The dividend proxy inside a structured note is a leveraged short in disguise — here is the index that de-levers itself.

A decrement index subtracts a fixed synthetic dividend from an ordinary index. When the underlying falls, the same points become a larger share of a smaller number, so the fee climbs at the worst possible time.

On this page 8 sections

That is the leveraged short in the headline. This page rebuilds 6 live decrement families to the published cent, 1,858 index-days replayed step by step, then authors a hybrid deduction that de-levers the fee once the level falls through a switch. A companion to the index-rebalancing and comovement studies.

cyclical bank · deep 2020 drawdown · cap engages

show the underlying Mar 2020

Illustrative path, shaped to each name’s real profile — the engine (percent, fixed-points and hybrid recursion) is exact.

The engine, in one line each

PercentIV_t = IV_t−1 · ( U_t/U_t−1 − r · Δ/365 )

Fixed pointsIV_t = IV_t−1 · U_t/U_t−1 − d · Δ/365

HybridIV_t = IV_t−1 · U_t/U_t−1 − min(d, ρ·IV_t−1) · Δ/365

The effective annual fee is d / IV for fixed points and min(d/IV, ρ) for the hybrid. Percent holds r = d / B₀, so all three begin at the same launch fee and diverge only as the level moves — the difference you see is the deduction rule alone. The hybrid min(d, ρ·IV) is an original design; the percent and fixed-points forms follow published methodology.

BNP Paribas decrement paths — illustrative path, indexed to 100 at launch. Decrement d = 6.97 points/year, hybrid cap ρ = 8.0%, switch level d/ρ = 87.1. Peak fixed-points fee 14.0%; hybrid cap binds on 659 days.
Year end	Underlying (=100 at launch)	Percent level	Fixed-points level	Hybrid level	Fixed-points effective fee
2018	82.0	76.5	75.2	75.8	9.27%
2019	97.9	85.2	82.0	83.5	8.50%
2020	120.4	97.6	91.9	95.0	7.58%
2021	159.3	120.6	113.3	117.5	6.15%
2022	142.7	100.7	95.7	99.3	7.29%
2023	142.3	93.7	88.2	91.8	7.91%
2024	150.3	95.6	89.5	93.4	7.79%

An exact decrement engine you can drive. Set the fixed deduction d and the percentage cap ρ, pick one of four calibrated names, and replay a crash. The gauge reads the effective annual fee and alarms once a fixed-points deduction would exceed its cap — the moment the hybrid switches to de-levering. Illustrative path; the recursion is exact.

How to read it: the flat line is the deduction in fixed points; the rising curve is that same deduction as a share of the level. Drag d and ρ, or replay the crash, and the two meet where the cap engages.

Three Proofs

Three proofs a structured-products desk hires for.

Replicate

6 families

1,858 index-days replayed step by step
matched to the published cent on 100% of them
worst single-step residual 6.4×10⁻¹² index points

Author

min(d, ρ·IV)

a hybrid deduction, its rulebook and a four-name calibration, authored for this study
rebuilt from the rulebook text alone — pro-forma-identical 12 / 12, bit-identical 20 / 21

Be honest

four chapters

where the cap does nothing, where it loses, where a governance trigger turns on a single day
every figure in-sample, priced rather than tuned away

The market

153 lines

single-stock decrement lines across 67 distinct names
150 fixed-points against 3 percentage
active lines only, so retired vintages fall out — a survivorship floor, a lower bound on the population

The Machine

First, the machine — rebuilt to the cent.

Replicate, then author. Before designing anything new, the six published families had to come out to the last decimal. The kernel is one line of published recursion; the work is in the conventions around it — per-index carry precision, commercial rounding on the printed decimal, zero look-ahead read off each index's own trading calendar. Every residual is classified instead of averaged away. Across 6 families and 1,858 index-days, every step reproduced the published cent, 100% of them, the worst miss just 6.4×10⁻¹² index points. Three of the families rebuild their whole history from a single seed to within 0.01 index points.

