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Volatility Drag and Leveraged ETF Decay

After reading this you will understand why a 3× fund does not deliver 3× the index over a year, how to estimate the gap from just the volatility and the leverage, and when leverage helps rather than hurts.

What volatility drag is

Suppose an index goes up 10% one day and down 10% the next. You might expect to break even. You do not. Start with $100. After the gain you have $110. After the 10% loss you have 110 × 0.90 = 99. You are down $1 even though the two percentage moves cancel on paper.

That $1 is volatility drag. Compounding rewards steady growth and punishes swings, because a loss needs a larger gain to recover from: down 10% then up 10% leaves you below where you started. The bigger the swings, the bigger the loss even when the average daily return is zero.

A daily-rebalanced leveraged ETF magnifies this. A 3× fund targets three times the index return each day, then resets. In the example above, day one it gains 30% (to $130), day two it loses 30% (to $91). The plain index is at $99; the 3× fund is at $91. Tripling the exposure tripled the daily moves, and the drag grew with the square of that scaling. This tool shows how much of your return the effect eats over a full horizon.

When to use this tool, and when not to

Use it to sanity-check what a leveraged ETF can realistically return before you hold one for months or years. The output is a distribution, not a single number, so it also tells you how wide the range of outcomes is and how often leverage ends up behind the plain index.

Do not use it as a forecast of any specific fund. The model assumes daily returns drawn from a lognormal process with constant volatility. Real markets have fat tails, volatility clusters, and funds face tracking error the model ignores. Treat the numbers as the physics of compounding under leverage, not a prediction.

Leveraged ETFs are built to deliver their multiple over a single day. Their own prospectuses say holding them longer produces returns that can differ sharply from the multiple times the index return. This tool quantifies that difference.

The formula and the intuition behind it

The core fact is that compounding turns an arithmetic mean return into a smaller geometric (compound) growth rate. If daily returns have arithmetic mean \mu per year and volatility \sigma per year, the long-run compound growth rate is approximately:

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

Here g is the growth rate you actually earn over many periods, \mu is the simple average of the returns, and \sigma^2 is the variance. The term \sigma^2/2 is the drag. With \mu = 0.07 and \sigma = 0.18, the drag is 0.18² / 2 = 0.0162, so 1.62% per year, leaving a compound rate near 5.38%.

Now add leverage L. A daily L× fund has arithmetic mean L\mu and volatility L\sigma. Substitute those in and you get the growth rate of the leveraged fund, before costs:

g_L \approx L\mu - \frac{(L\sigma)^2}{2} = L\mu - \frac{L^2\sigma^2}{2}

The mean scales with L, but the drag scales with L^2. Double the leverage and the drag quadruples. That single asymmetry is the whole story. Subtract the fund's expense ratio and its borrowing cost on the leveraged portion, and you have the estimate the tool reports alongside the simulation.

A worked example using the demo data

7% return, 18% volatility, 3×, 10 years

Load the demo: mu = 7, sigma = 18, leverage 3, expense 0.95%, borrow spread 1%, horizon 10 years. Work the closed-form pieces by hand.

  1. Plain index compound rate: 0.07 − 0.18²/2 = 0.07 − 0.0162 = 0.0538, so about 5.38% per year.
  2. 3× arithmetic mean: 3 × 0.07 = 0.21.
  3. 3× drag: (3 × 0.18)² / 2 = 0.54² / 2 = 0.2916 / 2 = 0.1458, so 14.58% per year lost to drag alone.
  4. Gross 3× growth: 0.21 − 0.1458 = 0.0642, about 6.42%.
  5. Subtract costs. Expense ratio is 0.95%. Borrowing applies to the 2 units borrowed at a 1% spread, so 2 × 1% = 2%. Net: 6.42% − 0.95% − 2% = 3.47%.

Read that carefully. The 3× fund is projected to compound at about 3.47% per year while the plain index compounds at 5.38%. You took triple the risk and ended below the unleveraged index. Over 10 years the index grows by 1.0538¹⁰ ≈ 1.69×, while the 3× fund grows by 1.0347¹⁰ ≈ 1.41×.

