Main Examination period 2019
MTH5124: Actuarial Mathematics I
Duration: 2 hours
until instructed to do so by an invigilator.
You should attempt ALL questions. Marks available are shown next to
the questions.
You are permitted to bring your own handwritten or typewritten notes into the exam. No more
than 5 sheets of double sided A4 pages of notes are permitted. You are NOT permitted to
bring any form of book or notes in excess of 5 pages into the exam; bringing any such
material into the exam is an assessment offence. Sharing material with other students
during the exam is NOT permitted and is an assessment offence.
Only non-programmable calculators that have been approved from the college list of
non-programmable calculators are permitted in this examination. Please state on your
answer book the name and type of machine used.
Complete all rough work in the answer book and cross through any work that is not to be
assessed.
Possession of unauthorised material at any time when under examination conditions is an
assessment offence and can lead to expulsion from QMUL. Check now to ensure you do not
have any notes, mobile phones, smartwatches or unauthorised electronic devices on your
person. If you do, raise your hand and give them to an invigilator immediately.
It is also an offence to have any writing of any kind on your person, including on your body. If
you are found to have hidden unauthorised material elsewhere, including toilets and
cloakrooms, it shall be treated as being found in your possession. Unauthorised material
found on your mobile phone or other electronic device will be considered the same as being in
possession of paper notes. A mobile phone that causes a disruption in the exam is also an
assessment offence.
Exam papers must not be removed from the examination room.
Examiners: A. Baule, C. Sutton
c© Queen Mary University of London (2019) Turn Over
Page 2 MTH5124 (2019)
Question 1. [18 marks]
(a) A lump sum of £5,000 is invested at a nominal interest rate of 6% p.a., compounded
monthly. Determine the accumulation of the lump sum after 25 years. [4]
(b) Determine the AER corresponding to the nominal interest rate i(4) = 4.5% p.a.. [4]
(c) You sign a mobile phone contract agreeing to pay £35 at the beginning of each month
for the next two years. Assuming an APR of 2% p.a., how much money do you need to
put into a bank account to cover all future payments? [5]
(d) In return for a loan of £200, a loan shark charges £10 per week payable when the loan is
paid back. What is the APR if you pay back the loan and fees after 1 year? [5]
Question 2. [18 marks]
John Smith is paying off a loan of £50,000 taken out exactly four years ago and being paid
annually in arrears for 8 years. The fourth payment has just been made. The effective annual
interest rate charged on the loan is 8.5%.
(a) Show that the annual payment is approximately £8,866 and the capital outstanding after
4 years is approximately £29,043. [7]
(b) Determine the schedule of payments for years 5-8. [7]
(c) John decides to pay back the capital outstanding after 4 years by taking out a new loan.
The new loan is paid in monthly instalments in advance over 3 years at an AER of
7.5%. What are the new monthly payments? [4]
Question 3. [15 marks]
An investor buys £10,000 nominal of a government bond on 1st March 2018. The bond is
redeemable at 110% of its nominal value on 1st September 2030. Coupons are paid twice
yearly in arrears on 1st March and 1st September at an annual rate of 1.5% of nominal.
(a) Assuming that the gross redemption yield on the bond at the time of purchase is 2.7%
per annum what is the price of the bond? [6]
(b) Assuming the same gross redemption yield of 2.7% per annum what is the price of the
bond on 1st January 2020? [5]
(c) Show, with explanation, that the investor’s net redemption yield will be greater than
2.0% per annum if the bond is held until maturity and the investor pays tax at the rate of
20% on all coupon payments. [4]
c© Queen Mary University of London (2019)
MTH5124 (2019) Page 3
Question 4. [20 marks]
(a) Use mortality given by table AMC00 ultimate values. You can find the table in the
appendix.
(i) Explain in words the meaning of the symbol 2|3q21 and calculate its value. [5]
(ii) Calculate the value of the symbol 2.5p45. State any assumptions made. [5]
(b) Consider the survival function
s(x) =
50−2x2
50
, for 06 x6 5.
(i) Show that s(x) is a valid survival function. [6]
(ii) Calculate the complete expectation of life at age x= 2 using this survival function. [4]
Question 5. [15 marks] John Brown plans to retire at age 63 with a pension of £24,000 per
year, paid annually in advance during his remaining lifetime. Assume an effective annual
interest rate of 4% and the mortality given by table AMC00 select values.