Then the rulebook was built a second time, in a clean room. A separate engine, working from the rulebook text alone and never the replication code, reproduced every real pro-forma series (12 / 12) and came out bit-identical on all but one run (20 / 21); the exception was a floating-point summation-order artifact rather than a disagreement about method. The hybrid itself has no third-party series to check against, because it is new. So its certification is the strongest a from-scratch design allows: two independent readings of one rulebook, agreeing to the digit.

Given only public inputs — the seed level, two consecutive published closes of the underlying benchmark, and the fixed-points decrement rate — one day of the index falls out to the cent. Work the right-hand side, then reveal the published base value.

seed 3225.79underlying price return (4560.43 / 4570.97)one day of decrement 120 / 365 published base value 3218.02

matches the published base to the cent

every input is a publicly disseminated, sparse two-decimal published level or a published rulebook constant -- no oracle series is used or needed to check this arithmetic by hand

6 indices certified rebuilt to the published cent

1,858 index-days reproduced per-step, zero exclusions

100% matched to the cent on every index-day

6.4×10⁻¹² worst single step index points, tick to tick

3 full-precision indices rebuild their entire single-seed history to ≤0.01 pts one seed, decades of ticks, one published hundredth of a point

Clean-room rebuild: 12 / 12 pro-forma identical · 20 / 21 bit-identical an independent engine, given only the issued rulebook text

the kernel · step() published recursion + hybrid cap

# one recursion, three deduction rules
def step(iv_prev, u_ratio, d, rho, r, dt, kind):
    if kind == "percent":       # published · flat rate
        drop = r * dt * iv_prev
    elif kind == "fixed":       # published · fixed points
        drop = d * dt
    else:                       # ODX-H · my hybrid cap
        drop = min(d, rho * iv_prev) * dt
    iv = iv_prev * u_ratio - drop
    return max(iv, 0.0)         # zero floor is absorbing

no external oracle ODX-H is an original design, so no third party publishes a reference series for it. Its certification is clean-room sibling-consistency — an independent rebuild that agrees with the reference implementation.

oracle-checked on one name The single-stock gross-return line is validated against an external oracle for TotalEnergies alone: return correlation 0.999960, median absolute difference 0.0002 bp, end-of-window drift +0.19%. The other three names are by-construction reconstructions from public data.

Every input in the identity is a publicly disseminated level or a published rulebook constant, so the base value can be checked by hand to the cent. The same conventions reproduce 1,858 index-days across six published decrement indices at 100%, and rebuild three full-precision histories from a single seed to within 0.01 points. The kernel is published recursion for the flat-rate and fixed-points forms; the min(d, rho·IV) cap is ODX-H, an original design.

Golden-anchor identity and replication certification for the decrement indices
Metric	Value
Seed level	3225.79
Underlying prior close	4570.97
Underlying next close	4560.43
Fixed decrement (points/yr)	120
Day-count	365
Published base value	3218.02
Indices certified	6
Index-days reproduced	1,858
Match to the published cent	100%
Worst single-step residual (pts)	6.4×10⁻¹²
Full-precision indices	3
Full-precision tolerance (pts)	0.01
Clean-room pro-forma identical	12 / 12
Clean-room bit-identical	20 / 21
TotalEnergies return correlation	0.999960
TotalEnergies median abs diff	0.0002 bp
TotalEnergies end-of-window drift	+0.19%

The 2020 Scissors

A fixed toll met a dividend ban.

In 2020 the mechanism and the market came apart. A fixed-points decrement subtracts its scheduled dividend every day, whether or not the underlying actually pays one. Then the ECB asked European banks to suspend their distributions through the pandemic, and the realized dividend on this basket all but disappeared: 2.39 index points gross, 1.97 net of withholding. The index kept deducting 50.14. The toll dwarfed the dividend it was standing in for.

A one-year gap does not stay one year. Run the counterfactual, deducting only the dividend that was actually paid, and by end-2023 the reconstructed index sits +37.88% lower on a net basis and +29.37% lower gross. Some of that gap is tax. The withholding wedge accounts for 22.5% of it, which is why the gross and net paths are drawn side by side.

