Leveraged ETFs, and the arithmetic nobody reads

The prospectus already told you

A leveraged ETF targets a multiple of the underlying index's return over one trading day. That is the entire mandate. Nothing in the product promises 3x over a week, a month, or a year, and the funds say so plainly in their own documents.

To hold that daily multiple, the fund rebalances its exposure at the end of every session. If the index rises, the fund must add exposure to keep the ratio; if the index falls, it must cut exposure. The mechanical consequence is that the fund buys after up days and sells after down days, which is exactly the wrong behaviour in a market that oscillates.

The worked two-day example

Take an index at 100 and a 3x fund at 100. Day one the index rises 10%; day two it falls 9.0909%, which returns it exactly to 100.

Day 2 fund value = 130.00 × (1 − 0.272727) = 130.00 × 0.727273 = 94.55 Index round trip: 0.00% 3x fund round trip: −5.45%

The index finished exactly flat. The 3x fund lost 5.45%. Nobody made a bad decision and no fee caused it. The loss is pure arithmetic: a −27.27% day requires a +37.5% day to recover, and the fund never got one.

The closed form, and why it is exact

That example is not a special case, and the drag on a flat round trip has an exact solution rather than an approximation. Let d be the up-day return and L the daily multiple:

A round trip that returns the index to flat needs a down leg of d ÷ (1 + d). Fund = (1 + Ld) × (1 − L·d ÷ (1 + d)) = (1 + Ld)(1 + d − Ld) ÷ (1 + d) = (1 + d − L(L − 1)d²) ÷ (1 + d) Round-trip return = − L(L − 1) × d² ÷ (1 + d) Check, L = 3 and d = 10%: −(3)(2)(0.01) ÷ 1.10 = −0.06 ÷ 1.10 = −5.4545% ✓ matches the table

Everything worth knowing about decay falls out of that one expression. Drag scales with d squared, so doubling daily volatility quadruples the bill. It scales with L(L−1), so a 3x fund carries 6 units of drag against a 2x fund's 2 — three times as much, not one and a half times. And it is completely independent of direction: the same formula applies whether the round trip goes up first or down first.

Two results in that table surprise people. First, a plain −1x inverse fund has exactly the same drag as a +2x fund, because L(L−1) equals 2 for both. An inverse ETF is not a mild product; the moment you invert, you inherit leveraged-style decay even at a stated multiple of one. Second, a −3x fund carries 12 units of drag, double the 3x long, and loses nearly 11% on a flat 10% round trip. Inverse-leveraged products are the most path-sensitive instruments on the list.

A useful sanity check on the squared term: at 10% swings the 3x loses 5.455% and at 1% swings it loses 0.059%, a ratio of 91.8 rather than the 100 the d² term alone suggests. The gap is exactly the (1 + d) denominator — 100 × (1.01 ÷ 1.10) = 91.8. The approximation people quote, drag ≈ ½ × L × (L − 1) × σ², is that same result with the denominator dropped, which is fine at small daily moves and increasingly optimistic at large ones.

Path, not direction: a five-day comparison

The daily-reset mechanic is not only a tax. Compare two five-day paths, both with more up days than down.

Smooth trend index = 1.02^5 = 1.10408 → +10.41% 3x = 1.06^5 = 1.33823 → +33.82% 3 × index return = 31.22% → fund beats it by 2.60 points Oscillating index = 1.02×0.98×1.02×0.98×1.02 = 1.01918 → +1.92% 3x = 1.06×0.94×1.06×0.94×1.06 = 1.05238 → +5.24% 3 × index return = 5.75% → fund trails it by 0.52 points

Both paths had three up days. The trending path had compounding working for the holder and beat the naive multiple; the oscillating path had it working against and fell short. Leveraged ETFs are instruments for short, directional, low-chop periods. They are not broken products; they are products whose behaviour depends on the path, not just the endpoint.

The fees you do not see on the fact sheet

Drag is the headline cost and it is not the only one. A leveraged fund also pays an expense ratio and an embedded financing cost, because a 3x fund holding three units of exposure per unit of net assets is funding the other two units at short rates through swaps or futures.

Illustrative 3x fund, figures chosen to show the shape of the arithmetic: expense ratio 0.95%/yr financing: 2 borrowed units × 4.5% short rate = 9.00%/yr total carry = 9.95%/yr ≈ 0.0395% per trading day

The drag column uses ½ × 3 × 2 × 0.02² = 0.12% per day. At that volatility drag is roughly three times the carry, which is the right order of priority: if you are worried about the expense ratio on a leveraged ETF, you are worried about the wrong number. The rates and expense ratio above are illustrative — check the actual prospectus and the current short rate, because the financing component moves with policy and was materially smaller in a zero-rate era than it is now.

