Leveraged ETFs: Convexity Bonus vs. Volatility Drag, and the Cheapest Way to Hold 1.25x

Marco Sammon opened the arithmetic on Rational Reminder episode 432 with a quiz. An index rises 10% one day and falls 10% the next. It ends down 1%. A fund that delivers twice the index’s daily return ends down 4%, where twice the index’s loss would be 2%. Chris Murray and Sammon, the authors of a 2026 study of U.S. leveraged ETFs from 2006 to 2025, split the extra two points into two pieces: a “convexity bonus” of 0.01 points and volatility drag of 2.01 points.¹ On that path the bonus is too small to see. On other paths it is larger than the drag, and the difference between the two decides whether a leveraged fund beats its index.

This guide explains both terms with the authors’ examples, shows how large each one became over real decades of U.S. market history, and then tests the practical question Ben Felix raised at the end of the episode and the guests left open: can an investor who wants about 1.25x exposure get it by mixing a 2x fund with an ordinary index fund, and how does that compare with borrowing on margin? Felix said a short backtest of his own, using Canadian funds with a few years of history, came within a few basis points a year; Murray said they would need to work out the math.¹

What the convexity bonus and volatility drag measure

A 2x daily-reset fund earns $1 + 2r$ on a day the index returns $r$. That one-day result can be written exactly as the index’s growth squared, minus a small correction:

Multiply that across every day of a holding period and the first factors combine into the index’s total growth, squared. If the index’s total return over the period is $R$, the fund before costs is worth

The naive expectation for a 2x fund is $1 + 2R$, twice the index’s return. The convexity bonus is how far the squared growth exceeds that:

It depends only on where the index ended, and it is positive whether the index went up or down. The volatility drag is the second factor. Each day removes a sliver equal to roughly the square of that day’s move, so it depends on how far the index traveled along the way and not on where it finished. For any leverage $L$, Murray and Sammon write the same split in log terms: the fund’s log return equals $L$ times the index’s log return, less about $\tfrac{1}{2}L(L-1)$ times the sum of squared daily returns.²

The table shows the three paths from the episode. Two 10% gains in a row give the index 21% and a naive 2x expectation of 42%; squaring 1.21 gives 46.41%, a bonus of 4.41 points, and the drag from two 10% days takes back 2.41, for 44%. In the round trip the index barely moves, so the bonus is 0.01 points and the drag decides the result. In the third path the index falls from $100 to $75 to $50. Someone who borrowed once to hold $2 of index per $1 of equity is wiped out at −50%. The daily fund cut its exposure after the first loss and ends at $16.67 per $100: the bonus adds 25 points to the naive −100% and the drag takes back 8.33.

Two cautions keep the bonus in proportion. It is measured against “two times the index return,” and no fund or margin account delivers that over a long horizon; a margin position bought once comes closest, but its leverage drifts down after gains and up after losses. And the comparison that matters to an investor is the leveraged fund against the plain index fund after borrowing costs, covered below. The existing daily-reset guide shows that resetting monthly instead of daily carries the same leading drag term; the reset schedule changes only how runs and reversals of daily moves are treated.

Over a decade, both numbers get large

Over a few days the bonus and the drag are a few points each. Over ten years they can be hundreds of points, and small differences in index volatility decide which one wins. The table below runs modeled daily-reset positions at 1.25x, 2x and 3x through three decades of U.S. market returns, with borrowing at the 1-month T-bill plus a spread and a 0.90% fee, the cost levels Murray and Sammon estimate for index leveraged ETFs.²

From 2010 through 2019 the market compounded at about 13.6% a year with 15% volatility. For the 2x position the bonus added 664 points to the naive +515%, the drag removed 266, borrowing and fees removed another 175, and the position ended up 738% against the market’s 258%. From 2000 through 2009 the market ended about where it started, down 3.9%, with 22% volatility. There was almost no bonus (0.15 points) to offset 36 points of drag, and the 2x position finished down 63%. At 3x the same decade took 91%.

