Do Daily Resets Hurt Leveraged ETFs? 100 Years of Data

ProShares UltraPro S&P500 (UPRO), which seeks three times the daily return of the S&P 500, returned 33.19% a year at net asset value from its June 2009 launch through September 2026.¹ In June 2009, the same month UPRO launched, FINRA told brokers that leveraged and inverse ETFs “typically are not suitable for retail investors who plan to hold them for more than one trading session, particularly in volatile markets.”² Both statements are accurate, and the reason both can be accurate is the subject of a debate that reopened this year.

Four 2026 working papers take up the question of what the daily reset actually costs, and they reach sharper conclusions than the 2009 warnings did. Jennifer Carpenter, Fangzhou Lu and Robert Whitelaw call “the false accusation that these funds erode value via ‘volatility decay’” a “major disservice to investors.”³ Ben Felix revisited his own earlier view on Rational Reminder episode 430 in October 2026 after reading the same research.⁴ The research’s central point, that the daily reset does not cause volatility drag, holds up on 100 years of daily U.S. market data, with one qualification: on the S&P 500 since 1990, resetting daily has cost a 2x position about a point a year compared with resetting monthly.

What a reset does to a two-day path

A 2x daily-reset fund holds $2 of index exposure for every $1 of shareholder equity at each day’s open. After a gain, equity has grown less than exposure has in dollar terms, so leverage drifts below 2x and the fund buys more exposure at the close. After a loss it sells. With index return $r_t$ on day $t$, leverage $L$, and financing and fees ignored, the fund’s value after $T$ days is

Two paths show the effect in both directions. If the index rises 10% and then falls 9.09%, it ends where it started. A 2x daily fund ends down 1.82% and a 3x fund down 5.45%. If the index rises 10% on two consecutive days, it gains 21%. The 2x daily fund gains 44%, while an investor who borrowed $1 to buy $2 of the index on day one and held it gains 42%. The reset cost money on the round trip and made money on the trend. The TQQQ vs. QQQ guide walks through longer versions of both paths with ProShares’ own volatility table.

Where volatility drag comes from

Marco Avellaneda and Stanley Zhang derived the standard approximation in 2010.⁵ Over a holding period, a leveraged fund’s value relative to the index’s value raised to the power $L$ shrinks with the realized variance $V_T$ of the index:

For $L = 2$ the exponent is $-V_T$, and for $L = 3$ it is $-3V_T$. That term is what most writing calls volatility decay. Chris Murray and Marco Sammon state where it comes from: “although maintaining the daily leverage target requires each LETF to increase exposure after gains and reduce exposure after losses, volatility drag results from compounding leveraged returns rather than from the daily reset itself.”⁶ Their appendix shows the same leading term appears “with any fixed rebalancing interval.”

The reason is arithmetic. If a fund resets weekly instead of daily, it resets one-fifth as often, but each period’s variance is about five times as large when returns are independent from day to day. The product, realized variance over the holding period, is about the same. Rebalancing less often changes when the drag is realized and leaves its expected size about the same.

Hendrik Bessembinder isolates what is left. For a two-day holding period, the expected gap between a buy-and-hold leveraged position and a daily-reset one, with daily means $\mu_1, \mu_2$, volatilities $\sigma_1, \sigma_2$ and correlation $\rho_{12}$ between the two days’ returns, is⁷

Since $L - L^2$ is negative for any leverage above 1, a positive mean or a positive correlation between consecutive days favors the daily reset, and a negative correlation favors holding still. His summary: volatility “increases average LETF returns in the presence of positive serial correlation and vice versa, and has no effect if returns are serially uncorrelated.”⁷

Volatility still hurts the typical outcome. It widens the gap between the mean and the median compound return because it makes outcomes more skewed. In Bessembinder’s example of a stock with 1.5% daily volatility and a zero mean return, a 3x position’s median one-year compound return is −39.78% while its mean is zero.⁷ A few large winners hold up the average while most holders lose. An investor gets one draw from that distribution, which is why the median matters more to a household than the mean.

