what is the interest rate necessary for an investment to quadruple after 5 years of continuous compound interest?
I keep getting r=11%
with interest rate r, continuous compounding for t years results in a growth factor of e^rt. So, you want
e^5r = 4
r = ln4/5 = .277, or 27.7%
Which formula were you using to get 11% ?
To find the interest rate necessary for an investment to quadruple after 5 years of continuous compound interest, we can use the formula for compound interest:
A = P * e^(rt)
Where:
A is the final amount (quadruple the initial amount)
P is the initial amount
e is Euler's number (approximately 2.71828)
r is the interest rate
t is the time in years
In this case, we want to find the interest rate "r". Given that the initial amount is P and the final amount is 4 times the initial amount, we have:
4P = P * e^(rt)
By dividing both sides by P, we get:
4 = e^(rt)
To solve for "r", we can take the natural logarithm of both sides:
ln(4) = rt * ln(e)
Since ln(e) is equal to 1, the equation simplifies to:
ln(4) = rt
Now we can divide both sides by t to isolate "r":
r = ln(4) / t
Plugging in the values, t = 5 (years), we can calculate the interest rate:
r = ln(4) / 5
Using a calculator, we find that ln(4) is approximately 1.38629:
r = 1.38629 / 5
This gives us the interest rate approximately equal to 0.277258, or 27.73% (rounded to two decimal places). Therefore, the interest rate necessary for the investment to quadruple after 5 years of continuous compound interest is approximately 27.73%.
To determine the interest rate necessary for an investment to quadruple after 5 years of continuous compound interest, we can use the compound interest formula:
A = P * (1 + r/n)^(n*t)
Where:
A is the future value of the investment.
P is the principal (initial investment).
r is the interest rate.
n is the number of times interest is compounded per year.
t is the time in years.
In this case, we want the investment to quadruple, which means the future value (A) will be four times the principal (P).
A = 4P
We also know that the investment is compounded continuously, which means n approaches infinity. In this case, we can use the continuous compound interest formula:
A = P * e^(r*t)
Where e is Euler's number (approximately 2.71828).
Now, let's substitute the known values into the formula:
4P = P * e^(r*5)
Dividing both sides by P:
4 = e^(r*5)
To solve for r, we need to take the natural logarithm (ln) of both sides:
ln(4) = ln(e^(r*5))
Using the logarithmic property ln(a^b) = b * ln(a):
ln(4) = r * 5 * ln(e)
Since ln(e) is equal to 1:
ln(4) = 5r
Finally, divide both sides by 5:
r = ln(4) / 5 ≈ 0.158
So, the approximate interest rate necessary for an investment to quadruple after 5 years of continuous compound interest is approximately 0.158 or 15.8%. Therefore, your answer of r=11% seems incorrect.