Time Is the Multiplier

The doubling rule, and how far it can be trusted

Compound growth is easier to reason about in doublings than in percentages. The rule of 72 — doubling time is roughly 72 divided by the annual return — gets you from a rate to a number of years in your head: 4% doubles in 18 years, 7% in about 10.3, 9% in 8. The exact figure is ln 2 ÷ ln(1 + r), and between about 4% and 12% the approximation is accurate to a few tenths of a year, which is exactly the range most honest long-run assumptions occupy. Doublings are useful because they turn a horizon into a count. Thirty years at 7% is about three doublings, so $10,000 becomes roughly $76,100: eight times the money, not triple. The same thirty years at 4% is under two doublings, giving about $32,400. Both are unremarkable, defensible returns, and one ends with 2.3 times the other — which is what makes the difference between 4% and 7% feel much larger than three percentage points. The rule also runs in reverse, and that is where it earns its place in a household plan. A 1% annual fee is a negative return applied to the whole balance, so it doubles *against* you on the same schedule: over thirty years a one-point drag consumes roughly a quarter of the final balance. The arithmetic does not know whether the rate is an asset return or a cost. $10,000, thirty years, four rates — 4% — under two doublings: $32,400 · 7% — three doublings: $76,100 · 9% — nearly four doublings: $132,700 · The 7% outcome against the 4%: 2.3× the money ← Doubling counts are the honest way to read a long projection: not “the market returns 7%”, but “this horizon is about three doublings, so the money grows eightfold”.

The last decade holds most of the money

A fifty-year projection is not fifty equal years. Each year applies its return to a larger balance than the year before, so the money added late dwarfs the money added early. In a 40-year, $400-a-month plan, the balance in year 25 is a small fraction of the balance in year 30, and more than half of the final total often arrives in the last third of the horizon. Two consequences follow, and they pull in opposite directions. The first is encouraging: contributions made now are worth several times their face value by the end, so the early years are the ones worth protecting. The second is brutal: the same logic means **starting late cannot be fixed by contributing more.** Doubling a contribution doubles the balance; adding ten years multiplies it. A household that begins at 45 and contributes aggressively is buying a much smaller multiple than one that began at 25 and contributed the same amount. The way to see it is to look at where the growth comes from rather than at the balance. Early in a plan, almost everything in the account is contribution; late in a plan, almost everything is growth, and the growth is larger than every contribution ever made. That crossover is what people mean when they say compounding “takes over” — and it is also why the volatility of the later years matters so much: the returns are being applied to a large number. Same contribution, different decade of the plan — Years 1–10 contributions and growth: a small share of the final balance · Years 25–30: often more than the first three decades combined ← · What the crossover means: growth eventually exceeds every contribution ever made Volatility drag is real: two years of +50% and −50% average 0% arithmetically and leave you down 25%, because 1.5 × 0.5 = 0.75. Average returns are not what a balance experiences — the geometric sequence is.

Time is the only input you cannot buy later

A plan has four inputs: the amount, the return, the cost and the years. Two of them are under your control (amount and cost), one is an assumption (return), and one is fixed by the calendar — the years are the only input that cannot be renegotiated, because they are already spent. That asymmetry is why a household should treat the early years as the expensive ones. It also gives the right answer to a common excuse. A 28-year-old who cannot save $500 a month can save $150, and the $150 started now is worth more than the $500 started at 40 — the contribution is three times smaller and the horizon is not three times longer, so the question is genuinely close in the middle of the range and only resolves in favour of starting early as the gap widens. None of this argues for heroics or for taking more risk than a plan can carry. It argues for a boring amount started immediately: automate it, leave it alone, and let the doublings do the work. A plan that begins with $150 and no drama beats a plan that begins in five years with $600 and a story. Where an unspent decade goes — Contributed, age 25–35 at $400 a month: $48,000 · Contributed, age 35–65 at $400 a month: $144,000 · Balance at 65, if the first saver stops at 35: can exceed the second saver’s ← · What the difference is made of: three decades of compounding on the earliest dollars The plan’s numbers are illustrative and use a 7% assumption with no fees or taxes. Changing the assumption moves the totals; it does not move the ordering, which is the teaching point.

Contributions, returns, and which one you control

The compounding arithmetic in this lesson is usually presented as a story about returns, and in the early years it is mostly a story about contributions. Separating the two is what turns the growth curve from a source of anxiety about picking the right investments into a list of things a household can actually do. The arithmetic is straightforward and worth doing once on your own numbers. A balance grows from two sources: the money added, and the return on what is already there. In the first decade, contributions dominate almost any realistic return — at a seven percent return, the first ten years of a regular contribution leave the account with more contributed than earned. By the third decade the position reverses, and the returns on the accumulated balance are larger than the contributions made in the same period. That crossover is the whole point of starting early, and it also means that the return you earn matters more later and the amount you save matters more now. The practical consequence is a priority order rather than an optimisation. In the early years, effort spent raising the contribution rate beats effort spent selecting investments, because the contribution is the larger term and it is entirely under the household’s control. The same logic makes the employer match the highest-return decision available: an immediate, certain return on the contributed amount that no market outcome can compete with. Only after the contribution rate is high does the choice among investments become the binding decision — and by then it is usually a choice between broad, cheap funds rather than a choice of securities. The compounding also has two silent costs, and they are worth computing because they compound exactly like the returns do. Fees subtract from the rate, and they do so every year, so a hundred basis points of extra cost over thirty years removes a substantial fraction of the terminal balance — a difference that appears trivial in a single year and enormous in a lifetime. Taxes do the same when they are triggered annually rather than deferred, which is why the wrapper decision interacts with compounding rather than sitting beside it. Neither is a market risk; both are certainties, which makes them the most reliable improvements available. And the input that cannot be recovered is time, which is what makes the interruptions expensive. A decade of not contributing, taken in the middle, cannot be repaired by contributing more afterwards at a normal rate — the arithmetic is not symmetric, because the missing decade also misses the compounding on the contributions that would have been made. That is why the practical advice that follows from this lesson is unglamorous: contribute an amount that can be sustained through a bad year, automate it so it does not require a decision, and raise it when income rises. The rate of return is uncertain and partly out of a household’s hands; the contribution rate is neither. • Contributions dominate the early years; returns dominate the later ones. • The employer match is a certain return that no market outcome competes with. • Fees and taxes compound like returns and are certainties rather than risks. • A missing decade cannot be repaired by saving more later, because the compounding is lost too. A useful split when reading your own accounts: for each, the amount contributed and the amount earned. Watching the earned figure overtake the contributed one is the only way the compounding becomes concrete rather than theoretical.

Sources

• The rule of 72 and the accuracy of the approximationStandard compounding arithmetic; Bogleheads wiki — rule of 72
• Why the last years dominate a long projectionBogleheads wiki — importance of saving early
• Expected returns and the limits of forecasting themVanguard capital market assumptions (revised annually)
• Volatility drag and the difference between arithmetic and geometric returnsStandard geometric-mean arithmetic; CFA Institute curriculum