Rebalance flow near the close

The nightly rebalance is not abstract. It is a real order that has to be executed, and its size has a closed form too.

Rebalance trade = L × (L − 1) × NAV × d a 3x fund with $1B NAV after a +2% index day = 3 × 2 × $1B × 0.02 = $120M of exposure to BUY into the close aggregate across an illustrative $30B of 3x long AUM, on a −3% day = 6 × $30B × 0.03 = $5.4B to SELL into the close

The direction is always the same as the day's move, which is why leveraged funds are structurally momentum-amplifying in the last half hour. The AUM figure above is illustrative — I do not have a verified current number and I am not going to invent one. The mechanism is what matters: it is real, it is mechanical, it is predictable, and it means the closing auction on a big directional day contains flow that has nothing to do with anyone's opinion of value.

Two practical consequences. Do not judge your fill quality against the close on a violent day, and do not assume the last thirty minutes carries the same information as the rest of the session.

Tracking error against the stated multiple

Over one day these funds do their job well; deviation from the stated multiple is typically small and comes from fee accrual, swap pricing, and the mechanics of tracking the index. Over any longer window, deviation is not error at all — it is the product working correctly. A 3x fund that returned 5.24% while the index returned 1.92% did not miss its target, because it never had a multi-day target.

This is why "the ETF is broken" complaints are usually a reading failure rather than a product failure. The one genuine warning sign is a fund whose single-day return persistently diverges from the multiple, which points at liquidity or swap-counterparty problems and is worth taking seriously. Compare the fund's daily return against the multiple for a week before you size anything meaningful in it.

When they fit and when they do not

Sizing must be tighter, not the same

The naive mistake is to size a 3x position the same way you size a normal one and treat the extra leverage as extra upside. Run the risk arithmetic instead. Take a $30,000 account risking 1%, a $300 budget, and a stop set at 5% of the underlying's price in every case.

The position must shrink by roughly the leverage factor for the risk to stay identical. That is the whole result, and it is the opposite of how leveraged products get used in practice, where the leverage is treated as a way to control more exposure with the same share count.

My own execution rules encode this as a hard half-cap on leveraged ETF positions: whatever normal sizing produces, a leveraged instrument gets half of it. That is a deliberately blunt rule, and blunt rules survive contact with a bad week better than clever ones do. The bottom row shows what it actually costs — 20 shares and $144 of risk instead of $295 — which is the price of not being the person who discovers path dependence with real money.

• Compute the stop from the fund's own ATR, not from the underlying's chart. The stop placement guide covers the true-range arithmetic, and gaps matter more here because a gap is multiplied by the leverage factor too.
• Set a maximum holding period in days and honour it, because both drag and carry are functions of time in the instrument, not of being right.
• Never hold two correlated leveraged positions and count them as separate risk. They are one position with a bigger number.
• Never treat an inverse fund as the safe version of a short. A −1x carries the same drag as a +2x, and a −3x carries double a +3x.
• Recompute size on the day you enter, not the day you found the idea, because leveraged ETF ATR moves fast.
• If the thesis genuinely needs weeks, use a defined-risk structure instead — vertical spreads have known decay rather than path-dependent decay.

One last framing. Leverage does not change expectancy, it multiplies both the expectancy and the variance, and variance is what determines whether you survive long enough to collect the expectancy. That asymmetry is the whole subject of drawdown recovery math — a 50% loss needs a 100% gain to undo — and it is why Kelly argues for smaller size, not larger, whenever the edge estimate is uncertain. Measure your leveraged trades as a separate tag in your journal and compare their realised R against your unleveraged trades. Position sizing arithmetic, break-even win rates, and risk of ruin are all in the free risk toolkit, and the general sizing framework is in position sizing per trade.

Educational content, not financial advice. No live profit-and-loss figures are claimed anywhere on this site; backtest and walk-forward results are always labelled as such. Full terms: /terms.html

Tools referenced in this guide

• Trader's risk toolkit — position sizer and risk-of-ruin simulation for tighter leveraged sizing.
• Kelly criterion guide — why estimated edges argue for smaller size, especially with leverage.
• TradeLog — tag leveraged trades separately and compare their real expectancy against unleveraged ones.