You have been accepted to study gourmet cooking at Le Cordon Bleu Culinary Institute in Paris, France.You will need $15 000 every six months (beginning six months from now)for the next three years to cover tuition and living expenses.Your parents have agreed to pay for your education.They want to make one deposit now in a bank account earning 6% interest, compounded semi-annually, so that you can withdraw $15 000 every six months for the next three years.How much must they deposit now?

Question

Quizplus · Accepted Answer

To find the present value of an annuity (the amount that needs to be deposited now), we use the formula for the present value of an annuity: $PV = PMT \times \left(\frac{1 - (1 + r)^{-n}}{r}\right)$, where $PMT$ is the payment amount per period, $r$ is the interest rate per period, and $n$ is the total number of payments.Given:- $PMT = \$15,000$- Annual interest rate = 6%, so the semi-annual rate $r = 0.06 / 2 = 0.03$ or 3%- Payments are made every six months for three years, so $n = 3 \times 2 = 6$Plugging these values into the formula:$$PV = \$15,000 \times \left(\frac{1 - (1 + 0.03)^{-6}}{0.03}\right)$$$$PV = \$15,000 \times \left(\frac{1 - (1.03)^{-6}}{0.03}\right)$$$$PV = \$15,000 \times 5.4172$$ (approx.)$$PV = \$81,258$$ (approx.)Therefore, they must deposit approximately \$81,258 now.

You Have Been Accepted to Study Gourmet Cooking at Le