Rule of 72 binomial

Question

Rule of 72?

So you have to work out the time it takes for an investment to double. With interest rate = x

So I get:

(1+x)^n = 2

Then I get how the 72 rule is formed through solving for n, giving n = ln2/x which is where the 72 comes from. As well as following the wiki page with Taylor series too.

But how would I go about showing the 72 rule through binomial expansion?

Answer 1

Here's a loose derivation. Starting with $\left(1+\frac r{100}\right)^t = 2,$ the binomial theorem gives us $$1 + t\frac r{100} + \frac{t(t-1)}2\left(\frac r{100}\right)^2 \approx 2$$

This is a quadratic in $t$; using the quadratic formula and simplifying the resulting fraction, we get $$t \approx -\frac{100}r + \frac12 + \sqrt{\frac{30\ 000}{r^2} - \frac{100}r + \frac14}$$

Now, since $r$ is small, the expression in the square root is dominated by its first term, so $$\begin{align} t &\approx -\frac{100}r + \frac12 + \sqrt{\frac{30\ 000}{r^2}}\\ &= -\frac{100}r + \frac12 + \frac{100\sqrt3}r\\ &= \frac12 + \frac{100}r(\sqrt3-1)\\ &= \frac12 + \frac{73.2\ldots}r \end{align}$$ which is close enough to $\frac{72}r$.

The above calculation involved a certain disregard for accuracy, but since the "rule of 72" is fairly loose anyway, then no harm done.

Answer 2

the rule of $72$ is a heuristic, and not a hard and fast rule of mathematics.

$(1+ x)^n = 2 \implies xn \approx 72\%$

e.g. $(1.08)^9 = 1.999$

What is the math?

$\lim_\limits{n\to\infty} (1+ \frac xn)^n = e^x$

When $n$ is large, and $nx = \ln 2\approx 0.69$, $(1+ x)^n = 2$

But for smaller values of $n$

$(1+ x)^n =$$ 1 + nx + \frac{n(n-1)}{2} x^2 + \frac{n(n-1)(n-2)}{3!} x^3 + \cdots\\ 1 + nx + \frac 12(1 - \frac{1}n) (nx)^2 + \frac 1{3!}(1-\frac 1n)(1-\frac 2n) (nx)^3 + \cdots$ $e^{nx} = 1 + nx + \frac 12 (nx)^2+ \frac 1{3!} (nx)^3+\cdots\\ (1+ x)^n < e^{nx}$

For finite $n, (1 +x)^n=2 \implies nx > \ln 2$