Question

Time Value of Money: 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

Answer 1

$13,420.57

Explanation:

Since interest is compounded annually, use the formula
`A=P(1+r)^t`.
A is the final Amount in the account;
P is the Principal amount (initial investment);
r is the interest Rate, as a decimal; and
t is the Time in years.

1. Substitute the right numbers in the right spots:

`A=P(1+r)^t`

`28974=P(1+.08)^(10)`

2. Simplify the right-hand side of the equation:

`28974=P(1.08)^(10)`

3. Solve for P:

`P=(28974)/(1.08^(10))`

4. Round to the nearest penny (hundredth): P≈13420.57

Answer 2

$13,420.57

Explanation:

To determine the single payment that could be made at the beginning of the first year to achieve the objective of having $28,974 at the end of 10 years with an 8 percent long-term certificate of deposit compounded annually, we can use the future value formula for compound interest:

`\sf FV = PV * (1 + r)^n`

where:

-
`\sf FV` is the future value of the investment (in this case, $28,974),

-
`\sf PV` is the present value or initial investment (what we want to find),

-
`\sf r` is the interest rate per compounding period (8 percent or 0.08), and

-
`\sf n` is the number of compounding periods (10 years).

Rearrange the formula to solve for
`\sf PV`:

`\sf PV = (FV)/((1 + r)^n)`

Now, plug in the values:

`\sf PV = (28,974)/((1 + 0.08)^(10))`

Calculate the expression:

`\sf PV = (28,974)/((1.08)^(10))`

`\sf PV = (28,974)/(2.158924997 )`

`\sf PV \approx 13420.56812`

`\sf PV \approx 13420.57 \textsf{( in 2 d.p.)}`

So, the single payment that could be made at the beginning of the first year to achieve the objective is approximately $13,420.57.