Mathematics of Finance Assignment Solution for BCO126 Course

This assignment solution addresses a Mathematics of Finance problem, focusing on calculating future values of investments and savings. The assignment involves analyzing the impact of monthly deposits, interest rates, and compounding periods on the final amount accumulated over a 20-year period. It explores scenarios with deposits made at the end and beginning of each month, demonstrating how the timing of deposits affects the overall returns. The solution also includes calculations for varying deposit amounts with a growth rate. The assignment covers concepts of compound interest, annuity valuation, and the practical application of financial formulas to determine the future value of savings, with detailed tables illustrating the calculations.

1
Mathematics of finance

2
Table of Contents
Question 1..................................................................................................................................3
a).............................................................................................................................................3
b)............................................................................................................................................3
c).............................................................................................................................................4
Question 2..................................................................................................................................4
a).............................................................................................................................................4
b)............................................................................................................................................5
c).............................................................................................................................................5
d)............................................................................................................................................5
Question 3..................................................................................................................................6
a).............................................................................................................................................6
b)............................................................................................................................................6
c).............................................................................................................................................6
Appendix....................................................................................................................................7

3
Question 1
a)
1 2 3 4 5 6 7 8 9 10
0
200
400
600
800
1000
1200
100
200.3
300.9009
401.8036027
503.0090135081
604.5180405486
24
706.3315946702
7
808.4505894542
81
910.8759412226
44
1013.608569046
31
Deposit
Closing balance
Interest
Period
Cash flows
Period(Month
end)
Opening
balance
Deposit Interes
t
Closing
balance
1 0 100 0 100
2 100 100 0.3 200.3
3 200.3 100 0.60 300.90
4 300.9009 100 0.90 401.80
5 401.8036 100 1.21 503.01
6 503.009 100 1.51 604.52
7 604.518 100 1.81 706.33
8 706.3316 100 2.12 808.45
9 808.4506 100 2.43 910.88
10 910.8759 100 2.73 1013.61
b)
The deposit is made on the monthly basis and that too for a period of 20 years till retirement.
Due to this the total tenure will be involving various cash flows which will be occurring. This
will be calculated with the following formula:
Number of Cash flows = Number of years * period in one year as per compounding
= 20 * 12
= 240

4
There will be total 240 cash flows which will be taking place till retirement.
c)
The deposits which are made will be maturing at the end of the duration and at that time there
will be a final amount which will be received. The amount will be involving the amount that
is deposited as principal and the interest which is earned on compounding basis. The
calculation is made and that is presented below:
Particulars Amounts
PMT 100
Years 20
payments in one
year
12
Nper 240
rate 3.60%
Final value of
deposits
35074.00
The money which will be available at the end amounts to €35074.
Question 2
a)
1 2 3 4 5 6 7 8 9 10
0
200
400
600
800
1000
1200
100.30
200.90
301.80
403.01
504.52
606.33
708.45
810.88
913.61
1016.65
Interest
Closing balance
Deposit
Period
Cash flows
Period(Month
beginning)
Opening
balance
Deposit Interest Closing
balance
1 0 100 0.3 100.3
2 100.30 100 0.60 200.90
3 200.90 100 0.90 301.80

5
4 301.80 100 1.21 403.01
5 403.01 100 1.51 504.52
6 504.52 100 1.81 606.33
7 606.33 100 2.12 708.45
8 708.45 100 2.43 810.88
9 810.88 100 2.73 913.61
10 913.61 100 3.04 1016.65
b)
In this case and the previous case there is an important difference in respect of the timing to
make deposits. The deposits in the previous case have been at the end of the period and in the
given case there is the payment in the beginning of the period. The other aspects in relation to
the amount deposited, rate and period are same but the change in the deposit timing will
affect the overall receipt which will be made.
c)
The deposits which are made at the beginning of the period will be more beneficial in
comparison to the ones which are made in the end. This is because of the interest which will
be received. The month end payments will lead to loss of the interest for the last instalment
which will be made which will be available if the payment is made at the beginning. So the
total amount which will be received is higher in case of beginning payments.
d)
The amount which will be available with the month beginning deposits is being calculated
below for proper understanding.
Particulars Amounts
PMT 100
Years 20
payments in one
year
12
Nper 240
rate 3.60%
Final value of
deposits
35179.22
There will be availability of €35179.22 at the end of the complete duration.

6
Question 3
a)
1 2 3 4 5
0
100
200
300
400
500
600
100 100.2 100.4004100.6012008100.8024032016100
200.5
301.5019
403.0076065
505.0190325211
Deposit
Interest
Closing balance
Period
Cash flows
Period(Month
end)
Opening
balance
Deposit Interes
t
Closing
balance
1 0 100 0 100
2 100 100.2 0.3 200.5
3 200.5 100.40 0.60 301.50
4 301.5019 100.60 0.90 403.01
5 403.0076 100.80 1.21 505.02
b)
The new deposit will be made at the end of the period but in that there will be regular
increment which will be made. The calculation has been made and is presented in the
appendix. The deposit is made which is increasing at 0.02% each month and this will provide
with the final value of €43692.01.
c)
In case the interest rate which is available and the growth rate are same then the future value
which will be available for the deposit will turn out to be 0. This is because there will be
discounting at 0% which will make all the values to be zero. The addition which will be made
will be eliminated with the interest rate and the net result will be zero.

