Compound growth with monthly contributions

Two engines drive the balance and they behave nothing alike. The starting amount compounds for the whole horizon; each monthly contribution compounds only for the time remaining after it arrives. The last contribution earns almost nothing, and the first earns almost everything a contribution can.

That is why the split matters. Total contributed is a decision about your budget and it is linear — double the monthly amount and it doubles. Total growth is a decision about time and rate, and it is not linear in either. Looking only at the ending balance makes both look like the same lever, and they are not.

The rate is annual and the contributions are monthly, so the two have to be reconciled before anything can be added up. The convention here divides the annual rate into monthly periods rather than compounding an effective annual rate downward. Which convention is used changes the answer, so it is stated rather than assumed.

Nothing here is a forecast. A constant rate is an arithmetic assumption, not a claim about any market: real returns arrive unevenly and the order they arrive in changes the outcome even when the average is identical. This answers what a steady rate would produce, which is a useful baseline and not a prediction.

The balance is driven by two mechanisms that share a rate and nothing else. The starting principal is exposed to the rate for the entire horizon — every period, from the first to the last, acts on it. Each monthly contribution is exposed only for the time remaining after it arrives, so the earliest deposit is worked on for almost the whole horizon and the final one is credited and immediately counted, having earned nothing at all.

This is why an existing balance carries more weight than its size suggests, and why the date a schedule of contributions begins matters more than the amount of each. A contribution is not a unit of savings; it is a unit of savings multiplied by the time it is given. Two people contributing identical totals over identical horizons finish with visibly different balances if one front-loads and the other does not.

The consequence for reading a projection is direct. Adding a period to the end of the horizon does not add an average period’s worth of growth — it adds a period acting on the largest balance the arrangement has ever held. Growth is therefore back-loaded by construction: the years that look uneventful early on are the ones building the base that the later years act upon.

Total contributed and total growth answer to different levers, and reading only their sum makes those levers look interchangeable. Contributing more is a budget decision with a proportional effect: the contributed half moves exactly as much as the deposit does. Starting earlier is a time decision with a compounding effect: the growth half moves by considerably more than the extra time, because the extra periods act on the largest balances rather than the smallest.

Presented separately, the two halves also make a projection auditable. If the growth half is doing most of the work in a plan, the plan is leaning on an assumed rate that is not yours to choose and may not arrive. If the contributed half is doing most of the work, the plan is essentially a savings account with extra steps, and its risk is that the contribution proves unsustainable. Which of those two a projection is depends entirely on the horizon, and only the split shows it.

This is the argument behind the familiar advice to start early rather than to save harder later — but stated as arithmetic rather than as exhortation. The calculator makes the claim testable: hold the total outlay fixed, move it earlier in the horizon, and read what happens to the growth half. Nothing here says which choice is right for a household; it says what each choice does.

Each period does the same small thing. The balance earns at the periodic rate; the return joins the balance rather than being set aside; the contribution arrives and joins too; and the enlarged balance faces the next period. Nothing dramatic happens in any single pass — the drama is entirely in the number of passes and in the fact that the base never resets.

Because the base never resets, the arrangement’s behaviour early and late in a horizon is qualitatively different even though the rule never changes. Early periods act on a small balance and produce visibly little; late periods act on the accumulated result of every period before them. A reader who judges a projection by its first years will conclude that compounding is overrated, and a reader who extrapolates its last years linearly will conclude that it is magic. Both are reading one part of the same curve.

The annual rate has to be reconciled with monthly contributions before any of this can be added up, and the convention used here divides the annual rate into monthly periods rather than compounding an effective annual rate downward. Which convention is used changes the answer, so it is stated rather than assumed — the same discipline the loan pages apply to a quoted rate.

A constant rate is an arithmetic convenience that makes the question answerable. It is not a claim that any account, fund or market delivers the same return every month. Real returns arrive unevenly, and the order in which they arrive changes the ending balance even when the average is identical — a poor stretch early in a horizon with small balances costs far less than the same stretch late, when the balance is at its largest. Nothing in a single-rate projection can express that.

What the projection is good for is comparison under a stated assumption. Two horizons at the same rate, two contribution levels over the same horizon, one plan started now against the same plan started later — those comparisons are meaningful precisely because the rate assumption is held fixed and visible on both sides. Treated that way it is a baseline. Treated as a prediction of a balance on a date, it is a number wearing more confidence than it owns.

And the figure is nominal. Returns here sit before inflation, tax and fees, so the result is a quantity of currency at a future date rather than an amount of buying power — a gap that grows with the horizon and belongs to the inflation and real-rate pages rather than to this arithmetic. This is an educational reference, not investment advice.

The engine below is the same one the calculator page runs, verified and mounted mid-lesson. It arrives pre-filled with the pack’s own declared example: a modest starting balance, a monthly contribution a household could actually sustain, an ordinary long-run rate and a horizon of several years. Read the three outputs together rather than the balance alone. Then double the horizon without touching anything else, and watch the contributed half merely double while the growth half does considerably more.

Read total contributed as what you chose, total growth as what the rate and the horizon did with it, and the future value as their sum — the least informative of the three on its own. All three are nominal, before inflation, tax and fees. Every figure is computed live by the verified engine from your inputs; this page stores none.