10.6 Ordinary Annuities, Sinking Funds, and Retirement Investments 643 To determine the accumulated amount of an ordinary annuity, we could make many individual calculations using the compound interest formula discussed in Section 10.3. A better way is to use a computer to generate a spreadsheet. A portion of such a spreadsheet is shown in Table 10.6. This spreadsheet was generated using the data from the example involving the Brabsons. Note that the Brabsons would make monthly payments for 30 years for a total of 360 payments. Also note that had the Brabsons simply put $100 in their cookie jar each month, their total amount at the end of 30 years would have been $100 360, × or $36,000. Instead, by investing in an interest-bearing annuity, they will have accumulated $83,225.86 after 30 years. This amount is circled in blue in Table 10.6. Table 10.6 Ordinary Annuity Growth Accumulated Amount in an Ordinary Annuity with $100 Monthly Investments with Interest Rate 5% Compounded Monthly Payment Number Monthly Investment Interest Accumulated Amount 1 $100.00 $ — $ 100.00 2 $100.00 $ 0.42 $ 200.42 3 $100.00 $ 0.84 $ 301.25 4 $100.00 $ 1.26 $ 402.51 5 $100.00 $ 1.68 $ 504.18 6 $100.00 $ 2.10 $ 606.28 7 $100.00 $ 2.53 $ 708.81 8 $100.00 $ 2.95 $ 811.76 9 $100.00 $ 3.38 $ 915.15 10 $100.00 $ 3.81 $ 1018.96 351 $100.00 $328.58 $79,287.40 352 $100.00 $330.36 $79,717.76 353 $100.00 $332.16 $80,149.92 354 $100.00 $333.96 $80,583.88 355 $100.00 $335.77 $81,019.65 356 $100.00 $337.58 $81,457.23 357 $100.00 $339.41 $81,896.63 358 $100.00 $341.24 $82,337.87 359 $100.00 $343.07 $82,780.94 360 $100.00 $344.92 $83,225.86 Another way, and perhaps a more efficient way, to calculate the accumulated amount in an ordinary annuity is to use the ordinary annuity formula. Ordinary Annuity Formula The accumulated amount, A, of an ordinary annuity with payments of p dollars made n times per year, for t years, at interest rate r as a decimal number, compounded at the end of each payment period is given by the formula A p r n r n 1 1 n t ( ) = ⎛ + ⎝ ⎞ ⎠ − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎛ ⎝ ⎞ ⎠ ⋅
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