How much would $500 invested at 5% interest compounded continuously be
worth after 8 years? Round your answer to the nearest cent.
A(t)- Pet
The Correct Answer and Explanation is :
To find the future value of an investment with continuous compounding interest, we can use the formula:
[
A(t) = Pe^{rt}
]
Where:
- ( A(t) ) is the amount of money accumulated after time ( t ),
- ( P ) is the principal amount (initial investment),
- ( r ) is the annual interest rate (as a decimal),
- ( t ) is the time the money is invested for (in years),
- ( e ) is the base of the natural logarithm (approximately equal to 2.71828).
Given:
- ( P = 500 ) (the initial investment),
- ( r = 0.05 ) (5% interest rate),
- ( t = 8 ) years.
Plugging in the values:
[
A(t) = 500 \times e^{0.05 \times 8}
]
Calculating ( e^{0.05 \times 8} ):
[
e^{0.4} \approx 1.49182
]
Now substituting this value back into the equation:
[
A(t) \approx 500 \times 1.49182 \approx 745.91
]
Thus, the amount after 8 years would be approximately $745.91.
Explanation
Continuous compounding occurs when interest is calculated and added to the principal continuously rather than at fixed intervals. This method of compounding results in higher returns compared to standard compounding (annually, semi-annually, etc.), as it allows for interest to be earned on previously accrued interest at every instant.
In this scenario, investing $500 at a 5% annual interest rate for 8 years means that the investment grows exponentially. The use of the natural base ( e ) in the formula reflects the idea that the accumulation of interest is ongoing.
As we calculated, the investment grows to $745.91 after 8 years due to the effects of continuous compounding. This demonstrates how powerful the effects of compounding can be over time. It encourages saving and investing early, as the earlier money is invested, the more time it has to grow. Thus, understanding the impact of different compounding methods is crucial for effective financial planning and investment strategies.