Track 0 · Ledger · lesson 7

The curve, before the maths

11 min

Ask someone to draw forty years of steady saving on a napkin and almost all of them draw a line sloping upwards. The line is a reasonable guess and it is wrong in one specific, expensive way.

It is wrong at the end. The first ten years of that picture are very nearly a straight line, which is why the intuition survives. The last ten are not, and the last ten are where most of the total shows up.

Growth applies to the growth.

A balance earns a return. Next year, that return earns a return too, alongside everything else. Nothing else is going on — there is no second mechanism and nothing clever is happening. But because the base keeps getting bigger, the amount added each year keeps getting bigger, and a quantity whose increments keep growing does not draw a line. It draws a curve.

This lesson is about the shape of that curve. The arithmetic behind it comes later, and it turns out to be about four lines long.

Where the money comes from

There are only two sources for the final number. There is everything that was put in, and there is everything the growth added. That is a complete list.

At the beginning, almost all of it is the first one. Money that arrived last month has had no time to grow, so it is worth what it was. As the years pass the second source overtakes the first, and after that it keeps pulling away, because it is growing on a base that includes all of its own previous work.

Predict

Someone saves the same amount every year for forty years at a steady rate of return. Which ten-year stretch adds the most to the final balance?

The first ten, because that money has the longest to growThe middle stretch, where the curve is steepestThe last ten, by a wide marginThey all add about the same, since the contribution never changes

Find the crossover

Below is the curve itself. Nothing is put in at the start; a fixed amount is added each year at a steady rate, and the chart draws two lines. One is the balance. The other is the running total of everything ever paid in.

The gap between them is the growth. Start at ten years and slide outwards.

Compound growth

Hypothetical model, not a forecast

Grows a starting balance plus a fixed yearly contribution at a constant rate.

It assumes

The return is exactly the same every year, which no real market has ever managed.
The same amount is paid in every year for the whole period, without a break.
Nothing is taken out, and nothing is charged in fees or tax.
Prices are ignored entirely, so these are not amounts of stuff.

It ignores

Tax on growth, income or withdrawal
Platform fees, spreads and trading costs
Inflation — every figure shown here is nominal
Any year in which the return happens to be negative

Slide the years out and find the first year at which the growth is worth more than everything ever paid in — the point where growth passes half of the balance.

Years10

Final balance: 26,362
Total paid in: 20,000
Growth: 6,362
Growth as a share of the final balance: 0.241

Balance
Total paid in

Two lines starting together, one straight and one curving away above it, with the gap between them widening steadily after about year twenty.
Year	Balance	Total paid in
0	0	0
1	2k	2k
2	4.1k	4k
3	6.4k	6k
4	8.7k	8k
5	11.3k	10k
6	14k	12k
7	16.8k	14k
8	19.8k	16k
9	23k	18k
10	26.4k	20k

Not there yet — keep moving the controls.

On these settings the crossover lands at about year twenty-three. Before it, the balance is mostly savings. After it, the balance is mostly growth, and the savings are the smaller half of a thing they started off being all of.

Now compare two points on that same curve.

At thirty years the balance is around 158,000. At forty years it is around 309,500. The final decade added about as much as the first three put together, and it did it with exactly the same yearly contribution as every other decade.

Nothing changed except the size of the base the rate was applied to. That is the entire mechanism, and it is why the picture looks so different depending on where you stop drawing it.

Check

Everything above is at a 6% return. If the return were 4% instead, what happens to the crossover year — the point where growth passes half the balance?

It arrives earlier, because the balance grows more slowly and contributions matter lessIt arrives laterIt does not move, because the crossover depends only on timeThere is no crossover at 4%, only at higher rates

What this shape does not promiseOptional

A smooth curve is a drawing, not a forecast. The rate here is the same every year, and no real return behaves that way. A real path with the same average wanders above and below this line, sometimes for a decade at a time, and it can be well below it at the exact moment somebody needs the money.

The shape is still worth knowing, because it is what the average produces and because it explains why the middle of a long run feels so unrewarding. But anyone who shows you this curve and calls it a plan has quietly swapped a picture of an average for a picture of an outcome.

Worth remembering

A balance growing at a steady rate draws a curve, not a line, because the growth earns growth of its own.
The final balance has exactly two sources: what was paid in, and what the growth added.
Early on the balance is mostly contributions; after a crossover point it is mostly growth.
Roughly: at a 6% return with steady yearly saving, growth passes half the balance somewhere around year twenty-three.
The last decade of a long run adds far more than the first, because the rate is applied to the largest base at the end.