7
Appendix
Question 3
Period(Month
end)
Opening
balance
Deposit Interest Closing
balance
1 0 100 0 100
2 100 100.2 0.3 200.5
3 200.5 100.40 0.60 301.50
4 301.501
9
100.60 0.90 403.01
5 403.007
6
100.80 1.21 505.02
6 505.019 101.00 1.52 607.54
7 607.538
1
101.21 1.82 710.57
8 710.566
7
101.41 2.13 814.11
9 814.106
9
101.61 2.44 918.16
10 918.160
4
101.81 2.75 1022.73
11 1022.72
9
102.02 3.07 1127.82
12 1127.81
6
102.22 3.38 1233.42
13 1233.42
1
102.43 3.70 1339.55
14 1339.54
8
102.63 4.02 1446.20
15 1446.19
8
102.84 4.34 1553.37
16 1553.37
3
103.04 4.66 1661.08
17 1661.07
6
103.25 4.98 1769.31
18 1769.30
8
103.45 5.31 1878.07
19 1878.07 103.66 5.63 1987.37
20 1987.36
7
103.87 5.96 2097.20
21 2097.19
8
104.08 6.29 2207.57
22 2207.56
6
104.29 6.62 2318.47

8
23 2318.47
4
104.49 6.96 2429.92
24 2429.92
3
104.70 7.29 2541.92
25 2541.91
6
104.91 7.63 2654.45
26 2654.45
3
105.12 7.96 2767.54
27 2767.53
9
105.33 8.30 2881.17
28 2881.17
3
105.54 8.64 2995.36
29 2995.36 105.75 8.99 3110.10
30 3110.1 105.97 9.33 3225.40
31 3225.39
5
106.18 9.68 3341.25
32 3341.24
9
106.39 10.02 3457.66
33 3457.66
2
106.60 10.37 3574.64
34 3574.63
7
106.82 10.72 3692.18
35 3692.17
7
107.03 11.08 3810.28
36 3810.28
3
107.24 11.43 3928.96
37 3928.95
7
107.46 11.79 4048.20
38 4048.20
2
107.67 12.14 4168.02
39 4168.01
9
107.89 12.50 4288.41
40 4288.41
1
108.10 12.87 4409.38
41 4409.38 108.32 13.23 4530.93
42 4530.92
8
108.54 13.59 4653.06
43 4653.05
8
108.75 13.96 4775.77
44 4775.77
1
108.97 14.33 4899.07
45 4899.06
9
109.19 14.70 5022.96
46 5022.95
6
109.41 15.07 5147.43
47 5147.43
2
109.63 15.44 5272.50
48 5272.50
1
109.85 15.82 5398.16

9
49 5398.16
4
110.07 16.19 5524.42
50 5524.42
4
110.29 16.57 5651.28
51 5651.28
3
110.51 16.95 5778.74
52 5778.74
3
110.73 17.34 5906.81
53 5906.80
6
110.95 17.72 6035.47
54 6035.47
5
111.17 18.11 6164.75
55 6164.75
2
111.39 18.49 6294.64
56 6294.63
9
111.62 18.88 6425.14
57 6425.13
8
111.84 19.28 6556.25
58 6556.25
2
112.06 19.67 6687.98
59 6687.98
4
112.29 20.06 6820.33
60 6820.33
4
112.51 20.46 6953.31
61 6953.30
6
112.74 20.86 7086.90
62 7086.90
2
112.96 21.26 7221.12
63 7221.12
5
113.19 21.66 7355.98
64 7355.97
6
113.41 22.07 7491.46
65 7491.45
8
113.64 22.47 7627.57
66 7627.57
3
113.87 22.88 7764.32
67 7764.32
3
114.10 23.29 7901.71
68 7901.71
2
114.32 23.71 8039.74
69 8039.74
1
114.55 24.12 8178.41
70 8178.41
3
114.78 24.54 8317.73
71 8317.73 115.01 24.95 8457.69
72 8457.69
5
115.24 25.37 8598.31
73 8598.30
9
115.47 25.79 8739.58

