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\}} \\ (4000)/(1000)=(1000e^(10r))/(1000) \\ 4=e^(10r) \\ \ln (4)=\ln (e)^(10r) \\ 10r=\ln (4) \\ r=(\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%