A large bonus over a good decade is also why the mean outcome of a leveraged fund looks better than the typical one. Hendrik Bessembinder shows that for a stock with 1.5% daily volatility and a zero expected return, a 3x position’s median one-year return is about −40% while its mean is zero.³ On the podcast, Murray made a related point about how funds get judged: if a fund is ranked on expected terminal wealth computed from its arithmetic average return, the leverage that maximizes it is infinite whenever the index has a positive expected return, while maximizing expected log wealth gives a moderate answer.¹

Why 1.25x carries about a sixth of the drag of 2x

Leverage adds expected return in proportion to the extra exposure, $L - 1$. Drag grows with $L(L-1)/2$, which is 0.16 at 1.25x, 1 at 2x and 3 at 3x. Going from 1x to 1.25x adds a quarter of the equity premium and about a sixth of the drag of a 2x fund; going from 2x to 3x adds the same premium as going from 1x to 2x and twice the drag. In a constant volatility model, the compound growth rate of a leveraged position is

which rises with leverage, peaks, and then falls. The peak sits at $L^{*} = (\mu - r_f - s)/\sigma^2$, the Merton share for an investor with log utility.⁴ With an equity premium of 4 points over borrowing costs and 16% volatility, that is about 1.6x; with a 2-point premium it is under 1x. The calculator draws the full curve for whatever inputs you set. A household with more risk aversion than log utility, or with income that falls in recessions, belongs below the peak; the Merton share guide covers how to size that.

The break-even return

Setting the leveraged growth rate equal to the index’s gives the compound return the index must earn for a leveraged position to keep up:

Murray and Sammon tabulate this hurdle by daily volatility with a 4.0% risk-free rate, a 0.5% spread and a 1.0% expense ratio. At 1% daily volatility, about a typical year for the S&P 500, the index must return 8.4% in a year for a 2x fund to match it and 8.9% for 3x. At 2% daily volatility the figures are 16.9% and 22.0%; at 5%, typical of the stocks that get single-stock leveraged funds, they are 98.4% and 169.8%.² Those hurdles include the fee. On the episode, Ben Felix suggested the 5% figure was gross of fees; the paper’s Table 1 includes a 1.0% expense ratio and the 0.5% spread.

The last term in the formula matters most for light leverage. A fund fee is charged on the fund’s whole value, but the only thing an investor is paying extra for is the borrowed part, which is $L - 1$ per dollar. Per borrowed dollar, a 0.90% fee costs 0.90% at 2x, 0.45% at 3x and 3.6% at 1.25x. That one ratio explains most of the comparison in the next two sections.

Three ways to hold 1.25x and what each borrowed dollar costs

An investor who wants 1.25x has three practical routes: hold most of the money in an index fund and a sleeve in a 2x or 3x fund on the same index, buy a fund that targets 1.25x on its own, or borrow on margin. Each charges for the borrowed quarter in a different way. A 2x fund holds $2 of index exposure, mostly through swaps, for every $1 of shareholder equity, pays a financing spread on the extra dollar, and charges its expense ratio on the $1 of equity. A sleeve built to borrow $0.25 therefore pays the fee on $0.25, so the fee per borrowed dollar equals the fee itself. In a 3x fund it is half the fee. In a 1.25x fund the fee falls on $1 of equity to borrow $0.25, four times the fee per borrowed dollar.