The Monte Carlo simulation with 252 daily steps per year and 2000 trials will report a median near these figures, plus the spread. Because leveraged outcomes are heavily right-skewed, the median sits well below the arithmetic mean: a few lucky paths pull the average up while the typical path lands lower.

Explore how leverage reshapes the outcome

With μ = 7% and σ = 18%, the projected annual compound rate is about 5.38% at 1×, 6.72% at 2× (before costs) and 6.42% at 3× (before costs). Raising volatility to 30% flips the ranking: 1× compounds at 2.5%, but 3× drags to −3.5% before any fees. Higher volatility punishes higher leverage first.

The peak of that curve sits at L^* = \mu / \sigma^2. For the demo numbers, 0.07 / 0.18² = 0.07 / 0.0324 ≈ 2.16. Even ignoring fees, the return-maximising leverage is about 2.16×, so 3× is already past the top of the hill.

Reading and interpreting the results

The tool gives you three things worth separating in your mind.

Median outcome
The middle path. Half of simulated runs end above it, half below. For skewed leveraged returns this is a more honest "typical" figure than the mean.
Percentile spread
The distance from, say, the 10th to the 90th percentile. Leverage widens this dramatically. A 3× fund can 5× your money or lose 80% of it over the same horizon depending on the path.
Probability of lagging the index
The fraction of trials where the leveraged fund finishes below the plain 1× index. When this exceeds 50%, leverage is more likely than not to have hurt you, even though its average return may look higher.
With μ = 7%, σ = 18% and the demo costs, 2× barely beats 1× and 3× falls well behind. The 2× figure is 2×0.07 − (2×0.18)²/2 − 0.95% − 1% = 5.24%.

Compare the median, not the mean, when judging a leveraged position. Mean returns of skewed distributions are dominated by a handful of extreme winners you are unlikely to experience.

Common mistakes

Assuming linearity. "3× the index" describes one day, not one year. Over a year the multiple you actually earn depends on the path the index took to get there.

Ignoring the squared term. People discount volatility drag because at 1× it is small (1.62% here). Under 3× leverage it becomes 14.58%, larger than the gross expected return itself.

Forgetting borrowing cost scales with leverage. A 3× fund borrows two units per unit of equity, so a 1% financing spread costs 2% of your capital per year, not 1%.

Reading a good backtest as proof. Leverage wins in strong, low-volatility uptrends and loses in choppy, sideways markets. A backtest over one calm decade tells you little about the next volatile one.

Related tools

To project unleveraged growth with regular contributions, use the Investment Growth Calculator. To convert a start and end value into a compound rate, use the CAGR Calculator. To see how ordinary fund fees compound over decades, see the Investment Fee Impact Calculator, and to strip inflation out of a return, use the Inflation-Adjusted Return Calculator. If you are weighing how leverage fits a broader mix of assets, the Efficient Frontier & Rebalancing Bonus tool shows how diversification and rebalancing interact.

Frequently asked questions

Why does a 3× fund sometimes still beat the index?

In a strong, steady uptrend with low volatility, the L\mu gain outweighs the L^2\sigma^2/2 drag. If the index rises 15% a year with only 10% volatility, 3× compounds far ahead. The loss cases come from volatile, directionless markets.

What leverage maximises long-run growth?

Ignoring fees, L^* = \mu / \sigma^2. For μ = 7% and σ = 18% that is about 2.16×. This is the same idea as the Kelly criterion. Fees and the risk of large drawdowns argue for holding less than L^*, not more.

Does volatility drag apply to a normal, unleveraged fund?

Yes. Every asset with volatility compounds below its arithmetic mean by roughly \sigma^2/2. At σ = 18% that is 1.62% per year. Leverage does not create the effect, it multiplies it by L^2.

Why simulate instead of just using the formula?

The formula gives a smooth median estimate. The Monte Carlo shows the full spread and the probability of lagging the index, which the closed form cannot. Leveraged returns are skewed, so knowing the range matters as much as the center.

Is holding a leveraged ETF for years always a mistake?

Not always, but the math tilts against it, and the fund's own prospectus warns that returns over long periods can diverge sharply from the daily multiple. This is an educational estimate, not financial advice. Speak with a qualified adviser before committing money.

Try it yourself in the tool

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