Leverage, Without the Ruin

AThe simulated worlds

The historical backtest gives me one path, and one path is nowhere near enough to choose a leverage rule. For the decision grid I generate fifteen-year paths with six hidden macro regimes: Expansion, LateCycle, Recession, Crisis, Stagflation, and Recovery.

One regime drives equity returns, volatility, short rates, and the yield curve at the same time, so a rate shock cannot accidentally arrive with boom-time equity returns and free financing. GARCH lets volatility stay hot after a shock instead of resetting the next month, Student-t innovations make large moves happen more often than a normal distribution allows, and the S&P 500 history supplies the ordinary return and volatility anchors.

Eleven versions of the future

A calibrated base model is still one view of the future. I don't want that view hidden inside the model, so I make it explicit. Sixty percent of the scenario weight encodes what I currently believe, including the AI melt-up, the possibility of a fast AI bust, a higher-growth gear change, a bad rate shock, and the small geopolitical catastrophe tail. The remaining 40% gives ordinary history and slower bear markets enough weight to beat me if the whole thesis is wrong.

I choose those weights before the run, and the model never estimates them for me. The two regimes that matter most to the left tail, Crisis and Stagflation, are also the two that 1993 to 2026 cannot calibrate honestly, because the sample has few true crises and no 1970s stagflation. I trust the ranking across worlds much more than the terminal dollar amount from their weighted mixture.F4

Keeping strategy comparisons matched

Every strategy starts with $100k, receives the same $2k monthly contribution, and sees the same seeded market path and financing rates. Signals are lagged, so no strategy trades on tomorrow's information, and I inspect the return, volatility, rate, and regime mix of the unlevered world before counting its result. The fan charts use 200 paths per scenario, or 2,200 total, while the decision grids use 1,000 to 2,000 per scenario.

Before I trust a simulated world, I check that recent variance predicts later variance without predicting later returns.F5 If that relationship disappears, the simulation has also removed the reason for varying leverage with volatility.

BThe strategy choices

The formula from Section 4 is short, but it still leaves several choices open:

L_t = clamp ( β · (r_target − r_f) / σ_t²,   L_min,   L_max )

I read r_f from the 3-month T-bill rate at each check rather than choosing it. I split the remaining choices into two groups: the ones that change the leverage target, and the ones that decide when that target becomes a trade.

Choices inside the formula

I also test a GARCH oracle that gets to see the simulator's true hidden conditional variance. I cannot see that in real life, so it only tells me how much a perfect volatility estimate would buy. Every strategy I could actually run uses observed returns.

Choices about execution

At this point I have a leverage target and a rule for when to trade it. I still need to turn the target into actual funds and account for the taxes created by rebalancing. Those are Appendices D and F.

CWhy I chose these values

Once the worlds were fixed, I changed one family of choices at a time while keeping the random seeds matched. I judged p10 wealth, median, p75, p90, drawdown, and the odds of finishing below contributions together rather than optimizing one headline number.

Choosing beta and target return

Beta and target return set the amount of risk before the execution details matter. Beta 0.6 bought meaningful upside over 0.5, while 0.7 added little to the center and made drawdowns materially worse. A 10% target return won p10 in 10 of 11 scenarios and the median in 8 of 11. Raising it from 10% to 12% pushed p75 from $3.72M to $4.44M and p90 from $8.13M to $11.08M, but p10 fell from $298k to $238k and p10 drawdown worsened from 67.7% to 74.8%. The median barely moved. That is too much left tail for upside that only shows up when my bullish scenarios get most of the weight.

Choosing the signal, cap, band, and cadence

With beta and target return fixed, the last implementation sweep crossed the volatility signal, cap, band, and cadence on the same 11,000 paths. The rows below were enough to make the choice:

The volatility estimate was the least dramatic decision. EWMA matched trailing 3M volatility in the bad paths and did better through the middle and upside. The GARCH oracle did better, as it should, but I cannot trade on hidden simulator state. Raising the cap from 2.5× to 3.0× bought enough p75 and p90 to justify the smaller p10 cost.

Monthly was the clearest choice, because quarterly lost money across the distribution and somehow had a deeper bad-path drawdown too. The band was much closer: checking monthly and trading every target change produced almost the same distribution, so I kept 0.25 as a behavioral and tax guardrail. Most monthly checks now end with no trade.

Snap re-levering is the one place my forward view directly changes the implementation. Moreira and Muir's estimates support cutting quickly after a variance shock and adding exposure back more slowly, while my scenario prior puts more weight on shallow dips that recover before a slow rule gets back in. The matched runs favored snapping to target, so I follow that result and I would revisit it first if my view changed toward long, grinding recoveries.

DInstruments and construction

A 1.7× target is only a number, so I still need to turn it into funds I can actually hold. These are the liquid S&P 500 building blocks I use, and the leveraged ones target a multiple of each day's return rather than a multiple of the return over my whole holding period.

Expense ratios are current net fund expenses. Dollar volume is the rounded 3-month average share volume multiplied by the current fund price.

SPUU is cheaper than SSO but trades about two orders of magnitude less each day. SPY, SPUU, and SPXL are possible same-multiple tax-loss-harvesting candidates, not funds I switch into casually based on the day's spread or tracking.