(a) Show that the expected present value of the pension at retirement is approximately
£331,561. [6]
(b) In order to cover the cost of the pension, John wants to make monthly payments in
arrears starting at age 35 until retirement. What are the monthly premiums assuming he
survives until retirement? [5]
(c) How does the expected present value in (a) change when he switches to a pension of
£2,000 paid monthly in advance during his remaining lifetime? Give reasons. [4]
Question 6. [14 marks]
On his 40th birthday, Adam Smith takes out a 15-year endowment policy with £25,000 death
benefit payable at the end of year of death and £25,000 survival benefit. The premium is
payable annually in advance. Assume an effective annual interest rate of 4% and the mortality
given by table AMC00 select values.
(a) Calculate the amount of the annual premium. [6]
(b) Calculate the expected present value of the survival benefit. [4]
(c) Calculate the expected present value of the death benefit. [4]
End of Paper – An appendix of 5 pages follows.
c© Queen Mary University of London (2019)
Page 4 MTH5124 (2019)
AMC00 Male Life Table
Age x I[x] I[x−1]+1 Ix Age x
17 9997.5094 10000.0000 17
18 9992.9305 9994.6901 9995.4200 18
19 9988.3337 9990.1025 9990.8321 19
20 9983.6991 9985.4871 9986.2163 20
21 9979.0567 9980.8438 9981.5827 21
22 9974.3666 9976.1827 9976.9213 22
23 9969.6487 9971.4740 9972.2222 23
24 9964.8832 9966.7276 9967.4854 24
25 9960.0701 9961.9336 9962.7010 25
26 9955.1996 9957.0921 9957.8691 26
27 9950.2618 9952.1832 9952.9698 27
28 9945.2469 9947.1972 9947.9933 28
29 9940.1847 9942.1340 9942.9398 29
30 9935.0655 9936.9939 9937.7794 30
31 9929.8794 9931.7671 9932.5024 31
32 9924.5767 9926.4238 9927.0892 32
33 9919.1182 9920.9344 9921.5201 33
34 9913.4841 9915.2596 9915.7755 34
35 9907.6152 9909.3799 9909.8162 35
36 9901.4922 9903.2460 9903.6126 36
37 9895.0760 9896.8385 9897.1357 37
38 9888.3275 9890.1087 9890.3363 38
39 9881.1880 9882.9977 9883.1559 39
40 9873.5893 9875.4570 9875.5558 40
41 9865.4534 9867.4085 9867.4578 41
42 9856.7223 9858.7745 9858.7942 42
43 9847.3090 9849.4875 9849.4875 43
44 9837.1169 9839.4312 9839.4312 44
45 9826.0499 9828.5291 9828.5291 45
46 9813.9735 9816.6562 9816.6562 46
47 9800.7439 9803.6786 9803.6786 47
48 9786.2090 9789.4437 9789.4437 48
49 9770.1885 9773.7708 9773.7708 49
50 9752.4550 9756.4712 9756.4712 50
51 9732.8031 9737.3192 9737.3192 51
52 9710.9228 9716.0627 9716.0627 52
53 9686.5083 9692.4138 9692.4332 53
54 9659.2100 9666.0213 9666.1182 54
55 9628.6170 9636.5205 9636.7719 55
56 9594.3554 9603.4960 9604.0069 56
57 9555.9547 9566.5317 9567.3964 57
58 9512.9284 9525.1558 9526.4767 58
Age x I[x] I[x−1]+1 Ix Age x
59 9464.7096 9478.8531 9480.7305 59
60 9410.6849 9427.0212 9429.5820 60
61 9350.1805 9369.0144 9372.4010 61
62 9282.4873 9304.1589 9308.5187 62
63 9206.8320 9231.7028 9237.1968 63