Scrub the review year

Arrow keys step years · Home / End jump to the ends

Review year

2020

fixed deduction50.14 pts

gross dividend2.39 pts

net dividend1.97 pts

over-deduction vs net−48.17 pts

over-deduction vs gross−47.74 pts

The 2020 scissors — a full toll against a suspended dividend.

Counterfactual — what the over-deduction compounds to

Net dividends are measured after withholding tax; gross before. Their difference is the withholding wedge — 15.0% of the net headline at end-2020, rising to 22.5% by end-2023.

A fixed index-point decrement collided with a dividend suspension. In 2020 the toll deducted 50.14 index points against 2.39 gross / 1.97 net realized; the counterfactual shows what deducting the realized dividend instead would have compounded to by end-2023. Aggregates only — no disseminated daily index level is shown. Underlying: a EURO STOXX Banks decrement index.

Fixed decrement versus realized dividend, and the compounding counterfactual, by review year
Year	Fixed deduction (pts)	Gross dividend (pts)	Net dividend (pts)	Over-deduction vs net (pts)	Over-deduction vs gross (pts)
2018(partial, 11 months)	45.62	35.20	26.99	−18.62	−10.42
2019	50.00	36.49	27.99	−22.01	−13.51
2020	50.14	2.39	1.97	−48.17	−47.74
2021	50.00	24.32	18.51	−31.49	−25.68
2022	49.86	30.51	23.23	−26.63	−19.35
2023	49.86	36.70	27.95	−21.91	−13.16

Counterfactual compounding — cumulative gap versus the fixed-deduction baseline
Anchor	Net-basis gap	Gross-basis gap	Withholding wedge (share of net headline)
end-2020	+19.09%	+16.23%	15.0%
end-2023	+37.88%	+29.37%	22.5%

The scissors: the fixed deduction against realized gross and net dividends, drawn as a pair, 2018–2023. In 2020 the distribution ban opens the blades — 50.14 index points deducted against 2.39 gross and 1.97 net. The counterfactual path compounds the gap to +37.88% net and +29.37% gross by end-2023; the shaded band is the withholding wedge, 22.5% of the gap. Aggregates only; no daily series is drawn.

Aging

The same deduction, three different fees.

The decrement charges a constant number of index points. What a note holder feels is that constant as a share of the level, and the two come apart the moment the level moves. The EURO STOXX Banks decrement launched at an effective 5.00% a year. The March 2020 crash pushed the level down far enough that the same points read 15.32%. By 2026, after the recovery, the quarterly census has it back near 3.01%. Same deduction, three different fees, set by nothing but where the level happened to sit. A fixed toll carries that 1/level convexity for free.

The drift is also a governance problem. In March 2021 the provider re-based the index off-calendar, resetting the effective rate to 9.50%, a discretionary move made below the level the rulebook itself set for a re-strike. The fee had swung by an order of magnitude and swung back, and a committee had to decide when. The hybrid takes that decision out of committee hands and writes it into the formula. The honest ledger picks up the other half of the question: would the rule ever have fired on its own?

Drag or use arrow keys · Home / End jump to the ends

Effective-rate census · 2020Q3

d · fixed toll 50 index points

unchanged since launch

e · effective rate 14.50%

Original strike 14.50%

Re-struck series 9.43%

The drift · fixed points, moving fee

struck5.00%

2020 trough15.32%

re-strike9.50%

20263.01%

Drag through the quarters, or use the arrow keys: the toll d holds at 50 index points while the effective rate e climbs in the 2020 crash and drifts back down.

A fixed points toll is a percentage of a moving index level: e = d / IV. As the level falls the fee rises along a 1/IV curve — the leverage the design later removes.

The 10% designed re-strike trigger never fired against the observed path; the March 2021 re-strike was an off-calendar decision at 9.50%, below the bar.

Own quarterly effective-rate census and event aggregates. Effective rate only; no index levels are shown.

