Final answer:
The future values of ordinary annuities and annuities due for different interest rates and time periods are calculated using the formula for compound interest. The example calculations demonstrate the substantial impact of compound interest over time, even when starting with small amounts of money. Annuities due calculations take into account the earlier payment timeline by multiplying an additional interest factor.
Step-by-step explanation:
The question involves calculating the future value of ordinary annuities and annuities due, using different interest rates and time periods. It requires understanding the concept of compound interest and applying the future value formula of annuities. Here are the solutions to the given scenarios:
a. Future value of $1,000 per year for 16 years at 14%:
FV = $1,000 × [(1 + 0.14)¹⁶- 1] / 0.14
FV = $1,000 × [5.5071 - 1] / 0.14
FV = $1,000 × [4.5071] / 0.14
FV ≈ $1,000 × 32.1936
FV ≈ $32,193.57
b. Future value of $500 per year for 8 years at 7%:
FV = $500 × [(1 + 0.07)⁸ - 1] / 0.07
FV = $500 × [1.7182 - 1] / 0.07
FV = $500 × [0.7182] / 0.07
FV ≈ $500 × 10.2600
FV ≈ $5,130.00
c. Future value of $1,000 per year for 8 years at 0%:
Since the interest rate is 0%, the future value is simply the sum of the payments.
FV = $1,000 × 8
FV = $8,000.00
When these are calculated as annuities due, an additional factor of (1 + interest rate) is multiplied to each calculation to account for the immediate interest accrual as payments are made at the beginning of each period.