Question

Otto invests $1,000 at age 20. Otto wants theinvestment to be worth $4,000 by age 30. Ifinterest is compounded continuously, what rateof return does Otto need to reach that goal?Enter your answer as a decimal rounded to thethousandths place.

Answer 1

$r\approx0.139$

1) Since this is a Continuously Compounded operation in a 10 yrs period, then we can write out the following equation:

$\begin{gathered} A=Pe^{rt} \\ \end{gathered}$

2) Plugging into the equation the given data and since Otto is 20 yrs old and he plans to get $4,000 in ten years, we can write out:

$\begin{gathered} 4000=1000e^{\mleft\{10r\mright\}} \\ \frac{4000}{1000}=\frac{1000e^{10r}}{1000} \\ 4=e^{10r} \\ \ln (4)=\ln (e)^{10r} \\ 10r=\ln (4) \\ r=\frac{\ln (4)}{10} \\ r=0.1386 \end{gathered}$

3) Thus the rate Otto needs is

$r=0.1386\approx0.139$

Note that since the 0.1386 the six here is greater than 5 then we can round up to the next thousandth, in this case: 0.139. For the 0.1386 is closer to 0.139 (0.004) than to 0.138 (0.006).

Or 13.9%