A 100-year test of reset frequency

The theory makes a testable prediction: daily resets should beat less-frequent resets when daily returns have tended to continue, and lose when they have tended to reverse. I tested it on the Kenneth French daily U.S. market series, which runs from July 1926 to May 2026 and includes the daily 1-month T-bill return used here as the cost of borrowing.⁸ Each portfolio starts at the same leverage. The daily version resets every close, the monthly version resets at each month end and lets leverage drift in between, and all of them pay the same T-bill rate on what they borrow, with no fee or spread, so the only difference between columns is the reset schedule.

The sign of the daily-minus-monthly column follows the sign of the autocorrelation column, as the theory predicts. From 1926 through the 1990s, daily market returns had positive lag-one autocorrelation in every decade and daily resets came out ahead. Since 2000 the autocorrelation has been negative and monthly resets came out ahead, by 1.6 points a year at 2x and 5.0 points a year at 3x in the 2020s through May.

The early positive autocorrelation overstates what a fund could have captured. The French series is a value-weighted index of all listed U.S. stocks, and before the 1980s many small stocks traded infrequently, so their recorded prices lagged the market by a day or more. That makes the index look as if it continued its moves when the underlying shares had already repriced. A fund trading at the close could not have earned that. The S&P 500 total return index, which is tradable through index funds and futures, gives a cleaner reading from 1988:⁹

• Daily lag-one autocorrelation was −0.05 in 1988 and 1989, +0.02 in the 1990s, −0.09 in the 2000s, −0.05 in the 2010s and −0.17 from 2020 through May 2026.
• From January 1988 through May 2026, financed at T-bill rates with no fees, a 2x daily reset compounded at 17.02% a year and a 2x monthly reset at 18.16%. At 3x the figures were 18.80% and 22.21%.
• Across all 317 rolling 10-year windows starting each month from January 1990 to May 2016, the 2x daily reset trailed the 2x monthly reset in every one. The median shortfall was 1.15 points a year, and the 10th to 90th percentile range was 0.60 to 2.09 points.

Over the full century the picture is mixed: across all 1,079 rolling 10-year windows in the French data, the 2x daily reset beat the monthly reset in 71%, almost all of them before 1990. For an investor today the recent decades are the more relevant evidence, and they show a cost of about a point a year at 2x from resetting daily rather than monthly. That matches what Bessembinder finds in single-stock funds, where individual stocks also tend to reverse at short horizons (next section but one).

Two cautions keep this from becoming an argument for monthly-reset leverage. First, the sign has flipped once in a century and could flip again; Carpenter, Lu and Whitelaw argue that rebalancing “can never destroy value in an efficient market,” and a cost that depends on a persistent pattern of reversals is a bet that the pattern continues.³ Second, less frequent resets move the wipeout risk. A 3x daily fund is wiped out only by a one-day index fall of about 33%. The worst day in the French data, October 19, 1987, was −17.4%. A 3x quarterly reset is wiped out by a 33% fall at any point within a quarter. In this test the 3x quarterly portfolio went to zero on November 12, 1929 in the French data and on November 20, 2008 in the S&P 500 data.

Against a position bought once on margin and never rebalanced, the daily reset looks better in the bad decades. With a 25% maintenance requirement, a buy-once 2x position opened at the start of the 1930s, 1970s or 2000s was margin called in December 1930, August 1974 or September 2001. Zugang Liu and Jia Wang compared leveraged ETFs with margin portfolios back to 1927 and conclude that “the LETFs are less risky during every bear market since 1927.”¹⁰ A daily-reset fund cuts exposure as it falls, while a fixed loan becomes a larger share of a shrinking position. Over horizons up to a year, Baolian Wang finds that daily-reset index funds track the buy-and-hold leveraged position closely, and that the large deviations are “positively skewed and generally favorable.”¹¹ Wang’s research was funded by ProShares Advisors.

Try it: Reset Frequency Lab

Choose any start year from 1926 and a holding period of 5 to 30 years. The lab runs the same leverage three ways on the actual daily path and reports the autocorrelation of that window, so you can see the sign of the daily-minus-monthly gap follow it.

