QuanterLab · Research

What the broker takes, Betting Against Beta, approximately: the institution’s machine against the retail hand, twenty sealed windows of the S&P 500

A point-in-time walk-forward investigation with hypotheses registered before scoring and the size of the search left unreconstructed rather than guessed.

By Serhat Girgin

Universe · S&P 500 (point-in-time constituents)

Method · Comparative: The institution: Frazzini-Pedersen halves at institutional financing vs The retail hand: 20/20 extreme tails at retail financing

Manipulated variable · WHO IS HOLDING THE TRADE. Arm A is the machine as published: the entire below-median half long against the entire above-median half short (leg_mode=half, rank-tapered to the median, both legs levered to beta-neutral), financed at institutional terms (prime netting, 50bps spread + 25bps GC borrow). Arm B is the same idea as a phone-and-brokerage account can hold it: the 20 most extreme betas a side, retail financing (no netting, 350bps + 150bps). Two declared fields apart: leg_mode and financing_profile.

Step size · 1 year per forward window

In-sample · 2 years before each anchor

Out-of-sample span · 2006-01-03 → 2025-12-31

Compiled · August 08, 2026

Search record · none (size unknown, see §2.3)

Abstract

Build the celebrated low-beta trade two ways and make them race for twenty sealed years. Arm A is our closest approximation of the machine Frazzini and Pedersen published: the entire below-median half of the S&P 500 held long, the entire above-median half held short, both legs rank-weighted toward the extremes and levered to a market beta of zero, financed at institutional terms, measured carrying cost, four points in twenty years. Arm B is the same idea as a person with a phone and a brokerage account can actually hold it: the twenty most extreme betas a side, margin at retail spreads, borrow fees on every short. Say it plainly before the numbers: this is an APPROXIMATION, not a replication, six disclosed deviations from the published construction, each with its reason and its direction of bias, open this paper. A beta-neutral book's proper benchmark is not the stock market it was built not to resemble; it is cash, zero, at this platform's zero risk-free convention. Against that benchmark: on price returns the institution's book lost 38.9% over twenty years; credit the dividend carry its yield-heavy long leg earns, bounded arithmetic at the measured leg scales, stated in the discussion, and twenty years of the machine land between −7% and +4%. Approximately zero, judged against the benchmark it was engineered to match. (The equal-weight index made +324.8% over the same years, not this book's benchmark, but the alternative every reader actually faced.) The retail version lost 95.3%, dividends move it to roughly −86%, with almost three-quarters of its starting capital consumed by financing and borrow costs alone, and even that flatters it: at the measured 2.41× gross, this book exceeds what a Regulation-T account may even open (deviation six). Across 5,011 paired days the gap between the two hands prices at 11.9 points a year of CAGR, the pairing statistic, not a success statistic; both books lost money. The record supports one sentence without qualification: whatever remains of this anomaly on the most liquid five hundred stocks, none of it is reachable by a retail investor.

Author’s note

This is the second paper in the series the low-beta record forced on us. The first asked what beta SELECTS and found the answer changed in 2020. This one asks what the celebrated construction PAYS once someone has to hold it, and prices the distance between the institution that wrote the paper and the reader holding a phone. We could not run the true construction: no small caps, no total returns, no monthly machinery, the ledger opens the paper for exactly that reason, and every deviation carries its bias direction so you can attack the right thing. What survived our honesty constraints earned its dividend carry and nothing else, and the retail rendering of it was destroyed by financing before selection ever mattered. If you take one number, take twelve: the points per year separating the same idea held by two different hands. The sealed windows, the leg decompositions and the financing meters are all one click deep on the canvas.

What this is not, six disclosed deviations from Frazzini–Pedersen (2014)

Read this before the results, because the results are only as good as these admissions. ONE, Universe. The paper builds on the full CRSP tape: NYSE, AMEX and NASDAQ, thousands of names, micro-caps included, and its own size-sorted tables put the largest Betting-Against-Beta premium in the smallest stocks. We run the point-in-time S&P 500, because it is the only universe this platform can hold survivorship-clean with verifiable historical membership; it is also, not incidentally, the only slice a retail reader can inspect name by name. Direction of bias: against our institutional arm, we are testing the machine where the literature itself says its fuel is thinnest. TWO, Returns. The platform computes price returns; a beta-neutral book is structurally long dividend yield, so this omission bleeds both arms and we CREDIT it in the headline as bounded arithmetic at the measured leg scales rather than bury it. THREE, Cadence. Quarterly point-in-time re-selection, not the paper's monthly, so that a 500-name universe re-screens fully and honestly within compute budgets rather than approximately at speed. Bias small, direction unclear. FOUR, Estimator. A 252-day OLS beta against SPY with the paper's own 0.4 shrink toward one, instead of their split correlation-and-volatility windows; published work finds estimator variants second-order for this trade. Within-leg weighting is a linear taper toward the extremes, at a median split, the same shape as the paper's rank weights. FIVE, Financing. The paper finances at the risk-free rate by assumption. We charge explicit terms on both arms, institutional netting at 50 basis points over plus 25 of borrow (measured cost across twenty years: four points), and the retail arm's 350-plus-150 unnetted terms, which are not a deviation at all: they are the experiment. SIX, Regulation T. The retail book's beta-neutral scales imply 2.41× gross exposure, and Regulation T caps an initial margin book at 2×: a real retail account could not even OPEN this book, it would be forced to deleverage toward 2×, changing the trade rather than merely taxing it. (Portfolio margin escapes the cap, but its minimums, typically around $125,000, are not 'a person with a phone'.) We ran the unconstrained 2.41× at the best published retail tier anyway, which means the direction of this bias FAVORS the retail arm: the −95.3% printed here is the OPTIMISTIC rendering, and a Reg-T-compliant version is strictly worse. Everything else, the sealed windows, the point-in-time membership, the frozen reports, the leg-level decomposition, is the platform's standard record and is inspectable on the canvas.

