You receive five annual payments of $10,000 followed by five
annual payments of $5,000. what is the future value of all payments
at the time of the last payment when the interest rate is 7%
Using the formula for the future value of an annuity:
FV = P * ((1 + r)^n - 1) / r
where:
P = payment amount
r = interest rate
n = number of payments
For the first set of payments:
FV1 = 10,000 * ((1 + 0.07)^5 - 1) / 0.07 = $63,589.32
For the second set of payments:
FV2 = 5,000 * ((1 + 0.07)^5 - 1) / 0.07 = $31,794.66
The total future value of all payments is:
FV = FV1 + FV2 = $95,383.98
To calculate the future value of these cash flows, we can use the formula for the future value of an ordinary annuity:
Future Value = P * [(1 + r)^n - 1] / r
Where:
- P is the payment amount per period ($10,000 for the first five payments and $5,000 for the next five payments),
- r is the interest rate per period (7% = 0.07), and
- n is the number of periods (10 in this case, since there are five $10,000 payments and five $5,000 payments).
First, let's calculate the future value of the first five payments:
Future Value of the first five payments = $10,000 * [(1 + 0.07)^5 - 1] / 0.07
Using a calculator:
Future Value of the first five payments = $10,000 * (1.07^5 - 1) / 0.07
Future Value of the first five payments ≈ $57,675.41
Next, let's calculate the future value of the next five payments:
Future Value of the next five payments = $5,000 * [(1 + 0.07)^5 - 1] / 0.07
Using a calculator:
Future Value of the next five payments = $5,000 * (1.07^5 - 1) / 0.07
Future Value of the next five payments ≈ $24,672.98
Now, we can add these two future values together to find the total future value of all payments:
Total Future Value = Future Value of the first five payments + Future Value of the next five payments
Total Future Value ≈ $57,675.41 + $24,672.98
Total Future Value ≈ $82,348.39
Therefore, the future value of all payments at the time of the last payment is approximately $82,348.39.