The Boring Math of Compound Returns (And Why Everyone Gets It Wrong)

Compound interest is the kind of topic that gets called “the eighth wonder of the world” in motivational graphics. The actual math is mundane. The most common rules of thumb are subtly wrong. And the most-shared retirement charts make assumptions about return paths that real markets rarely deliver.

This isn’t a takedown of compounding — it’s a takedown of how compounding gets explained. The reality is calmer and more useful.

The basic formula isn’t the whole story

The textbook compound formula is final = principal × (1 + r)^n. That works perfectly if every year delivers exactly r.

Markets don’t. Returns vary. The arithmetic average of yearly returns and the actual compound return are not the same number.

Arithmetic vs geometric mean

Take an investment that returns +50% in year one and -50% in year two.

• Arithmetic mean: (50% + (-50%)) / 2 = 0%. “Average return was zero.”
• Geometric mean (actual outcome): start with $100, end at $150, then $75. The compound return is −25% over two years — about −13.4% per year.

The geometric mean is always ≤ the arithmetic mean, with equality only when every return is identical. This gap is called volatility drag or the variance penalty.

For a roughly stable broad-market portfolio, the gap is small but not negligible — typically around 1–2% per year. For a volatile asset (an individual small-cap, a leveraged ETF, a single cryptocurrency), the gap can be much larger.

Why this matters in practice

Most “average return” figures you see in retail finance content are arithmetic. The wealth you’d actually have is geometric. Over decades, the difference compounds in the wrong direction.

A reasonable approximation: geometric ≈ arithmetic − (variance / 2). That subtraction is the variance penalty in dollar form. Over twenty or thirty years, it’s the difference between a clean projection and the path you’d actually live through.

The Rule of 72 (and where it fails)

“Years to double = 72 / interest rate.” At 8%, money doubles in about 9 years. The rule is a Taylor-series approximation that’s accurate around 6–10% and breaks down at higher rates.

• At 12%: rule says 6 years, actual ~6.1. Fine.
• At 20%: rule says 3.6 years, actual ~3.8. Off.
• At 50%: rule says 1.44 years, actual ~1.7. Misleading.

For low rates, the rule slightly overestimates the doubling time. The “Rule of 70” is closer for very low rates; the “Rule of 72” is the practical compromise because 72 has more divisors and gives nicer mental arithmetic.

Sequence of returns risk

Two retirees with identical average returns can end up with very different outcomes if the returns arrive in different orders.

If you retire just before a bad decade for equities and are drawing down a portfolio, the drawdowns compound against you in a way that doesn’t reverse just because returns recover later. The withdrawals from a depressed portfolio book the losses permanently.

This is why fixed-rate retirement projections (“assume 7% return forever”) overstate the safety of withdrawal rates. The same 7% delivered with realistic year-to-year variance gives a wider distribution of outcomes, with the bad tail materially worse than the smooth path implies.

The fix isn’t to abandon compounding — it’s to use stochastic projections (Monte Carlo) instead of single-path ones, especially near and during the withdrawal phase.

What the “start early” charts get right and wrong

The right part: time in the market matters. Contributing consistently from age 25 to 35 (then nothing) often ends up with more than contributing the same amount per year from 35 to 65, assuming the same return path.

The misleading part: that calculation assumes a flat compound return for the entire period. Real return paths have decades- long regimes — some decades have been great for US equities, some have been bad. The “start early” math is correct in expectation; the actual outcome for any specific cohort depends heavily on which decades they were contributing during.

What this means for decisions

• Don’t conflate the average return you read in marketing copy with the wealth path you’d actually experience. Volatility drag is small per year and large over decades.
• Don’t use Monte Carlo for everything, but don’t use single-path 7%-forever models when the answer matters. The smoothness hides risk.
• Time in the market is a real lever, but the magnitude depends on the regime you’re contributing during, not just on the number of years.

Compounding is real, useful, and reliable. The mystification around it is the problem.