64 9122.3956 9150.8360 9157.6369 64
65 9028.3257 9060.7100 9069.0001 65
66 8923.7129 8960.4327 8970.3747 66
67 8807.6361 8849.0571 8860.8106 67
68 8679.1539 8725.6105 8739.3111 68
69 8537.3365 8589.1250 8604.8568 69
70 8381.2654 8438.6022 8456.4058 70
71 8210.1056 8273.0800 8292.9181 71
72 8023.0608 8091.6666 8113.3848 72
73 7819.4936 7893.5124 7916.8461 73
74 7598.8879 7677.9138 7702.4263 74
75 7360.9680 7444.3113 7469.3894 75
76 7105.6585 7192.3651 7217.1705 76
77 6833.2053 6921.9772 6945.4296 77
78 6543.2750 6633.3614 6654.1244 78
79 6234.6920 6326.2215 6343.5631 79
80 5905.8869 6000.3050 6014.4463 80
81 5557.1270 5655.4241 5667.9541 81
82 5191.0860 5293.5691 5305.7888 82
83 4811.5773 4917.5469 4930.2080 83
84 4422.4381 4531.0383 4544.0494 84
85 4027.2597 4137.4915 4150.7347 85
86 3630.1730 3740.8772 3754.2191 86
87 3235.7358 3345.6473 3358.9374 87
88 2848.8406 2956.5986 2969.6769 88
89 2474.5254 2578.7306 2591.4291 89
90 2117.7709 2217.0436 2229.1940 90
91 1876.3069 1887.7528 91
92 1571.4202 92
93 1283.8047 93
94 1027.5920 94
95 804.3661 95
96 614.5164 96
97 457.2229 97
98 330.5502 98
99 231.6268 99
100 156.9026 100
c© Queen Mary University of London (2019)
MTH5124 (2019) Page 5
AMC00 Male Life Functions (4%)
Age x D[x] D[x−1]+1 Dx Age x
17 5132.45 5133.73 17
18 4932.79 4933.66 4934.02 18
19 4740.89 4741.73 4742.07 19
20 4556.43 4557.25 4557.58 20
21 4379.15 4379.93 4380.25 21
22 4208.74 4209.50 4209.82 22
23 4044.95 4045.69 4045.99 23
24 3887.51 3888.23 3888.53 24
25 3736.19 3736.89 3737.18 25
26 3590.73 3591.42 3591.70 26
27 3450.92 3451.58 3451.85 27
28 3316.52 3317.17 3317.43 28
29 3187.33 3187.96 3188.22 29
30 3063.17 3063.76 3064.00 30
31 2943.81 2944.37 2944.59 31
32 2829.08 2829.61 2829.80 32
33 2718.77 2719.27 2719.43 33
34 2612.72 2613.19 2613.32 34
35 2510.74 2511.19 2511.30 35
36 2412.68 2413.11 2413.20 36
37 2318.39 2318.80 2318.87 37
38 2227.70 2228.10 2228.15 38
39 2140.47 2140.86 2140.90 39
40 2056.56 2056.95 2056.97 40
41 1975.83 1976.22 1976.23 41
42 1898.16 1898.55 1898.56 42
43 1823.41 1823.81 1823.81 43
44 1751.46 1751.87 1751.87 44
45 1682.20 1682.63 1682.63 45
46 1615.52 1615.96 1615.96 46
47 1551.29 1551.75 1551.75 47
48 1489.41 1489.90 1489.90 48
49 1429.78 1430.30 1430.30 49
50 1372.29 1372.86 1372.86 50
51 1316.85 1317.47 1317.47 51
52 1263.36 1264.03 1264.03 52
53 1211.71 1212.45 1212.46 53
54 1161.83 1162.65 1162.66 54
55 1113.60 1114.52 1114.55 55
56 1066.96 1067.98 1068.03 56
57 1021.82 1022.95 1023.04 57
58 978.09 979.35 979.49 58
Age x D[x] D[x−1]+1 Dx Age x
59 935.71 937.11 937.29 59
60 894.58 896.14 896.38 60
61 854.65 856.37 856.68 61
62 815.83 817.73 818.11 62
63 778.05 780.16 780.62 63
64 741.27 743.58 744.13 64
65 705.41 707.94 708.59 65
66 670.42 673.18 673.92 66
67 636.25 639.24 640.09 67
68 602.85 606.08 607.03 68
69 570.19 573.65 574.70 69
70 538.24 541.92 543.07 70
