Logjam 4: Compound uselessness
Elsewhere [Poweringup] I have elaborated how compound interest arises and how it relates to exponential behaviour. Here I will look at problems such as: If I want a 30% yield on an investment after 10 years, what sort of annual interest rate should I set or seek? In mathematical terms we a looking for a solution x to:
\[\left( {1 + x} \right)^{10} = 1 + 30 = 1.3\]
Well you could just try a series of values for x, hopefully using ones intelligence to get closer and closer. This is called “shooting”, and is a respectable method in the absence of anything better. As part of shooting you might apply a bracketing method along with some interpolation, guessing lower and upper bounds (1%? and 3%) and zeroing in.
However, here logarithms can save us all that. We have from applying logarithms to both sides:
\[\left. 10\log\left( {1 + x} \right) = \log 1.3\Longrightarrow\log\left( {1 + x} \right) = \frac{\log 1.3}{10} \right.\]
Antilogging in whatever base gives:
\[1 + x = 1.02658\ldots\]
So the desired interest rate would be around 2.7%, not much different from the upper bound of 3%. However, the difference is more significant if you want to double your money — i.e. 100% yield:
\[\left. \log\left( {1 + x} \right) = \frac{\log 2}{10}\Longrightarrow 1 + x = 1.0718 \right.\]
You can reduce the annual rate to 7.2%, rather than the naive 100%/10=10%. For a bank’s PR/advertising, this looks far less menacing to the intended victims.