A 25% sleeve in ProShares Ultra S&P500 (SSO), with a 0.84% net expense ratio, costs about the T-bill rate plus 1.3% per borrowed dollar; Direxion’s 2x S&P 500 fund (SPUU), at 0.60%, about T-bills plus 1.1%; a 12.5% sleeve in ProShares UltraPro S&P500 (UPRO) about the same as SSO.⁵⁶⁷ Both ProShares prospectuses state that financing costs “are not reflected in the table,” so the spread column is an estimate from the funds’ realized returns.²

Funds that target 1.25x directly have struggled to survive at fees low enough to compete. Direxion launched the Daily S&P 500 Bull 1.25X Shares (LLSP) in January 2015, renamed it the Portfolio+ S&P 500 ETF in February 2018 with a 0.30% net expense ratio, raised its target to 135% of the index by 2020, and liquidated it in October 2020.⁸ (On the episode, Felix recalled it closing in 2018, the year of the rename.) In Canada, the Global X Enhanced S&P 500 Index ETF (USSL) targets 125% through cash borrowing, with a 0.35% management fee, a 1.19% management expense ratio as of June 30, 2026, and about C$15.6 million in net assets on October 28, 2026.⁹ Charged on the whole fund, its management fee alone works out to 1.4% a year per borrowed dollar. Its product sheet does not say whether the expense ratio includes the interest on the borrowing.

Margin rates span a wide range. On October 28, 2026, Interactive Brokers’ IBKR Pro charged 5.38% on the first $100,000 borrowed; Schwab and Fidelity posted 10.575% for balances of $100,000 to $249,999, and Vanguard 10.75%, against a 13-week T-bill at 4.11%.¹⁰ At those rates, borrowing at a large retail broker costs about five times as much per borrowed dollar as a 2x-fund sleeve. Investors approved for futures or index options can borrow more cheaply still: CME reports that the financing implied by E-mini S&P 500 futures averaged 0.62 points over 3-month SOFR from the December 2022 roll into 2025, and reached 0.81 points at the June 2026 roll.¹¹ Box spreads on the S&P 500 priced at 0.35 to 0.37 points over Treasury yields on average from 2004 to 2018.¹² The box spread guide covers the mechanics and risks of that route.

100 years of holding 1.25x

To test the blend, I simulated each way of holding 1.25x U.S. equity exposure on daily data from July 1926 to May 2026, using the same Kenneth French market series and T-bill rates as the reset-frequency test, and repeated it on the S&P 500 total return index from 1988. Leveraged-fund costs are Murray and Sammon’s estimates for index funds; the index fund costs 0.03%; margin is rebalanced to 1.25x at each month end.

Over the full century every 1.25x route except high-cost margin beat the unlevered index, the 1.25x fund only barely. The worst drawdown, from 1929 to 1932, was 84% for the index and 90% to 91% for the 1.25x routes. The blend added about a point a year over the index.

Across 10-year windows starting each month:

• The 2x-fund blend beat the 1.25x fund charging 0.90% in every window, by a median of 83 basis points a year. The blend’s weighted fee is about 0.25% of the portfolio, against 0.90% for the single fund.
• In 10-year windows starting before 1990, the blend beat margin at T-bills plus 1.27% in 99% of cases; in windows starting from 1990 it trailed that margin rate in 98% of windows, by a median of 21 basis points. Most of that swing comes from the pattern the reset-frequency guide documents: before about 2000, daily U.S. market moves tended to continue, which favors a daily reset; since then they have tended to partly reverse, which favors resetting less often. The 2x sleeve resets daily inside itself, while a margin position at 1.25x is rebalanced monthly.
• Margin at T-bills plus 6.5%, about the October 2026 posted rate at Schwab, Fidelity and Vanguard for a $100,000 balance, beat the unlevered index in 36% of windows since 1926 and 39% since 1990.
• Against the plain index fund, the blend came out ahead in 81% of 10-year windows since 1990, by a median of 75 basis points a year, and its worst window returned -5.8% a year against -3.2% for the index.

