Risk compensation shows up in the average return, but what investors receive is the compound return. Volatility stands between the two: for the same average return, the greater the volatility, the less is left after compounding, and most people end up below the average. This is not some additional leakage; it is a mathematical property of compounding itself. It is amplified in fixed-leverage products, and in a regularly rebalanced portfolio it can work the other way and become a benefit.
An investment rises 50% in the first year and falls 50% in the second. What is the average annual return? Adding the two years' returns and averaging gives 0%. Yet 100 becomes 150 and then 75, a real loss of 25% over the two years. The first is called the arithmetic mean; the second is the geometric mean, that is, the compound return. What investors actually receive is the geometric mean.
Up 50%, down 50%: what is the average?
Each year the value either rises x% or falls x%, alternately. Adjust x and compare the arithmetic mean with the compound return actually received.
The gap between the arithmetic and the geometric mean comes from volatility: the greater the volatility, the wider the gap, which is roughly half the square of the volatility. For an asset with 20% annualized volatility, the compound return is about 2 percentage points below the arithmetic average return; at 40% volatility it is about 8 percentage points below. This is "volatility drag".
Same average return: more volatility, less compounding
Both assets have an arithmetic mean return of 8% a year. Adjust the volatility of the one on the right and see the gap in the typical (median) outcome after 20 years.
This means that two assets cannot be compared by their average returns alone. Of two assets with the same average return, the more volatile one ends up worse after long-term compounding. It is also why diversification, by lowering volatility, raises the compound return of a portfolio.
Volatility drag has a more intuitive form: after a loss, a larger gain is needed to get back to the starting point. A 10% fall needs an 11% rise to recover; a 50% fall needs a 100% rise; an 80% fall needs a 400% rise. Falls and rises are measured from different bases.
Easy to fall, hard to climb back
Drag the size of the loss and see how large a gain is needed to get back to the start.
Over longer horizons, the distribution of compound returns becomes increasingly skewed: a few exceptionally good outcomes pull the average up, while most people's outcomes fall below it. Hold a highly volatile asset for 30 years, and the average outcome may look impressive, yet most paths end below that average. Presenting a long-term product only by its average return will leave most users' expectations unmet.
Most people sit below the average
A simulation of 2,000 investors, each holding the same asset. Choose the holding period and volatility to see the distribution of ending values, the mean and the median, and how many people end up below the mean.
Daily 2x and 3x leveraged ETFs are often thought to suffer a kind of "decay". In a published article, the author used a simple example to show that compounding itself is not the problem; the deviation comes from buying on the way up and selling on the way down in order to maintain a fixed leverage.
Suppose you have $100, borrow another $100, and buy $200 of an asset, which is 2x leverage. On the first day the asset rises 10%: the position becomes $220, of which your own share is $120, and leverage falls to about 1.83x. To restore 2x you must borrow another $20 and buy, which is adding to the position at a high price. On the second day the asset falls 10%: the position loses $24, all of it borne by you, so your own capital falls to $96 and leverage rises to 2.25x. To restore 2x you must sell $24 of the position to repay debt, which is cutting the position at a low price. Over the two days the underlying fell only 1%; the "2x" intuition says you should lose 2%, but you actually lose 4%.
Change the market path, and the direction of the deviation changes
An ordinary investor who buys a leveraged ETF and simply holds it does not personally buy high and sell low. The fund makes these adjustments every day, and the effect passes to holders through the net asset value, which is why it is hard to notice. A more accurate word than "decay" is "deviation", and the deviation itself is neutral: in a market trending steadily up or down it can be favourable; in a market that swings back and forth it is unfavourable. Cheng and Madhavan (2009) and Avellaneda and Zhang (2010) analysed this path dependence systematically.
For the same reason, trying to "exploit" the deviation of leveraged ETFs for arbitrage is logically equivalent only to doing the opposite yourself, selling high and buying low, and selling high and buying low does not guarantee a profit.
A diversified portfolio held in fixed proportions, for example half stocks and half bonds, does exactly the opposite when it is rebalanced periodically: it sells the asset that has risen more and buys the asset that has fallen more, that is, it sells high and buys low. In a market that swings back and forth, this lifts the portfolio's compound return above the weighted average of its assets' compound returns; Booth and Fama called this the "diversification return" in 1992. In a one-way trend, by contrast, it lags a portfolio that is never adjusted.
Rebalancing: the same mechanism in reverse
Put 50 into each of two assets with the same long-run trend but alternating moves. Compare “buy and never adjust” with “reset to half and half every period”.
Fixed leverage and fixed proportions are two directions of the same mathematical property: the first buys high and sells low and loses in choppy markets; the second sells high and buys low and gains in choppy markets. Neither is a free lunch; each is a different bet on the shape of market movements.
What investors receive is the compound return, yet products often display the average return. Products should explain the gap between the two on their own initiative.
Express past performance as an annualized compound return; when the average and compound returns differ widely, explain why.
Let users know before they buy that a 50% drawdown needs the value to double to break even.
Do not present long-term outcomes only through the average case; show the typical outcome and the poorer cases.
Explain that results drift away from the “multiple” when held for more than a day, and that the direction of the deviation depends on the shape of the market path; it is not simply “decay”.