An annuity has end of quarter payments for 15 years, and the payment at the end
ID: 2725731 • 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.I've already computed the effective interest rate I=1.5075125 but I cannot figure out the balance.
the correct answer in the back of the book is $251,477.70 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.
I've already computed the effective interest rate I=1.5075125 but I cannot figure out the balance.
the correct answer in the back of the book is $251,477.70
I've already computed the effective interest rate I=1.5075125 but I cannot figure out the balance.
the correct answer in the back of the book is $251,477.70
Explanation / Answer
The balance is calculated with the use of following table:
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
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