The answer is: (a) The effective quartely interest rate is about 1.50751%, and t

Question

The answer is:

(a) The effective quartely interest rate is about 1.50751%, and the accumu- lated value is $251,477.70.

(b) Again, the accumulated value is $251,477.70.

(1) An annuity has end of quarter payments for fifteen 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. (a) Find the effective interest rate for a quarter, and use it to compute the balance in the savings account immediately after the last payment. (b) Use Fact (4.4.3) to recalculate the balance in the savings account imme- diately after the last payment. Make sure that your answer agrees with your answer to part (a), and note which method you found easier. FACT 4.4.3 Let k divide n. The accumulated value at the time of the last payment of an annuity lasting n interest periods, having a payment at the end of each k interest periods, and having the j -th payment be for an amount j is The value of this annuity one payment period before the first payment is

Explanation / Answer

The balance is calculated with the use of following table:

Quarter Effective Interest Rate (B) Payment at the End of Jth Quarter (C) Payment (A*B) FV (Payment*(1+Effective Interest Rate)^(60 – Quarter))
1 1.51% 100 100 241.76
2 1.51% 100 200 476.35
3 1.51% 100 300 703.91
4 1.51% 100 400 924.61
5 1.51% 100 500 1,138.60
6 1.51% 100 600 1,346.03
7 1.51% 100 700 1,547.04
8 1.51% 100 800 1,741.79
9 1.51% 100 900 1,930.41
10 1.51% 100 1,000 2,113.05
11 1.51% 100 1,100 2,289.83
12 1.51% 100 1,200 2,460.90
13 1.51% 100 1,300 2,626.38
14 1.51% 100 1,400 2,786.41
15 1.51% 100 1,500 2,941.10
16 1.51% 100 1,600 3,090.58
17 1.51% 100 1,700 3,234.97
18 1.51% 100 1,800 3,374.40
19 1.51% 100 1,900 3,508.97
20 1.51% 100 2,000 3,638.79
21 1.51% 100 2,100 3,763.99
22 1.51% 100 2,200 3,884.67
23 1.51% 100 2,300 4,000.93
24 1.51% 100 2,400 4,112.88
25 1.51% 100 2,500 4,220.62
26 1.51% 100 2,600 4,324.26
27 1.51% 100 2,700 4,423.89
28 1.51% 100 2,800 4,519.60
29 1.51% 100 2,900 4,611.49
30 1.51% 100 3,000 4,699.66
31 1.51% 100 3,100 4,784.20
32 1.51% 100 3,200 4,865.18
33 1.51% 100 3,300 4,942.71
34 1.51% 100 3,400 5,016.86
35 1.51% 100 3,500 5,087.71
36 1.51% 100 3,600 5,155.36
37 1.51% 100 3,700 5,219.87
38 1.51% 100 3,800 5,281.33
39 1.51% 100 3,900 5,339.82
40 1.51% 100 4,000 5,395.40
41 1.51% 100 4,100 5,448.15
42 1.51% 100 4,200 5,498.15
43 1.51% 100 4,300 5,545.46
44 1.51% 100 4,400 5,590.15
45 1.51% 100 4,500 5,632.29
46 1.51% 100 4,600 5,671.95
47 1.51% 100 4,700 5,709.19
48 1.51% 100 4,800 5,744.07
49 1.51% 100 4,900 5,776.65
50 1.51% 100 5,000 5,807.00
51 1.51% 100 5,100 5,835.17
52 1.51% 100 5,200 5,861.23
53 1.51% 100 5,300 5,885.23
54 1.51% 100 5,400 5,907.22
55 1.51% 100 5,500 5,927.26
56 1.51% 100 5,600 5,945.40
57 1.51% 100 5,700 5,961.69
58 1.51% 100 5,800 5,976.19
59 1.51% 100 5,900 5,988.94
60 1.51% 100 6,000 6,000.00
Total $2,51,477.70

Notes:

1) The payment for each quarter will be calculated by multiplying the quarter with 100. For Instance, the payment for 1st, second and third quarter will be 100 (100*1), 200 (100 *2) and 300 (100*3) and so on.

2) The future value of the payment made in each quarter will be calculated with the use of following formula:

Future Value = FV*(1+Effective Rate of Interest)^(60 - Quarter)

For Instance the future value of payment made in 1st and second quarter will be calculated as follows:

Future Value (Quarter 1 Payment) = 100*(1+1.1.5075125 or 1.51%)^(60 - 1) = $241.76

Future Value (Quarter 2 Payment) = 200*(1+1.51%)^(60 -2) = $476.35

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