Irene plans to retire on January 1, 2020. She has been preparing to retire by making annual deposits, starting on January 1, 1980, of 2300 dollars into an account that pays an effective rate of interest of 8.4 percent. She has continued this practice every year through January 1, 2001. Her goal is to have 1.35 million dollars saved up at the time of her retirement. How large should her annual deposits be (from January 1, 2002 until January 1, 2020) so that she can reach her goal

$16,876

Explanation:

first we have to calculate how much money Irene saved until January 1, 2001:

P = PMT ×   [(1 + r)ⁿ - 1] / r

P = 2,300 ×   [(1 + 8.4%)²² - 1] / 8.4% = $134,089

if she stops making any more payments, in 19 years those $134,089 will be worth:

FV = PV x (1 + r)ⁿ

FV = 134,089 x (1 + 8.4%)¹⁹ = $620,797

that means she still needs to get $1,350,000 - $620,797 = $729,203

we can use the first formula to determine the payments she will need to make during the next 19 years:

P = PMT ×   [(1 + r)ⁿ - 1] / r

PMT = P /  {[(1 + r)ⁿ - 1] / r}

PMT = 729,203 / {[(1 + 8.4%)¹⁹ - 1] / 8.4%} = 729,203 / 43.21 = $16,876

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