What leverage costs at any reset frequency

Under the standard approximations, the long-run compound growth rate of a portfolio held at constant leverage is

where $\mu$ is the index’s expected annual arithmetic return, $\sigma$ its annual volatility, $r_f$ the T-bill rate, $s$ the financing spread on borrowed exposure and $f$ the fund’s fee. Setting $g_L$ equal to the unlevered growth rate $\mu - \sigma^2/2$ gives the arithmetic return the index must earn for leverage to break even on compound growth:

With the 3-month T-bill at 4.22% on October 7, 2026,¹² a 0.5% spread, a 0.9% fee and 17.1% volatility (the French series’ long-run figure), the 2x hurdle is about 10.0% a year. For 3x, with a 0.9% spread, it is about 11.4%. The U.S. market’s arithmetic average calendar-year return from 1927 through 2025 in the same data was 12.1%. A forecast of lower stock returns, higher volatility or higher bill yields closes that gap quickly, and the hurdle rises point for point with the T-bill rate. These are approximations under constant parameters, not forecasts.

Murray and Sammon publish break-even returns by daily volatility, with a 4.0% risk-free rate, a 1.0% expense ratio and a 0.5% spread. At 1% daily volatility, roughly a typical year for the S&P 500, the underlying must return 8.4% for a 2x fund to match it and 8.9% for 3x. At 2% daily volatility, typical of a volatile single stock, the figures are 16.9% and 22.0%; at 5%, 98.4% and 169.8%.⁶

The financing cost does not appear in the expense ratio. The current ProShares UltraPro QQQ summary prospectus says: “The Fund pays transaction and financing costs associated with the purchase and sale of securities and derivatives. These costs are not reflected in the table or the example above.”¹³ Murray and Sammon estimate the spread over the risk-free rate from the gap between each fund’s realized return and its stated objective. For broad equity-index funds weighted by market value, they find 0.5% a year for 2x products and 0.9% for 3x. For a 2x equity-index fund, their annual cost of the borrowed half of the exposure at the end-of-sample risk-free rate is “4.09% + 0.4% + 0.90% = 5.39%,” using the equal-weighted 0.4% spread.⁶ As a check, I compared UPRO’s adjusted price history from June 25, 2009 through May 29, 2026 with a frictionless 3x daily reset of the S&P 500 total return financed at T-bill rates. UPRO compounded at 33.50% a year in that window against 36.58% for the frictionless version, a gap of about 3.1 points a year that covers its fee and a financing spread of roughly 0.7% on the two borrowed units.

What the live funds show about start dates

UPRO’s 33.19% a year starts three months after the March 2009 market bottom and runs through a period with T-bill rates near zero for most of its first decade. Other start dates tell different stories:

• ProShares Ultra S&P500 (SSO), a 2x fund launched in June 2006, has returned 15.78% a year at NAV through September 2026.¹⁴ On adjusted prices it fell 84.7% from October 2007 to March 2009.
• UPRO fell 76.8% from February 19 to March 23, 2020, on adjusted prices.
• ProFunds UltraBull (ULPIX), a 2x S&P 500 mutual fund launched in November 1997, returned 9.66% a year through June 2026 against 9.43% for the S&P 500, with a 1.47% expense ratio on its investor class.¹⁵ On adjusted prices it was behind the S&P 500 total return on about 84% of trading days since launch, and at the end of 2008 it held about 35% as much wealth as the index. It matches the 1990s 2x fund described on Rational Reminder, which did not name it.

The modeled portfolios in the French data show how long the deep drawdowns lasted. Net of a 0.90% fee and the Murray and Sammon spreads, the 2x daily portfolio took until August 1951 to regain its September 1929 peak and the 3x until September 1955; the unlevered market recovered in February 1945. From the March 2000 peak, the market took about 6.6 years to recover, the 2x portfolio about 14 years and the 3x about 17.8 years.

