How your savings rate sets the timeline

The axis is a share of take-home pay, not an amount. Save a share $s$ each year and you spend $1 - s$, so the finish line is $M(1-s)$ where $M$ is the target measured in times yearly spending. Deposits of $s$ compounding at a real return of $r$ reach it after

$$t = \frac{\ln\left(1 + M(1-s)r/s\right)}{\ln(1+r)}$$

Pay appears on both sides and cancels, which is why a bigger pay packet is not on the chart at all. On the default setting, 20 percent saved at a 5 percent real return towards 25 times spending, the curve reads 36.7 years. At 50 percent it reads 16.6 years, and at 80 percent it reads 5.6 years.

Drag the dot across the low end and the step is tall. From 10 percent to 15 percent cuts 8.5 years. From 20 to 25 cuts 4.8. The same 5 points from 50 to 55 cuts 2.2 years, and from 70 to 75 it cuts 1.7. Nothing changed except where you started.

The reason is the ratio $M(1-s)/s$, the target measured in years of saving. At a 10 percent savings rate the pot has to cover 225 years of deposits, at 20 percent 100 years, at 50 percent 25 years and at 80 percent 6.25 years. Halving the savings rate from 20 percent to 10 percent more than doubles that pile of deposits, from 100 years to 225, because the numerator rises as the denominator falls. Growth then absorbs most of the damage, since a longer run gives compounding more to work with: the wait itself goes from 36.7 years to 51.4, up by about 40 percent rather than by 125. Softened, not flattened.

Raising the savings rate does two jobs at once. It raises the deposit, and it lowers the finish line, because the target is a multiple of what you spend. At 20 percent saved the target is 20 times take-home pay. At 50 percent it is 12.5 times. Cutting spending is the only move that pulls both ends of the problem towards each other.

The same idea in years: one year of saving covers 0.25 years of spending at a 20 percent rate, and a full year of spending at a 50 percent rate. It also means a pay rise moves nothing on this chart so long as the share saved holds. The deposit rises with the pay, the target rises with the spending, and the timeline sits exactly where it was. A pay rise that goes entirely to spending is the other case, and it moves the wrong way: the amount saved has not changed while the pay has, so the share saved falls, the target rises, and the run gets longer.