Leverage Decay Calculator
Leverage decay is the gap between a leveraged return and the simple multiple of the unleveraged one. It comes from compounding on a path rather than on a total, it grows with volatility and with leverage, and the drawdown scales faster than the return does.
Leveraged return versus the simple multiple
Defaults are this site's own measured series: a 3.61% unleveraged return at an implied volatility of 7.9%.
The estimate uses the standard volatility-drag approximation, L·r − ½·L·(L−1)·σ². At the defaults it reproduces this site’s measured 2x and 3x results to within 0.03 percentage points, which is why those defaults are the ones shown.
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How the number is built
A leveraged position multiplies each period’s move, then compounds the result. That is not the same as multiplying the total, and the difference is the decay.
The standard approximation is one line:
Leveraged return ≈ L·r − ½·L·(L−1)·σ²
where L is the leverage, r the unleveraged return and σ the volatility. The first term is what people expect. The second is the correction, and it is driven by volatility squared — so it grows much faster than volatility itself.
Notice L·(L−1). At 1x that term is zero and there is no decay at all. At 2x it is 2, at 3x it is 6, at 4x it is 12. Doubling the leverage from 2 to 4 multiplies the drag term six-fold.
A worked example
Take this site’s measured series: an unleveraged return of 3.61% at an implied volatility of 7.9%, held at 3x.
Simple multiplication says 3 × 3.61 = 10.83%. That is the figure most people carry in their head and it is the one the calculator labels “what simple multiplication says”.
The drag term is ½ × 3 × 2 × 7.9² ÷ 100 = 1.87 percentage points.
So the estimate is 10.83 − 1.87 = 8.96%. The measured result on the actual series was 8.93% — the approximation lands within 0.03 points of it.
The same formula at 2x gives 6.60% against a measured 6.61%. One volatility figure reproduces both measured results, which is the reason to trust the model rather than just the individual numbers.
What this site actually measured
Unleveraged, the shared series finished +3.61% with a worst drawdown of 3.76%. At 2x the result was
6.61% against a naive 7.22%, with drawdown at 7.45%. At 3x it was 8.93% against a naive 10.82%, with
drawdown at 11.08%. The figures are in research/series-measurements.json.
Read them as pairs and the asymmetry is the whole story. The drawdown at 3x is 2.95 times the unleveraged drawdown — almost exactly the multiple. The return is 2.47 times — noticeably short of it.
The downside scales faithfully and the upside does not. That single sentence is what the entire calculator exists to make concrete, and it is the opposite of how leverage is usually described.
The mechanism is the path, not the destination. Two series ending at the same price produce different leveraged results if one took a choppier route, because each period’s multiplied move compounds on the previous one.
The point where more leverage returns less
Because the drag term grows with L² and the return term only with L, there is a leverage beyond which adding more reduces the expected result. Differentiating the approximation gives that point as r ÷ σ² + 0.5, which the calculator reports.
At the defaults it is 6.28x. That is well above anything a retail leveraged product offers, which is worth saying plainly rather than implying that 3x is somehow past the peak on this series.
But the figure moves fast with volatility. Double the volatility to 15.8% and the peak falls to about 1.95x — below the leverage of every 2x and 3x product on the market. On a volatile instrument the peak can sit below 2, and that is the situation the formula is genuinely warning about.
Which makes volatility the input that matters most. Change the return and the answer moves linearly. Change the volatility and it moves with the square.
The original data
Of the 24,971 unique videos in research/search-study-corpus.jsonl, zero have an
instruction-shaped title about leverage decay or volatility drag. Leverage appears in 17
instruction-shaped titles at a median of 16,703 views across 17 channels. The counts come from
site/rank_tools2.py, which deduplicates by video id.
Seventeen videos on leverage and none on what it does over time. The mechanism that determines whether a leveraged position works over any period longer than a day is absent from a corpus of nearly twenty-five thousand videos, while the products themselves are sold to retail buyers as long-term holdings.
The answer to the question on that chart is that adding at 3x after an 11% fall is the decay working against you twice. The position needs a larger recovery than the fall to get back to even, and the leveraged version needs a larger one still — because each period’s move is multiplied on the way back up as well as on the way down.
When it fails
The approximation degrades over long horizons and at high volatility. It is a second-order expansion, so it is accurate for modest values and drifts when either input gets large. At 3x on a very volatile instrument held for years, treat the output as directional rather than precise — the sign and the rough magnitude are right; the decimals are not.
The second failure is that this model says nothing about being closed out. A gap can breach a maintenance requirement between prints, and no return calculation applies to a position that no longer exists.
A third is applying it to a product that rebalances differently. A daily-rebalanced fund and a margin position behave differently, and the approximation assumes constant leverage.
A fourth is using a volatility figure you guessed. The answer moves with the square of it, so an estimate that is out by half produces an answer that is out by four times as much.
A fifth is ignoring the cost of maintaining the leverage. A round trip on this site’s shared series is 2% of a median bar’s range, borrowing accrues interest daily, and neither appears in the formula above.
And a sixth is reading the peak-leverage figure as a recommendation. It is the point where the expected return stops rising, not a level anyone should trade at — the drawdown at that leverage would be unmanageable long before the return stopped improving.
Related
Leverage trading covers what borrowing does to a position and the mechanics underneath the decay. Leveraged ETF is the packaged version, where daily rebalancing makes this effect structural. And drawdown is the measurement that scales faithfully with leverage while the return does not.
The pair of numbers that settled this for me was 8.93 against 11.08 — the return at 3x and the drawdown at 3x. The pain nearly trebled and the reward did not quite double. Once you see leverage as something that scales the downside faithfully and the upside imperfectly, choosing a smaller multiple stops feeling timid and starts looking like arithmetic.
— Michael Whitman
This page is educational, not financial advice. Test every idea on your own charts before risking money.