1 Methodology

Two arms, two declared fields apart, sealed at registration. Both arms select from point-in-time S&P 500 membership, estimate each name's market beta as of the anchor (252 trading days of daily returns against SPY, ordinary least squares, Vasicek-shrunk 0.4 toward one, the same shrink weight Frazzini–Pedersen use), rank the cross-section, and build a long/short book with both legs scaled by 1/β to an ex-ante market beta of zero, rank-tapered toward the extremes. Arm A splits at the MEDIAN of whatever the era's universe resolves, 344 names held in 2006 growing to 504 by 2025, the faithful reading of the published construction on this universe, and pays institutional financing: short proceeds net against the margin loan, a 50 basis-point spread on any residual borrowing, 25 basis points of general-collateral short borrow. Arm B holds the twenty most extreme betas a side, the concentrated book a small account builds when it cannot hold five hundred positions, and pays retail terms: no netting of short proceeds (Regulation-T treatment), a 350 basis-point margin spread at the BEST published retail tier, 150 basis points of borrow. Both arms re-select point-in-time quarterly inside one-year out-of-sample windows, twenty sealed windows walking 2006–2025, each registered before it ran. The benchmark is RSP, the equal-weight S&P 500; Sharpe ratios use a zero risk-free rate, platform convention, disclosed. Every window reports the book's LONG leg and SHORT leg separately, daily P&L split by position sign, financing charged to the long leg, borrow to the short, so the reader can see what each half of the trade did on its own, and the self-audit flag on that decomposition reconciles in all forty arm-windows.

Transaction costs are not modelled in this study; all results are gross of costs.

2 Results

2.1 Headline

The institution: Frazzini-Pedersen halves at institutional financing, pooled Sharpe

-0.07

5012 OOS bars

The retail hand: 20/20 extreme tails at retail financing, pooled Sharpe

-0.21

5012 OOS bars

P(The institution: Frazzini-Pedersen halves at institutional financing beats The retail hand: 20/20 extreme tails at retail financing)

92.2%

5011 paired bars · CAGR gap (The institution: Frazzini-Pedersen halves at institutional financing − The retail hand: 20/20 extreme tails at retail financing) +11.9 pp

Figure 1. Both arms stitched through the identical windows, The institution: Frazzini-Pedersen halves at institutional financing (-39.0%), The retail hand: 20/20 extreme tails at retail financing (-95.3%), benchmark grey (+325.6%). Dotted verticals mark the step boundaries; the dashed horizontal is break-even. These figures compound each arm's own stitched daily series; the pooled statistics in the text inner-join both arms' trading days, one session apart, both are printed from the frozen record.

Figure 2. The same walk, re-based to 1.00x at the first window starting in 2020, 6 of the 20 windows above. The institution: Frazzini-Pedersen halves at institutional financing (-42.0%), The retail hand: 20/20 extreme tails at retail financing (-85.4%), benchmark grey (+67.9%). This is a subset of Figure 1, not a correction to it. The era boundary here is pinned by the author at 2020, a break this study's own record shows, not a chart-scaling choice, and the era rows below put a number on the two periods it separates. The full record is what the study claims.

2.2 Per-step results

**Table 1.** One row per step, raw out-of-sample results.
#	Out-of-sample window	The institution: Frazzini-Pedersen halves at institutional financing SR	The retail hand: 20/20 extreme tails at retail financing SR
1	2006-01-03 → 2006-12-29	1.22	-0.15
2	2007-01-03 → 2007-12-31	-0.47	-0.44
3	2008-01-02 → 2008-12-31	-0.04	-0.35
4	2009-01-02 → 2009-12-31	-0.46	-0.13
5	2010-01-04 → 2010-12-31	-0.87	-0.81
6	2011-01-03 → 2011-12-30	1.40	0.80
7	2012-01-03 → 2012-12-31	-0.28	-0.67
8	2013-01-02 → 2013-12-31	-0.06	-0.94
9	2014-01-02 → 2014-12-31	1.46	1.45
10	2015-01-02 → 2015-12-31	0.44	-0.51
11	2016-01-04 → 2016-12-30	-0.46	-0.53
12	2017-01-03 → 2017-12-29	0.09	0.05
13	2018-01-02 → 2018-12-31	0.23	0.19
14	2019-01-02 → 2019-12-31	0.66	0.40
15	2020-01-02 → 2020-12-31	-0.64	-0.45
16	2021-01-04 → 2021-12-31	-0.11	-0.12
17	2022-01-03 → 2022-12-30	0.75	0.30
18	2023-01-03 → 2023-12-29	-1.29	-1.96
19	2024-01-02 → 2024-12-31	0.09	0.02
20	2025-01-02 → 2025-12-31	-0.50	-0.43

Figure 3. The institution: Frazzini-Pedersen halves at institutional financing: every step's out-of-sample curve overlaid, each rebased to 1× at its own start. Read alongside Table 1: consistent shape across steps is the walk-forward's evidence; a single lucky leg is not.

Figure 4. The retail hand: 20/20 extreme tails at retail financing: the same windows, the other arm. Compare shape-for-shape with the previous figure: the two arms trade the identical out-of-sample legs.

2.3 Search accounting

No search record exists for this design. It was not promoted from a recorded evolving search, so the number of alternatives tried before it, on paper, in another tool, or in the author's head, is unknown. Unknown is a different fact from one: a study with no lineage is not a strategy with one trial, it is a strategy with an unrecorded number of them. Accordingly no count of alternatives tried is claimed, and nothing in this paper is corrected for a search that was never recorded; the number that stands is the raw out-of-sample result plus this disclosure. The registered per-step record below (§4) still guarantees each window's hypothesis was hashed and registered before that window was scored.

2.4 The comparison

Both arms trade the same registered windows, so their returns can be PAIRED: inside each window the two return series are inner-joined date by date and the difference r_{The institution: Frazzini-Pedersen halves at institutional financing} − r_{The retail hand: 20/20 extreme tails at retail financing} is the object under test. Because this is ONE pre-declared contrast, frozen at registration before any window was scored, the paired statistic needs no multiple-testing deflation, and the per-arm pooled numbers above are likewise uncorrected, this design has no recorded search to correct against (§2.3). The paired contrast is the one statistic here that a missing search record does not weaken: it was declared in advance, and it is scored on the difference rather than on either arm's level.

