Gretta wants to retire in thirteen years. At that time, she wants to be able to withdraw $12,500 at the end of each six months for thirteen years. Assume that money can be deposited at 12 percent per year, compounded semiannually. What exact amount will Gretta need in thirteen years?

Quick answer:

On the thirteenth year, Gretta should have an amount of $162,539.58 in her account to be able to withdraw $12,500 at the end of each six months starting from the thirteenth year to twenty-sixth year.

Expert Answers

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For this problem, a periodic payment has to be made. So to solve this, a formula for annuity should be applied.

It says that on the thirteenth year, Gretta wants to retire. Starting from this year until the twenty-sixth year, she wants to withdraw $12,500 from her account at the end of each six months.

To determine the amount of money she should have in her account on the thirteenth year (start the year of this annuity), the formula for the present value of annuity should be applied. Take note that present value of annuity refers to the amount needed at the beginning to be able to fund future periodic payments. The formula is

`P = (PMT * [1-(1+r/n)^(-nt)])/(r/n)`

where P is the present value, PMT is the amount of equal payments, r is the rate in decimal form, n is the number of compounding periods and t is the number of years.

The periodic payment is PMT = $12500. The interest rate is 12 percent, so r = 0.12. The rate is compounded semiannually, so n = 2. And the number of years is t = 13. Plugging in these known values, the formula becomes

`P= (12500 *[1 -(1+0.12/2)^(-2*13)])/(0.12/2)`

This will simplify to

`P=(12500*(1- 1.06^(-26)))/0.06`

`P=162 539.58`

Therefore, Gretta should have $162,539.58 in thirteen years.

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