Tools

Leverage Decay Horizon Calculator

A fund states a daily multiple of an index. Over what holding period does it stop delivering that multiple, and what does the answer actually depend on?

A fund promising twice an index promises it over a day. Held longer, it drifts away from twice the index, and the gap has four separate causes — only one of which behaves like a fee. The largest is proportional to the square of volatility, which is a quantity nobody showed you when the multiple was quoted. This page asks you for it and tells you how long the fund takes to fall as far behind as you say counts.

What this tool does not decide

What the index will do. It does not appear in the arithmetic at all — the underlying's own return cancels out, which is exactly what makes it possible to state a horizon without forecasting a price.
What volatility will be. The figure you enter is a scenario you are choosing. Realised volatility is not knowable in advance, and no answer here is a date.
What any fund returned or will return. This page publishes a gap against a benchmark and holds no expected return, no distribution and no probability.
Whether to hold a leveraged fund, or for how long. A horizon is something to compare against your own, not a verdict on a holding.
Which fund you hold. This tool names no fund, ticker, index or issuer, ships none and fetches none. Every figure is yours.
Anything about tax, in any jurisdiction.

Your figures

The interactive calculation runs in your browser and needs local JavaScript, which is currently switched off. Activating the Calculate button does nothing at all: no request is sent and nothing is submitted.

Everything else on this page works without it. The model, the formulas, the timing convention, the worked example, the assumptions and the limitations are all in the methodology below, and the companion explainer covers the mechanism in full.

The fund, and the scenario you are asking about

Seven figures, all yours. Nothing is sent anywhere, nothing is stored, and no fund, ticker, index, price or volatility is looked up. There is no field for what the index does, because it cancels out of the arithmetic entirely — that is what makes it possible to answer this question without forecasting a price.

The stated daily multiple

2 for a double-long fund, -2 for a double-short, 3 for a triple-long. This is the multiple the fund states over a single day, and it is not a position size or a borrowing ratio. A multiple between 0 and 1 is admitted and behaves differently from all the others: such a fund runs ahead of its benchmark rather than behind it.

The volatility you are asking about

The annualised volatility of the underlying over the period. This is the quantity nobody showed you when the multiple was quoted, and it is the one the gap depends on most sharply — the volatility term is proportional to its square, so doubling this quadruples that part of the answer. It is a scenario you are choosing, not a measurement and not a forecast.

The reference financing rate

The rate at which the leverage is financed. On a long fund it is a cost; on an inverse fund it is a credit, because a short position earns on the cash it holds. Enter 0 to see the volatility term on its own.

The fund’s expense ratio

The fund’s own charge, as stated in its documents. This is the one of the four causes that behaves the way a reader expects a cost to behave: a straight line, the same every year, independent of everything else.

The cost of borrowing the underlying

Only meaningful on an inverse fund, which has to borrow what it is short. Enter 0 for any fund with a positive multiple — a long fund does not borrow the underlying, and the relationship this page rests on states that assumption explicitly rather than quietly ignoring a number you entered.

How far behind counts as “stopped”

Your own threshold: how far behind the stated multiple the fund has to fall before you would say it has stopped delivering it. There is no default here and no suggested figure, because this is a judgement rather than a fact, and a number filled in for you would be an opinion published as a horizon.

The horizon you are asking about

How long you are considering holding, in years, and it may be a fraction. It does not change the rate at which the gap opens; it decides where you stand along it, and it is the range the table below is sampled over.

Result

There is no such thing as “the” decay rate for leveraged funds. The gap between a fund and the multiple it states has four separate causes, and only one of them is the one people mean: the volatility term is proportional to the square of volatility, so doubling the volatility quadruples it, and it is not the same size for a double-long and a double-short on the same underlying. The other three are financing, the fund’s charges, and the cost of borrowing the underlying, and each moves independently of the others. Every figure on this page is conditional on the five assumptions you entered. Change any of them and the answer changes with it, which is why no single annual figure can be published as a general rule.

Enter your seven figures above and select Calculate. Nothing is sent anywhere: the calculation runs in this browser, and no value is stored, shared or placed in the address bar.

From volatility — the only cause that is not a straight line: Not yet calculated
From financing the borrowing: Not yet calculated
From the fund’s own charges: Not yet calculated
From borrowing the underlying: Not yet calculated
The four together, as a rate a year against the multiplied underlying: Not yet calculated
Which way the gap runs: Not yet calculated
Where the fund stands at your horizon, as a fraction of the multiplied underlying: Not yet calculated
How far behind that is, at your horizon: Not yet calculated
How long until the shortfall you named is reached: Not yet calculated

This relationship is exact rather than approximate, and it matters what it is exact in. Given the volatility that was actually realised over the period, the gap between a continuously rebalanced fund and the multiplied underlying is an identity — no distribution is assumed, volatility is not assumed constant, and no return is being forecast. What is approximate is the daily rebalancing a real fund does rather than the continuous rebalancing the identity describes, and how large that difference is depends on the fund and the period. This page publishes the exact relationship and names the approximation rather than folding one into the other.

