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# A Deferred Annuity

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A young married couple intends to save some money for their retirement at age 65. They began their retirement plan by depositing a fixed amount of money annually for 5 years, and they would skip the next 5 years. They would continue with this scheme for 40 years. In other words, 5 years on and 5 years off. If they intend to save \$500,000 and can earn 5% compounded annually, how much money do they expect to deposit each active year until their retirement at 65?. Thank you for any help.

Aug 6, 2017

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A problem, such as this, requires a multi-step solution.
First, find the FV of \$1 per period for the first 5 years @ 5% using this common formula: FV =\$1 x (1.05^5 - 1) / 0.05=\$5.53. Then project this amount forward for the second 5-year period using this second common FV formula: FV = PV x 1.05^5 =\$5.53 x 1.2762815625=\$7.05. Then continue this process for eight 5-year periods. You should get the following amounts:
At the end of the first 5-year period =\$5.53
At the end of the second 5-year period=\$7.05
At the end of the third 5-year period =\$14.53
At the end of the fourth 5-year period=\$18.54
At the end of the fifth 5-year period =\$29.19
At the end of the sixth 5-year period=\$37.25
At the end of the seventh 5-year period=\$53.07
At the end of the eighth 5-year period =\$67.73
This last amount shown to 10 decimal places=\$67.7308673760.
Since the young couple's wish is to have \$500,000 at 65, then we just divide: \$500,000 / \$67.7308673760 =\$7,382.16 - which is their expected annual payment into their retirement fund.

Aug 7, 2017