Buying the Discount: A Statistical Test of Momentum-Based Allocation

01Abstract

02The strategy in plain terms

The strategy rests on one idea: buy more when the asset is trading at a discount, and bank gains when it is trading at a premium. “Discount” and “premium” are defined by a proprietary momentum-based allocation model that reads how stretched price is relative to its own recent behavior. We label a deep-discount reading a Sigma discount and a stretched reading a Sigma premium. The rule is fixed for every asset and every era -- no per-asset tuning. Sigma is evaluated independently on four timeframes -- 1-Day, 3-Day, 5-Day, and Weekly bars -- so a brief one-day washout and a months-long cyclical bottom register as different, separately actionable events.

Two methods follow from that single signal:

1. Accumulate. Deploy a budget into Sigma discounts as they appear and hold to today. This is the cost-basis play: it converts the same dollars into more units by concentrating purchases at depressed prices.
2. Take-profit rotation. Scale into discounts and scale out into premiums, banking realized cash along the way. This is the cash-flow play: it turns volatility into withdrawals you can actually spend.

Both are deliberately simple. The whole argument of this paper is that a plain, fixed rule -- not a complex, fitted model -- already extracts a measurable, statistically defensible edge. Section 03 specifies exactly how the shipped product turns these two methods into two runnable accounts.

03The two accounts, as shipped

The product runs the two methods as two separate accounts with two separate mandates, driven by the same signal. Keeping them separate is deliberate: the mandates disagree on purpose (a premium is a sell signal for one and a non-event for the other), and mixing them in one pot makes both unmeasurable.

Long-term accumulation

The long-term account sizes every buy by timeframe confluence. Each discounted timeframe carries a fixed confluence weight -- 3-Day = 2, 5-Day = 3, Weekly = 4, out of 9 -- and at each daily candle close, while the discount stays lit, the account deploys that share of its remaining dry powder. With the 5-Day and Weekly timeframes discounted together, for example, the combine rule deploys (3 + 4) / 9 ≈ 78% of whatever dry powder remains; with all three weighted timeframes lit it deploys all of it. The 1-Day timeframe carries a weight of zero: a daily-only discount is too frequent to be meaningful for accumulation, so the long-term rule sits it out. Positions are never sold; premiums are a no-op. Idle dry powder waits in a money-market-style yield until the next discount. The weights are fixed in advance and identical across assets -- sizing conventions, not tuned parameters.

Swing rotation

The swing account scales in on a per-timeframe ladder -- 2% of the account per 1-Day discount candle, 6% per 3-Day, 10% per 5-Day, and 14% per Weekly -- rarer, longer-timeframe discounts command a bigger bite. Premiums trim 4% of the position per premium candle, converting strength into realized, spendable cash that redeploys into the next discount. The account rides cycles rather than holding through them.

The two mandates produce different daily instructions from identical information. A premium on an asset you hold long-term means “do nothing”; the same premium in the swing account means “take some profit.” The product’s daily board therefore shows each asset twice when you run both mandates -- once per account, each with its own action.

04Per-cycle results

We report the result the way an allocator lives it: one market cycle at a time, with each cycle valued at its top. Every comparison is run on the same capital, the same idle-cash yield, and the same transaction fees for both methods, so neither gets a funding or cost advantage. The six headline assets span 30 cycles in total, and Sigma’s accumulation beat DCA in 21 of them.

Crypto headline set

Equities and index

† Multiple is valued at the cycle top (hindsight). XIRR is the annualized money-weighted return over the cycle. “n/a - no discount fired” marks cycles where Sigma made essentially no buys, so it has no comparable average cost basis.

The receipts in these tables are from the result set computed July 19, 2026 (price history through July 17, 2026). Two things follow when you cross-check against the live terminal. Scope: ALL figures here are a dated snapshot — the live pages recompute daily from the current data feed, so in-progress cycles drift as history accrues, and even completed-cycle figures can shift slightly when data providers revise past prices. Lens: these are the funded, fee-charging per-timeframe backtests; the Explorer’s headline “average cost vs DCA” line is a different, labeled lens (an entry-price-only average across whichever timeframes you have selected, no funding schedule) — the two are not meant to match digit for digit.

05Where the edge lives: average cost

The per-cycle multiples are the scoreboard; the average cost per unit is the engine. The mechanism is direct: in a cycle with a genuine deep discount, Sigma pays a lower average cost than steady DCA, so the same dollars buy more units — and more units is exactly what produces the higher multiple and the higher annualized return. This is the cleanest way to see the edge, because average cost is not inflated by the start point or the holding horizon the way a whole-chart terminal multiple is.

On Bitcoin Sigma bought +635% more units per dollar than DCA over its full history; on Ethereum, +161% more. That lower cost basis is the mechanical source of the multiple advantage, and it is the channel that holds wherever a real discount existed. Where there was no flaggable discount, Sigma paid more — and the average-cost chart shows exactly that, asset by asset, cycle by cycle.