This computes a gap and nothing else. It does not say where the underlying went, what the fund returned, or what either will do — the return of the underlying cancels out of the arithmetic entirely, which is why a horizon can be stated without anyone forecasting a price. There is no distribution, no probability, no expected return and no drift anywhere in it, and there is no field on the model that could hold one.

The volatility is yours, not a measurement and not a prediction. Every figure here answers “if volatility over the period turns out to be this, how far behind does the fund end up” — and realised volatility is not knowable in advance, so the horizon below is a scenario and not a date. It is also not a recommendation: whether a leveraged fund suits a holding period is a question about what you are trying to do, and nothing on this page decides it.

How this is worked out

There is no single decay rate, and this is why

The gap between a leveraged fund and the multiple it states has four separate causes. Three of them behave the way a reader expects a cost to behave — a straight line in their own driver, the same every year. One does not.The volatility term is proportional to the square of volatility, so doubling the volatility quadruples it, and its size depends on the multiple in a way that is not symmetric: a double-short's volatility term is three times a double-long's on the same underlying.

Five funds under one set of costs — a financing rate of 4% and charges of 0.95% — and the holding period each takes to fall 10% behind its own stated multiple. Nothing varies down this table except the multiple and the volatility.
The fund	Volatility	From volatility	From financing	From charges	Total a year	Until 10% behind
Double-long, 2×	20%	−4.000%	−4.000%	−0.950%	−8.950%	1.18 years
Double-long, 2×	40%	−16.000%	−4.000%	−0.950%	−20.950%	0.50 years
Double-short, −2×	20%	−12.000%	+12.000%	−0.950%	−0.950%	11.09 years
Triple-long, 3×	20%	−12.000%	−8.000%	−0.950%	−20.950%	0.50 years
Half, 0.5×	20%	+0.500%	+2.000%	−0.950%	+1.550%	Never — it runs ahead

Two rows are worth reading twice. The double-short has the largest volatility term of the four decaying funds and the smallest total gap, because a short position earns the financing a long one pays — a single netted headline figure would have hidden both facts at once.The fund at half the index does not decay at all: below a multiple of one the volatility term changes sign, and the fund runs ahead of its own benchmark rather than behind it. There is no horizon for it, and this page says so rather than printing a very large number.

The relationship, and what is exact about it

Writing β for the stated daily multiple, σ for annualised volatility,r for the financing rate, f for the expense ratio and λ for the cost of borrowing the underlying, the fund's value as a fraction of the β-power of its own underlying grows at:

g = (1 − β) × r  −  f  +  β × λ  −  (β² − β) × σ² ÷ 2

and the fraction itself is exp(g × t) after t years. The horizon this page publishes inverts it:

years to a shortfall of x   =   ln(1 − x) ÷ g

Where the underlying went does not appear. It cancels out of the arithmetic entirely, which is what makes it possible to answer this question without forecasting a price — and it is why there is no field on this page for what the index does.

The research this rests on, and the line it draws

The relationship above is equation (10) of M. Avellaneda and S. Zhang,Path-dependence of Leveraged ETF returns (Courant Institute, 2009; SIAM Journal on Financial Mathematics, 2010). Its section 2.2 draws the line this page is built on in one sentence: the discrete daily form "is an approximation which is valid for Δt ≪ 1 whereas (10) is exact if the ETF price follows an Itô process."

The Itô assumption is weaker than it sounds, and that is the finding. Volatility and drift are "assumed to be random and non-anticipative", and a footnote adds that they are not assumed to be deterministic functions or constants. So the exactness requires no constant volatility, no lognormality and no forecast of anything — it is a statement about the volatility that actually happened, which is why this page can ask you for one rather than telling you what it will be.

That distinction also settles a contradiction in the evidence behind this page. Published long-run figures for leveraged funds disagree with each other by very large margins, and at least one paper reports errors of over a hundred percentage points annually in some of them. Those figures are about the expected return of a fund, conditional on an index's mean and standard deviation — a forecast over a distribution that has not happened yet.This page publishes no expected return and holds no distribution. It reports a gap against a benchmark, given a variance you state, and there is no field on the model that could hold anything else.