Effective decrement rate e = d / IV (percent), quarterly census, by series
Quarter	Original strike e (%)	Re-struck series e (%)
2018Q1	5.63%	3.92%
2018Q2	6.29%	4.37%
2018Q3	6.59%	4.56%
2018Q4	8.19%	5.62%
2019Q1	7.76%	5.30%
2019Q2	8.02%	5.44%
2019Q3	8.20%	5.53%
2019Q4	7.53%	5.04%
2020Q1	13.71%	9.11%
2020Q2	12.19%	8.01%
2020Q3	14.50%	9.43%
2020Q4	11.05%	7.10%
2021Q1	9.48%	6.04%
2021Q2	9.00%	5.69%
2021Q3	8.56%	5.36%
2021Q4	8.53%	5.30%
2022Q1	9.58%	5.91%
2022Q2	10.67%	6.52%
2022Q3	11.16%	6.74%
2022Q4	9.28%	5.55%
2023Q1	8.85%	5.25%
2023Q2	8.23%	4.84%
2023Q3	8.04%	4.69%
2023Q4	7.67%	4.43%
2024Q1	6.58%	3.77%
2024Q2	6.54%	3.73%
2024Q3	6.24%	3.53%
2024Q4	6.21%	3.49%
2025Q1	4.90%	2.74%
2025Q2	4.44%	2.47%
2025Q3	3.84%	2.13%
2025Q4	3.40%	1.87%
2026Q1	3.76%	2.07%
2026Q2	3.01%	1.64%

The effective rate of one fixed-points family, quarterly. Struck at 5.00%, it climbs to 15.32% at the April 2020 trough as the level collapses, then decays toward 3.01% as the level recovers. The marked re-base in March 2021 resets the rate to 9.50%, taken off-calendar below the governance trigger. Effective rate only; no index levels are shown.

The Hybrid

The fix is one line.

The flaw and the fix share one formula. Write the daily deduction as min(d, ρ·IV). While the level is high it subtracts the fixed d index points, exactly as before. Once the level falls through the switch IV* = d/ρ, it switches to the percentage cap ρ·IV and starts to de-lever. Above the switch the note keeps its predictable point-deduction; below it the fee tracks the level down instead of climbing into the crash. The discretion a committee exercised in 2021 now lives in the rulebook. On BNP Paribas, calibrated at d = 7.12 and ρ = 8%, the cap engages once the index sits more than 12.9% below its launch level.

Each name in the four-name family sets its own switch depth from its own dividend and price: TotalEnergies (d = 4.50, ρ = 8%, switch at 17.3% from launch), BNP Paribas (7.12, 8%, 12.9%), Enel (0.50, 6%, 17.1%), and ASML (8.39, 4%, 88.0%, where the switch sits so far below the traded range that the cap never wakes up). So the cap is a hedge for the names that draw down onto it and a dead letter for the ones that never do. That is the honest ledger's first entry.

Cap ρ 8%published

Drag, or arrow keys · Home / End jump to the ends

The fix is one line · BNP Paribas

min(d, ρ·IV) — fixed d 7.12 (6.97%), cap ρ 8%

switch 87% of launch · −13%

cap-binding index-days 900

give-back vs the fixed toll +6.66 pts

Published cap ρ 8%: the design becomes live below a 12.9% drawdown.

Give-back is versus the plain fixed toll, so it can only help. Against a plain percentage decrement the sign can flip over the full window — the honest ledger shows where.