71 506.97 510.86 512.08 71
72 476.36 480.44 481.73 72
73 446.42 450.65 451.98 73
74 417.14 421.48 422.82 74
75 388.54 392.94 394.26 75
76 360.64 365.04 366.30 76
77 333.47 337.80 338.95 77
78 307.04 311.27 312.24 78
79 281.31 285.44 286.22 79
80 256.22 260.32 260.93 80
81 231.82 235.92 236.44 81
82 208.22 212.33 212.82 82
83 185.58 189.66 190.15 83
84 164.01 168.03 168.52 84
85 143.61 147.54 148.01 85
86 124.47 128.26 128.72 86
87 106.68 110.30 110.74 87
88 90.31 93.73 94.14 88
89 75.43 78.60 78.99 89
90 62.07 64.98 65.34 90
91 52.88 53.20 91
92 42.58 92
93 33.45 93
94 25.74 94
95 19.38 95
96 14.23 96
97 10.18 97
98 7.08 98
99 4.77 99
100 3.11 100
c© Queen Mary University of London (2019) Turn Over
Page 6 MTH5124 (2019)
AMC00 Male Life Functions (4%)
Age x N[x] N[x−1]+1 Nx Age x
17 121013.22 121014.86 17
18 115879.55 115880.77 115881.13 18
19 110945.59 110946.76 110947.11 19
20 106203.56 106204.70 106205.03 20
21 101646.03 101647.13 101647.45 21
22 97265.82 97266.89 97267.20 22
23 93056.05 93057.08 93057.39 23
24 89010.09 89011.10 89011.39 24
25 85121.59 85122.57 85122.86 25
26 81384.45 81385.41 81385.69 26
27 77792.78 77793.72 77793.99 27
28 74340.96 74341.87 74342.13 28
29 71023.58 71024.44 71024.70 29
30 67835.43 67836.24 67836.49 30
31 64771.51 64772.26 64772.48 31
32 61827.01 61827.70 61827.89 32
33 58997.30 58997.93 58998.09 33
34 56277.95 56278.53 56278.66 34
35 53664.69 53665.23 53665.34 35
36 51153.45 51153.95 51154.04 36
37 48740.30 48740.77 48740.84 37
38 46421.48 46421.92 46421.97 38
39 44193.38 44193.79 44193.82 39
40 42052.51 42052.91 42052.93 40
41 39995.55 39995.95 39995.96 41
42 38019.32 38019.72 38019.72 42
43 36120.76 36121.17 36121.17 43
44 34296.94 34297.35 34297.35 44
45 32545.06 32545.48 32545.48 45
46 30862.41 30862.85 30862.85 46
47 29246.43 29246.89 29246.89 47
48 27694.65 27695.14 27695.14 48
49 26204.72 26205.24 26205.24 49
50 24774.37 24774.94 24774.94 50
51 23401.47 23402.08 23402.08 51
52 22083.94 22084.61 22084.61 52
53 20819.83 20820.58 20820.58 53
54 19607.27 19608.12 19608.13 54
55 18444.47 18445.44 18445.47 55
56 17329.76 17330.87 17330.93 56
57 16261.53 16262.80 16262.89 57
58 15238.27 15239.71 15239.85 58
Age x N[x] N[x−1]+1 Nx Age x
59 14258.53 14260.18 14260.36 59
60 13320.96 13322.83 13323.07 60
61 12424.28 12426.38 12426.69 61
62 11567.26 11569.63 11570.01 62
63 10748.78 10751.44 10751.90 63
64 9967.77 9970.73 9971.28 64
65 9223.22 9226.50 9227.15 65
66 8514.21 8517.82 8518.56 66
67 7839.85 7843.79 7844.64 67
68 7199.32 7203.60 7204.55 68
69 6591.87 6596.47 6597.52 69
70 6016.77 6021.68 6022.82 70
71 5473.35 5478.53 5479.75 71
72 4960.98 4966.38 4967.67 72
73 4479.04 4484.61 4485.94 73
74 4026.96 4032.62 4033.96 74
75 3604.16 3609.81 3611.14 75
76 3210.07 3215.62 3216.88 76
77 2844.13 2849.44 2850.58 77
78 2505.65 2510.66 2511.63 78
79 2193.87 2198.61 2199.39 79
80 1907.94 1912.56 1913.17 80
