Question

What is the answer to the problem, and how do i solve?

Answer 1

Continuous Compound Interest

The formula to calculate the future value (FV) of an investment, given an initial investment P and an interest rate r is:

`FV=P\cdot e^(rt)`

If we have a fixed initial investment and the interest rate is halved (r/2), the new final value is:

`FV^(\prime)=P\cdot e^{(r)/(2)t}`

The ratio between both FV's is:

`\begin{gathered} (FV)/(FV^(\prime))=\frac{P\cdot e^(rt)}{P\cdot e^{(r)/(2)t}} \\ (FV)/(FV^(\prime))=e^{(r)/(2)t} \end{gathered}`

For r = 0.184 and t= 50:

`\begin{gathered} (FV)/(FV^(\prime))=e^(0.092\cdot50) \\ (FV)/(FV^(\prime))=e^(4.6) \\ (FV)/(FV^(\prime))=99.48 \end{gathered}`

The reduction of the final value is close to 1/100. This is due to the nature of the exponential function, which grows much faster than any proportional function.