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:
[tex]FV=P\cdot e^{rt}[/tex]
If we have a fixed initial investment and the interest rate is halved (r/2), the new final value is:
[tex]FV^{\prime}=P\cdot e^{\frac{r}{2}t}[/tex]
The ratio between both FV's is:
[tex]\begin{gathered} \frac{FV}{FV^{\prime}}=\frac{P\cdot e^{rt}}{P\cdot e^{\frac{r}{2}t}} \\ \frac{FV}{FV^{\prime}}=e^{\frac{r}{2}t} \end{gathered}[/tex]
For r = 0.184 and t= 50:
[tex]\begin{gathered} \frac{FV}{FV^{\prime}}=e^{0.092\cdot50} \\ \frac{FV}{FV^{\prime}}=e^{4.6} \\ \frac{FV}{FV^{\prime}}=99.48 \end{gathered}[/tex]
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.