Murray and Sammon’s aggregate numbers show both sides. Long broad-equity-index leveraged ETFs produced about $107.6 billion of dollar-weighted investor gains in their matched sample, against $61.1 billion had the same dollars gone into the unlevered underlying assets at the same times.⁶ Yet among equity-index leveraged funds with at least three years of history, which include sector and international funds, only 43.5% beat their underlying index. Investors in these funds tended to buy after losses and sell after gains, and the authors find those flows “do not predict future returns.” The authors tie the aggregate gain to their sample period: “broad equity returns were strong over our sample period, volatility was relatively low, and financing costs were lower.”

Single-stock leveraged ETFs are a separate product

Murray and Sammon find that issuers launch single-stock leveraged funds on stocks near the top of the volatility and recent-return distributions: the median underlying stock sat near the 94th percentile of volatility and the 92nd percentile of six-month return at launch. The estimated financing spread for 2x single-stock funds is 6.8% to 9.8% a year over the risk-free rate, against 0.4% to 0.5% for 2x index funds, and their all-in annual cost estimate is 11.92%.⁶ A ProShares semiannual report for November 2025 lists swap financing rates on its 2x Coinbase fund ranging from 7.89% to 13.39% across four counterparties.¹⁶

Bessembinder’s companion paper measures how much of the shortfall comes from the reset itself. For long single-stock funds in his main sample, launched from mid-2022, the shortfall against a buy-and-hold leveraged benchmark averaged 0.75% a month: 0.20 points from daily rebalancing and 0.55 from fees, financing and other frictions. Long funds listed from April 2024 trailed by 1.47% a month, 0.42 from rebalancing and 1.04 from frictions.¹⁷ The rebalancing cost there comes from reversals; he finds the stocks’ returns were negatively autocorrelated at weekly and monthly horizons, and in a 1974 to 2024 backtest that high-volatility periods for individual stocks were also periods of more frequent reversals.

Single-stock funds also carry a failure mode that index funds have so far avoided. On July 14, 2026, Lucid shares fell more than 50% during the session, the swap counterparty for the GraniteShares 2x Long LCID Daily ETF (LCDL) closed out the swap, and the fund’s net asset value per share went to −$0.016. GraniteShares said “there are no residual assets available for distribution to shareholders.”¹⁸ On December 2, 2025, SEC staff told Direxion that it would not review 29 proposed 3x funds, including funds on individual stocks and on bitcoin and ether, and asked the trust to revise them to fit Rule 18f-4 or withdraw them.¹⁹ The rule limits a fund’s value-at-risk to 200% of its reference portfolio’s, with an exemption for leveraged funds that already operated above that level on October 28, 2020, which is why existing 3x index funds such as UPRO continue to trade.²⁰

The lifecycle case for leverage and what it assumes

The academic case for a young investor to borrow comes from treating future earnings as an asset. Zvi Bodie, Robert Merton and William Samuelson showed in 1992 that a worker with safe, bond-like earnings and little financial wealth holds too little stock as a share of total wealth and may optimally use leverage.²¹ Ian Ayres and Barry Nalebuff turned that into a rule: start near 200% stocks and glide down to 50% by retirement. In their historical simulations it matched the mean retirement wealth of a constant 74% stock allocation with a standard deviation about 21% lower.²² The Personal Leverage guide covers their argument and the margin mechanics in detail.

The August 2026 revision of Aizhan Anarkulova, Scott Cederburg and Michael O’Doherty’s “Beyond the Status Quo” puts a price on the borrowing.²³ In their lifecycle model, at a spread of 6.50% over bills, “close to the median broker margin spread in April 2024,” the optimal household does not borrow at all. At 1.40%, “equal to the lowest broker margin spread in April 2024,” gross exposure ranges from 132% to 200% of wealth over the lifecycle. At 0.37%, a rate implied by derivatives prices, borrowing hits the 200% cap at every age. The answer turns on the borrowing rate. A 2x broad-index ETF charges about 0.5% over T-bills on the borrowed half plus a 0.9% fee on the whole, which works out to about 1.4% over bills per borrowed dollar, close to the paper’s middle case.

