Instrument 2 · exponential growth

The curve that is always steeper than it looks

Anything that grows by a percentage rather than a fixed amount has a doubling time, and the doubling time does not care how far you have already come. On a normal axis it looks like a wall appearing out of nowhere. On a log axis it is a straight line.

The balance

Where the time goes

the balance when it doubles if it added the same amount every year

The same statement, with the numbers in it

A_{t} = A_{0} {(1 + \frac{r}{n})}^{n t}

2,000 = 1,000 {(1 + \frac{0.07}{1})}^{10.2}

The doubling time comes straight out of it: t = ln 2 / ln(1 + r) — and note that it contains no A₀. Where you start is irrelevant to how long the next doubling takes.

Fine tune

Growth per year 7%

Compounded per year 1

Years 20

Starting amount 1,000

The balancelinear axis

Where the time goes

The balance