How often to rebalance the blend

A blend drifts. After a strong quarter the 2x sleeve has grown faster than the index fund and total exposure rises above 1.25x; after a weak one it falls below. To second order, a blend restored to its target weights has the same expected drag as a fund held at constant 1.25x, because the drift the investor allows between rebalances is the same kind of drift a monthly reset allows. On 400 simulated 30-year paths with independent daily returns, 10% expected return and 17% volatility, the 75/25 blend rebalanced quarterly grew 0.9 basis points a year faster than constant 1.25x on average, with a spread of 3.6 basis points across paths. The historical tracking gap depends on the rebalancing rule:

Quarterly rebalancing kept the blend closest to constant 1.25x. Rebalancing only when exposure drifted more than 0.05 from target needed about one trade every two years at 2x and had a larger year-to-year gap, which works in either direction. A 3x sleeve is cheaper per borrowed dollar but drifts about four times as often and swings further between rebalances, and a 3x fund can lose a third of its value on a day the index falls 11%. Over the full century the 3x-sleeve blend returned slightly more; since 1990 the two sleeves were within a few basis points of each other.

Taxes

Inside an IRA or 401(k), the blend has the field mostly to itself: margin is generally unavailable there, rebalancing the sleeve creates no tax, and fund distributions are not taxed when paid. In a taxable account, each rebalance after a strong quarter sells part of the sleeve at a gain. The 0.05 band rule, which traded about once every two years at 2x on the S&P 500 data, and steering new contributions toward whichever side is underweight both cut the number of taxable sales.

For SSO, at least, distributions have cost holders little. Its summary prospectus reports a 10-year annualized return to December 31, 2025 of 22.70% before taxes, 22.55% after taxes on distributions, and 19.80% after taxes on distributions and a sale at the end, computed at the highest historical federal rates.⁵ Margin interest is deductible only up to net investment income, and only for taxpayers who itemize.¹³ Futures and SPX box spreads are Section 1256 contracts, marked to market at year end and taxed 60% long-term and 40% short-term.¹⁴

When leverage is the wrong answer

Every result above assumes the investor holds through 1929, 1974, 2008 and 2020 without selling and without needing the money. That assumption fails most often for the households the lifecycle argument for leverage is aimed at. Patrick Adams, using U.S. Census Bureau panels of individual tax returns from 1998 to 2023 for working-age households in the top fifth of earners for their age, finds that a $1,000 annual drop in wage or business income leads households to cut saving and draw down liquid assets (stocks and bonds held in taxable accounts) by $500 to $850 a year, and that the largest average outflows from stocks came around the recessions of the early and late 2000s, “when both aggregate stock prices and average nonfinancial income fell sharply.”¹⁵ In his calibrated model, households that find it costly to cut spending hold less than a third of the stock share the frictionless model recommends. Murray and Sammon made the same point on the episode: job losses and market declines tend to arrive together, and some human-capital risk cannot be diversified.¹ The lifecycle leverage re-test shows how much the case for borrowing young depends on the borrowing rate.

Two product categories fail the arithmetic outright. Single-stock leveraged funds track stocks whose AUM-weighted daily volatility was 3.71% in 2025 against 1.35% for index funds, which multiplies the drag term by about 7.6, and Murray and Sammon estimate their financing spreads at 6.8% to 9.8% over the risk-free rate.² Inverse funds carry the same drag: for $L = -1$, $L(L-1)/2$ equals 1, the same as a 2x long fund. In Murray and Sammon’s sample, inverse funds took in $93.4 billion of net flows and produced investor losses of $80.3 billion.² The authors caution that some of those losses may have been offset by gains elsewhere in hedged portfolios, and that stocks rose unusually fast during their sample.

If an ordinary stock allocation already reaches your goals, adding leverage raises the average outcome and widens the spread of outcomes around it. A household that needs a specific amount by a specific date bears the bad end of that spread, and is better off without the leverage.