The assumption that carries the most weight is safe earnings. The same paper models a positive correlation between earnings and domestic stock returns as an extension, and notes published estimates for highly educated workers in the 0.3 to 0.5 range. Even a correlation of 0.1 moves the model’s youngest households out of domestic stocks entirely.²³ A software engineer whose bonus, equity grants and job security fall with the stock market holds human capital that looks more like stock than like a bond. For that worker, a leveraged stock fund adds to a bet the paycheck already makes.

Taxes

A fund trading every day does not mean its shareholders realize gains every day. Trades inside the fund create gains and losses at the fund level, and the fund distributes net gains only if it has any after offsetting losses and carryforwards. UPRO, SSO and the ProShares UltraPro QQQ (TQQQ) did not appear in ProShares’ year-end capital gain distribution notices for 2019, 2021, 2023, 2024 or 2025, and ProShares paid no capital gain distributions on any of its ETFs for 2020.²⁴ Past years do not bind future ones; a long bull market can exhaust loss carryforwards, so check the issuer’s distribution history before holding one in a taxable account.

Borrowing on margin is treated differently. Interest on money borrowed to buy investments is deductible only up to net investment income, with the excess carried forward, and only for taxpayers who itemize.²⁵ Financing paid inside a leveraged ETF simply reduces the fund’s return. Inside an IRA or 401(k), where margin is generally unavailable, a leveraged ETF is one of the few ways to hold leverage at all, and the tax questions disappear.

Who might use a leveraged index fund

A leveraged fund changes how much equity risk a portfolio carries per dollar invested. A 2x fund does not have to be the whole position. Holding 75% in an ordinary index fund and 25% in a 2x fund of the same index gives about 1.25x exposure; 50/50 gives about 1.5x. That mix drifts as the two funds diverge and needs rebalancing on a rule, such as quarterly or when exposure moves outside a set band. The investors for whom the evidence supports a modest sleeve like that share most of these traits:

• Income that would likely hold up in a recession, and an emergency fund that would not have to be raised by selling stocks.
• A long horizon, no large planned withdrawals for a decade or more, and a written plan they have already followed through at least one bear market.
• Room to hold the fund in a tax-advantaged account, using a broad-market index rather than a sector or single stock.
• A plan that would still work if the leveraged sleeve fell close to 90% and took 14 years to recover, as the modeled 2x portfolio did from March 2000.

Most investors are better served without it. Someone who can reach their goals with an ordinary stock allocation gains little from raising expected wealth and takes on a larger chance of a bad outcome; a higher mean with a lower median is a poor trade for a household that needs a specific amount at a specific date. The same applies to anyone near retirement or a home purchase, anyone whose income moves with the stock market, and anyone attracted to the funds by a recent chart.

Key takeaways

• Volatility drag comes from leverage. Any fixed reset interval carries the same leading drag term; Murray and Sammon say so directly.
• Reset frequency matters through autocorrelation. When daily moves continue, daily resets help; when they reverse, daily resets cost money.
• Recent decades favored monthly resets. On the S&P 500 since 1990, a 2x daily reset trailed a 2x monthly reset in every rolling 10-year window, by a median 1.15 points a year, financed at T-bill rates with no fees.
• Daily resets limit the worst cases. A daily-reset fund cannot be margin called and sheds exposure in a crash; a 3x fund reset quarterly would have been wiped out in 2008.
• The live record depends on the start date. UPRO made 33.19% a year from June 2009; a 2x S&P 500 mutual fund launched in 1997 is barely ahead of the index after 28 years.
• Single-stock leveraged funds are a separate product, with financing spreads near 7% to 10%, high volatility drag and swap-termination risk.
• If your plan works without leverage, keep it that way. For a long-horizon investor with stable income who wants more equity exposure, a small broad-index 2x sleeve in a tax-advantaged account is the defensible form.

Frequently asked questions

Do leveraged ETFs decay to zero over time?

No. A broad-index leveraged fund grows when the index return is high enough relative to its volatility and costs, as UPRO did from 2009. It loses value over time when the index is flat or volatile, and it can lose nearly everything in a deep bear market: a modeled 2x daily fund fell 98% from 1929 to 1932.

Is the daily reset what makes leveraged ETFs risky?

