Note · Opinion, not rated
The average return is not the return you get.
An investment that gains 60% one year and loses 40% the next has averaged 10% a year and lost you money. The gap between the average return and the one you compound at has a formula, and leverage squares it.
The arithmetic
An average return adds the yearly returns and divides. Your money does something else: it multiplies them. A loss always needs a larger gain to undo it, so whenever returns vary, the compound return you actually receive is lower than the average. Two years, four ways:
| Two years | Average return | Annualized return | What happened to your money |
|---|---|---|---|
| +50%, then -50% | 0% | -13.4% | -25.0% |
| +30%, then -10% | +10% | +8.2% | +17.0% |
| +60%, then -40% | +10% | -2.0% | -4.0% |
| +10%, then +10% | +10% | +10.0% | +21.0% |
The last three rows all average 10%. Only the one with no volatility delivers it. The +60% and -40% pair averages 10% a year and leaves you 4% poorer. Both numbers are true. Only one of them is yours.
So when a fund advertises an average annual return, ask for the annualized one. They are different numbers, and the difference is a measure of how much of the advertised return you would not have received.
The half-variance rule
The gap has a simple approximation: the typical compound return is the average return minus half the variance, the square of the volatility. Hold the expected return at 8% and raise the volatility:
| Annual volatility | Typical compound return |
|---|---|
| 0% | +8.00% |
| 10% | +7.46% |
| 15% | +6.79% |
| 20% | +5.86% |
| 30% | +3.25% |
| 50% | -4.69% |
Volatility is not just discomfort. It is a cost that comes straight off the compound return, even when the expected return does not change at all. At 20% volatility the typical outcome compounds at 5.9%. At 50%, an asset with an 8% expected return typically loses money. That is the arithmetic behind the cost of concentration in your employer already owns enough of you: a single stock carries far more volatility than an index, so its typical outcome is worse even at the same expected return.
Where it bites: daily leveraged funds
Most leveraged and inverse exchange-traded funds reset daily. The SEC's investor education office says so plainly, and adds that their performance over longer periods can differ significantly from the stated multiple, an effect magnified in volatile markets. Here is why. A fund that delivers three times the index's return every day multiplies the average by three and the variance by nine.
The cleanest demonstration is a market that goes nowhere. The index rises 10% one day and falls back to exactly where it started the next:
| Fund | Day one | Day two | After two days |
|---|---|---|---|
| 1x | +10.0% | -9.09% | 0.00% |
| 2x | +20.0% | -18.18% | -1.82% |
| 3x | +30.0% | -27.27% | -5.45% |
The index is flat. The 3x fund is down 5.5%, and it would keep losing for as long as the market kept chopping sideways.
Ten years of a 3x fund
Now a realistic market. We simulated 20,000 ten-year paths for an index with an 8% expected return and 18% volatility, and ran 1x, 2x and 3x daily funds on exactly the same paths, before any fees or borrowing costs. Daily returns are drawn from a normal distribution, which understates how sharply real markets can jump, and sharp jumps hurt leveraged funds most.
| Fund | Average outcome | Typical outcome | Bad case (10th) | Good case (90th) | Lost money | Ended behind 1x |
|---|---|---|---|---|---|---|
| 1x | 2.15x | 1.84x | 0.89x | 3.81x | 14.5% | – |
| 2x | 4.65x | 2.43x | 0.57x | 10.46x | 22.1% | 31.3% |
| 3x | 10.01x | 2.33x | 0.26x | 20.86x | 31.3% | 42.0% |
Read the first two columns together. The average 3x outcome is 10.01x your money. That number is real, it is what you would get averaged across every path, and it is the one that makes the product look good. The typical outcome, meaning the median path, is 2.33x, below the 2x fund's 2.43x. The average is carried by a small number of enormous paths: the good case is 20.86x. Meanwhile the bad case is 0.26x, a loss of nearly three quarters, 31% of paths lost money over a full decade, and 42% ended behind the plain index they were built on.
Leverage multiplies the average and squares the drag. The typical holder lives with the drag.
And that is before costs
Before costs, at these inputs, the typical 3x outcome still beats the index, by 2.6 percentage points a year. That edge is fragile in two ways.
It depends on calm markets. The same closed-form rule that the simulation confirms shows how fast it goes as volatility rises:
| Annual volatility | 1x typical | 2x typical | 3x typical |
|---|---|---|---|
| 10% | +7.5% | +14.3% | +20.4% |
| 15% | +6.8% | +11.5% | +13.8% |
| 18% | +6.3% | +9.3% | +8.9% |
| 25% | +4.7% | +2.9% | -4.9% |
| 30% | +3.2% | -2.6% | -16.0% |
Above about 20% annual volatility, the typical 3x outcome falls below the unleveraged index even before costs, and markets are not always that calm.
And a 3x fund has to pay, one way or another, for the two extra dollars of exposure it holds per dollar of yours. If the borrowed portion costs 4% a year, an illustrative figure, the typical annual return at 18% volatility is 6.3% for the index, 5.1% for 2x and 0.7% for 3x. Over ten years that is 1.84x your money for the plain index and 1.07x for the fund with three times the exposure. Fees would come on top.
What to take from it
Use annualized returns, never averages, when comparing anything held for more than a year. The average describes the returns. The annualized figure describes your money.
Treat volatility as a cost, not just as a risk. Two investments with the same expected return are not equally good if one is twice as volatile. The calmer one typically compounds faster.
Understand what a daily leveraged fund is designed for: a day. Held for years, it is a bet on low volatility as much as on direction, and the typical result can be worse than not using leverage at all.
Sources
- U.S. Securities and Exchange Commission, Office of Investor Education and Advocacy, Leveraged and Inverse ETFs, for the daily reset and the warning that performance over longer periods can differ significantly from the stated multiple, magnified in volatile markets.