Key takeaways

• The convexity bonus is how far the index’s growth raised to the power $L$ exceeds $L$ times the index return. It grows with the size of the index’s total move, up or down.
• Volatility drag grows with the sum of squared daily moves and with $L(L-1)/2$. It comes from holding leverage constant at any reset interval.
• Over 2010 to 2019 the bonus outweighed the drag for 2x and 3x positions; over 2000 to 2009, a flat and volatile decade, the drag dominated and a modeled 2x position lost about 63%.
• A fund fee costs $f/(L-1)$ per borrowed dollar, so lightly levered funds need very low fees. A 75/25 blend of an index fund and a 2x fund beat a modeled 1.25x fund charging 0.90% in every 10-year window since 1926.
• Since 1990, margin at about T-bills plus 1.3% beat the blend by about 0.2 points a year; margin at large-broker rates trailed the unlevered index in most 10-year windows.

Frequently asked questions

What is the convexity bonus in a leveraged ETF?

It is the amount by which the index’s growth multiple raised to the fund’s leverage exceeds the naive “leverage times the index return.” For a 2x fund it equals the square of the index’s total return over the period. Murray and Sammon named it on Rational Reminder episode 432. It comes from keeping leverage constant, so a position bought once on margin does not get it.

Is volatility drag caused by the daily reset?

No. Any position held at a constant leverage multiple has the same leading drag term whether it resets daily, weekly or monthly. The reset interval changes whether runs of same-direction days help or reversals hurt.

Can I mix a 2x ETF with an index fund to get 1.25x?

Yes. 25% in a 2x fund and 75% in an index fund of the same index gives 1.25x at each rebalance. Rebalanced quarterly, the blend tracked a constant 1.25x position within about 0.1 point a year on the S&P 500 from 1988, before costs. Rebalancing realizes gains in a taxable account, so the blend fits best in an IRA or 401(k).

Is margin cheaper than a leveraged ETF?

It depends on the broker. Through a 25% sleeve of SSO, each borrowed dollar costs about the T-bill rate plus an estimated 0.5% financing spread plus the 0.84% fee, or T-bills plus about 1.3%. In October 2026 that matched IBKR Pro’s rate on the first $100,000 and was about a fifth of the spread in the posted rates at Schwab, Fidelity and Vanguard for a $100,000 balance.

Why do single-stock leveraged ETFs need such large returns?

Drag scales with the square of volatility, and the stocks that get these funds are among the most volatile in the market. At 5% daily volatility a 2x fund needs a 98% one-year return just to match the stock, before the 6.8% to 9.8% financing spreads Murray and Sammon estimate for these funds.

Related guides

• Do Daily Resets Hurt Leveraged ETFs?: 100 years of daily versus monthly resets, fund costs, single-stock funds and the regulatory history.
• Personal Leverage: margin mechanics, margin calls and forced selling.
• TQQQ vs. QQQ: how a 3x Nasdaq-100 fund compounds over longer periods.
• Should Young Investors Use Leverage?: the Ayres and Nalebuff strategy re-tested at real borrowing rates.
• Do You Need Return Stacking?: leverage spread across stocks and bonds instead of more stocks.
• The Merton Share: sizing total risk from expected return, volatility and risk aversion.

Method note

Market returns are the Kenneth French daily value-weighted U.S. market (Mkt-RF plus RF), July 1, 1926 to May 29, 2026, with borrowing at the daily RF series (the 1-month T-bill) plus each route’s spread. The cross-check uses Yahoo Finance’s ^SP500TR series from January 1988 with the same T-bill series. A blend holds an unlevered core and a sleeve in a daily-reset fund; the sleeve resets to its leverage every day, pays the T-bill return plus its spread on its loan and its fee on its equity, and cannot fall below zero; the core pays 0.03% a year. Blends return to target weights at the close of the last trading day of each month, quarter or year, or, under the band rule, at the close of any day when total exposure is more than 0.05 from 1.25. The hypothetical 1.25x fund resets daily with a 0.90% fee and 0.5% spread. Margin positions reset to 1.25x at each month end; a 25% maintenance call would need a 73% fall within one month, which never occurred. Rolling windows start on the first trading day of each month. Decade decompositions are exact identities on daily data; the calculator uses the continuous-time approximation with constant inputs. No taxes, trading costs or tracking error are modeled. Code: nwtracker/leverage_paths.py (blend_path) and scripts/leverage_blend_backtest.py in the Summitward repository; results in leverage-blend-results.json.