The leverage makes them risky. The daily reset changes the shape of the outcomes: it adds exposure after up days and cuts it after down days. That helped in trending markets and in deep bear markets compared with a fixed loan, and has cost about a point a year at 2x compared with a monthly reset since 1990, when daily S&P 500 moves tended to partly reverse.

Would a weekly or monthly reset product be better?

In the S&P 500 data since 1990 a monthly reset would have earned more, but that edge depends on short-term reversals persisting, and a less frequent reset needs a smaller move within the reset period to wipe out a 3x position. Few funds offer it; investors who want it usually rebalance a margin or futures position themselves.

How much does a 2x S&P 500 fund cost to hold?

The expense ratio is about 0.9%, and financing is extra. Murray and Sammon estimate the spread over T-bills at about 0.5% for 2x broad-index funds, so the full annual cost of the extra exposure is roughly the T-bill rate plus 0.5% plus the fee, which they put at 5.39% at end-of-sample rates.

Is a leveraged ETF better than buying on margin?

They fail differently. A margin position can be called and sold at the bottom, and you can owe more than you invested; a leveraged ETF cannot be called and loses at most what you put in. Counting its fee, a 2x index fund borrows at about the cheapest broker margin rate and well below a typical one, though the interest is not deductible. In every U.S. bear market since 1927, Liu and Wang found the ETF structure carried less downside risk.

Should young investors use leverage?

Lifecycle models say a worker with safe earnings and little savings can benefit from it, if borrowing is cheap. At typical broker margin rates the Anarkulova, Cederburg and O’Doherty model chooses no leverage. The case also weakens quickly when income moves with the stock market, which is common for workers in finance and technology.

Related guides

• TQQQ vs. QQQ: how a 3x daily fund compounds, with the issuer’s volatility table.
• Personal Leverage: margin mechanics, margin calls and the lifecycle argument.
• HEDGEFUNDIE’s Excellent Adventure: a leveraged stock and bond portfolio and the correlation it relied on.
• The Korea Leveraged ETF Crash: single-stock leverage on real price paths, with a path lab.
• Portable Alpha and Return Stacking: funds that stack two return streams instead of multiplying one, with rebalancing schedules from daily to quarterly.
• The Rebalancing Bonus: the same variance arithmetic applied to rebalancing a diversified portfolio.
• The Merton Share: how much stock exposure the theory calls for, and how sensitive the answer is.

Method note

Market returns are the Kenneth French daily value-weighted U.S. market (Mkt-RF plus RF) from July 1, 1926 to May 29, 2026, 26,253 trading days, and borrowing accrues the daily RF series (the 1-month T-bill). Each reset portfolio holds fixed exposure financed by a fixed loan between resets; the loan accrues RF plus the spread, the fee accrues daily on equity, and equity is floored at zero. Monthly resets occur on the last trading day of each calendar month and quarterly resets on the last of each quarter. The buy-once margin position is liquidated at the close on the first day equity falls below 25% of exposure and is held unlevered thereafter. Decade and rolling comparisons of reset frequency use no fee and no spread. Net-of-cost figures use a 0.90% fee and spreads of 0.5% (2x) and 0.9% (3x), the AUM-weighted expense ratio and market-cap-weighted funding spreads Murray and Sammon report for equity-index funds. Rolling windows start on the first trading day of each month. The S&P 500 cross-check uses Yahoo Finance’s ^SP500TR series from January 1988 with the same T-bill series; the UPRO comparison uses Yahoo Finance adjusted prices. No taxes, trading costs or tracking error are modeled. The code is nwtracker/leverage_paths.py and scripts/leveraged_reset_backtest.py in the Summitward repository, and the calculator runs the same equations in the browser.