In the table: Arm A = The institution: Frazzini-Pedersen halves at institutional financing · Arm B = The retail hand: 20/20 extreme tails at retail financing.

**Table 2.** Window-by-window paired comparison. Δ is the growth gap (Arm A − Arm B) over the window's paired dates.
#	Window	Paired bars	Arm A	Arm B	Δ	Leader
1	2006-01-04 → 2006-12-29	250	+9.9%	-11.6%	+21.5 pp	Arm A
2	2007-01-04 → 2007-12-31	250	-3.4%	-7.4%	+4.0 pp	Arm A
3	2008-01-03 → 2008-12-31	252	-2.9%	-22.6%	+19.7 pp	Arm A
4	2009-01-05 → 2009-12-31	251	-17.3%	-27.3%	+9.9 pp	Arm A
5	2010-01-05 → 2010-12-31	251	-6.9%	-13.4%	+6.6 pp	Arm A
6	2011-01-04 → 2011-12-30	251	+17.5%	+17.2%	+0.3 pp	Arm A
7	2012-01-04 → 2012-12-31	249	-2.6%	-11.6%	+9.0 pp	Arm A
8	2013-01-03 → 2013-12-31	251	-0.7%	-13.4%	+12.7 pp	Arm A
9	2014-01-03 → 2014-12-31	251	+9.5%	+26.6%	-17.1 pp	Arm B
10	2015-01-05 → 2015-12-31	251	+3.8%	-14.8%	+18.6 pp	Arm A
11	2016-01-05 → 2016-12-30	251	-7.3%	-22.6%	+15.3 pp	Arm A
12	2017-01-04 → 2017-12-29	250	+0.4%	-0.9%	+1.3 pp	Arm A
13	2018-01-03 → 2018-12-31	250	+2.1%	+0.6%	+1.5 pp	Arm A
14	2019-01-03 → 2019-12-31	251	+8.1%	+7.8%	+0.3 pp	Arm A
15	2020-01-03 → 2020-12-31	252	-23.6%	-54.5%	+31.0 pp	Arm A
16	2021-01-05 → 2021-12-31	251	-2.9%	-9.3%	+6.3 pp	Arm A
17	2022-01-04 → 2022-12-30	250	+12.5%	+4.0%	+8.5 pp	Arm A
18	2023-01-04 → 2023-12-29	249	-18.7%	-52.3%	+33.5 pp	Arm A
19	2024-01-03 → 2024-12-31	251	+0.2%	-4.6%	+4.8 pp	Arm A
20	2025-01-03 → 2025-12-31	249	-14.6%	-25.2%	+10.6 pp	Arm A

Paired Sharpe of the difference track: 0.28 · block bootstrap (2000 paths, block 10, seed 1234): P(The institution: Frazzini-Pedersen halves at institutional financing beats The retail hand: 20/20 extreme tails at retail financing) = 92.2%.

Window win-rate. The institution: Frazzini-Pedersen halves at institutional financing led 19 of 20 windows (95.0%), The retail hand: 20/20 extreme tails at retail financing led 1 , and the mean window gap of +9.92 pp points the same way. Widest single window: 2023 at +33.5 pp.

**Table 3.** The same comparison split at 2020. Pooling the whole walk into one row hides which side of the split the difference came from.
Period	Windows	The institution: Frazzini-Pedersen halves at institutional financing	The retail hand: 20/20 extreme tails at retail financing	Mean gap	The institution: Frazzini-Pedersen halves at institutional financing led
All windows	20	-1.84%	-11.77%	+9.92 pp	19/20
Before 2020	14	+0.73%	-6.67%	+7.40 pp	13/14
2020 onward	6	-7.85%	-23.65%	+15.80 pp	6/6

Figure A1, mean window return per period. The institution: Frazzini-Pedersen halves at institutional financing above, The retail hand: 20/20 extreme tails at retail financing below, with the gap at right. The pooled bar and the post-2020 bar are the same comparison over different periods.

The two eras disagree by 8.40 pp. The pooled figure is therefore not a standing property of either method, it is dominated by the later period. Read the two rows, not the average.

3 The circuit

The strategy is a circuit of platform primitives, frozen when the study is registered. Below is the circuit as wired on the canvas, the objective it encodes and how the search runs through it, followed by the mathematics each primitive actually computes, the same formulas the execution engine runs. The complete parameterisation is preserved in the study ledger (Appendix A).

The hypothesis under test

A COMPARATIVE study, The institution: Frazzini-Pedersen halves at institutional financing vs The retail hand: 20/20 extreme tails at retail financing, walked on the same registered out-of-sample windows. The institution: Frazzini-Pedersen halves at institutional financing: S&P 500, selected by statistical / factor criteria, rebalanced quarterly across the selected basket, and run out-of-sample from the anchor, whatever the design estimates from history is re-estimated at each anchor from pre-anchor data only, and the walk advances through registered out-of-sample windows; its disposition is the realized forward path versus the benchmark. The retail hand: 20/20 extreme tails at retail financing: S&P 500, selected by statistical / factor criteria, rebalanced quarterly across the selected basket, and run out-of-sample from the anchor, whatever the design estimates from history is re-estimated at each anchor from pre-anchor data only, and the walk advances through registered out-of-sample windows; its disposition is the realized forward path versus the benchmark. The arms differ in: 2 places, Long / Short, leg_mode: half → fixed; Portfolio Forward Test, financing_profile: institutional → retail. NOTE: with more than one difference, an out-of-sample gap cannot be attributed to any single change. The contrast under test: whether The institution: Frazzini-Pedersen halves at institutional financing generates better risk-adjusted returns than The retail hand: 20/20 extreme tails at retail financing over the identical out-of-sample windows.

