№ IV — MEASURED + MODELED
Daily leverage and path dependence
Modest leverage has a serious academic case under specific objectives and implementation assumptions. Daily-reset products add fees, borrowing costs, and path dependence. The model below separates the leverage idea from the retail wrapper used to obtain it.
Nominal units only. Every instrument here compares a strategy against the index under identical draws, so the deflator cancels — there is no real/nominal toggle.
median growth at L = 3.0×: 4.8%/yr · 1× index at 3bp: 7.0%/yr
Leverage can increase expected growth over part of the modeled range. Fees, borrowing costs, volatility, and daily resetting narrow that range.
The two-day reset
The underlying rises 10%, then falls 1/11 — a round trip to where it began. The 2× fund does not come back.
Same path, different result: daily resetting leaves a 1.82% loss over the two-day round trip.
…inverse and leveraged ETFs that are reset daily typically are unsuitable for retail investors who plan to hold them for longer than one trading session, particularly in volatile markets.
Notes on method— formulas, RNG, closed-form cross-checks
Units. Nominal only. Every instrument here compares a strategy against the index under identical draws, so the inflation deflator cancels exactly; there is no real/nominal toggle. Returns are lognormal in log space; geometric inputs convert with ln(1+g).
Y1 — the relabeling machine. Deterministic, annual, per $1. The payer’s share price grows (1+g) then pays a δ dividend; the seller sells the same fraction δ of shares (basis $1/share, never repurchased). Pre-tax terminal wealth is identical to machine precision. Tax: payer τ_d·dividend; seller τ_g·proceeds·(1 − 1/Q). DRIP leg: W ← W·(1+g)·(1 − δ·τ_d); the annual drag is δ·τ_d. Withdrawals and taxes are applied at year end, so the model does not represent intra-year tax timing.
Y2 — covered calls. Monthly. m_mo = ln(1+g_m)/12, s_mo = σ_m/√12. Premium from Black–Scholes with σ_iv = σ_m + vrp, r = g_cash, T = 1/12. Per month the covered factor is (min(G,k) + c)·(1−fee)^(1/12) under the same draw as the index. Live cross-check: E[G] = 1.00673, E[min(G,k)] = 0.98468, P(capped) = 54.9%, premium = 2.08%/mo. MC: 4,000 base → 8,000 effective (antithetic), CRN with the index.
Y3 — leverage. Continuous frontier growth(L) = L·m + (L − L²)σ²/2 − fee(L) − max(0, L−1)·(g_cash + 0.5%), fee(L) = 3bp at 1× else 0.95%. Growth-maximizing L in this continuous approximation: frictionless 3.14, with the stated frictions 1.78. This is an objective-specific model result, not a recommended allocation. Live cross-check: 1× 6.97%, 2× 6.74%, 3× 4.78% per year. The daily-reset truth simulation uses 252 steps/yr, w ← max(0, w·(1 + L·(e^{r_d} − 1) − cost_d)), seed 42, antithetic. Growth-rate: 0.0467 log/yr at 3×.
Gap-risk caveat. Overnight gaps beyond −1/L are not modeled; real leveraged funds can and do terminate. The two-day reset exhibit is exact: +10% then −1/11 returns the underlying to 1.0000 while the 2× fund lands at 0.9818.
Tax constants. Qualified-dividend and long-term capital-gains rates share brackets (0/15/20%, 2025 IRS thresholds); both sliders default to 15%. The model does not select a filing status, account type, state tax, NIIT status, or holding-period qualification. Basis step-up at death per IRC §1014 is relevant only to the no-liquidation comparison. This is a simplified federal-tax illustration, not tax advice.13
Market-data methodology. Product-table figures are computed from dividend/split-adjusted price series and corroborated against totalrealreturns.com and financecharts.com, pinned to their as-of dates.38