Tax-loss harvesting

The second ticker at each leverage level may be useful when harvesting a loss, but the IRS gives no ETF-specific safe harbor for funds tracking the same index. Publication 550 applies the substantially-identical test across accounts, including an IRA, so I treat any swap as a tax-advice question. That means checking purchases and dividend reinvestment across every account during the full 30-day window before and after the sale. I credit no harvesting benefit in the simulation.

Turning a target into holdings

Below 1×, I hold L in VOO and leave 1−L in T-bills. Between 1× and 3×, I use the two endpoints and hold (3−L)/2 in VOO and (L−1)/2 in UPRO. At a 2× target, for example, 50% VOO contributes 0.5× exposure and 50% UPRO contributes 1.5×, which adds to 2×.

The endpoint barbell leaves less capital sitting inside an expensive leveraged fund than an adjacent VOO/SSO/UPRO construction. At 2×, half the portfolio pays UPRO's fund expense while the whole portfolio would pay SSO's, and both constructions finance the same extra 1× notional. Their leverage also drifts differently between monthly checks, so I tested the complete constructions rather than comparing expense ratios on paper.

Taxable changes the answer because keeping a VOO/UPRO barbell at 2× means selling whichever side has run ahead. I ran the pure S&P 500 constructions again through the corrected tax ledger, this time as a fixed 2× position:

So I use the endpoint barbell where rebalancing is sheltered and buy-and-hold leveraged funds where selling would repeatedly realize gains. The table only isolates the S&P 500 instrument choice, while the complete taxable VOO* implementation also has account and financing details that I do not publish here.

LETFs vs. LEAPS

I modeled deep in-the-money LEAPS because both instruments are ways to borrow against the index, and I wanted to choose the risk distribution before choosing the wrapper. Daily-reset LETFs pay through financing, expenses, and realized-variance drag, with the path-dependent part roughly −½ L(L−1) σ²T. Daily compounding helps in a smooth trend and hurts in a choppy flat market.C11

The LEAPS model buys calls about 20% in the money with two years left, rolls them annually, and puts the unspent cash in T-bills. The loss floor improves the ugly paths, while implied volatility, skew, bid/ask spreads, theta, and the roll all take their share.

One LEAPS contract is not path dependent over its one- or two-year life: its payoff depends on where the index ends up, not the route it took there. But decade-long exposure requires a sequence of contracts, and each roll depends on the index level and option price at that moment, which makes the full strategy path dependent.

LEAPS won p10 in 8 of 11 scenarios and improved p10 drawdown by 2.9 points. LETFs won the median, p75, and p90, including a 23% advantage at p90. At my risk tolerance, I take LETFs.

I only trust this result for this SPY rule and the long-dated IV, skew, spread, and roll behavior I assumed. The taxable engine also only supports LETFs. I removed LEAPS from the active research grid because modeling sparse deep-ITM liquidity, strike breaches, crash-time rolls, and the holding period honestly would require live-chain calibration and a separate tax ledger. Any real LEAPS implementation starts with live quotes and a written roll rule.

EThe VOO* completion

The simulations and every result in this essay use plain VOO, so they are evidence for the leverage rule on the S&P 500 rather than a backtest of the completed VOO* portfolio. VOO* is how I apply the rule: VOO exposure plus a small cap-weighted basket of foreign mega-caps such as TSMC, ASML, Samsung, and Tencent, because the S&P 500's U.S. incorporation rule leaves out several companies that sit directly inside the compute stack.

The foreign shares are ordinary 1× securities, but I hold them inside a leveraged household portfolio and finance them as part of the whole construction, so describing the basket itself as unleveraged would be wrong. The exact account placement and financing are outside this essay.

FTaxes, done honestly

Tax is the strongest argument for fixed leverage. I can buy SSO once, hold it, and defer the gain until liquidation, while the dynamic rule has to trim appreciated shares whenever volatility pushes its target down.

Why the first tax result was wrong

My first taxable run made the dynamic strategy look bad enough to flip the decision, with a weighted median around $465k. When I traced the median path month by month, the ledger had taxed every winning sale while throwing away the realized losses, used New York's 10.9% rate even though I live in New Jersey, and sold more shares to pay the tax without recording those funding sales as taxable events.

On that one path it counted $1.824M of gross gains, ignored $881k of realized losses, and charged $927k. Same-year netting alone cut the rough estimate to $535k before fixing the state rate or the tax-on-tax-payment sales, so the result was unusable and I rebuilt the ledger.

What the corrected ledger does

Six lot-level tests cover short-term loss netting, federal carryforwards, New Jersey's lack of carryforwards, holding periods, tax-funding sales, and contribution-funded costs. I also audited representative paths transaction by transaction, and their pre-tax histories reproduce the compiled simulation before any tax is applied.

The tax-aware version also allows the VOO/SSO/UPRO/T-bill mix to drift inside the leverage band instead of trading every sleeve back to an exact weight.

The tax-aware strategy recovers most of the avoidable damage. Dynamic still trails fixed 1× at p10, then beats it from the median through p90. Fixed 2× and 3× keep much more of the right tail through deferral and pay for it with far worse p10 outcomes.

This simulation covers capital gains only. It leaves out progressive brackets and NIIT thresholds, wash-sale adjudication, the federal $3k ordinary-loss deduction, dividends, T-bill distributions, charitable gifting, and basis step-up, so I do not credit any of those as free alpha.