10
74 8739.57
6
115.70 26.22 8881.50
75 8881.49
7
115.93 26.64 9024.08
76 9024.07
6
116.17 27.07 9167.31
77 9167.31
4
116.40 27.50 9311.21
78 9311.21
4
116.63 27.93 9455.78
79 9455.77
9
116.86 28.37 9601.01
80 9601.01
1
117.10 28.80 9746.91
81 9746.91
2
117.33 29.24 9893.49
82 9893.48
5
117.57 29.68 10040.73
83 10040.7
3
117.80 30.12 10188.66
84 10188.6
6
118.04 30.57 10337.26
85 10337.2
6
118.27 31.01 10486.55
86 10486.5
5
118.51 31.46 10636.52
87 10636.5
2
118.75 31.91 10787.17
88 10787.1
7
118.98 32.36 10938.52
89 10938.5
2
119.22 32.82 11090.56
90 11090.5
6
119.46 33.27 11243.29
91 11243.2
9
119.70 33.73 11396.72
92 11396.7
2
119.94 34.19 11550.85
93 11550.8
5
120.18 34.65 11705.68
94 11705.6
8
120.42 35.12 11861.22
95 11861.2
2
120.66 35.58 12017.46
96 12017.4
6
120.90 36.05 12174.42
97 12174.4
2
121.14 36.52 12332.09
98 12332.0 121.39 37.00 12490.47

11
9
99 12490.4
7
121.63 37.47 12649.57
100 12649.5
7
121.87 37.95 12809.39
101 12809.3
9
122.12 38.43 12969.93
102 12969.9
3
122.36 38.91 13131.20
103 13131.2 122.60 39.39 13293.20
104 13293.2 122.85 39.88 13455.93
105 13455.9
3
123.10 40.37 13619.39
106 13619.3
9
123.34 40.86 13783.59
107 13783.5
9
123.59 41.35 13948.53
108 13948.5
3
123.84 41.85 14114.22
109 14114.2
2
124.08 42.34 14280.64
110 14280.6
4
124.33 42.84 14447.82
111 14447.8
2
124.58 43.34 14615.74
112 14615.7
4
124.83 43.85 14784.42
113 14784.4
2
125.08 44.35 14953.85
114 14953.8
5
125.33 44.86 15124.04
115 15124.0
4
125.58 45.37 15294.99
116 15294.9
9
125.83 45.88 15466.71
117 15466.7
1
126.08 46.40 15639.19
118 15639.1
9
126.33 46.92 15812.44
119 15812.4
4
126.59 47.44 15986.47
120 15986.4
7
126.84 47.96 16161.27
121 16161.2
7
127.09 48.48 16336.85
122 16336.8
5
127.35 49.01 16513.21
123 16513.2
1
127.60 49.54 16690.35

12
124 16690.3
5
127.86 50.07 16868.28
125 16868.2
8
128.11 50.60 17047.00
126 17047 128.37 51.14 17226.51
127 17226.5
1
128.63 51.68 17406.81
128 17406.8
1
128.88 52.22 17587.92
129 17587.9
2
129.14 52.76 17769.83
130 17769.8
3
129.40 53.31 17952.54
131 17952.5
4
129.66 53.86 18136.05
132 18136.0
5
129.92 54.41 18320.38
133 18320.3
8
130.18 54.96 18505.52
134 18505.5
2
130.44 55.52 18691.47
135 18691.4
7
130.70 56.07 18878.25
136 18878.2
5
130.96 56.63 19065.84
137 19065.8
4
131.22 57.20 19254.27
138 19254.2
7
131.49 57.76 19443.51
139 19443.5
1
131.75 58.33 19633.59
140 19633.5
9
132.01 58.90 19824.51
141 19824.5
1
132.28 59.47 20016.25
142 20016.2
5
132.54 60.05 20208.84
143 20208.8
4
132.81 60.63 20402.28
144 20402.2
8
133.07 61.21 20596.55
145 20596.5
5
133.34 61.79 20791.68
146 20791.6
8
133.60 62.38 20987.66
147 20987.6
6
133.87 62.96 21184.49
148 21184.4
9
134.14 63.55 21382.19

13
149 21382.1
9
134.41 64.15 21580.74
150 21580.7
4
134.68 64.74 21780.16
151 21780.1
6
134.95 65.34 21980.45
152 21980.4
5
135.22 65.94 22181.60
153 22181.6 135.49 66.54 22383.63
154 22383.6
3
135.76 67.15 22586.54
155 22586.5
4
136.03 67.76 22790.33
156 22790.3
3
136.30 68.37 22995.00
157 22995 136.57 68.98 23200.56
158 23200.5
6
136.85 69.60 23407.00
159 23407 137.12 70.22 23614.35
160 23614.3
5
137.39 70.84 23822.58
161 23822.5
8
137.67 71.47 24031.72
162 24031.7
2
137.94 72.10 24241.76
163 24241.7
6
138.22 72.73 24452.70
164 24452.7 138.50 73.36 24664.56
165 24664.5
6
138.77 73.99 24877.33
166 24877.3
3
139.05 74.63 25091.01
167 25091.0
1
139.33 75.27 25305.61
168 25305.6
1
139.61 75.92 25521.14
169 25521.1
4
139.89 76.56 25737.59
170 25737.5
9
140.17 77.21 25954.96
171 25954.9
6
140.45 77.86 26173.28
172 26173.2
8
140.73 78.52 26392.52
173 26392.5
2
141.01 79.18 26612.71
174 26612.7
1
141.29 79.84 26833.84
175 26833.8 141.57 80.50 27055.92