An annuity has end of quarter payments for 15 years and the payment at the end o
ID: 2725685 • Letter: A
Question
An annuity has end of quarter payments for 15 years and the payment at the end of the j-th quarter (j=1,2,....,60) is $100j. The payments are made directly into a savings account with a nominal interest rate of 6% payable monthly and they are left in the account. find the effective interest rate for a quarter and use it to compute the balance in the savings account immediately after the last payment.the correct answer is 1.50751% for the effective quarterly interest and $251,477.70 for the accumulated value An annuity has end of quarter payments for 15 years and the payment at the end of the j-th quarter (j=1,2,....,60) is $100j. The payments are made directly into a savings account with a nominal interest rate of 6% payable monthly and they are left in the account. find the effective interest rate for a quarter and use it to compute the balance in the savings account immediately after the last payment.
the correct answer is 1.50751% for the effective quarterly interest and $251,477.70 for the accumulated value
the correct answer is 1.50751% for the effective quarterly interest and $251,477.70 for the accumulated value
Explanation / Answer
Quarters (j)
Payment
No. of quarters left
FV factor @1.50751%
FV
1
$ 100.00
59
2.417643715
$ 241.76
2
$ 200.00
58
2.381738765
$ 476.35
3
$ 300.00
57
2.346367047
$ 703.91
4
$ 400.00
56
2.311520642
$ 924.61
5
$ 500.00
55
2.277191749
$ 1,138.60
6
$ 600.00
54
2.243372681
$ 1,346.02
7
$ 700.00
53
2.210055868
$ 1,547.04
8
$ 800.00
52
2.17723385
$ 1,741.79
9
$ 900.00
51
2.144899279
$ 1,930.41
10
$ 1,000.00
50
2.113044916
$ 2,113.04
11
$ 1,100.00
49
2.081663628
$ 2,289.83
12
$ 1,200.00
48
2.050748391
$ 2,460.90
13
$ 1,300.00
47
2.020292283
$ 2,626.38
14
$ 1,400.00
46
1.990288485
$ 2,786.40
15
$ 1,500.00
45
1.96073028
$ 2,941.10
16
$ 1,600.00
44
1.93161105
$ 3,090.58
17
$ 1,700.00
43
1.902924277
$ 3,234.97
18
$ 1,800.00
42
1.874663536
$ 3,374.39
19
$ 1,900.00
41
1.846822502
$ 3,508.96
20
$ 2,000.00
40
1.819394942
$ 3,638.79
21
$ 2,100.00
39
1.792374714
$ 3,763.99
22
$ 2,200.00
38
1.765755769
$ 3,884.66
23
$ 2,300.00
37
1.739532148
$ 4,000.92
24
$ 2,400.00
36
1.713697979
$ 4,112.88
25
$ 2,500.00
35
1.68824748
$ 4,220.62
26
$ 2,600.00
34
1.663174951
$ 4,324.25
27
$ 2,700.00
33
1.63847478
$ 4,423.88
28
$ 2,800.00
32
1.614141436
$ 4,519.60
29
$ 2,900.00
31
1.590169473
$ 4,611.49
30
$ 3,000.00
30
1.566553522
$ 4,699.66
31
$ 3,100.00
29
1.543288296
$ 4,784.19
32
$ 3,200.00
28
1.520368588
$ 4,865.18
33
$ 3,300.00
27
1.497789265
$ 4,942.70
34
$ 3,400.00
26
1.475545272
$ 5,016.85
35
$ 3,500.00
25
1.45363163
$ 5,087.71
36
$ 3,600.00
24
1.432043432
$ 5,155.36
37
$ 3,700.00
23
1.410775845
$ 5,219.87
38
$ 3,800.00
22
1.389824108
$ 5,281.33
39
$ 3,900.00
21
1.369183529
$ 5,339.82
40
$ 4,000.00
20
1.348849488
$ 5,395.40
41
$ 4,100.00
19
1.328817432
$ 5,448.15
42
$ 4,200.00
18
1.309082877
$ 5,498.15
43
$ 4,300.00
17
1.289641404
$ 5,545.46
44
$ 4,400.00
16
1.27048866
$ 5,590.15
45
$ 4,500.00
15
1.251620358
$ 5,632.29
46
$ 4,600.00
14
1.233032274
$ 5,671.95
47
$ 4,700.00
13
1.214720244
$ 5,709.19
48
$ 4,800.00
12
1.196680171
$ 5,744.06
49
$ 4,900.00
11
1.178908015
$ 5,776.65
50
$ 5,000.00
10
1.161399797
$ 5,807.00
51
$ 5,100.00
9
1.144151597
$ 5,835.17
52
$ 5,200.00
8
1.127159554
$ 5,861.23
53
$ 5,300.00
7
1.110419864
$ 5,885.23
54
$ 5,400.00
6
1.093928778
$ 5,907.22
55
$ 5,500.00
5
1.077682605
$ 5,927.25
56
$ 5,600.00
4
1.061677707
$ 5,945.40
57
$ 5,700.00
3
1.045910502
$ 5,961.69
58
$ 5,800.00
2
1.030377459
$ 5,976.19
59
$ 5,900.00
1
1.0150751
$ 5,988.94
60
$ 6,000.00
0
1
$ 6,000.00
$ 251,477.56
The above table shows the calculation.
