How long will it take for an investment to double in value if it earns 9.5% compounded continuously?
For r=0.095, with normal compound interest, compounded yearly, the number of years to double at a rate of r% is ln(2)/ln(1+r)=ln(2)/ln(1.095)=7.638 years.
from A=P(1+r)^n
take ln both sides,
ln(A/P)=n ln(1+r)
n=ln(2)/ln(1+r)
With continuous compounding,
A=Pern
Take logs
ln(A/P)=ern
ln(2)=rn
n=ln(2)/r
=ln(2)/0.095
=7.296 years
To find out how long it will take for an investment to double in value with continuous compounding at a rate of 9.5%, you can use the formula:
t = ln(2) / (r)
Where:
t = Time in years
ln = Natural logarithm
2 = Desired doubling factor
r = Annual interest rate expressed as a decimal
Substituting the values into the formula, we have:
t = ln(2) / (0.095)
Using a calculator, we can calculate:
t ≈ 7.27 years
Therefore, it will take approximately 7.27 years for the investment to double in value with continuous compounding at a rate of 9.5%.
To determine how long it takes for an investment to double in value when earning a continuously compounded interest rate of 9.5%, you can use the formula for continuous compounding:
A = P * e^(rt),
where:
A = the future value of the investment,
P = the initial investment amount,
e = the base of the natural logarithm (approximately 2.71828),
r = the interest rate, and
t = the time period.
In this case, we want to find the time required for the investment to double, so we set A = 2P (twice the initial investment) and plug in the given interest rate (r = 0.095) into the formula. Then we can solve for t:
2P = P * e^(0.095t).
Dividing both sides of the equation by P, we get:
2 = e^(0.095t).
To solve for t, we need to take the natural logarithm (ln) of both sides:
ln(2) = ln(e^(0.095t)).
Using the property of logarithms that ln(e^x) = x, we simplify further:
ln(2) = 0.095t.
Finally, we isolate t by dividing both sides by 0.095:
t = ln(2) / 0.095.
Using a calculator, we find:
t ≈ 7.3 years.
Therefore, it will take approximately 7.3 years for the investment to double in value when earning a continuously compounded interest rate of 9.5%.