What inflation does to buying power

The falling line is money held as cash. After $t$ years at an inflation rate $i$ it buys $(1+i)^{-t}$ of what it buys today, so the line never quite reaches zero and never stops falling either. The second line is the same money in an account paying $s$ a year, whose buying power is $\left(\frac{1+s}{1+i}\right)^t$. Both leave today at 100 percent, and the gap between them is what the interest gave back.

The vertical scale halves at every gridline instead of stepping evenly, because a constant percentage lost each year is a constant fraction rather than a constant amount. On that scale both lines come out straight, and the falling line crosses exactly one gridline per half-life.

$$t_{1/2} = \frac{\ln 2}{\ln(1+i)}$$

The years it takes to lose half of today's buying power depend only on the rate, never on how much money is sitting there. At 2 percent that is about 35 years, at 3 percent about 23.4 years, at 5 percent about 14.2 years, and at 8 percent about 9 years.

The second half then takes as long as the first, because the same fraction goes every year rather than the same amount. After two half-lives roughly a quarter is left, after three roughly an eighth. That is why a rate that sounds mild does real work over a working life: at 3 percent, money earning nothing keeps about 41.2 percent of what it buys after 30 years and about 17 percent after 60.

An account is ahead only when the rate it pays is above inflation, and the figure that settles it is the real rate rather than the advertised nominal rate:

$$r = \frac{1+s}{1+i} - 1$$

Subtracting inflation from the headline rate is close but not exact. The shortcut misses by the real rate multiplied by inflation, so it flatters an account that is already ahead and overstates the loss on one that is behind.

On the default setting, 2 percent paid against 3 percent inflation, the real return is about -0.97 percent a year rather than the -1 percent the shortcut gives. That reads as almost nothing, and it still halves buying power in about 71 years rather than the 23.4 years cash takes. The interest slowed the loss without stopping it. Drag the account line above the dashed line and it is genuinely ahead; below it, the balance grows while what it buys shrinks. Tax on the interest sits outside the chart, and it raises the rate an account has to pay to break even.