Volatility drag: why averages overstate growth

Averages are worked out by adding and dividing. Balances are worked out by multiplying. Those two operations part company the moment the numbers involved are not all the same, and volatility drag is the size of the disagreement.

Turn each period's return into a growth factor: plus 60 percent is 1.60, minus 37.5 percent is 0.625. A $10,000 holding that gains 60 percent reaches $16,000, and a 37.5 percent fall from $16,000 puts it back at $10,000. An ending balance is the product of the factors, so the rate that describes it is found by multiplying and then taking a root:

$$1 + g = \left[\prod_{t=1}^{n}(1 + r_t)\right]^{1/n}$$

That is the geometric mean, and it is the only average that reproduces an ending value. The arithmetic mean adds the returns and divides by $n$: for this pair it gives $(60 - 37.5)/2 = 11.25$ percent a year, while $1.60 \times 0.625 = 1$ gives a compound rate of zero. Both figures are computed correctly from the same two returns. Only one of them is a rate at which anything grew.

The relationship between them is not a tendency with exceptions. For any set of non-negative growth factors the geometric mean is at most the arithmetic mean, and the two are equal only when every factor is identical. A compound return is therefore never above the average of the returns it came from, and it is below whenever those returns moved at all. Annualised return is the compound figure. A number labelled average return may be either one.

For exactly two periods the drag has a closed form. Write $\mu$ for the average of the two returns and $\sigma$ for their standard deviation, which for two numbers is half the distance between them. Then the compound growth factor satisfies:

$$(1 + g)^2 = (1 + \mu)^2 - \sigma^2$$

Nothing is approximated there. The spread of the returns, which is what volatility measures, is subtracted as a squared term: it costs nothing when the two returns match, and rises steeply as they part.

Hold the average at 10 percent a year and widen the spread:

Every row averages 10 percent. The bottom row loses money. And the gap is not proportional to the spread: tripling the half-spread from 10 points to 30 multiplies the gap by about nine.

Past two periods the exact identity gives way to an approximation carrying the same message, with $\sigma^2$ as the variance of the periodic returns:

$$g \approx \mu - \frac{\sigma^2}{2}$$

It is close over ordinary yearly swings and drifts once they get large. On the bottom row it predicts a gap of 12.5 points against a true 12.02, and on this page's opening pair it predicts minus 0.63 percent against a true zero.

Ask what it takes to undo a loss and the same asymmetry appears with no averaging involved at all. A fall of $L$ leaves $1 - L$ per dollar, and climbing back to 1 from there takes a gain of:

$$\frac{L}{1 - L}$$

The gain is larger than the loss because it is measured against the smaller balance the loss left behind. That denominator is the whole mechanism, and it also fixes the boundary: as $L$ approaches 1 the gain needed runs away to infinity, and at a total loss the formula stops working altogether, because a percentage of nothing is nothing and no gain of any size recovers it.

The fourth row is this page's opening example read backwards, and it is why that pair of returns nets to nothing.

What the table adds is where the damage concentrates. Moving from a 50 percent fall to a 60 percent fall adds ten points of loss and adds fifty points to the recovery needed. The far end of a deep drawdown therefore costs far more to undo than the first part of the same fall, which is why one severe year pulls a compound record down much harder than several mild ones that add up to the same arithmetic average.

This also settles a common objection to the whole idea. Nobody has to believe anything about markets for the drag to apply. It is a property of multiplication, and it would hold on a spreadsheet of made-up numbers.

A fund built to return a fixed multiple of an index's daily move resets its exposure every day. It hits the multiple on each day's return, and those days compound on one another, so what it delivers over a year is a different question.

One up-then-down pair makes the mechanism exact. If the index rises $x$ and then falls $x$, a fund at $L$ times the daily move finishes at:

$$(1 + Lx)(1 - Lx) = 1 - L^2x^2$$

Take $x$ as 10 percent. The index itself, where $L = 1$, ends the pair at 0.99, down 1 percent. A fund at twice the daily move ends at $1.20 \times 0.80 = 0.96$, down 4 percent. At three times it ends at $1.30 \times 0.70 = 0.91$, down 9 percent. Within the pair the return term scales with $L$ and the drag term with $L^2$, so doubling the exposure quadruples the penalty. That clean square is a property of the single pair, not of a long run of them: compound enough pairs and the fund's loss runs into its own floor of 100 percent, which no multiple can push it past.

Repeat that pair ten times and the index is down 9.56 percent while the fund is down 33.52 percent, not the 19.12 percent that doubling the index's fall would suggest. The worked examples below carry that arithmetic.

The effect does not run one way only. Three straight days of plus 10 percent leave the index at 1.331, up 33.1 percent, and the same fund at $1.20^3 = 1.728$, up 72.8 percent, more than twice the index's gain. A trend compounds in the fund's favour for the reason chop compounds against it. The multiple is fixed to the day, and which side of it the longer result lands on is a property of the path rather than something to assume in advance.

Two costs sit on top of this arithmetic: the financing inside the fund, and its ongoing charge. Both reduce the return in every market, up, down or flat. In the United States, prospectuses for these funds state the objective as a daily one and say results over longer holding periods can differ from the stated multiple.

Two statements get mixed together here, and only one of them is an identity.

The first is about a run of returns that has already happened. There the compound rate is below the arithmetic average whenever the returns varied, with no assumptions attached. It is arithmetic, not a theory about how markets behave.

The second is about a return that has not happened yet. Suppose each period's return is drawn independently from the same distribution with arithmetic mean $\mu$. The expected ending balance after $n$ periods is still $(1 + \mu)^n$, because expectations multiply for independent draws. What falls short of $\mu$ is the typical path rather than the average of all paths. An ending balance is a product of random factors, so its logarithm is a sum and its distribution is skewed to the right: a small number of very good paths carry the mean, while the middle of the distribution grows at roughly $\mu - \sigma^2/2$. Volatility drag describes that middle.

The distinction has a practical edge. It does not follow that cutting volatility raises compound growth, because most ways of cutting it also cut $\mu$, and holding cash cuts both. What raises compound growth is lowering dispersion without giving up as much expected return, which is the mechanical case for diversification: combining holdings that do not move together leaves a portfolio's expected return at the weighted average of its parts while its variance lands below the weighted average of theirs. The compound rate picks up that difference. Risk and return sets out the other half of the trade, which is what the extra dispersion is being paid for.

Since two averages exist and they disagree, the first question to ask about any performance figure is which one is in front of you.

The test is whether the number reproduces the ending balance. Take the growth factors, multiply them, take the $n$th root and subtract 1: that is the compound figure, and money grown at it lands exactly where the account landed. Add the yearly returns and divide by $n$ instead and the result lands somewhere else, always higher when the years differed.

In the United States, a fund's published average annual total return is the compound annualised figure, worked out from the beginning and ending value over the stated period, so the drag is already inside it. A number someone produced by adding a column of yearly percentages is a different object wearing the same word. Where a figure comes from a marketing page rather than a regulated performance disclosure, the working is worth asking for.

The arithmetic mean is not a mistake everywhere. It is the right input when the question is what a single next period is expected to return, and one-period models take arithmetic inputs by construction. It becomes wrong the moment it is presented as growth.

Two related habits are worth the same care. A monthly figure turned into a yearly one by multiplying by twelve is an arithmetic step laid over a compound process. And a compound return is still a nominal one, so subtracting inflation is a second adjustment on top, which the real return calculator handles properly rather than by simple subtraction. This page is educational material rather than advice about any particular holding.