You want to make an investment in a continuously compounding account over a period of 100 years. What interest rate is required for your investment to double in that time period? Round the logarithm value and the answer to the nearest hundredth. I got C, is this correct?
0.7 %
6.9 %
2.3 %
23 %
The rule of 72 says that the time in years it takes to double your money is 72 divided by the interest rate in percent. However, it applies to compound interest, but it still gives an approximate number of
100=72/x or x=0.72%.
To calculate accurately, we use
Future amount
=Pe^(rt)
P=current investment
e=Natural log constant=2.7182818284...
r=annual interest rate
t=time in years.
So we have
2P=Pe^(rt)
e^(rt)=2
Take (natural) log on both sides
rt=log(2)
r=log(2)/100=0.006931
To find the interest rate required for your investment to double in a continuously compounding account, you can use the formula for continuous compounding:
A = P * e^(rt)
Where:
A = the final amount (double the initial investment, in this case)
P = the principal investment (the initial amount)
e = Euler's number (approximately 2.71828)
r = the interest rate (expressed as a decimal)
t = the time period (100 years, in this case)
In this case, you want to find the interest rate (r) required for the investment to double, so we will set A = 2P. Therefore:
2P = P * e^(rt)
Simplifying, we have:
2 = e^(rt)
To solve for r, we can take the natural logarithm (ln) of both sides:
ln(2) = ln(e^(rt))
Using the logarithm property, we can bring down the exponent:
ln(2) = rt * ln(e)
Since ln(e) is equal to 1, the equation becomes:
ln(2) = rt
To solve for r, divide both sides of the equation by t:
r = ln(2) / t
Substituting the values into the formula:
r = ln(2) / 100
Now we can calculate the interest rate required:
r ≈ 0.00693
Converting to a percentage:
r ≈ 0.693%
Rounding to the nearest hundredth, the interest rate required for your investment to double in 100 years is approximately 0.69%.
Now let's compare the options you provided:
- 0.7% is close to the calculated interest rate, but not exact.
- 6.9% is significantly higher than the calculated interest rate.
- 2.3% is significantly lower than the calculated interest rate.
- 23% is significantly higher than the calculated interest rate.
Therefore, the correct answer is not option C (2.3%).