Arithmetic vs Geometric Mean

You already know how to take an average. Add up the numbers, divide by how many there are, done — you’ve been doing it since primary school. So here’s an uncomfortable truth to open with: that average, the familiar one, is the wrong tool for the single most important question in investing — how much did my money actually grow? Reach for it on a multi-year track record and it will lie to you, cheerfully and by a lot, always in the flattering direction.

This isn’t a rounding quibble or an edge case. It’s a structural feature of compounding, and the entire Kelly-and-growth arc you’re about to climb rests on getting it straight. There are two averages in finance — the arithmetic mean and the geometric mean — and they answer two different questions. Confuse them and every backtest, every fund comparison, every “average annual return” you ever read becomes a small act of self-deception.

The hook: two means, one truth

The setup. Start with 100 dollars. Year one returns +50%, so you climb to 150 dollars. Year two returns −50%, so you fall to… not back to 100, but to 75 dollars. The arithmetic average of the two returns is $(+50\% + -50\%)/2 = 0\%$ — a perfect wash, says the schoolbook average. Yet your account is down a very real 25%.

Where did the money go? The −50% in year two was applied to 150 dollars, not to your original 100. Half of 150 is 75, so you lost 75 dollars — but the +50% in year one had only made you 50 dollars. The loss bit a bigger apple than the gain grew. Percentages are deceptive because the base keeps moving underneath them, and a loss always strikes after a gain has fattened the base it attacks.

Define the problem. When returns compound — when each period’s result rides on top of the last — the simple average of the period returns does not equal the rate at which your wealth actually grew. The average says 0%; your wealth says −25% (which is about −13.4% per year, compounded). We need an average that respects compounding. That average has a name, and the average you already know has a much narrower job than you thought.

Arithmetic mean — the average you already know

Analogy. The arithmetic mean is the “fair share” average. Five friends pool their lunch receipts, divide by five, and everyone pays the same — nobody cares about order or compounding, just the flat central value of a pile of numbers. That’s exactly the right instinct for a single, isolated draw.

Definition. The arithmetic mean of returns $r_1, r_2, \dots, r_n$ is their sum divided by the count:

$\bar{r}_{\text{arith}} = \frac{1}{n}\sum_{i=1}^{n} r_i = \frac{r_1 + r_2 + \cdots + r_n}{n}.$

For our +50% / −50% pair: $\bar{r}_{\text{arith}} = (0.50 + (-0.50))/2 = 0$. Flat zero.

What it’s genuinely good for. The arithmetic mean is the expected value of the next single period’s return. If each year is an independent draw from the same distribution, then your best guess for one upcoming year — the bet you’d make on a single roll — is the arithmetic mean. For one-period bets, one-shot wagers, or “what return should I expect next month, considered in isolation,” it is exactly the right number. It is also what every standard expected-value calculation, every $E[r]$ you met in portfolio theory, is built from.

Why it’s the wrong tool for multi-period wealth. The arithmetic mean adds returns, but wealth multiplies them. Your money doesn’t go up by the sum of the percentages; it goes up by their product of growth factors. Averaging by addition when reality multiplies is a category error — it answers the wrong question. The arithmetic mean systematically overstates what a compounding series actually delivered, every single time there’s any variation in the returns.

Geometric mean — the only average that compounds

Analogy. Imagine flattening a bumpy mountain road into a single, perfectly even ramp that climbs from the same start to the same finish over the same distance. The road went up, down, up, down — but the ramp is the one steady grade that gets you to the identical altitude. The geometric mean is that ramp: the single constant per-period rate that, applied every period, lands you on exactly the ending wealth your bumpy real returns produced.

Definition. The geometric mean return is the constant rate $g$ such that growing by $g$ every period reproduces the actual ending wealth. With per-period returns $r_1, \dots, r_n$, each contributes a growth factor $(1 + r_i)$, and:

$g = \left(\prod_{i=1}^{n}(1 + r_i)\right)^{1/n} - 1 = \sqrt[n]{(1+r_1)(1+r_2)\cdots(1+r_n)} - 1.$

You multiply all the growth factors, take the $n$-th root (the “un-compounding” step), and subtract 1 to get back to a return. This is exactly what CAGR — the compound annual growth rate you’ve heard named — is: CAGR is the geometric mean of annual returns, no more and no less.

Worked example — three years. Suppose a fund returns +20%, then −10%, then +15%. Start with 100 dollars and walk the money forward:

So 100 dollars became 124.20 dollars. Now the two averages:

• Arithmetic mean: $(0.20 + (-0.10) + 0.15)/3 = 0.25/3 = 0.0833$, i.e. +8.33% per year.
• Geometric mean: $g = (1.20 \times 0.90 \times 1.15)^{1/3} - 1 = (1.242)^{1/3} - 1 = 1.0749 - 1 = 0.0749$, i.e. +7.49% per year.

Check the geometric mean against reality: growing 100 dollars at 7.49% for three years gives $100 \times 1.0749^3 = 100 \times 1.242 = 124.20$ dollars — the exact ending wealth. The arithmetic mean’s 8.33% would have falsely predicted $100 \times 1.0833^3 = 127.13$ dollars, nearly 3 dollars too generous. Geometric is less than arithmetic (7.49% < 8.33%), and only the geometric figure tells the truth about the money.

