Etf Mechanics

ETF mechanics

An ETF is a fund with a share price. The index is a formula. The two differ by five things a price series never shows: the daily reset of a leveraged product, the gap between NAV and price, cash that leaves as distributions, holdings that change, and fees. Every computed number below is printed by scripts/leveraged_reset.py (numpy/pandas, seed 0). Issuer and regulator facts carry the date they were read.

1. 🚨 "3x the index" is a one-day statement

✅ The issuer's own words (proshares.com, read 2026-09-08): TQQQ "seeks daily investment results, before fees and expenses, that correspond to three times (3x) the daily performance of the Nasdaq-100 Index"; SQQQ is the same sentence with "three times the inverse (-3x)". Both pages add: "For any holding period other than a day, your return may be higher or lower than the Daily Target. These differences may be significant."

✅ FINRA Regulatory Notice 09-31 (2009-06-11): "Most leveraged and inverse ETFs 'reset' daily". Its example — index 100 → 101 → 100 costs an inverse ETF 0.02%; 100 → 110 → 100 costs it 1.82% — is reproduced exactly by the script (§A: −1x on "+10% then back to flat" = −1.82%).

A flat year is not free — but the honest size is smaller than the folklore

✅ §B: a random 252-day path whose realized vol is pinned and whose index return is exactly 0:

The popular "up 10%, down 10%" examples use daily moves of 5–10%; a real S&P year at 16% vol costs a 3x product 7.4%, not 50%. What decides the number is realized variance, and it grows with the square of leverage: the −1x product loses about as much as 2x, the −3x loses twice what 3x loses.

The analytic form, and what it does and does not tell you

W_L ≈ (1 + R)^L · exp(−(L² − L)/2 · Σr²) — the product's wealth is the index wealth to the power L, times a drag set by the realized variance of the path. For L = 3 the exponent is −3 × Σr². ✅ §C: across 2,000 random flat 25%-vol years the second-order form is off by a mean −0.030 pp and at most 0.382 pp against a −17.10% drag — the approximation is good.

🚨 It is a statement about the path already taken. Σr² is realized variance, so the formula predicts nothing about next year unless you also forecast the variance — which is the whole problem, restated.

"Decay" is the wrong word: the gap changes sign

✅ §D, one random shape at 16% vol, index return pinned per row:

Near zero the variance drag dominates; in a strong trend compounding dominates and the 3x product beats three times the index — in both directions (a −30% year loses 68%, not 90%). The daily reset is path dependence, not a fee.

Holding-period sensitivity — and it is the median that suffers, not the mean

✅ §E, 20,000 zero-drift Monte Carlo paths, gap = 3x return − 3 × index return:

The mean gap is ~0 at every horizon (E[Π(1 + 3r)] = 1 when E[r] = 0); at 40% vol and one year the median 3x outcome is −51.06% while the mean is +0.86%. Volatility drag lowers the typical outcome, not the expected one — a right-skewed lottery, which is why "it always decays" and "expected return is 3x" are both wrong and both commonly said.

The expense ratio is the small cost

✅ §F, the flat 16%-vol year: variance drag alone −7.40%; add financing of the borrowed 2 × NAV at 4.00%/yr → −14.53% (−7.13 pp); add a 0.82%/yr expense ratio → −15.23% (−0.70 pp). The 4.00% is an input — the fund pays what its swap counterparties charge, and that is not published as one number. ✅ TQQQ's own holdings page (as of 2026-09-04) lists Nasdaq-100 index swaps with ten bank counterparties, each shown at 16–30% exposure, over a partial equity basket; ✅ yfinance's funds_data.top_holdings shows 3 lines totalling 24.1% of TQQQ — the leverage is invisible there.

What this does NOT show: no swap spread, rebalancing cost or tracking noise, so every cost figure is a lower bound; the drag numbers transfer to any product with the same realized variance; the sign of the gap does not transfer anywhere.

2. NAV vs price

NAV is (Σ shares × price + cash − liabilities) / shares outstanding, struck once a day from the holdings file. ✅ §G, five lines: NAV 59.7696, close 59.79, premium +0.034% = +3.4 bp. An authorized participant's round trip — creation fee 1.7 bp + basket half-spread 2.0 bp + hedge 1.0 bp = 4.7 bp — is wider than that, so nobody creates or redeems, and the premium is noise. (Those three inputs are chosen for the example; measure yours.) Arbitrage keeps the premium inside the cost of the basket, which for a liquid US-equity ETF is a few basis points.

✅ What the issuer must publish — Rule 6c-11(c)(1), 17 CFR 270.6c-11 (text read 2026-09-08 at the LII mirror; ecfr.gov and federalregister.gov refused automated fetches; substance confirmed in SEC press release 2019-190, 2019-09-26): each business day, before the opening of regular trading, every portfolio holding (ticker, CUSIP, description, quantity, weight); the prior day's NAV, market price and premium/discount; a table and line graph of premiums/discounts for the last calendar year; the median bid-ask spread over the prior 30 calendar days sampled every 10 seconds; and an explanation if the premium or discount exceeded 2% for more than seven consecutive trading days. This is the data you use to check an ETF, and it is free.