What you enter

The stated daily multiple: The multiple the fund states over one day. Not a position size
The volatility: Annualised, as a percentage. Yours to state — a scenario, not a measurement
The financing rate: Per year. A cost on a long fund and a credit on an inverse one
The expense ratio: Per year, from the fund’s own documents
The borrow cost: Per year, and zero on any long fund. Refused rather than ignored there
The shortfall: How far behind counts as “stopped”. Yours, and there is no default
The horizon: In years, and it may be a fraction. It decides where you stand, not the rate

The limits this rests on

Every one of these comes from the paper the relationship is taken from, or from what this implementation does and does not do. A figure published without them would be a stronger claim than the research supports.

The relationship published here is the continuous one, and it is exact if the price follows an Itô process. The daily form a real fund actually follows is an approximation to it, valid where the time step is small. This page publishes the exact relationship and names the approximation rather than folding one into the other.
Exactness is exactness in the volatility that was realised. It is not a claim that any particular volatility will occur, and the figure you enter is a scenario you are choosing.
The source assumes explicitly that a fund with a positive multiple has no cost of borrowing the underlying. This calculator enforces that rather than warning about it.
Volatility and drift are not assumed constant and not assumed deterministic — the source says so in a footnote, and it is what makes the identity usable without a forecast. Nothing here requires lognormality.
The fund is assumed to deliver its stated method. Tracking error, creation and redemption mechanics, and any deviation between a fund and its own prospectus sit outside this arithmetic.
Transaction costs inside the fund beyond the stated expense ratio are not modelled, and on a high-volatility underlying they are not negligible.

What is not modelled

where the underlying goes, which cancels out of the arithmetic and is deliberately absent from every result
daily rather than continuous rebalancing, which is the approximation a real fund makes to this relationship
the probability of any volatility, which is not a quantity anywhere on this page
tracking error, creation and redemption, and any difference between a fund and its own stated method
transaction costs inside the fund beyond the stated expense ratio
any fund, issuer, ticker, index or price — none is named and none is fetched
what a leveraged fund is suitable for, and for how long
tax, in any jurisdiction

Compounding is not symmetric — a fall and the gain that undoes it are different sizes — and that asymmetry is the same property that puts a variance term into the relationship above.The explainer on why losses require larger gains covers it in the simplest case, and needs no JavaScript at all.

Go deeper

The nearest published mechanism
Why Losses Require Disproportionately Larger Gains
Why a percentage loss and the same percentage gain never cancel, why getting back to the old balance is not the same as getting back to the original plan, and why the size of a loss is not a reason to keep holding.
Compounding is not symmetric, and that asymmetry is the same property that puts a variance term into the relationship on this page. The explainer covers it in the simplest case, with no leverage in it at all.
Read the explainer: Why Losses Require Disproportionately Larger Gains

Calculation model and corrections

Calculation model: Leverage Decay Horizon v1.0
Last reviewed: 27 August 2026
What is measured: The gap between the fund and the stated multiple of its own underlying — a fraction of a benchmark, never a return. Where the underlying went cancels out of the arithmetic entirely and appears nowhere in the result
What is exact: The continuous relationship, given the volatility actually realised. No distribution is assumed, volatility is not assumed constant, and nothing is forecast. What is approximate is a real fund rebalancing daily rather than continuously
The volatility: Yours. It is the scenario you are asking about, not a measurement, not an estimate this page produces and not a prediction
The rates: Per year, continuously compounded, as decimal fractions. No day count is stated and none is needed, because nothing here is discretised
The variance term: Proportional to the square of volatility, and the sign depends on the multiple. Negative above one and below zero; positive strictly between them, where a fund runs ahead of its benchmark rather than behind it
The borrow cost: Zero on a long fund, and refused rather than ignored if entered. The primary source states that assumption explicitly, and a long fund does not borrow the underlying
The shortfall: Yours, and there is no default. What counts as a multiple having stopped being that multiple is a judgement, and a number chosen for you would be an opinion published as a horizon
Excluded: Tracking error, creation and redemption, transaction costs beyond the stated expense ratio, any probability, and tax
Rounding: Display only; intermediate values remain unrounded

Correction history

The research corpus behind this model contains a contradiction large enough to sink a published surface: a December 2025 paper reporting errors of over one hundred percentage points annually in provider-published long-run figures. It was read rather than set aside, and the reason it does not apply is a distinction the model makes unrepresentable. That paper is about the *expected* return of a leveraged fund conditional on an index’s mean and standard deviation — a forecast over a distribution that has not happened. This page publishes a pathwise identity in a variance that has, and there is no field anywhere on it that could hold an expected return.
The four causes of the gap were originally going to be published as a single annual rate, which is how every observed source states them. That was rejected before implementation and the reason is recorded rather than the choice being quietly made: three of the four are straight lines in their own driver and one is proportional to the square of volatility, so a single netted figure would reproduce exactly the presentation this candidate exists to correct — and would hide that a double-short decays three times as fast as a double-long on the same underlying.