FV factor = ( 1+ quarterly rate)^(No. of quarters left)
Adding all the FVs, we will get $251477.56
Quarters (j)
Payment
No. of quarters left
FV factor @1.50751%
FV
1
$ 100.00
59
2.417643715
$ 241.76
2
$ 200.00
58
2.381738765
$ 476.35
3
$ 300.00
57
2.346367047
$ 703.91
4
$ 400.00
56
2.311520642
$ 924.61
5
$ 500.00
55
2.277191749
$ 1,138.60
6
$ 600.00
54
2.243372681
$ 1,346.02
7
$ 700.00
53
2.210055868
$ 1,547.04
8
$ 800.00
52
2.17723385
$ 1,741.79
9
$ 900.00
51
2.144899279
$ 1,930.41
10
$ 1,000.00
50
2.113044916
$ 2,113.04
11
$ 1,100.00
49
2.081663628
$ 2,289.83
12
$ 1,200.00
48
2.050748391
$ 2,460.90
13
$ 1,300.00
47
2.020292283
$ 2,626.38
14
$ 1,400.00
46
1.990288485
$ 2,786.40
15
$ 1,500.00
45
1.96073028
$ 2,941.10
16
$ 1,600.00
44
1.93161105
$ 3,090.58
17
$ 1,700.00
43
1.902924277
$ 3,234.97
18
$ 1,800.00
42
1.874663536
$ 3,374.39
19
$ 1,900.00
41
1.846822502
$ 3,508.96
20
$ 2,000.00
40
1.819394942
$ 3,638.79
21
$ 2,100.00
39
1.792374714
$ 3,763.99
22
$ 2,200.00
38
1.765755769
$ 3,884.66
23
$ 2,300.00
37
1.739532148
$ 4,000.92
24
$ 2,400.00
36
1.713697979
$ 4,112.88
25
$ 2,500.00
35
1.68824748
$ 4,220.62
26
$ 2,600.00
34
1.663174951
$ 4,324.25
27
$ 2,700.00
33
1.63847478
$ 4,423.88
28
$ 2,800.00
32
1.614141436
$ 4,519.60
29
$ 2,900.00
31
1.590169473
$ 4,611.49
30
$ 3,000.00
30
1.566553522
$ 4,699.66
31
$ 3,100.00
29
1.543288296
$ 4,784.19
32
$ 3,200.00
28
1.520368588
$ 4,865.18
33
$ 3,300.00
27
1.497789265
$ 4,942.70
34
$ 3,400.00
26
1.475545272
$ 5,016.85
35
$ 3,500.00
25
1.45363163
$ 5,087.71
36
$ 3,600.00
24
1.432043432
$ 5,155.36
37
$ 3,700.00
23
1.410775845
$ 5,219.87
38
$ 3,800.00
22
1.389824108
$ 5,281.33
39
$ 3,900.00
21
1.369183529
$ 5,339.82
40
$ 4,000.00
20
1.348849488
$ 5,395.40
41
$ 4,100.00
19
1.328817432
$ 5,448.15
42
$ 4,200.00
18
1.309082877
$ 5,498.15
43
$ 4,300.00
17
1.289641404
$ 5,545.46
44
$ 4,400.00
16
1.27048866
$ 5,590.15
45
$ 4,500.00
15
1.251620358
$ 5,632.29
46
$ 4,600.00
14
1.233032274
$ 5,671.95
47
$ 4,700.00
13
1.214720244
$ 5,709.19
48
$ 4,800.00
12
1.196680171
$ 5,744.06
49
$ 4,900.00
11
1.178908015
$ 5,776.65
50
$ 5,000.00
10
1.161399797
$ 5,807.00
51
$ 5,100.00
9
1.144151597
$ 5,835.17
52
$ 5,200.00
8
1.127159554
$ 5,861.23
53
$ 5,300.00
7
1.110419864
$ 5,885.23
54
$ 5,400.00
6
1.093928778
$ 5,907.22
55
$ 5,500.00
5
1.077682605
$ 5,927.25
56
$ 5,600.00
4
1.061677707
$ 5,945.40
57
$ 5,700.00
3
1.045910502
$ 5,961.69
58
$ 5,800.00
2
1.030377459
$ 5,976.19
59
$ 5,900.00
1
1.0150751
$ 5,988.94
60
$ 6,000.00
0
1
$ 6,000.00
$ 251,477.56