One line, four names — behaviour at each published cap

TotalEnergies

cap ρ8%

switch depth17.3%

bind days147

BNP Paribasshown above

cap ρ8%

switch depth12.9%

bind days900

Enel

cap ρ6%

switch depth17.1%

bind days0

cap never engages

ASML

cap ρ4%

switch depth88.0%

bind days0

cap never engages· designed-inert

BNP Paribas hybrid cap-sweep — cap ρ, cap-binding index-days, and hybrid-minus-fixed give-back
Cap ρ	Switch depth (drawdown from launch)	Cap-binding index-days	Give-back vs fixed toll (index points)
4%	above launch	2,492	+39.96
5%	above launch	2,365	+28.69
6%	above launch	1,948	+19.07
8%	13%	900	+6.66
10%	30%	206	+2.29
12%	42%	90	+0.84

Annex A family — behaviour at each published cap
Name	Fixed d (index points)	Published cap ρ	Switch depth	Cap-binding days
TotalEnergies	4.50	8%	17.3%	147
BNP Paribas	7.12	8%	12.9%	900
Enel	0.50	6%	17.1%	0
ASML	8.39	4%	88.0%	0

The hybrid deduction against the index level. Above the switch IV* = d/ρ the deduction is the flat fixed-points line; below it the deduction bends into the percentage cap ρ·IV and de-levers with the market. Drag ρ to move the switch. The overlaid drawdown is BNP Paribas on public closes; the cap binds on the shaded below-switch days, first reached 12.9% below launch.

The Honest Ledger

Where the design does nothing, or loses.

01 Where it does nothing

Start where the cap buys nothing. ASML pays a small dividend against a high price, so its switch sits far below anything the stock has ever traded. Across every tested cap and margin the hybrid never binds: 0 bind days, so it reduces to the plain fixed-points index. To reach the switch the stock would have to fall 88.0% from launch, a depth it never comes near. For a low-yield name the cap is inert by construction. The ledger says so plainly.

02 Where it loses

Same engine, same name, opposite sign. On BNP the hybrid ends ahead of plain percent across a recovery-dominated window, by +1.11 index points. Give it a longer window, with more time spent below the switch, and it ends behind, the gap swinging to −0.72. The published cap sits above BNP's equivalent percentage rate, so it costs money in the good regime and only pays in the bad one. The design is a trade. The ledger prints both sides.

03 Where it turns on one day

The governance layer is honest about its own trigger. Against the observed path, the designed re-strike condition never fires at all. On one calculation day the rule basis and the review-day basis land on opposite sides of the threshold, a hair apart — the kind of edge case a rulebook has to name rather than hope away. So the 2021 re-strike was a human call, made while the rule sat silent.

04 A design input, disclosed

One selection input is disclosed instead of smoothed over. The margin that governs the cap comes from a dispersion statistic on the calendar-year dividend. The raw input, 0.402, clears the published threshold and picks the high-dispersion margin; a cadence-corrected input, 0.238, would fall short of it. The same calendar quirk also inflates a headline dividend sum. Both readings sit in the ledger, flagged as an artifact and kept there.

Everything on this page is in-sample, calibrated at a 2026 cutoff on the windows shown. There is no out-of-sample test and no live track record. A from-scratch methodology has no outside series to be judged against, so the strongest certification available is two independent engines agreeing on one rulebook. That is the only claim it makes.

Name

Window

Arrow keys move within each group · Home / End jump to the ends

BNP Paribas · window 2016–26

2016-01-01 – 2026-06-30

Same engine, same name — the sign flips by window.

2019–21 +1.11 hybrid ahead

2016–26 −0.72 hybrid behind

Published cap ρ 8% sits above BNP’s equivalent percent rate 6.97%; below the switch it deducts more than pure percent. 900 of 2,685 steps sat below the switch (switch depth 12.9%).

The same inertness holds for a diversified basket: its dividend-to-level ratio stays low and stable, so the fixed-points design ages gracefully and the cap has nothing to remove.

Pick a name and a window. For BNP the hybrid cap min(d, ρ·IV) beats plain percent in one window and loses in the other; for the low-yield names it never engages.

Also in the ledger · CV-threshold sensitivity

The raw calendar-year coefficient of variation (0.402) clears the published 0.25 threshold and selects BNP for the high-dispersion margin; a cadence-corrected input (0.238) would fall short. Disclosed as a design-input sensitivity.

Also in the ledger · Understated-d cadence artifact

BNP’s selected d (7.12) looks understated against the raw 2025 calendar-year dividend sum (7.38), which double-counts a cadence transition. Read against the honest forward run-rate (5.16), d clearly exceeds it — the raw flag is a calendar artifact.