Why gains and losses are brutally asymmetric

Here’s the engine underneath everything above: losses and gains of the same percentage are not equal opponents. A loss is the heavier puncher, because after it shrinks your base, every recovering gain has less to work with.

Analogy. Falling down a hole and climbing back out aren’t symmetric jobs. Drop 50% of your height into a pit and you’re at half-height — to get back to the top you must now double, a +100% climb, because you’re climbing relative to the diminished bottom, not the original top. The deeper the hole, the more wildly disproportionate the climb out.

Definition. To recover from a loss of $L$ (as a fraction), the gain $G$ you need satisfies $(1 - L)(1 + G) = 1$, which solves to:

$G = \frac{L}{1 - L}.$

A 50% loss needs $G = 0.5 / 0.5 = 1.0 = +100\%$. The recovery gain always exceeds the loss, and explodes as the loss deepens:

A 10% dip is a shrug — get 11% back and you’re whole. But a 90% wipeout demands a tenfold +900% rally just to return to even, which essentially never happens. This asymmetry is precisely why the geometric mean gets dragged below the arithmetic mean: a big loss doesn’t just subtract — it shrinks the base that all future returns compound on, and no equal-sized gain can repair the damage.

The gap is volatility — and only volatility

You’ve now seen the shortfall in three separate examples. Here is the unifying law: the geometric mean equals the arithmetic mean only when every return is identical — when there is zero variation. The instant returns start to scatter, the geometric mean falls below the arithmetic mean, and the more violently they scatter, the wider that gap yawns open.

Analogy. Two roads run from the same town to the same town. One is dead flat; the other rollercoasters up and down to the same net elevation. The flat road and the rollercoaster cover the same horizontal distance, but the rollercoaster burns more fuel and arrives later — all that thrashing up and down is pure waste. Volatility is that wasted motion. A steady 7% every year and a wild +30%/−16%/… that averages to 7% do not deliver the same wealth; the wild one delivers less, and the wildness is the only reason.

The relationship, in words. The shortfall of the geometric mean below the arithmetic mean is driven almost entirely by variance — by how spread out the returns are. The next lesson makes this exact with the famous approximation that the geometric mean is roughly the arithmetic mean minus half the variance of the returns. For now, hold the qualitative shape: zero volatility means zero gap; double the volatility and the drag grows much faster than double, because variance scales with the square of the swings. Volatility isn’t just risk you feel in the moment — it is a permanent, compounding tax on your growth rate.

The chart below makes it tangible. An asset swings up by $+g\%$ and down by $-g\%$ in alternation, so its arithmetic mean per period is exactly 0% — the flat line. Yet the realized, compounded wealth curve sinks below the starting line and keeps sinking. Drag the swing slider: at zero swing the two coincide, and as you widen the swing the gap (the drag) balloons.

When it matters

This is not an academic distinction — using the wrong mean misleads in concrete, money-losing ways. Reach for the geometric mean (CAGR) whenever you are describing or comparing what actually happened to wealth over multiple periods:

• Reading a track record. A fund advertising its “average annual return” has very likely quoted the arithmetic mean — the flattering one. The return an investor who stayed in actually earned is the geometric mean, which is always lower for any fund with real volatility. The gap can be several percentage points a year, and over a decade that’s a different retirement.
• Comparing two funds. Fund A: arithmetic mean 10%, but wild swings. Fund B: arithmetic mean 9%, but smooth. The smoother fund can easily have the higher geometric mean and end with more money, because it pays less volatility drag. Comparing on arithmetic means alone can rank them backwards.
• Backtests and historical series. Any compounding series — a strategy’s equity curve, an index’s history — must be summarized geometrically. Average the daily or annual returns arithmetically and you’ll report growth the strategy never delivered.

Use the arithmetic mean only for genuinely single-period, non-compounding questions: the expected return of one upcoming bet, the input to a one-period expected-value calculation, or building blocks like $E[r]$ before you let them compound.

Putting it together

There are two averages, and they answer two questions. The arithmetic mean (sum over $n$) is the expected return of a single next period — right for one-shot bets, wrong for compounding wealth, and always the larger figure. The geometric mean — multiply the growth factors, take the $n$-th root, subtract 1 — is the one constant rate that reproduces your actual ending wealth; it is CAGR. The two are equal only when every return is identical; the moment returns scatter, the geometric mean drops below, because equal-percentage losses demand disproportionately larger gains to recover ($L/(1-L)$). That shortfall is volatility drag, it grows with the variance of returns, and it’s why a fund’s quoted “average return” is a flattering ceiling rather than the truth about your money.

Next up — Volatility drag, made precise — we turn the qualitative “the gap grows with variance” into the clean approximation that the geometric mean is about the arithmetic mean minus half the variance. That little $-\sigma^2/2$ term is the mathematical heart of volatility drag, and it’s the bridge to understanding why, later, the Kelly criterion cares so obsessively about not betting too big.