81 1647.13 1651.72 1652.24 81
82 1410.71 1415.31 1415.80 82
83 1197.92 1202.49 1202.97 83
84 1007.84 1012.34 1012.82 84
85 839.45 843.83 844.31 85
86 691.61 695.84 696.30 86
87 563.10 567.14 567.58 87
88 452.62 456.42 456.84 88
89 358.78 362.31 362.70 89
90 280.12 283.35 283.71 90
91 218.05 218.37 91
92 165.17 92
93 122.59 93
94 89.14 94
95 63.39 95
96 44.02 96
97 29.78 97
98 19.60 98
99 12.52 99
100 7.75 100
c© Queen Mary University of London (2019)
MTH5124 (2019) Page 7
AMC00 Male Life Functions (4%)
Age x M[x] M[x−1]+1 Mx Age x
17 478.10 479.31 17
18 475.89 476.71 477.05 18
19 473.75 474.54 474.88 19
20 471.68 472.45 472.77 20
21 469.68 470.42 470.74 21
22 467.74 468.47 468.77 22
23 465.87 466.57 466.86 23
24 464.05 464.73 465.02 24
25 462.28 462.94 463.22 25
26 460.56 461.21 461.48 26
27 458.89 459.52 459.78 27
28 457.25 457.86 458.12 28
29 455.66 456.25 456.50 29
30 454.11 454.67 454.91 30
31 452.60 453.13 453.34 31
32 451.12 451.62 451.80 32
33 449.65 450.12 450.27 33
34 448.18 448.63 448.76 34
35 446.72 447.14 447.25 35
36 445.24 445.65 445.74 36
37 443.76 444.15 444.22 37
38 442.25 442.64 442.69 38
39 440.72 441.10 441.13 39
40 439.16 439.53 439.55 40
41 437.54 437.92 437.93 41
42 435.88 436.26 436.26 42
43 434.15 434.54 434.54 43
44 432.35 432.75 432.75 44
45 430.47 430.88 430.88 45
46 428.50 428.92 428.92 46
47 426.42 426.87 426.87 47
48 424.23 424.70 424.70 48
49 421.91 422.41 422.41 49
50 419.43 419.98 419.98 50
51 416.80 417.39 417.39 51
52 413.98 414.62 414.62 52
53 410.95 411.66 411.66 53
54 407.70 408.49 408.50 54
55 404.20 405.08 405.10 55
56 400.43 401.41 401.46 56
57 396.37 397.46 397.55 57
58 392.01 393.21 393.34 58
Age x M[x] M[x−1]+1 Mx Age x
59 387.30 388.64 388.82 59
60 382.24 383.72 383.95 60
61 376.79 378.43 378.73 61
62 370.93 372.74 373.11 62
63 364.64 366.64 367.09 63
64 357.89 360.09 360.62 64
65 350.67 353.07 353.70 65
66 342.95 345.57 346.29 66
67 334.71 337.55 338.37 67
68 325.95 329.02 329.93 68
69 316.66 319.94 320.95 69
70 306.83 310.32 311.42 70
71 296.46 300.15 301.32 71
72 285.56 289.42 290.66 72
73 274.15 278.16 279.44 73
74 262.26 266.38 267.67 74
75 249.92 254.10 255.37 75
76 237.17 241.36 242.57 76
77 224.08 228.21 229.31 77
78 210.67 214.70 215.64 78
79 196.93 200.88 201.63 79
80 182.84 186.76 187.35 80
81 168.47 172.39 172.90 81
82 153.96 157.90 158.37 82
83 139.50 143.41 143.88 83
84 125.24 129.10 129.56 84
85 111.32 115.08 115.54 85
86 97.87 101.50 101.94 86
87 85.02 88.49 88.91 87
88 72.90 76.17 76.57 88
89 61.63 64.67 65.04 89
90 51.30 54.08 54.42 90
91 44.49 44.80 91
92 36.23 92
93 28.74 93
94 22.32 94
95 16.94 95
96 12.54 96
97 9.04 97
98 6.33 98
99 4.29 99
100 2.81 100
c© Queen Mary University of London (2019) Turn Over
Page 8 MTH5124 (2019)
AMC00 Male Life Functions (4%)
Age x A[x] Ax a¨[x] a¨x Age x
17 0.09315 0.09337 23.578 23.572 17