Figure 5. The frozen circuit, every node a primitive, every wire a typed data-flow; the two arms are colour-coded (The institution: Frazzini-Pedersen halves at institutional financing green, The retail hand: 20/20 extreme tails at retail financing blue, shared feeds neutral). Each box is one step of the strategy; data flows along the wires left to right, and no box can see data dated later than the box feeding it. The whole diagram was frozen when the hypothesis was registered. Click any node to open what that step ran with and what it produced.

Envelopes show counts, ratios, dates, and the parameters the author chose. Full price and per-name data series are not republished: the underlying market data is licensed to QuanterLab. Point figures quoted in the prose, a named holding's return over a stated span, are summary facts derived from public market prices, not redistributed series.

What each part does

Universe, The starting set of tickers, resolved point-in-time from the index change-log, so names delisted or removed later still compete on the dates they traded.

Price Loader, Bulk OHLCV fetch for the whole universe, point-in-time, no future bars.

Filter Beta, CAPM market beta, how hard this stock moves when the market moves.

Long Short Select, Both ends of the ranking in one book, long one tail, short the other.

Portfolio Backtest, Replay the portfolio forward, rebalanced, point-in-time. No transaction-cost overlay is wired in this circuit, so these results are gross of costs.

Portfolio Forward Autopsy, The post-mortem, where the forward test’s return actually came from.

The objective and the search

The institution: Frazzini-Pedersen halves at institutional financing

Universe	S&P 500 index constituents.
Selection	metric across Market beta (β) → long the bottom 20 and short the top 20 by Market beta (β), rank-tapered toward the extremes, each leg levered by 1/β to an ex-ante market-neutral book.
Validation & out-of-sample	portfolio forward test (buy-and-hold book) (1y horizon from the anchor, quarterly rebalance).
Other components	Select: Long / Short.

The retail hand: 20/20 extreme tails at retail financing

Universe	S&P 500 index constituents.
Selection	metric across Market beta (β) → long the bottom 20 and short the top 20 by Market beta (β), rank-tapered toward the extremes, each leg levered by 1/β to an ex-ante market-neutral book.
Validation & out-of-sample	portfolio forward test (buy-and-hold book) (1y horizon from the anchor, quarterly rebalance).
Other components	Select: Long / Short.

What differs between the arms, 2 differences; more than one thing changes at once:

paramLong / Short, leg_mode: half → fixed
paramPortfolio Forward Test, financing_profile: institutional → retail

Reader's note. With 2 simultaneous differences, an out-of-sample gap between the arms cannot be attributed to any single change, the arms are compared as whole packages, and any causal reading of one ingredient is unsupported by this design.

No transaction-cost elements are wired into this circuit; results are gross of costs.

Show the mathematics, 6 primitives, formulas and parity notes

3.1 Universe

The starting set of tickers, resolved point-in-time from the index change-log, so names delisted or removed later still compete on the dates they traded.

Before any math, you need a list of stocks. An index preset (S&P 500, Nasdaq-100, Dow 30) is reconstructed as it stood ON your anchor date by replaying the historical add/drop change-log backwards, so a 2018 backtest sees the 2018 membership, not today's winners.

Point-in-time membership

Start from today's constituents and un-apply every membership change after the anchor t:

\mathcal{U}(t) = \mathcal{U}_{\text{now}} \;\ominus\; \{\text{adds after } t\} \;\oplus\; \{\text{drops after } t\}

Constituents resolved from the index change-log; the same point-in-time set the factor + screening modules use.

3.2 Price Loader

Bulk OHLCV fetch for the whole universe, point-in-time, no future bars.

Momentum, volatility, trend, every price-based metric needs history. This loads open/high/low/close/volume for all names in parallel, clipped so nothing after the anchor can leak in. The lookback window is derived automatically from the deepest metric you wired.

The window is derived, not guessed

It loads exactly enough history for the hungriest downstream metric plus a warm-up buffer:

W = \max_k(\text{lookback}_k) + \text{buffer}, \qquad \text{bars} \le \text{anchor } t

3.3 Filter Beta

CAPM market beta, how hard this stock moves when the market moves.

Regress the stock’s daily returns on the market proxy’s over a trailing window. The slope is beta: 2.0 doubles the market’s move, 0.5 halves it, negative moves against it. One caution the estimate itself cannot give you: beta is measured AGAINST the index, so when one factor dominates index variance, a high beta is a high loading on that factor, whatever it happens to be that decade.

The regression slope

\hat\beta_i = \frac{\operatorname{Cov}(r_i, r_m)}{\operatorname{Var}(r_m)}

Daily simple returns, inner-joined dates, population moments (cov and var share the ddof, or the ratio drifts from OLS). Default window 252 trading days against SPY; fewer than 60 overlapping days returns no estimate rather than a guess.

Optional Vasicek shrinkage

\beta^{shrunk}_i = w\cdot 1 + (1-w)\,\hat\beta_i

Pulls extreme estimates toward 1.0 (Frazzini–Pedersen use w = 0.4 toward the cross-section). Default w = 0 keeps raw OLS, studies that rank on the extremes usually want the extremes unshrunk, and say so.

Reading it

The metric attaches to each stock for Composite Σ / Top-N ranking, “keep lowest” builds the defensive book, “keep highest” the aggressive one. The serial gate is OFF by default: this filter measures and ranks; it drops nothing unless you arm the gate.

Computed as-of the run anchor from anchor-clipped prices only, point-in-time at first selection and at every re-selection; the pure function is pinned by test_beta_filter (2×-market series → β = 2.0 to nine places).

3.4 Long Short Select

Both ends of the ranking in one book, long one tail, short the other.

The academic long-minus-short spread as a tradable book: long N names at one end of the ranking, short N at the other, `long_side` choosing which end is long (lowest = the Betting-Against-Beta reading of a beta ranking). Default legs are equal-weighted and the book runs at 100% of capital, no leverage. The Frazzini–Pedersen construction turns on two knobs: rank weighting tapers each leg toward its extreme, and β-neutral levers the low-beta leg up and scales the high-beta leg down so the book’s ex-ante market beta is zero, which is precisely where financing terms start to matter.

