You need $28,974 at the end of 10 years, and your only investment outlet is an 8 percent long-term certificate of deposit (compounded annually). With the certificate of deposit, you make an initial investment at the beginning of the first year.
A. What single payment could be made at the beginning of the first year to achieve this objective?
B. What amount could you pay at the end of each year annually for 10 years to achieve this same objective?
To calculate the answers to these two questions, we need to use the formula for the future value of an investment:
Future Value = Present Value x (1 + Interest Rate)^Number of Periods
A. To find the single payment required at the beginning of the first year, we need to solve for the Present Value. The Future Value is given as $28,974, the Interest Rate is 8% (or 0.08 as a decimal), and the Number of Periods is 10 years. Plugging these values into the formula, we have:
$28,974 = Present Value x (1 + 0.08)^10
We can rearrange the equation to solve for the Present Value by dividing both sides by (1 + 0.08)^10:
Present Value = $28,974 / (1 + 0.08)^10
Using a calculator, we find that the Present Value required is approximately $14,190.41.
Therefore, the single payment that could be made at the beginning of the first year to achieve the objective of $28,974 at the end of 10 years is $14,190.41.
B. To find the amount that could be paid at the end of each year for 10 years, we need to solve for the annuity payment. Again, the Future Value is $28,974, the Interest Rate is 8%, and the Number of Periods is 10 years. Plugging these values into the annuity formula, we have:
$28,974 = Payment x [((1 + 0.08)^10 - 1) / 0.08]
We can rearrange the equation to solve for the Payment by multiplying both sides by 0.08 and dividing by ((1 + 0.08)^10 - 1):
Payment = ($28,974 x 0.08) / ((1 + 0.08)^10 - 1)
Using a calculator, we find that the annuity payment required is approximately $2,245.70.
Therefore, the amount that could be paid at the end of each year annually for 10 years to achieve the objective of $28,974 at the end of 10 years is $2,245.70.