✅ Issuer pages, figures as of 2026-09-04, read 2026-09-08:

🚨 The stale-NAV trap: an international "premium" is mostly the clock

EWJ's holdings stopped trading in Tokyo hours before the US close; the NAV is struck from those last prices (✅ iShares states FX is taken "as of the close of business on the New York Stock Exchange"; ⚠️ whether it fair-values the Tokyo closes is not stated on the page). The ETF trades at the US-hours fair value. The difference prints as a premium, and it is not mispricing.

✅ §G simulates it (home-hours moves 0.8%/day, US-hours moves 0.6%/day): the premium's std is 0.59% — it is the US-hours move — and corr(premium today, NAV return tomorrow) = +0.59 while corr(premium today, price return tomorrow) = 0.00. A backtest that "buys the discount and waits for NAV to catch up" is right that NAV catches up and wrong that you can trade at NAV. Judge an international ETF's premium against its own history table (the 6c-11 disclosure above), not against IVV's.

Fixed income: the NAV is the stale price, not the ETF

Bond NAVs come from evaluated (vendor) prices (✅ iShares: "The vendor price is not necessarily the price at which the Fund values the portfolio holding"), and many bonds never trade on a given day. In stress the ETF trades continuously and the NAV lags. ⚠️ March 2020 (press reports via search, 2026-09-08): LQD traded at discounts of roughly 3–5% around 2020-03-20 and at a ~3% premium on 2020-03-23 after the Fed's announcement. A "5% discount" on a bond ETF in a crisis is information about the NAV, not an arbitrage — and the 6c-11 history table is where you see how often it happens.

⚠️ iNAV / IIV: Rule 6c-11 did not require an intraday indicative value; exchange listing rules required a 15-second IIV until they were amended (law-firm summaries of the adopting release, 2019). Where it exists it is computed from last prints, so it carries the same staleness as NAV.

3. Distributions and the phantom ex-date drop

Three kinds of cash leave a fund: income, capital-gains distributions (✅ IRS Topic 404: "always reported as long-term capital gains"), and return of capital, which ✅ "reduces the adjusted cost basis of your stock" and becomes a capital gain once basis reaches zero. ✅ Under Rule 19a-1 (17 CFR 270.19a-1, LII mirror) a payment not wholly from net income must come with a written statement of what portion is income, realized gains, or paid-in capital — that notice is where "12% yield" turns out to be 5% income and 7% of your own money back.

A distribution moves the price by its full amount whatever its tax label. ✅ §H: 1.00/share with 0.60 ROC moves the price by the full 1.00; only 0.40 is income. An unadjusted close series therefore shows a drop that never happened to a holder:

• ✅ 10 years, 1.8%/yr paid quarterly plus one 3% capital-gains payout: total-return CAGR +1.80%, unadjusted-price CAGR −0.24% — a 2.04 pp/yr gap for 2.01%/yr paid out.
• ✅ The capital-gains day prints −4.21% on a true −0.76% day.
• ✅ Ordinary quarterly dividends are not outliers (1 of the 10 worst unadjusted days), but for an income product (4.8% vol, 0.35% monthly payout = 1.2 daily sigmas) 5 of the 10 worst days are ex-dates. A stop-loss or reversal signal on that series fires on payouts.

Use a total-return series, backward-adjusted, and never a forward-adjusted one — the conventions, the yfinance Adj Close == Close tell and the per-library defaults are in ../research-integrity-guards/references/adjustment-conventions.md; this skill does not repeat them. ✅ Verified live with yfinance 1.7.0 (2026-09-08): Ticker("IVV").history(period="2y", auto_adjust=False, actions=True) carries Dividends, Stock Splits and Capital Gains columns — 8 dividend rows, 0 capital-gains rows; on the latest ex-date (2026-06-15) Adj Close == Close == 756.35, while 2025-12-16 shows Close 679.84 vs Adj Close 676.22. That is the present-anchored series the reference file warns about. ⚠️ Equity index ETFs rarely distribute capital gains because in-kind redemption removes low-basis lots — a mechanism, not measured here; IVV's zero rows are consistent with it, not proof.

4. Holdings files and reconstitution

Where a holdings file comes from. ✅ Under 6c-11 it is the issuer's daily website file (§2). Read 2026-09-08: IVV offers a CSV of 504 holdings as of 2026-09-04; SPY a daily XLSX; TQQQ/SQQQ a CSV as of 2026-09-04. ⚠️ Vanguard's ETF share classes (VOO and siblings) do not rely on 6c-11, and Vanguard's filings describe holdings for those funds as monthly with a 15-day lag — the phrase "complete portfolio holdings as of the end of the most recent month" returns 15 hits in Vanguard Index Funds' 485BPOS filings on EDGAR full-text search (latest 2025-04-29), but the full sentence was not read; investor.vanguard.com renders nothing without a browser. Treat "daily holdings" as issuer-specific until you have the file with its as-of date in hand.