Rank taper within a leg (weighting: rank)

w_i \;=\; \frac{N + 1 - k_i}{\sum_{j=1}^{N} j}, \qquad k_i = \text{rank from the extreme, } 1..N

The most-extreme name carries weight N/(1+…+N), the least-extreme 1/(1+…+N), Frazzini–Pedersen’s rank weighting restricted to the selected tail. Equal weighting is the default and the legacy behaviour.

β-neutral leg scaling (beta_neutral: on)

\text{long scale} = \frac{1}{\beta_L}, \quad \text{short scale} = \frac{1}{\beta_H}, \qquad \beta_{\text{leg}} = \sum_i w_i \beta_i

Each leg’s weighted-average beta is levered/de-levered to 1, so long-β minus short-β ≈ 0 ex ante. Needs a Market Beta filter upstream (it reads that metric); supersedes short_gross; leg β floored at 0.25 and book gross capped at 3×, both disclosed in the payload when they bind.

What the leverage costs

A β-neutral book’s gross exposure is 1/β_L + 1/β_H ≈ 2.2× of capital, the excess is borrowed. The Portfolio Forward Test charges that loan per its financing profile (institutional nets short proceeds against the borrowing; retail cannot), plus short borrow on the short leg. The construction is the same either way; the financing terms decide what survives.

3.5 Portfolio Backtest

Replay the portfolio forward, rebalanced, point-in-time.

Holds the basket and rebalances on schedule, re-selecting and re-optimizing point-in-time at each rebalance (so it only ever uses information available then), and reports the equity curve, Sharpe, drawdown and trade stats, optionally net of cost and risk overlays.

Compounded equity

E_t = E_{t-1}\big(1 + \mathbf w_{t}^{\top}\mathbf r_t - \text{costs}_t\big)

Drawdown

\text{DD}_t = \frac{E_t}{\max_{\tau\le t}E_\tau} - 1, \qquad \text{MaxDD} = \min_t \text{DD}_t

Financing a levered book

\text{charge}_t \;=\; \text{loan}_t \cdot \frac{\text{spread}}{252}, \qquad \text{loan}_t = \begin{cases}\max(0,\,-\text{cash}_t) & \text{institutional (netted)}\\ \max(0,\,\text{long MV}_t - E_t) & \text{retail (no netting)}\end{cases}

A levered long/short book (the β-neutral Long/Short Select) borrows its excess notional. WHO you are decides the loan: a prime broker nets short-sale proceeds against the margin loan, a fully-netted BAB book carries almost none, while a retail margin account cannot net, so the same book borrows the long leg’s excess over equity. Profiles: institutional = 50 bps spread + 25 bps GC short borrow; retail = 350 bps + 150 bps (the BEST retail tier, so any verdict is conservative); custom = your own knobs, spread on negative cash. Charged daily, reported as financing_drag_pct, never silent.

3.6 Portfolio Forward Autopsy

The post-mortem, where the forward test’s return actually came from.

Runs after the Portfolio Forward Test and dissects its realized path: per-rebalance contributions, winners and losers, exposure and cash periods, and how the realized route compares to what the risk cones projected. It computes nothing new about the future, it explains the past the book just lived.

Reading it

Depth I–IV: headline attribution, per-segment breakdown, per-name contributions, and the calibration ledger (projected cone vs realized, segment by segment). In a study, this is the node that fills the appendices.

4 Projection calibration

Not applicable to this study: the Monte Carlo projection model scores long-only books and does not score levered long/short books, so no rebalance in this walk carries a cone or a VaR verdict. No calibration claim is made anywhere in this paper.

5 Discussion

5.1 Findings

1, The headline, with dividends counted, against the right benchmark. A book built to a market beta of zero is benchmarked against cash, zero at this platform's convention, not against the equity index it was engineered not to resemble. Price returns first, because that is what the platform computes: the institution's approximation lost 38.9% over twenty years (−2.5%/yr); the retail hand lost 95.3% (−14.3%/yr); for context, not comparison, the equal-weight index made +324.8% (+7.5%/yr). (Units, once, for the whole paper: 'points' means percentage points of starting capital, and leg or cost contributions are SUMMED across windows, not compounded, the compounded book return is always stated separately.) But a beta-neutral book is long the yield-heavy half of the market at 1.32× and short the near-zero-yield half at 0.78× (measured average scales), so omitting dividends bleeds it by construction, roughly 2.1 to 2.7 points a year at stated yield assumptions. Credited, the institution's twenty years land between −7% and +4%: approximately zero, four points of measured carrying cost included. The same credit at the retail book's scales (1.80× long on ~3%-yielding staples tails) is worth ~5 to 6 points a year and still only moves it from −95.3% to roughly −86%. Zero for the institution; ruin for the retail hand.

2, The approximation tracks the thing it approximates, which is what makes the zero credible. Window by window, Arm A wears the published factor's fingerprint: destroyed in the 2009 junk rally (−17.3%), strong in 2011 (+17.5%) and 2014 (+9.5%), defensive when the market fell in 2018 (+2.1% against −10.2%) and 2015 (+3.8% against −4.2%), −23.6% in 2020, the year the live institutional factor itself printed near −25% (AQR Data Library, Betting Against Beta USA monthly series, calendar-2020 compounded), and positive in 2022 (+12.5%), the one recent year the regime reversed, with the SHORT leg contributing +18.9. A broken implementation does not reproduce the factor's good years, bad years and hedge years in the right places for twenty years.

3, Financing is free at the top and fatal at the bottom. The institutional arm's margin financing cost 0.03 points TOTAL across twenty years, netting short proceeds against the borrowing makes a beta-neutral book nearly self-financing, exactly the property the published construction assumes, plus 4.0 points of general-collateral borrow. The retail arm, denied netting, paid 54.2 points of margin financing plus 18.2 points of borrow: 72 points of starting capital summed across the twenty windows (charged year by year on an equity base those same charges were shrinking, summed, not compounded), on carrying costs alone, before a single selection decision had a chance to be right or wrong. The rates used are the BEST published retail tier; a mainstream brokerage schedule is worse. This is not an implementation detail of the trade. For a retail hand, it IS the trade.

