80 96, 634.14 0.26695 0.27842 0.29028 6.633033303 60
59 96, 870.22 96, 617.70 96, 305.75 0.27831 0.29017 0.30243 6.682597561 61
60 96, 568.13 96, 287.48 95, 940.60 0.29005 0.30230 0.31495 6.72467771 62
61 96, 232.34 95, 920.27 95, 534.43 0.30217 0.31481 0.32785 6.758828664 63
62 95, 858.91 95, 511.80 95, 082.53 0.31467 0.32770 0.34113 6.784618387 64
63 95, 443.51 95, 057.36 94, 579.73 0.32755 0.34097 0.35477 6.801632525 65
64 94, 981.34 94, 551.72 94, 020.33 0.34080 0.35459 0.36878 6.809479354 66
65 94, 467.11 93, 989.16 93, 398.05 0.35441 0.36858 0.38313 6.807795016 67
66 93, 895.00 93, 363.38 92, 706.06 0.36838 0.38292 0.39783 6.796248988 68
67 93, 258.63 92, 667.50 91, 936.88 0.38270 0.39760 0.41285 6.774549735 69
Page 7 of 10 Turn the page over
Module Code: MATH352001
x l[x] l[x]+1 lx+2 A[x] A[x]+1 Ax+2 (IA)x+2 x+ 2
68 92, 551.02 91, 894.03 91, 082.43 0.39736 0.41260 0.42818 6.742450463 70
69 91, 764.58 91, 034.84 90, 133.96 0.41234 0.42790 0.44379 6.699754898 71
70 90, 891.07 90, 081.15 89, 082.09 0.42762 0.44349 0.45968 6.646322984 72
71 89, 921.62 89, 023.56 87, 916.84 0.44319 0.45935 0.47580 6.582076391 73
72 88, 846.72 87, 852.03 86, 627.64 0.45902 0.47545 0.49215 6.507003704 74
73 87, 656.25 86, 555.99 85, 203.46 0.47509 0.49177 0.50868 6.421165157 75
74 86, 339.55 85, 124.37 83, 632.89 0.49138 0.50826 0.52536 6.324696777 76
75 84, 885.49 83, 545.75 81, 904.34 0.50785 0.52492 0.54217 6.21781377 77
76 83, 282.61 81, 808.54 80, 006.23 0.52447 0.54169 0.55906 6.100813016 78
77 81, 519.30 79, 901.17 77, 927.35 0.54121 0.55854 0.57599 5.974074522 79
78 79, 584.04 77, 812.44 75, 657.16 0.55802 0.57544 0.59293 5.838061685 80
79 77, 465.70 75, 531.88 73, 186.31 0.57488 0.59234 0.60984 5.693320261 81
80 75, 153.97 73, 050.22 70, 507.19 0.59175 0.60920 0.62666 5.540475924 82
81 72, 639.81 70, 359.94 67, 614.60 0.60857 0.62598 0.64336 5.380230363 83
82 69, 916.06 67, 455.99 64, 506.50 0.62532 0.64264 0.65990 5.213355855 84
83 66, 978.12 64, 336.53 61, 184.88 0.64193 0.65913 0.67622 5.040688357 85
84 63, 824.72 61, 003.79 57, 656.68 0.65838 0.67540 0.69229 4.86311913 86
85 60, 458.83 57, 464.98 53, 934.73 0.67462 0.69142 0.70806 4.681585034 87
86 56, 888.53 53, 733.29 50, 038.65 0.69060 0.70714 0.72349 4.49705762 88
87 53, 128.02 49, 828.70 45, 995.64 0.70629 0.72252 0.73853 4.310531237 89
88 49, 198.45 45, 778.84 41, 841.05 0.72163 0.73752 0.75317 4.123010406 90
89 45, 128.71 41, 619.51 37, 618.56 0.73660 0.75211 0.76735 3.935496741 91
90 40, 955.90 37, 394.80 33, 379.88 0.75115 0.76624 0.78104 3.748975755 92
91 36, 725.43 33, 156.84 29, 183.78 0.76526 0.77989 0.79423 3.564403889 93
92 32, 490.61 28, 964.73 25, 094.33 0.77889 0.79303 0.80688 3.382696112 94
93 28, 311.50 24, 882.75 21, 178.30 0.79201 0.80564 0.81897 3.204714446 95
94 24, 252.97 20, 977.73 17, 501.76 0.80460 0.81769 0.83049 3.031257728 96
95 20, 381.98 17, 315.59 14, 125.89 0.81665 0.82918 0.84143 2.863052897 97
96 16, 763.92 13, 957.13 11, 102.53 0.82814 0.84009 0.85177 2.70074804 98
97 13, 458.37 10, 953.56 8, 469.73 0.83905 0.85041 0.86153 2.54490735 99
98 10, 514.62 8, 342.12 6, 248.17 0.84938 0.86014 0.87068 2.396008116 100
99 7, 967.35 6, 142.47 4, 438.80 0.85913 0.86928 0.87925 2.254439739 101
