What History Did to Every Leverage Multiple from 1.00× to 3.00×

The usual leverage essay names a number. Two times. One and a half. Whatever multiple made the last backtest look clever. We wanted the opposite of that: a picture of the whole surface, fine enough that a reader can see the shape instead of a headline. So we held one stake fixed — $10,000, no contributions, no withdrawals — and walked constant leverage from 1.00× to 3.00× in steps of 0.01. Two hundred and one overlays. One market history. Download the dataset (CSV).

Each overlay is reset every month. The financed return is the equity return times L, minus the cash yield on the borrowed L − 1. If a month would take the account through zero, it stays at zero. That is a futures-style overlay, not a daily-reset product, and the difference matters. The tape is monthly S&P 500 prices from 1928 through 2025-08. Each complete calendar year is scaled so the product of those monthly prices matches Damodaran’s S&P 500 total return for that full year — intra-year months therefore depend on that year’s annual total. 2025 is a partial year through August and is not Damodaran-scaled. Endings are deflated by CPI. Financing is the historical T-bill yield, with no borrow spread. Pre-tax. No trading bps. Neither is a broker’s margin call above the month that would have wiped the account. We did not stress those frictions or a second country’s tape (Q2 and Q4 not run).

Long-term stock market leverage, one tape, 201 multiples

Long-term stock market leverage is usually sold as a single claim: the premium is large enough, and the horizon long enough, that borrowing to hold more equity is just a louder version of staying invested. The arithmetic is not that kind. Leverage multiplies the month you got, including the month that almost never happens and then does. A −30% month is unpleasant at 1.00×. At 3.00× it is −90% before the next open. The question is not whether the average equity premium is positive. It is what the path does to a constant multiple that refuses to de-lever.

That is why this study is a historical sweep and not a Monte Carlo fan of invented worlds (Q1, simulated orderings, was not run). A generator can be told to be polite. The Monte Carlo vs historical backtesting piece is the argument for running both; this page is the historical half, with leverage as the only moving part. 1,172 months. 97.7 years. The same crash sequence for every multiple.

The fan: 1928 to 2025

The chart above is the cousin of the worlds fan in the Monte Carlo personal-finance article. There, each faint line was one simulated future. Here, each faint line is one leverage multiple living through the future we already got. The dashed line is 1.00× — cash in the account, nothing borrowed. The two darker traces are 2.00× and 3.00×, so the eye has anchors.

On this one path, unlevered equity turned $10,000 into $5.11 million of real purchasing power, a 6.6% real compound return, with an 83.7% nominal peak-to-trough drawdown. 2.00× finished at $60.0 million — about 11.7 times the unlevered pile — at a 9.3% real compound and a 98.8% nominal drawdown. 3.00× finished at $5.65 million, only 11% more than 1.00×, after a 99.99% nominal drawdown that is a rounding error away from zero. The full-path real terminal, the number a one-chart thread would screenshot, is largest at 2.10× ($61.3 million) — an in-sample argmax on this tape, not a rule.

That last sentence is a fact about 1928–2025. It is not a recommendation. The fan already shows why. Higher multiples rise faster in the long postwar climb and then almost disappear in 1929–32 and again in 2008. Several of the upper lines cross back through the 1.00× line and stay there for decades. The picture is the finding: leverage does not scale the same life. It writes a different one.

2x leverage historical backtest across start years

A 2x leverage historical backtest on a single start year is a story about that year. The United States produced 58 overlapping January starts that still have a full 40 years of tape in front of them. Line those windows up without sharing years and you get 2 independent 40-year lives. We ran all 201 multiples through every overlapping window. The next chart is the 0.01 sweep read the other way: leverage on the x-axis, ending real wealth on the y-axis, and three lines for the 5th, 50th and 95th percentiles of those overlapping windows.

The three lines do not peak at the same multiple. Those crests are in-sample argmax on this tape, not a holdout rule. The 5th percentile — the left tail, the overlapping windows that include the ugly decades — is largest at 2.02×, where it prints $128,526 against $72,315 at 1.00×. The median is largest at 2.85× ($892,989 versus $131,197 unlevered). The 95th percentile is still rising at 3.00×. A person who quotes only the median is quoting overlapping windows that liked leverage. A person who quotes only the 5th percentile is quoting a more cautious slice of the same overlapping set. They are both on the same CSV.

Drawdowns split the surface more sharply than endings. At 1.00×, 3.4% of the 58 overlapping windows saw an 80% nominal peak-to-trough loss (two starts). At 2.00× the share is 17.2%. From 2.42× through 3.00× it is 100% of those overlapping windows: every sampled 40-year start that shares years with 1929 or 2008 went through an 80%+ hole. That is not 58 independent careers. Exact monthly ruin — a single month that takes 1+r through zero — never printed. The worst month on this tape is 1931-09, when the index returned -29.6%. A 3.00× overlay lost -88.8% that month and survived it. A -33% month would have closed the 3.00× account. The monthly tape does not contain one. Daily data in October 1987 is a different experiment, and we did not run it.