4, The short leg is the structural bleed, for both hands. Decomposed over twenty years, summed window contributions, per the units note in finding 1, the institution's long leg made +162 points while its short leg lost −199, which reconciles: +162 − 199 = −37 summed, against −38.9% compounded. The retail book's tails did far worse on both sides, long +28, short −263, a summed −235 that compounds to −95.3% because each year's loss fell on an already-shrunken base. The short half of this trade is a standing short of whatever has been driving index variance, financials into 2009, the semiconductor-and-platform complex after 2019 (the companion paper's composition ledger names the names), and holding that short for two decades is what the celebrated backtests quietly require. The one year it pays is the crash: in 2008 Arm A's short leg made +36.8 while its long leg lost −39.7, the hedge doing precisely what it was built for. Twenty years of insurance premium, one payout.

5, Concentration is its own tax, before financing, measured inside this study's own sealed record. Compare the two long legs alone: the institution's diversified low half earned +162 summed points; the retail book's twenty-name low-beta tail earned +28. The short sides: −199 for the half, −263 for the tail. Same ranking, same dates, same direction, same financing charged to the other meter, the tail is just twenty stocks doing what twenty stocks do, on both sides of the book at once. Because the two arms of this study differ in construction AND financing, the decomposition needed a third measurement, and we ran one: the IDENTICAL halves book walked over the same twenty windows with the financing profile as the only declared variable, a supporting run, registered and sealed the same way, made after the two arms above and reported here rather than separately. Single-variable, it gives the clean split: the proper construction at retail terms lost 60.8% against the institution's 38.9%, so retail financing alone costs about 2 points a year of CAGR on the proper book, charged in every one of the twenty windows, never once absent. Subtract that from the 11.9-point institution-versus-retail gap and roughly 10 points a year remain as the price of CONCENTRATION: the twenty-name tails, not the financing terms, do most of the damage to the retail rendering. The broker takes his cut; the twenty-stock book does the dying.

6, The gap between the hands is priced, and it is not noise. Across 5,011 paired out-of-sample days the institution beats the retail implementation by 11.9 points a year of CAGR, wins the block-bootstrap pairing with 92.2% probability (paired Sharpe +0.28), and does it while holding the SAME signal on the SAME dates. Everything separating the two curves is construction and financing, the parts of the published trade that papers assume and accounts must rent. Novy-Marx & Velikov (2022) argued the premium's paper form leans on exactly those parts; this record prices the lean at retail: about twelve points a year.

7, Era honesty, same boundary as the companion paper. Before 2020 the institution's approximation was the thin, positive thing the large-cap literature describes: +5.3% on price returns over fourteen years, roughly +2 points a year once dividends are credited, a modest carry trade that cleared its costs at institutional terms and nothing more. From 2020 the six windows that broke the companion study's naive books broke this one too: −42% for the era even at the full construction, because beta-neutral or not, the short half was the index's engine. The era cut is the same composition-anchored 2020 boundary, disclosed as interpretation; the sealed object is the twenty-window walk.

5.2 Interpretation

Why publish an approximation at all? Because the gap between a published factor and a holdable portfolio is exactly where retail money gets hurt, and measuring that gap requires building both ends honestly and saying loudly which end is which. The five deviations ledgered at the top of this paper are not apologies; four of them are the platform's honesty constraints (a survivorship-clean universe we can actually verify point-in-time, price returns we bound rather than invent, a re-selection cadence we can compute without sampling shortcuts, a beta estimator whose every input is inspectable), and the fifth, financing, is the experiment itself. Each one is stated with its direction of bias, and the two that bias against the institution arm (universe and dividends) are QUANTIFIED: the small-cap slice we exclude is where Frazzini–Pedersen's own size tables put the largest premium, and the dividend carry we cannot compound is credited into the headline as a bound. A reader who wants to attack this paper should start from its own ledger, we got there first, on purpose.

The dividend arithmetic, in full, because without it the record is attackable and with it the conclusion sharpens. The institution's book runs, on measured averages, 1.32× long and 0.78× short. Assume the low-beta half yields 2.4–2.8% a year (staples, utilities, the pantry half of the index) and the high-beta half 1.2–1.4%: the foregone net carry is 1.32×yield_L minus 0.78×yield_H, between 2.1 and 2.7 points a year. Compounded across the twenty years, that moves −38.9% to the range −7% to +4%. The retail tails run 1.80× long on a twenty-name staples book yielding 3.0–3.5% against 0.61× short on near-zero yielders: 5.0 to 6.0 points a year of credit, and −95.3% still only becomes roughly −86%, because no carry survives a 72-point financing bill on a shrinking equity base. These are stated-assumption bounds on the sealed price-return record, not computed total returns; the assumptions and the arithmetic are reproducible from the window table and the measured scales printed in every report.

What should a reader take from a twenty-year zero at the BEST terms? Not that Frazzini–Pedersen were wrong: their factor, on their universe, with small caps and total returns and institutional funding, is not what we tested, the ledger says so first. What this record tests is the version of the idea that survives contact with the most liquid five hundred stocks and honest accounting: it earned, approximately, its dividend carry, a real but thin edge that only exists at financing terms no retail account is offered, on a book no retail account can hold, with a short leg no retail investor should want. If there is an edge somewhere in betting against beta today, it lives below this universe's market-cap floor, inside institutional financing, behind monthly rebalancing machinery, three doors, none of which open from a phone. It is a sad arithmetic and we would rather publish it than sell its opposite: measuring where the edges are NOT reachable is most of what honest retail research is for.

The record's one unqualified positive belongs to the naive reader's defense: every mechanism here behaved as designed. The hedge paid in the crash year, the financing charged what the profile said it would, the legs reconciled to the book in all forty arm-windows, and the approximation reproduced the published factor's good and bad years in the right places. When a construction this honest lands on zero, the zero is information.

No search record exists for this study: the design was not promoted from a recorded evolving search, so the number of alternatives tried before it is UNKNOWN, which is a different fact from one. No count of alternatives tried is claimed, and nothing is corrected for a search that was never recorded; the number that stands is the raw out-of-sample result plus this disclosure. The out-of-sample windows are historical.