Sources

1. Rational Reminder, Episode 432, “How Leveraged ETFs Really Work (Marco Sammon & Chris Murray),” October 22, 2026, with transcript. rationalreminder.ca
2. Chris Murray and Marco Sammon, “The Costs and Benefits of Leveraged ETFs,” working paper, August 24, 2026 (Equations 3 and 4 and Appendix A.5 on volatility drag; Table 1 break-even returns; Table 2 funding spreads; Table 7 investor profits). SSRN 7133021
3. Hendrik Bessembinder, “Volatility and Returns to Leveraged ETFs,” working paper, 2026. SSRN 7376118
4. Robert C. Merton, “Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case,” Review of Economics and Statistics 51, no. 3 (1969): 247–257. doi.org
5. ProShares Ultra S&P500 (SSO), summary prospectus, September 28, 2026 (net expense ratio 0.84% after a 0.04% waiver; financing-cost statement; average annual total returns before and after taxes). SEC EDGAR
6. Direxion Daily S&P 500 Bull 2X Shares (SPUU), summary prospectus, February 27, 2026 (net expense ratio 0.60% under a cap that excludes swap financing costs). SEC EDGAR
7. ProShares UltraPro S&P500 (UPRO), summary prospectus, September 28, 2026 (expense ratio 0.88%). SEC EDGAR
8. Direxion Shares ETF Trust, summary prospectus for the Direxion Daily S&P 500 Bull 1.25X Shares, January 7, 2015; supplement renaming it Portfolio+ S&P 500 ETF, January 31, 2018; summary prospectus with a 135% target, February 28, 2020; liquidation supplement, August 28, 2020 (liquidation date October 2, 2020). SEC EDGAR (liquidation notice)
9. Global X Canada, Global X Enhanced S&P 500 Index ETF (USSL) product sheet, as at October 28, 2026 (125% target through cash borrowing; management fee 0.35%; MER 1.19% as of June 30, 2026; net assets C$15,569,102). globalx.ca
10. Margin rate pages, accessed October 28, 2026: Interactive Brokers (IBKR Pro, benchmark 3.88% plus 1.5% on the first $100,000); Schwab (base 10.25% plus 0.325% for $100,000 to $249,999); Fidelity (10.575% for $100,000 to $249,999); Vanguard (10.75% for $100,000 to $249,999). U.S. Treasury daily bill rates, 13-week coupon equivalent 4.11% on October 28, 2026. interactivebrokers.com
11. CME Group, “Quantifying and Hedging Equity Financing Risk,” May 2, 2025, and Paul Woolman, “Funding the U.S. Equity Market to New Heights,” July 28, 2026. cmegroup.com
12. Jules H. van Binsbergen, William F. Diamond and Marco Grotteria, “Risk-Free Interest Rates,” NBER Working Paper 26138, 2019 (Table 1); published in the Journal of Financial Economics, 2022. NBER
13. IRS Publication 550, Investment Income and Expenses (investment interest). irs.gov
14. IRS Form 6781, Gains and Losses From Section 1256 Contracts and Straddles. IRS (PDF)
15. Patrick Adams, “Stocks for the Long Run or Liquidity? Tax Data Evidence and Portfolio Choice Implications,” working paper, MIT Sloan, January 7, 2026 (Sections 3 and 4; model results pp. 40–41). Not reviewed by the Census Bureau. patrick-adams.com

Disclosure

The historical tables and the calculator are the author’s own calculations on public data under the stated assumptions; they describe what happened on past paths and do not forecast any fund or period. Fund and broker figures are the issuers’ own on the dates given. Nothing here is investment advice.