How much leverage survives a crash

How much leverage survives a crash is the question the long fan buries in a log scale. Zoom to January 1929 through December 1933 and start every multiple at the same $10,000. The unlevered line bottoms near $2,563. 2.00× bottoms near $251. 3.00× bottoms near $5. All three are still open. They are not the same investor. One of them is looking at a statement that is a few hundred dollars on a $10,000 start. Survival in the narrow sense — the account did not print zero — is a low bar.

The 2008 window is the same geometry on a shorter clock. From October 2007 to March 2009 the unlevered line gives back about half. The 2.00× and 3.00× lines give back most of what the previous decade had built, then spend years rebuilding a hole the 1.00× line never dug. That is the sequence-risk version of leverage, and it is why this page is a cousin of the crash-the-year-you-retire study rather than a cousin of a CAGR table. Order is the whole event.

What a leveraged ETF long term overlay is not

A leveraged ETF long term holding is usually a daily-reset fund. SSO, UPRO, TQQQ and the rest rebalance every session. Volatility drag on a daily reset is not the same object as the monthly overlay above, and 1929 did not have those tickers. Treating this fan as a backtest of UPRO in 1931 would be a lie. What we ran is closer to a futures overlay or a margin account that is put back to a target multiple once a month and charged the T-bill rate on the borrow. Even that is cleaner than a household would get. There is no spread over T-bills, no trading bps, no tax on the financing, no margin call, no behaviour. Those Q4 frictions were not re-run.

Two other silences. We did not glide. A lifecycle that starts near 2.00× at 25 and arrives at 1.00× by 65 is a different study — closer to the question Ayres and Nalebuff asked, and closer to how human capital actually works. We also did not withdraw. Leverage into retirement stacks this surface on top of sequence risk. That is a second paper, not a footnote. The staying-invested analysis is the behavioural half of the same tape: the hole is not only math. It is the month a person stops being willing to keep the multiple on.

What we are not claiming

We are not claiming that 2.10× is the right long-term stock market leverage, or that 2.02× is, or that 2.00× is the internet’s lucky number made respectable. Those crests are in-sample argmax on this tape. The full path, the overlapping-window median and the overlapping-window left tail disagree about where the surface crests, and the nominal drawdown column turns hostile well before the median does. The honest sentence is smaller: on this US monthly tape, with T-bill financing and monthly reset, modest constant leverage raised overlapping-window median and left-tail 40-year real endings up to around 2×, and past about 2.42× every overlapping sampled start took an 80% nominal drawdown. That is not a claim about other return orderings, other countries, daily 1987, or a household that pays a borrow spread.

The United States is one country. Monthly data is one resolution. A constant multiple is one policy. Anyone who needs a number for a life has to run their own stake, their own contributions, and their own willingness to sit through a statement that has fallen by four-fifths. The CSV is public. The product’s backtest tab will not yet let you type 2.10× — this page is a research sweep, not a screenshot of a feature. The method is the same idea: hold the path still, change one thing, look at the whole surface.

Notes on the figures. Start: $10,000, no contributions, no withdrawals. Sweep: 1.00×–3.00× inclusive, step 0.01 (201 multiples). Information set: constant L, reset from the contemporaneous month only — the overlay does not see future returns, latent regimes, or full-sample moments. Tape: monthly S&P 500 price returns (Stooq) from 1928-01-01 to 2025-08-01. Each complete calendar year is scaled so the product of monthly prices matches Damodaran’s S&P 500 total return for that full year (intra-year months therefore depend on that year’s annual total). 2025 is a partial year through August and is not Damodaran-scaled. Financing and CPI: historical-bundle v1 cash yield (3-month T-bills) and CPI-U, contentHash ffff7127cae8b20fef40d0a573f4d16c6eac75edeb6a9b77de3a5a539e629188. Endings are real. Max drawdown is nominal peak-to-trough. Overlay reset monthly; a month with 1+r ≤ 0 is monthly ruin. Rolling windows: every January start with 40 complete years (58 overlapping; 2 independent). Q1 (simulated orderings): not run. Q2 (borrow spread, daily tape, other countries): not run — this US monthly T-bill overlay. Q3 (history): this study. Q4 (frictions): tax off (pre-tax); no borrow spread; no trading bps; no broker margin call. This is a monthly-rebalanced constant-leverage overlay, not a daily-reset leveraged ETF. This article is educational analysis, not investment advice, and does not recommend any security or any leverage multiple.