🚨 yfinance does not give you a holdings file. ✅ 1.7.0 (released 2026-08-26), run 2026-09-08: Ticker(t).funds_data.top_holdings returns 10 rows for IVV (37.7% of the fund), 3 for TQQQ (24.1%), 10 for EWJ, columns Name and Holding Percent, no as-of date, no history. It is Yahoo's "top ten today". ✅ The iShares ...ajax?fileType=csv URL that circulates in scrapers returned the product web page, not a CSV, from this environment on 2026-09-08 — the download link is a page, not an API. Download from the page, keep the as-of date, and snapshot it (../market-data-engineering/SKILL.md §8).

🚨 Today's file is a current list, and using it for history is survivorship bias

✅ §J, 300 stocks with identical expected returns, 30% idiosyncratic vol, cap-weighted top-100 index rebalanced monthly for 10 years: point-in-time membership CAGR +5.34%; today's 100 members held throughout +8.74% — +3.40 pp/yr of pure selection, with 35 of today's 100 names not in the index at the start. Nothing here has any alpha; the gap is the file. Real churn and dispersion differ, so measure it on your index, but the sign never changes. This is gate 1 of ../research-integrity-guards/SKILL.md — a holdings file is exactly "a ticker list built from today's constituents". Build membership from dated files (../research-integrity-guards/scripts/ pit_universe.py) and record the snapshot id.

Reconstitution is a schedule, and the schedule is a trade

• ✅ FTSE Russell (LSEG press release 2026-05-22): the Russell US indexes are reconstituted semi-annually, June and December, from 2026 (previously annual); rank day 2026-04-30, effective after the close on Friday 2026-06-26; $217.2 billion traded at the June 2025 reconstitution close.
• ⚠️ S&P 500: rebalanced quarterly after the close on the third Friday of March, June, September and December, with constituent changes made as needed and announced days ahead (spglobal.com returned 403 to every fetch; from search snippets of the S&P U.S. Indices methodology).
• ⚠️ Petajisto, Journal of Empirical Finance 18(2), 2011 (abstract read 2026-09-08): the index-turnover cost of buying additions after they have run up and selling deletions after they have fallen has a lower bound of 21–28 bp/yr for the S&P 500 and 38–77 bp/yr for the Russell 2000 over 1990–2005, peaking in 2000.

Two dates matter. Prices move on the announcement; index funds trade at the close of the effective date. A backtest that adds a name at the effective-date close is paying what the index funds paid, which is the right price for an index-tracking strategy and the wrong one for a strategy that claims to front-run it.

5. Expense ratio, tracking difference, tracking error

• Tracking difference (TD) = fund return − index return over a period. This is the money.
• Tracking error (TE) = annualized standard deviation of the periodic return differences. Dispersion around the TD, not a cost.

✅ §I, one 10-year index path, three funds:

Fund A has zero tracking error and loses 9% of terminal wealth; fund B has the largest tracking error and costs a tenth of that. A screen sorted on TE picks the wrong fund. Two details the table also shows: A's TD is −1.05%, not −0.95%, because the fee accrues on wealth that grew; and TE makes the TD you measure noisy — a 0.50% TE over 10 years is ±0.16%/yr in the realized TD, so B's −0.11% is its 0.03% fee plus noise. ✅ 0.95%/yr for 10 years at a 0% index return is −9.10% of terminal wealth.

✅ Expense ratios from issuer pages, 2026-09-08: SPY 0.0945% (gross), IVV 0.03%, QQQ 0.18%, LQD 0.14%, EWJ 0.49%, TQQQ 0.82% net / 0.97% gross, SQQQ 0.95% net / 0.99% gross. For a leveraged product the expense ratio is the small line — §1 shows financing at 4% costing ten times the fee.

6. Rolling-futures ETFs: already measured elsewhere

UNG, USO, VIXY and their kin hold futures and pay the roll. Do not backtest the spot series and trade the ETF: ../../../fin-futures-fx/skills/futures-continuous-contracts/SKILL.md §5 measured UNG −23.25 pp/yr against NG=F and VIXY −45.9 pp/yr against ^VIX, with GLD as the near-zero control. Everything about roll yield, contango and continuous contracts lives there, not here.

7. Where this sits

• ../market-data-sourcing/SKILL.md — getting the price series, and which vendor adjusts what.
• ../research-integrity-guards/SKILL.md §1 — survivorship, the general form of §4 above; and ../research-integrity-guards/references/adjustment-conventions.md — the rules §3 relies on.
• ../../../fin-futures-fx/skills/futures-continuous-contracts/SKILL.md §5 — futures ETFs.
• ../portfolio-and-risk/SKILL.md — once you have a correct total-return series, the metrics.

8. Scripts

scripts/leveraged_reset.py — sections A–J: the two-day reset arithmetic, the pinned flat and trending years, the analytic-vs-exact check, the 20,000-path holding-period table, financing and expense drag, NAV/premium and the stale-NAV correlations, the phantom ex-date drop, TD vs TE, and the today's-holdings look-ahead. numpy and pandas only, seed 0, ASCII output, about ten seconds.