100 5, 833.20 4, 354.49 3, 022.58 0.86830 0.87784 0.88724 2.120504722 102
101 4, 108.86 2, 958.13 1, 962.49 0.87690 0.88582 0.89465 1.994421504 103
102 2, 771.25 1, 915.52 1, 207.79 0.88492 0.89323 0.90150 1.87632892 104
103 1, 780.05 1, 175.35 699.91 0.89239 0.90010 0.90781 1.766292023 105
104 1, 082.34 678.82 379.08 0.89931 0.90643 0.91360 1.664308939 106
105 618.75 366.26 190.28 0.90571 0.91224 0.91888 1.570318406 107
106 330.04 183.07 87.69 0.91159 0.91756 0.92367 1.484207571 108
107 162.85 83.97 36.71 0.91698 0.92240 0.92800 1.405819641 109
108 73.61 34.96 13.80 0.92189 0.92678 0.93188 1.334960925 110
109 30.16 13.06 4.59 0.92634 0.93072 0.93534 1.271406814 111
110 11.06 4.32 1.33 0.93036 0.93426 0.93840 1.214906225 112
111 3.58 1.24 0.33 0.93396 0.93739 0.94107 1.165184082 113
112 1.01 0.31 0.07 0.93716 0.94016 0.94339 1.121941457 114
113 0.24 0.06 0.01 0.93998 0.94257 0.94537 1.08485318 115
114 0.05 0.01 0.00 0.94244 0.94464 0.94703 1.053563039 116
115 0.01 0.00 0.00 0.94455 0.94641 0.94841 1.0276772 117
Page 8 of 10 Turn the page over
Module Code: MATH352001
x a¨[x] a¨[x]+1 a¨x+2 (Ia¨)[x] (Ia¨)[x]+1 (Ia¨)x+2 x+ 2
30 19.38449 19.30892 19.23034 330.37389 326.62691 322.78419 32
31 19.30979 19.23066 19.14838 326.64260 322.78973 318.84013 33
32 19.23156 19.14871 19.06258 322.80588 318.84584 314.78826 34
33 19.14965 19.06292 18.97277 318.86251 314.79415 310.62762 35
34 19.06390 18.97313 18.87880 314.81141 310.63373 306.35745 36
35 18.97415 18.87917 18.78049 310.65164 306.36380 301.97715 37
36 18.88024 18.78088 18.67766 306.38244 301.98377 297.48630 38
37 18.78201 18.67807 18.57014 302.00322 297.49322 292.88473 39
38 18.67927 18.57058 18.45776 297.51358 292.89198 288.17248 40
39 18.57184 18.45822 18.34031 292.91336 288.18010 283.34985 41
40 18.45956 18.34081 18.21763 288.20260 283.35788 278.41741 42
41 18.34224 18.21815 18.08951 283.38162 278.42589 273.37601 43
42 18.21969 18.09007 17.95577 278.45100 273.38499 268.22685 44
43 18.09172 17.95637 17.81621 273.41163 268.23638 262.97143 45
44 17.95814 17.81686 17.67065 268.26469 262.98157 257.61163 46
45 17.81876 17.67135 17.51889 263.01173 257.62244 252.14969 47
46 17.67340 17.51965 17.36074 257.65462 252.16123 246.58827 48
47 17.52187 17.36156 17.19602 252.19564 246.60062 240.93045 49
48 17.36397 17.19691 17.02453 246.63745 240.94367 235.17974 50
49 17.19952 17.02551 16.84612 240.98317 235.19392 229.34013 51
50 17.02835 16.84718 16.66060 235.23632 229.35536 223.41609 52
51 16.85028 16.66175 16.46782 229.40091 223.43245 217.41258 53
52 16.66514 16.46908 16.26762 223.48144 217.43017 211.33509 54
53 16.47277 16.26899 16.05987 217.48289 211.35402 205.18964 55
54 16.27303 16.06137 15.84443 211.41077 205.21001 198.98279 56
55 16.06579 15.84608 15.62122 205.27112 199.00472 192.72165 57
56 15.85091 15.62302 15.39012 199.07053 192.74525 186.41388 58
57 15.62831 15.39210 15.15109 192.81612 186.43928 180.06769 59
58 15.39789 15.15325 14.90407 186.51557 180.09501 173.69186 60
59 15.15960 14.90644 14.64906 180.17711 173.72124 167.29568 61
60 14.91340 14.65165 14.38606 173.80954 167.32725 160.88898 62
61 14.65927 14.38890 14.11512 167.42215 160.92287 154.48209 63
62 14.39724 14.11822 13.83632 161.02479 154.51844 148.08579 64
