This article explains compounding and volatility drag as concepts. It is educational and general, not personalised investment advice, and nothing here is a recommendation to buy, sell, size, or hedge any security. The illustrations use relative figures to make the arithmetic clear; they are not projections, and past performance does not predict future results.
There are two ways to average a stream of returns, and they answer two different questions. One tells you the typical single-period return. The other tells you what you actually ended up with after the periods compounded on top of each other. They are almost never equal, and the gap between them has a name: volatility drag. It is the reason a portfolio can post a healthy average return and still grow more slowly than a steadier one that averaged the same, and it is one of the most under-appreciated forces in long-horizon investing. Here is what the two averages are, the simple identity that links them, why losses hurt more than equal-sized gains help, and the important exception where strongly upside-skewed returns break the rule in the investor’s favour.
The short version
The arithmetic mean return is the simple average of the period returns; the geometric (compound) return is the constant rate that turns the starting value into the ending value. Because returns compound multiplicatively, the geometric return is always less than or equal to the arithmetic one, roughly the arithmetic mean minus half the variance. That subtracted term is “volatility drag.”
Two averages, two questions
Suppose an asset gains 50 percent one year and loses 50 percent the next. The arithmetic mean of those two returns is zero: (+50 minus 50), divided by two. It is tempting to read that as breaking even. It is not. A unit of value grows to 1.5, then falls by half to 0.75. The position is down 25 percent, and the compound annual rate that produces 0.75 over two years is about negative 13.4 percent a year. The arithmetic mean answered “what was the typical yearly return?” The geometric mean answered the question that actually pays for groceries: “at what steady rate did the money actually grow?”
The distinction is not academic hair-splitting. The arithmetic mean is the right tool for a single upcoming period, it is the expected value of one draw. The geometric mean is the right tool for a sequence, because wealth is a product of gross returns (1 plus each period’s return), not a sum of them. Multiply a run of numbers together and the low outcomes drag the product down out of proportion to how much the high ones lift it. That asymmetry is the whole story, and it is baked into the arithmetic of multiplication itself.
The identity: geometric is arithmetic minus half the variance
The link between the two averages is close to an identity for the modest returns typical of diversified investing. Written in the usual shorthand, the geometric return g relates to the arithmetic return a and the variance of returns (sigma squared) as:
g ≈ a − ½ σ²
Read it slowly, because it says something important. The compound growth that is actually kept equals the average return minus a penalty, and that penalty is half the variance of the returns. Variance is volatility squared, so the drag rises with the square of how much the returns bounce around. Double the volatility and the drag roughly quadruples. Two strategies can share an identical arithmetic mean and still deliver very different compound results purely because one took a bumpier path to get there. Volatility is not just a measure of comfort or risk tolerance; it is a direct, quantifiable tax on long-run growth. What volatility is, and how it is measured, is the subject of what is volatility.
The approximation is a second-order Taylor expansion of the log-return, and it is closest when returns are small and roughly symmetric. For large or badly skewed returns it drifts, and one direction of that drift is the exception that makes this whole topic interesting, covered below.
Why losses hurt more than equal gains help
The most concrete face of volatility drag is the asymmetry of recovery. A loss and a same-sized gain are not mirror images, because the gain has to work on a smaller base. A 10 percent loss requires about an 11.1 percent gain to get back to even. A 20 percent loss requires 25 percent. A 50 percent loss requires a full 100 percent gain, a doubling, just to return to the starting point. The deeper the hole, the more disproportionate the climb out, and the relationship is convex: the required gain accelerates as the loss grows.
| Loss taken | Gain required to break even |
|---|---|
| 10% | 11.1% |
| 20% | 25% |
| 1/3 (33%) | 50% |
| 50% | 100% |
| 75% | 300% |
| 90% | 900% |
This is why drawdowns are so corrosive to compounding and why steadier paths matter so much more than a casual look at average returns suggests. A strategy that avoids the deep holes does not need heroic rebounds to keep growing. One that periodically halves is a full doubling away from recovery, whatever the “down 50 and then up 50” intuition suggests. The same logic runs underneath why capping the depth of losses is worth so much, a theme shared with risk of ruin, where the concern is not just slow growth but the possibility of a hole too deep to climb out of at all.
The same average, two different endings
Put the identity to work on two hypothetical paths that share the exact same arithmetic mean return, and the drag becomes vivid. Both average 5 percent a year. One is calm and posts something close to 5 percent every year. The other is wild: it swings to large gains and large losses that happen to average 5 percent. Because the wild path carries far more variance, it pays a far larger drag penalty, and it compounds to less terminal wealth than the calm path, despite the two being indistinguishable on the arithmetic average that a headline usually quotes. The steadier path wins because it surrendered less to the half-variance tax.