All figures are in-sample: parameters were calibrated at the 2026-06-30 cutoff on windows overlapping the ranges shown. No out-of-sample or live track-record claim is made.

The honest ledger for the ODX four-name family. For BNP Paribas the hybrid cap min(d, ρ·IV) ends ahead of plain percent (ODX-P) in a recovery-dominated 2019–21 window (+1.11 index points) and behind it over the longer 2016–26 window (−0.72), because the published 8% cap sits above BNP’s own 6.97% equivalent rate. For ASML and Enel the cap never engages, so ODX-H equals the uncapped fixed index and the design buys nothing; the same inertness holds for a diversified basket. All figures are in-sample.
Name	Published cap ρ	Equivalent rate r	Switch depth	Bind days · 2016–26	Hybrid − fixed sibling (pts)	Hybrid − plain percent · 2019–21 (pts)	Hybrid − plain percent · 2016–26 (pts)
BNP Paribas	8%	6.97%	12.9%	900	+6.66	+1.11	−0.72
TotalEnergies	8%	6.61%	17.3%	147	+0.49	reported for BNP	reported for BNP
Enel	6%	4.98%	17.1%	0	0.00	reported for BNP	reported for BNP
ASML	4%	0.49%	88.0%	0	0.00	reported for BNP	reported for BNP

BNP Paribas, window 2016–26. Hybrid ends behind plain percent by minus 0.72 index points. The published cap rho 8% sits above BNP's equivalent percent rate 6.97%, so below the switch it deducts more than pure percent. Over 900 of 2,685 steps the level sat below the switch, switch depth 12.9%. Same engine, same name, opposite sign: hybrid ends ahead by plus 1.11 in 2019 to 2021 and behind by minus 0.72 in 2016 to 2026.

A window-by-name reading of the hybrid against plain percent. On BNP the end-level gap flips sign between the recovery window (+1.11) and the longer window (−0.72); on ASML the cap never binds (0 days at any tested cap). In-sample throughout.

Paper & Method

How it was built, and what it cannot claim.

Six published decrement families replayed step by step against their own trading calendars, each single step reproduced to the published cent under per-index carry precision and commercial rounding.
A hybrid deduction, min(d, ρ·IV), authored on top of the published recursion, with a rulebook and a four-name calibration written for this study.
An independent clean-room engine built from the rulebook text alone, reproducing every real pro-forma series and bit-identical on all but one run.
Every design cost and every inert case carried in an honest ledger, priced in-sample and disclosed instead of tuned away.

What this can't claim

01 In-sample only: Parameters are calibrated at a 2026 cutoff on the windows shown; there is no out-of-sample test and no live track record.
02 The hybrid has no external oracle: It is original, so no third-party series exists to validate it; certification is clean-room sibling-consistency between two independent engines, the strongest claim a from-scratch methodology allows.
03 One name is oracle-checked: The single-name reconstructions are validated against public data on TotalEnergies only; the other three share the same recipe by construction.
04 The census counts active lines: The 153-line single-stock market is an active-lines snapshot, so retired vintages are excluded — a survivorship floor and a lower bound on the population.
05 The replication proof uses the public golden anchor: It reproduces publicly disseminated levels and published constants; it never touches an employer-internal daily series.
06 Precision regimes differ: Three full-precision families rebuild their whole single-seed history to 0.01 index points; the fixed-tick families are step-exact but accumulate published-rounding drift over decades of single-seed replay, disclosed as a regime artifact.
07 The re-strike is the provider's: The March 2021 re-base was a provider decision; the design's own governance trigger never fires against the observed path.
08 Dividend gaps are shown net and gross: Every counterfactual is drawn on both bases with the withholding wedge labelled, so no single figure overstates the effect.

The working paper — the full calibration and the complete design ledger — is available on request while it is being finalized.

Request the working paper — fredhli@outlook.com

Companion studies: The Index Effect Before the Announcement · The Second Moment of the Index Effect