63 14.12736 13.83972 13.54979 154.62778 148.12474 141.71131 65
64 13.84972 13.55351 13.25568 148.24192 141.75300 135.37026 66
65 13.56444 13.25975 12.95420 141.87843 135.41483 129.07459 67
66 13.27169 12.95864 12.64561 135.54890 129.12216 122.83651 68
67 12.97168 12.65045 12.33019 129.26527 122.88721 116.66844 69
68 12.66467 12.33547 12.00830 123.03972 116.72238 110.58291 70
69 12.35097 12.01406 11.68035 116.88464 110.64020 104.59250 71
70 12.03093 11.68661 11.34678 110.81252 104.65322 98.70970 72
Page 9 of 10 Turn the page over
Module Code: MATH352001
x a¨[x] a¨[x]+1 a¨x+2 (Ia¨)[x] (Ia¨)[x]+1 (Ia¨)x+2 x+ 2
71 11.70495 11.35359 11.00812 104.83588 98.77394 92.94687 73
72 11.37350 11.01550 10.66491 98.96716 93.01470 87.31608 74
73 11.03709 10.67291 10.31778 93.21865 87.38753 81.82901 75
74 10.69629 10.32644 9.96740 87.60234 81.90411 76.49687 76
75 10.35171 9.97676 9.61449 82.12983 76.57561 71.33021 77
76 10.00402 9.62458 9.25981 76.81221 71.41257 66.33889 78
77 9.65395 9.27067 8.90416 71.65994 66.42480 61.53188 79
78 9.30225 8.91584 8.54841 66.68276 61.62127 56.91722 80
79 8.94973 8.56093 8.19341 61.88954 57.00997 52.50189 81
80 8.59722 8.20681 7.84008 57.28817 52.59784 48.29171 82
81 8.24560 7.85440 7.48934 52.88550 48.39069 44.29128 83
82 7.89575 7.50460 7.14211 48.68723 44.39307 40.50392 84
83 7.54859 7.15834 6.79934 44.69784 40.60828 36.93161 85
84 7.20502 6.81655 6.46193 40.92050 37.03827 33.57497 86
85 6.86596 6.48015 6.13079 37.35707 33.68364 30.43327 87
86 6.53231 6.15002 5.80679 34.00808 30.54362 27.50445 88
87 6.20493 5.82705 5.49077 30.87267 27.61612 24.78512 89
88 5.88469 5.51204 5.18352 27.94869 24.89776 22.27067 90
89 5.57237 5.20579 4.88575 25.23268 22.38390 19.95536 91
90 5.26873 4.90900 4.59814 22.71998 20.06878 17.83235 92
91 4.97447 4.62234 4.32126 20.40476 17.94558 15.89391 93
92 4.69020 4.34637 4.05562 18.28021 16.00655 14.13150 94
93 4.41647 4.08161 3.80166 16.33856 14.24315 12.53592 95
94 4.15375 3.82845 3.55971 14.57128 12.64622 11.09749 96
95 3.90242 3.58723 3.33001 12.96923 11.20605 9.80615 97
96 3.66279 3.35819 3.11273 11.52274 9.91263 8.65166 98
97 3.43504 3.14146 2.90794 10.22185 8.75572 7.62373 99
98 3.21930 2.93713 2.71563 9.05637 7.72505 6.71212 100
99 3.01561 2.74516 2.53572 8.01611 6.81042 5.90684 101
100 2.82392 2.56545 2.36804 7.09095 6.00184 5.19818 102
101 2.64410 2.39784 2.21237 6.27099 5.28965 4.57686 103
102 2.47597 2.24210 2.06843 5.54664 4.66457 4.03408 104
103 2.31927 2.09792 1.93589 4.90874 4.11782 3.56158 105
104 2.17370 1.96497 1.81439 4.34862 3.64118 3.15168 106
105 2.03893 1.84289 1.70352 3.85814 3.22697 2.79731 107
106 1.91458 1.73125 1.60288 3.42975 2.86816 2.49204 108
107 1.80025 1.62965 1.51201 3.05652 2.55830 2.23002 109
108 1.69554 1.53764 1.43049 2.73213 2.29158 2.00601 110
109 1.60004 1.45478 1.35785 2.45087 2.06278 1.81535 111
110 1.51334 1.38063 1.29366 2.20762 1.86724 1.65391 112
111 1.43504 1.31475 1.23747 1.99785 1.70086 1.51803 113
112 1.36476 1.25672 1.18882 1.81756 1.56002 1.40453 114
113 1.30212 1.20611 1.14726 1.66322 1.44157 1.31061 115
114 1.24676 1.16250 1.11232 1.53178 1.34275 1.23380 116
115 1.19833 1.12546 1.08348 1.42056 1.26116 1.17190 117
Page 10 of 10 End of Paper.
学霸联盟