Sources

1. ProShares, UltraPro S&P500 (UPRO) fund page (inception June 23, 2009; NAV total return since inception 33.19% annualized as of September 30, 2026; expense ratio 0.89%). proshares.com
2. FINRA Regulatory Notice 09-31, “Non-Traditional ETFs,” June 11, 2009. finra.org
3. Jennifer N. Carpenter, Fangzhou Lu and Robert F. Whitelaw, “In Defense of Leveraged and Inverse Funds,” working paper, August 18, 2026. Whitelaw lists an affiliation with VistaShares, an ETF issuer, on the title page. nyu.edu (PDF)
4. Rational Reminder, Episode 430, “The Truth About Leveraged ETFs,” October 8, 2026. rationalreminder.ca
5. Marco Avellaneda and Stanley Zhang, “Path-Dependence of Leveraged ETF Returns,” SIAM Journal on Financial Mathematics 1 (2010): 586-603. doi.org
6. Chris Murray and Marco Sammon, “The Costs and Benefits of Leveraged ETFs,” working paper, August 24, 2026 (Table 1 break-even returns; Table 2 expense ratios and funding spreads; Table A2 fund-level outperformance; Appendix A.5 reset interval). ssrn.com
7. Hendrik Bessembinder, “Volatility and Returns to Leveraged ETFs,” working paper, August 31, 2026. ssrn.com
8. Kenneth R. French Data Library, Fama/French 3 Factors [Daily]. dartmouth.edu
9. S&P 500 Total Return Index (^SP500TR), daily closes from January 1988, via Yahoo Finance; author’s calculation. finance.yahoo.com
10. Zugang Liu and Jia Wang, “Leveraged Exchange-Traded Funds Versus Investing on Margin,” The Journal of Investing 31, no. 5 (2022): 55-70. doi.org
11. Baolian Wang, “Multi-day Return Properties of Leveraged Index ETFs,” working paper, May 30, 2025 (funded by ProShares Advisors, LLC). ssrn.com
12. Summitward Markets, U.S. Treasury yield curve snapshot as of October 7, 2026 (3-month bill 4.22%). summitward.com/markets
13. ProShares Trust, UltraPro QQQ summary prospectus, Form 497K filed September 28, 2026. sec.gov
14. ProShares, Ultra S&P500 (SSO) fund page (inception June 19, 2006; NAV total return since inception 15.78% annualized as of September 30, 2026). proshares.com
15. ProFunds, UltraBull ProFund fact sheet, as of June 30, 2026 (inception November 26, 1997; investor class 9.66% since inception against 9.43% for the S&P 500; expense ratio 1.47%). profunds.com (PDF)
16. ProShares Trust, single-stock funds semiannual financial statements, November 30, 2025 (Ultra COIN swap agreements, rate paid by counterparty). proshares.com (PDF)
17. Hendrik Bessembinder, “Rebalancing, Frictional Costs, and Returns to Levered Single-Stock ETFs,” working paper, August 31, 2026 (Table 3, long funds). ssrn.com
18. GraniteShares, “GraniteShares 2x Long LCID Daily ETF (Nasdaq: LCDL) Fund Delisting, Negative NAV,” July 2026. graniteshares.com
19. SEC Division of Investment Management, letter to counsel for Direxion Shares ETF Trust, December 2, 2025. sec.gov (PDF)
20. 17 CFR 270.18f-4, Use of derivatives transactions by funds (relative VaR limit; paragraph (c)(5) for leveraged/inverse funds operating before October 28, 2020). law.cornell.edu
21. Zvi Bodie, Robert C. Merton and William F. Samuelson, “Labor Supply Flexibility and Portfolio Choice in a Life Cycle Model,” Journal of Economic Dynamics and Control 16 (1992): 427-449. nber.org
22. Ian Ayres and Barry Nalebuff, “Diversification Across Time,” The Journal of Portfolio Management 39, no. 2 (2013): 73-86; simulation figures from the 2010 working paper. yale.edu (PDF) doi.org
23. Aizhan Anarkulova, Scott Cederburg and Michael S. O’Doherty, “Beyond the Status Quo: A Critical Assessment of Lifecycle Investment Advice,” working paper, August 27, 2026 (Internet Appendix C.9 on leverage; Section 5.4 on income correlation). ssrn.com
24. ProShares, ETF announcements, year-end capital gain distribution estimates (2019 through 2025). proshares.com
25. IRS Publication 550, Investment Income and Expenses (investment interest). irs.gov

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 figures are the issuers’ own on the dates given. Nothing here is investment advice.