Multiply, Do Not Add

§ 01Why the sum lies

Suppose a quantity grows by 10 percent in one period and then by 10 percent in the next. Addition says the total change is 20 percent. Multiplication says it is 21 percent. The difference looks trivial, and over two periods it nearly is. But the mechanism behind that extra 1 percent is the whole story of compounding, and over many periods the divergence between the additive guess and the multiplicative reality becomes enormous.

The correct calculation runs like this. A 10 percent gain means multiplying by 1.10. Two consecutive 10 percent gains mean multiplying by 1.10 twice: 1.10 × 1.10 = 1.21. Three periods: 1.10 × 1.10 × 1.10 = 1.331. The sequence is a product, not a sum, and every step is applied to the total accumulated so far, not to the original quantity. That is the mechanism. The gain in each period earns its own gain in the next, and so on, iteratively, for as many periods as the process runs.

The additive error is not foolishness — it is a reasonable heuristic for small changes over short horizons, where the cross-terms are genuinely negligible. The trouble is that human intuition applies the same heuristic everywhere, including over long horizons where it fails badly. After thirty periods of 10 percent growth, multiplication gives a factor of 17.4. Simple addition would suggest a factor of 4. The gap is not a rounding error; it is the compounding process doing exactly what it is designed to do.

§ 02When factors mix

The same multiplication logic applies when growth rates vary from period to period, or when gains and losses alternate. Each period contributes its own factor, and the total outcome is the product of all of them in sequence. A 20 percent gain followed by a 20 percent loss does not return to the starting point. The gain multiplies by 1.20; the loss multiplies by 0.80. The product is 0.96 — a net loss of 4 percent, even though the two percentage changes were equal in size and opposite in sign.

The sequence is a product, not a sum, and every step is applied to the total accumulated so far, not to the original quantity.

This asymmetry is built into the multiplicative structure and cannot be avoided by choosing a different accounting convention. A fall and a rise of identical percentage magnitude are not symmetrical events when their effects compound. A fall needs a bigger rise to undo it — the recovery percentage must be calculated from a smaller base.

The practical implication for any sequence of growth factors is that order does not matter for the final total (multiplication is commutative), but the magnitudes of losses and gains matter asymmetrically. Large negative factors are disproportionately damaging because they compress the base on which subsequent positive factors operate.

§ 03What average to use

Because compound growth is a product of factors, the appropriate average for describing it is also multiplicative. The geometric mean of a set of growth factors — calculated by multiplying them all together and taking the nth root, where n is the number of factors — gives the single constant factor that would produce the same final result. The arithmetic mean, which simply adds and divides, will always produce a number equal to or higher than the geometric mean when the factors vary at all, and the gap between them grows with the variability of the series.

Confusing the two averages produces systematically optimistic descriptions of what a compounding process delivers. An arithmetic average of periodic growth rates, applied additively, overstates the outcome that multiplication actually achieves over the same periods. The degree of overstatement depends on how widely the period-by-period rates vary: a perfectly smooth process with constant rates produces no gap; a volatile process produces a large one. This relationship — where variance in the factors reduces the geometric mean relative to the arithmetic — is not an aberration but a direct consequence of the mathematical structure of multiplication applied repeatedly to a changing base.

The machinery, then, is simple to state. Write each period's change as a factor. Multiply the factors. Read the result. Any method that skips the multiplication step and substitutes addition is solving a different, easier problem — and compounding is not that problem.