Compute the future value of $6,750, which is invested for 65 years at an annual rate of return of 8%.
The correct answer and explanation is:
The future value (FV) of an investment can be calculated using the formula: FV=PV×(1+r)tFV = PV \times (1 + r)^t
Where:
- PVPV = Present Value (initial investment) = 6,750 dollars
- rr = annual interest rate (expressed as a decimal) = 0.08
- tt = number of years = 65
Substitute the values: FV=6750×(1+0.08)65FV = 6750 \times (1 + 0.08)^{65}
First, calculate the growth factor: 1+0.08=1.081 + 0.08 = 1.08
Then raise it to the power of 65: 1.08651.08^{65}
Using a calculator: 1.0865≈374.4861.08^{65} \approx 374.486
Now multiply by the initial investment: FV=6750×374.486≈2,527,813FV = 6750 \times 374.486 \approx 2,527,813
Future Value ≈ $2,527,813
Explanation
The future value formula accounts for compound interest, which means the investment earns interest on both the initial amount and the accumulated interest over time. Here, the 8% annual return compounds every year for 65 years. Because of this compounding effect, even a moderate interest rate can lead to a very large future value when the investment horizon is long.
The factor (1+r)t(1 + r)^t shows how much $1 invested today would grow over the time period. In this case, $1 grows roughly 374 times over 65 years at 8% interest. Multiplying this by the initial $6,750 yields the total accumulated amount.
Long-term investments benefit greatly from compound interest, which makes investing early and maintaining the investment a powerful strategy for wealth growth. Even though 8% might seem modest, the time span of 65 years creates an exponential growth effect, turning thousands of dollars into millions.
This demonstrates the importance of time in investing. The longer the money is invested at a positive rate of return, the greater the power of compounding to multiply the initial investment, resulting in significant wealth accumulation over decades.