The broker’s arithmetic, why the retail version cannot work

Walk the retail book through one year at its measured shape: 1.80× long, 0.61× short, gross around 2.4×, which, deviation six notes, already exceeds the 2× a Regulation-T account may open; hold the thought that everything following is the OPTIMISTIC case. The long leg's excess over equity is borrowed at a margin spread, 350 basis points at the best published retail tier, several hundred more at mainstream brokers, and short proceeds do not net against the loan the way a prime-brokerage account nets them. That asymmetry alone charged this book 54 points of starting capital across twenty years, against the institutional arm's 0.03. Add 150 basis points of borrow on a short leg that must be held continuously, 18 more points, and the retail implementation has spent nearly three-quarters of its capital on plumbing before its first good or bad selection. The trade the literature defends converts a risk-adjusted edge into raw return through cheap leverage; priced at retail, the converter consumes the edge and then the principal. This is why the landing sentence of this paper is not about the anomaly. It is about the reader: if an edge remains, it is not reachable from where you are standing.

Twenty sealed years, three lines: the institutional approximation (full halves, netted financing), the retail rendering (twenty-name tails, retail terms, see deviation six on Reg-T), and, for context only, the equal-weight index: a zero-beta book’s benchmark is cash, not equities. Price returns; the dividend bound in the discussion moves the institutional line to approximately flat and barely moves the retail one.

The institution’s book, decomposed: each window’s long-leg and short-leg contribution (financing charged to the long leg, borrow to the short). The long half earned +162 points across twenty years; the standing short of the index’s upper half cost −199. The one year the short leg is the hero, 2008, is the year the hedge existed for.

5.3 Limitations

This paper approximates Frazzini–Pedersen (2014); it does not replicate it, and the six deviations are ledgered with reasons at the top. In brief, with directions: (1) UNIVERSE, point-in-time S&P 500 only, because it is the only universe this platform can hold survivorship-clean with verifiable membership; the published construction spans the full CRSP tape including the small and micro caps where its own size-sorted tables locate the largest premium. Bias: against Arm A, materially. (2) RETURNS, price returns platform-wide; the dividend carry a beta-neutral book earns is credited as a stated-assumption BOUND in the headline and discussion, not compounded daily. Bias: against both arms, quantified. (3) CADENCE, quarterly point-in-time re-selection versus monthly in the paper, an honest-compute choice; direction small and unclear. (4) ESTIMATOR, 252-day OLS beta against SPY, Vasicek-shrunk 0.4, versus the paper's split correlation/volatility windows; Novy-Marx & Velikov find estimator variants second-order. Within-leg weights are a linear rank taper toward the extremes, which at a median split is the published rank weighting's shape. (5) FINANCING, the published construction assumes financing at the risk-free rate; our institutional profile charges explicit netted terms (measured cost: four points in twenty years), and the retail profile is the experiment. (6) REGULATION T, the retail book's 2.41× measured gross exceeds the 2× a Reg-T account may open; it runs unconstrained here, so the printed retail outcome is the optimistic bound and a compliant, force-deleveraged version is strictly worse. Additional honesty notes: Sharpe ratios use a zero risk-free rate throughout; the risk-cone calibration ledger that accompanies long-only studies is absent here because the projection model does not score levered long/short books, no calibration claim is made; books occasionally held fewer names than targeted when a selected name lacked a usable entry bar, disclosed per window; the 2020 era boundary is interpretation, anchored to index composition in the companion selection-content paper, while the sealed object is the twenty-window walk itself; and finding 5's financing-only split comes from a third sealed walk made to support this paper, outside the two-arm registration above and labelled as such wherever it appears.

References

QuanterLab reference architecture

Gelman, A., & Loken, E. (2013). The garden of forking paths: Why multiple comparisons can be a problem, even when there is no “fishing expedition.” Working paper, Columbia University.
Harvey, C. R., Liu, Y., & Zhu, H. (2016). … and the Cross-Section of Expected Returns. Review of Financial Studies, 29(1), 5–68. doi:10.1093/rfs/hhv059
Lo, A. W. (2002). The Statistics of Sharpe Ratios. Financial Analysts Journal, 58(4), 36–52. doi:10.2469/faj.v58.n4.2453

Author’s references?

Frazzini, A., & Pedersen, L. H. (2014). Betting Against Beta. Journal of Financial Economics, 111(1), 1–25.
Novy-Marx, R., & Velikov, M. (2022). Betting Against Betting Against Beta. Journal of Financial Economics, 143(1), 80–106.
Black, F. (1972). Capital Market Equilibrium with Restricted Borrowing. Journal of Business, 45(3), 444–455.
Asness, C., Frazzini, A., Gormsen, N. J., & Pedersen, L. H. (2020). Betting Against Correlation: Testing Theories of the Low-Risk Effect. Journal of Financial Economics, 135(3), 629–652.
Vasicek, O. A. (1973). A Note on Using Cross-Sectional Information in Bayesian Estimation of Security Betas. Journal of Finance, 28(5), 1233–1239.
CRSP US Stock Databases, universe definition underlying Frazzini–Pedersen (2014); NYSE/AMEX/NASDAQ, all capitalizations.
Published retail margin rate schedules of major U.S. brokers (2024–2025), vs. overnight benchmark rates, the 350bps 'best retail tier' spread used in Arm B.
AQR Capital Management, Data Library, Betting Against Beta: Equity Factors, USA, monthly (accessed 2026). Calendar-2020 compounded return of the USA BAB series, cited in finding 2.
QuanterLab (2026). Your beta is holding something different than it was, the companion selection-content study; composition ledger and era boundary.

Appendix A Reproducibility in QuanterLab

Each step is backed by a frozen run report. The study is re-derivable from the ledger below.