18 0.09647 0.09669 23.492 23.486 18
19 0.09993 0.10014 23.402 23.396 19
20 0.10352 0.10373 23.309 23.303 20
21 0.10725 0.10747 23.211 23.206 21
22 0.11114 0.11135 23.110 23.105 22
23 0.11517 0.11539 23.005 23.000 23
24 0.11937 0.11959 22.896 22.891 24
25 0.12373 0.12395 22.783 22.777 25
26 0.12826 0.12848 22.665 22.659 26
27 0.13297 0.13320 22.543 22.537 27
28 0.13787 0.13809 22.415 22.410 28
29 0.14296 0.14318 22.283 22.277 29
30 0.14825 0.14847 22.146 22.140 30
31 0.15375 0.15396 22.003 21.997 31
32 0.15946 0.15966 21.854 21.849 32
33 0.16539 0.16558 21.700 21.695 33
34 0.17154 0.17172 21.540 21.535 34
35 0.17792 0.17809 21.374 21.370 35
36 0.18454 0.18471 21.202 21.198 36
37 0.19141 0.19157 21.023 21.019 37
38 0.19853 0.19868 20.838 20.834 38
39 0.20590 0.20605 20.647 20.643 39
40 0.21354 0.21369 20.448 20.444 40
41 0.22145 0.22160 20.242 20.238 41
42 0.22963 0.22978 20.030 20.026 42
43 0.23810 0.23826 19.809 19.805 43
44 0.24685 0.24702 19.582 19.578 44
45 0.25590 0.25608 19.347 19.342 45
46 0.26524 0.26543 19.104 19.099 46
47 0.27488 0.27509 18.853 18.848 47
48 0.28483 0.28506 18.594 18.589 48
49 0.29508 0.29533 18.328 18.321 49
50 0.30564 0.30591 18.053 18.046 50
51 0.31651 0.31681 17.771 17.763 51
52 0.32768 0.32801 17.480 17.472 52
53 0.33915 0.33953 17.182 17.172 53
54 0.35091 0.35135 16.876 16.865 54
55 0.36297 0.36347 16.563 16.550 55
56 0.37530 0.37589 16.242 16.227 56
57 0.38791 0.38859 15.914 15.897 57
58 0.40079 0.40158 15.580 15.559 58
Age x A[x] Ax a¨[x] a¨x Age x
59 0.41391 0.41483 15.238 15.214 59
60 0.42728 0.42834 14.891 14.863 60
61 0.44087 0.44209 14.537 14.506 61
62 0.45467 0.45607 14.179 14.142 62
63 0.46866 0.47025 13.815 13.774 63
64 0.48281 0.48462 13.447 13.400 64
65 0.49711 0.49916 13.075 13.022 65
66 0.51154 0.51384 12.700 12.640 66
67 0.52608 0.52863 12.322 12.256 67
68 0.54069 0.54352 11.942 11.869 68
69 0.55536 0.55847 11.561 11.480 69
70 0.57005 0.57345 11.179 11.090 70
71 0.58476 0.58843 10.796 10.701 71
72 0.59945 0.60338 10.414 10.312 72
73 0.61411 0.61827 10.033 9.925 73
74 0.62870 0.63306 9.654 9.541 74
75 0.64322 0.64772 9.276 9.159 75
76 0.65765 0.66222 8.901 8.782 76
77 0.67197 0.67653 8.529 8.410 77
78 0.68613 0.69062 8.161 8.044 78
79 0.70005 0.70445 7.799 7.684 79
80 0.71360 0.71800 7.446 7.332 80
81 0.72672 0.73123 7.105 6.988 81
82 0.73942 0.74413 6.775 6.652 82
83 0.75173 0.75668 6.455 6.326 83
84 0.76365 0.76884 6.145 6.010 84
85 0.77518 0.78060 5.845 5.704 85
86 0.78629 0.79195 5.556 5.409 86
87 0.79698 0.80287 5.279 5.125 87
88 0.80724 0.81336 5.012 4.853 88
89 0.81705 0.82340 4.757 4.592 89
90 0.82642 0.83299 4.513 4.342 90
91 0.84213 4.105 91
92 0.85081 3.879 92
93 0.85905 3.665 93
94 0.86683 3.462 94
95 0.87417 3.272 95
96 0.88106 3.092 96
97 0.88751 2.925 97
98 0.89351 2.769 98
99 0.89904 2.625 99
100 0.90404 2.495 100
End of Appendix.
c© Queen Mary University of London (2019)