This is the practical punchline that makes the concept worth understanding. When two records show the same mean return, the smoother one is more comfortable to hold and, on the mathematics above, likely to have compounded to more. It is also the reason a mean return quoted without any measure of its volatility is close to meaningless for judging long-run outcomes, a point that echoes through how results should be read honestly, where return, risk-adjusted return, and drawdown belong together rather than cherry-picked apart.
The exception: upside skew can compound ABOVE the shortcut
Here is where the textbook rule earns an asterisk, and it is the most interesting part. The “minus half the variance” shortcut assumes returns are roughly symmetric, shaped like the familiar bell curve. Real return distributions often are not. When a return stream is strongly upside-skewed, meaning its big surprises cluster on the positive side while its losses stay shallow and bounded, the simple variance-drag formula understates the compound return. The realized geometric growth can land above what the Gaussian approximation predicts, not below.
The intuition follows straight from the asymmetry of recovery above, run in reverse. Volatility drag is punishing when the volatility is symmetric, because the deep down-moves demand those disproportionate recoveries. But if a strategy’s volatility is mostly to the upside, lots of variance expressed as occasional large gains, with the downside kept shallow, then much of that variance is no longer the corrosive kind. The formula charges a drag for all variance equally; a positively skewed distribution does not deserve the full charge, because its large moves are the helpful ones. Not all volatility is created equal. The shape of the distribution, its third moment, or skewness, matters, not just its width.
This is precisely the pattern visible on Shishin’s published track-record analysis. The realized volatility there is high, but it is mostly upside volatility: the distribution is skewed toward large gains while the drawdowns are comparatively contained. As a direct consequence, the realized compound return lands above the shortfall that a naive “arithmetic mean minus half the variance” calculation would predict from the headline volatility alone. It is a clean, real-world illustration of the skew exception: a high-variance record that compounds better than the Gaussian drag rule says it should, because the variance is the right shape. The “smoothed” partial-trim overlay the system runs is aimed at exactly this property, clipping the character of the volatility rather than merely reducing its size.
How a rules-based process treats this
A systematic, non-advisory process does not get to wish the arithmetic away, so it tends to treat the shape of returns as a first-class concern rather than an afterthought. Two records with the same average look different to it: the one with shallower losses and upside-skewed variance compounds better, so a rules-based approach watches the depth of drawdowns and the skew of the distribution, not the mean alone. That is a descriptive statement about how compounding mathematics constrains any long-horizon strategy, not a suggestion about what any individual should do. The practical lever most investors have over the drag is the depth of their losses, which is why volatility-aware exits and disciplined scaling out of extended positions keep recurring in the systematic-trading literature: their job is to keep the down-moves shallow enough that the recovery arithmetic stays forgiving.
The limits
A few cautions keep this from being over-applied. The half-variance identity is an approximation, accurate for the small, roughly symmetric returns of broad portfolios and looser for large or heavily skewed ones, so it is a lens, not a law. Variance and skewness are themselves estimated from a finite, noisy history, and both can shift with the market regime, so a drag or a skew measured over one period is not a constant. And the skew exception cuts both ways: just as upside skew can lift compound growth above the shortcut, downside skew (rare but severe losses, the fat left tail) can push it well below, which is the more dangerous case and the one behind most blow-ups. The safe reading is directional: volatility drags on compounding, deeper losses drag disproportionately, and the shape of the tail decides whether the simple formula overstates or understates the true result.
So why do the two returns differ?
Because wealth compounds multiplicatively while the arithmetic mean adds. The geometric return you actually keep is the arithmetic mean minus a penalty of roughly half the variance, so volatility is a direct tax on long-run growth, and losses cost more than equal gains return because the rebound works on a smaller base. The steadier of two paths with the same average tends to compound to more. The one genuine escape hatch is the shape of the distribution: variance expressed as upside skew, with the downside kept shallow, is far less corrosive than the symmetric kind, and can compound above what the simple drag formula predicts. The takeaway is a way of reading records: average return alone is not enough, and how a strategy earns its return, calm or wild, skewed up or skewed down, is as important as how much it earns.
Sources & further reading
- Fernholz, R. & Shay, B. (1982). “Stochastic Portfolio Theory and Stock Market Equilibrium.” Journal of Finance, 37(2), 615 to 624: the formal treatment of the variance drag on compound growth.
- Bernstein, W. & Wilkinson, D. (1997). “Diversification, Rebalancing, and the Geometric Mean Frontier.” An accessible derivation of the geometric-minus-half-variance relationship.
- Shishin, track-record analysis, a live illustration of the skew exception, where high but upside-weighted volatility compounds above the Gaussian variance-drag prediction, with the underlying record independently attested at /verify.
- Related concepts: what is volatility, risk of ruin, and the Kelly criterion, which uses this exact geometric-growth logic to frame sizing.