#	Commit	Report	Anchor	OOS window
1	ff74f535a2a9	633	2006-01-01	2006-01-03 → 2006-12-29
2	d2870523b587	634	2007-01-01	2007-01-03 → 2007-12-31
3	7ade42bfb42b	635	2008-01-01	2008-01-02 → 2008-12-31
4	43e28755c3f2	636	2009-01-01	2009-01-02 → 2009-12-31
5	f692c8391339	637	2010-01-01	2010-01-04 → 2010-12-31
6	561b54eb9c88	638	2011-01-01	2011-01-03 → 2011-12-30
7	ac9119844d25	639	2012-01-01	2012-01-03 → 2012-12-31
8	0fe32ed6f8f2	640	2013-01-01	2013-01-02 → 2013-12-31
9	f196ed684ae1	641	2014-01-01	2014-01-02 → 2014-12-31
10	34a3cded70b3	642	2015-01-01	2015-01-02 → 2015-12-31
11	e0bec307e70e	643	2016-01-01	2016-01-04 → 2016-12-30
12	4dae8b9b0296	644	2017-01-01	2017-01-03 → 2017-12-29
13	fc3355e7a4b0	645	2018-01-01	2018-01-02 → 2018-12-31
14	2e194900f882	646	2019-01-01	2019-01-02 → 2019-12-31
15	90a241ca94bb	647	2020-01-01	2020-01-02 → 2020-12-31
16	86b38173f204	648	2021-01-01	2021-01-04 → 2021-12-31
17	2eb44e86375b	649	2022-01-01	2022-01-03 → 2022-12-30
18	e42ff6f9e534	650	2023-01-01	2023-01-03 → 2023-12-29
19	598e46b0ffae	651	2024-01-01	2024-01-02 → 2024-12-31
20	d4b98f2cc621	652	2025-01-01	2025-01-02 → 2025-12-31

Appendix A2 Registration record

What this record does and does not establish. Every window in this study is historical: the data existed before the study began, so this is sequential sealing on past windows, not pre-registration in the clinical-trial sense, and no procedure could make it so. What the platform does enforce is order, each step's specification was frozen and hashed before that step was scored, and the walk cannot advance past a step that was never run or close one with a result registered for a different window. The two timestamp columns below are the evidence: read them together and each seal precedes its own run, and each run precedes the next seal. A study whose seals all post-date its runs would show it here. Wall-clock spacing between seals varies with the author's schedule and queue latency; the ordering, not the tempo, is the claim.

“A COMPARATIVE study, The institution: Frazzini-Pedersen halves at institutional financing vs The retail hand: 20/20 extreme tails at retail financing, walked on the same registered out-of-sample windows. The institution: Frazzini-Pedersen halves at institutional financing: S&P 500, selected by statistical / factor criteria, rebalanced quarterly across the selected basket, and run out-of-sample from the anchor, whatever the design estimates from history is re-estimated at each anchor from pre-anchor data only, and the walk advances through registered out-of-sample windows; its disposition is the realized forward path versus the benchmark. The retail hand: 20/20 extreme tails at retail financing: S&P 500, selected by statistical / factor criteria, rebalanced quarterly across the selected basket, and run out-of-sample from the anchor, whatever the design estimates from history is re-estimated at each anchor from pre-anchor data only, and the walk advances through registered out-of-sample windows; its disposition is the realized forward path versus the benchmark. The arms differ in: 2 places, Long / Short, leg_mode: half → fixed; Portfolio Forward Test, financing_profile: institutional → retail. NOTE: with more than one difference, an out-of-sample gap cannot be attributed to any single change. The contrast under test: whether The institution: Frazzini-Pedersen halves at institutional financing generates better risk-adjusted returns than The retail hand: 20/20 extreme tails at retail financing over the identical out-of-sample windows.”

The same hypothesis was registered independently at every step, hashed before each step's out-of-sample window was scored:

**Table 4.** Registration audit, one row per registered step, with the time each specification was frozen and the time its window was scored. The hypothesis is identical on every row by design: it was registered once and re-registered unchanged at each anchor. Rows that differ would mean the specification moved mid-walk, which is the thing this record exists to rule out. The timestamps are the separate claim: each seal precedes its own run, and each run precedes the next seal.
#	Anchor	Registered at (UTC)	Run completed (UTC)
1	2006-01-01	2026-08-08 12:53:05	2026-08-08 12:54:01
2	2007-01-01	2026-08-08 12:54:06	2026-08-08 12:56:07
3	2008-01-01	2026-08-08 12:56:12	2026-08-08 12:57:53
4	2009-01-01	2026-08-08 12:57:58	2026-08-08 13:00:19
5	2010-01-01	2026-08-08 13:00:24	2026-08-08 13:02:04
6	2011-01-01	2026-08-08 13:02:10	2026-08-08 13:04:10
7	2012-01-01	2026-08-08 13:04:15	2026-08-08 13:05:55
8	2013-01-01	2026-08-08 13:06:01	2026-08-08 13:08:01
9	2014-01-01	2026-08-08 13:08:06	2026-08-08 13:09:47
10	2015-01-01	2026-08-08 13:09:52	2026-08-08 13:10:52
11	2016-01-01	2026-08-08 13:10:57	2026-08-08 13:12:38
12	2017-01-01	2026-08-08 13:12:43	2026-08-08 13:14:43
13	2018-01-01	2026-08-08 13:14:48	2026-08-08 13:16:29
14	2019-01-01	2026-08-08 13:16:34	2026-08-08 13:18:15
15	2020-01-01	2026-08-08 13:18:20	2026-08-08 13:20:00
16	2021-01-01	2026-08-08 13:20:05	2026-08-08 13:22:06
17	2022-01-01	2026-08-08 13:22:11	2026-08-08 13:23:51
18	2023-01-01	2026-08-08 13:23:56	2026-08-08 13:25:37
19	2024-01-01	2026-08-08 13:25:42	2026-08-08 13:27:22
20	2025-01-01	2026-08-08 13:27:28	2026-08-08 13:29:28

Appendix B Per-step diagnostics

What each step's run actually did beyond its return: capital allocation across lanes and regimes, the portfolio book's rebalancing and cost drag, and how positions were sized. Harvested from the frozen run reports, present where the circuit produced them.