How long would it take an investment to double if at 6.5% compounded continuously?
The correct answer and explanation is:
Correct Answer:
It would take approximately 10.66 years for the investment to double at a 6.5% interest rate compounded continuously.
Explanation:
To determine how long it takes for an investment to double under continuous compounding, the formula used is:
A = P * e^(rt)
Where:
- A is the final amount
- P is the initial principal
- r is the interest rate (as a decimal)
- t is the time in years
- e is Euler’s number (approximately 2.71828)
If the investment doubles, then A = 2P. Substituting this into the formula:
2P = P * e^(0.065t)
Divide both sides by P:
2 = e^(0.065t)
To solve for t, take the natural logarithm (ln) of both sides:
ln(2) = 0.065t
Using a calculator, ln(2) ≈ 0.6931:
0.6931 = 0.065t
Now solve for t:
t = 0.6931 / 0.065
t ≈ 10.66 years
This result shows how long it takes for the value of an investment to double when interest is compounded continuously at a fixed annual rate. Continuous compounding means that interest is being added at every possible moment, resulting in slightly faster growth than traditional compounding intervals like yearly or monthly.
The higher the interest rate, the faster the investment will double. This doubling time is often estimated using the Rule of 72, where 72 is divided by the interest rate. For 6.5%, 72 ÷ 6.5 ≈ 11.08 years. Although close, this rule gives only an estimate, while the formula using continuous compounding provides a more accurate value.