Two conditional side-channels: drawdown and realized cash

Beyond the cost-basis engine, two further channels show up — both conditional, not universal. Buying lower often leaves Sigma less underwater versus its own cost basis than DCA (strong on BTC and ETH), but on NVIDIA Sigma is about 18.0% worse on this measure and on the S&P 500 mildly worse, because steady DCA already avoids being all-in at the top. Separately, the take-profit rotation banks spendable cash by trimming into premiums — on Bitcoin alone it realized $186,119 on a $100,000 starting budget while still holding a position. (The underwater-vs-cost measure used in this research channel is a per-position diagnostic; Section 10 explains how it relates to the two account-level drawdown lenses the product reports, and why the product numbers read shallower.)

06The annualized (XIRR) view

Per-cycle multiples reward long cycles — a 25-year hold compounds into a huge number that says more about duration than about timing. The annualized money-weighted return (XIRR) normalizes that. It answers “what rate per year did each dollar actually earn?” and it puts a 2-year cycle and a 25-year cycle on the same footing.

The effect is clarifying. Apple’s long final cycle (2000-2026 (in progress)) reads a headline multiple in the four figures, but annualized it is 32%/yr for Sigma versus 23%/yr for DCA — a believable, real-world edge rather than an unintelligible terminal number. Across cycles the XIRR ranking tracks the multiple ranking, which is the point: the edge is the same edge whether you measure it as units, as a multiple, or as a compound rate.

07Is the cost-basis edge real?

The toughest comparison is not DCA — it is luck. For each asset we draw 1,000 random buying schedules that deploy the identical budget on randomly chosen dates, and ask how often random timing matches or beats Sigma. We run this test on units-per-dollar (cost basis), because that is the clean, un-inflated measure of the edge: it does not depend on the start point or the holding horizon.

The empirical p-value uses the add-one convention the engine computes: p = (k + 1) / (n + 1), where k is how many of the n = 1,000 random schedules matched or beat Sigma. The +1 keeps a finite simulation honest -- it can never report an impossible p = 0 -- so beating all 1,000 schedules reports p = 1/1,001 ≈ 0.000999, the smallest value this test can return, and p = 0.002 means exactly one schedule matched or beat Sigma (999 of 1,000 fell short). Read every p in this paper and on the live terminal against that convention: a surface showing p = 0.002 is a “beat 999 of 1,000” result, not a “beat every one” result.

Across the headline set Sigma’s units-per-dollar beats a 1,000-run random-timing null with p ≤ 0.002 on 6 of 6 assets. The edge is not a single lucky path, and it survives multiple-testing correction: each significant asset clears the Bonferroni threshold of 0.05/6 = 0.0083. We deliberately do not headline the whole-chart terminal multiples (e.g. a deploy-at-inception, hold-to-today Bitcoin figure): those are inflated by the start point and the horizon, and they obscure rather than reveal where the edge lies.³

08The honest conditional edge

The edge is real, but it is conditional, and the precise shape of that condition is the most important thing in this paper. Of the 30 cycles across the headline set, Sigma beat DCA in 21. It is dominant where genuine cyclical deep discounts exist and mixed where they do not.

The split is not random, and the grouping criterion is what caused each loss, not the bare win rate. Sigma wins cleanly on the assets that have real, recurring discount cycles — Bitcoin (3/3), Ethereum (2/2), XRP (1/2), and Apple (6/6): on those assets it won every cycle in which a discount actually fired. XRP’s single loss is the boundary condition, not a mis-timed buy — a cycle in which no discount fired at all (the “n/a - no discount fired” row in the Section 04 tables), so the account sat in yielding cash while DCA kept buying. It is mixed on NVIDIA (1/4) and the S&P 500 (8/13), where Sigma also lost cycles in which it did deploy and still paid a higher average cost than DCA — the genuinely adverse case — and, as disclosed below, it lost outright on a launch-era coin we tested outside this set.

09Benchmark fairness by design

A timing edge is easy to fake by quietly handicapping the benchmark. Every comparison in this paper and in the product is therefore built on one rule: the benchmark gets everything Sigma gets. Concretely:

• Same money, same costs, same yield. DCA runs the identical budget and contribution schedule, pays the same per-side transaction fee on every buy, and earns the same money-market-style yield on any idle cash. No free-cash or free-fees asymmetry on either side.
• Matched market exposure. Sigma rarely deploys 100% of its budget -- it holds dry powder between discounts. So the DCA benchmark deploys the same time-averaged fraction of each contribution and holds the remainder in the same yielding cash. Without this, DCA would be handed extra time-in-market beta and the comparison would measure exposure, not timing.
• A DCA that isn’t a strawman. The per-cycle DCA is capped: it stops buying above the previous cycle’s all-time high rather than blindly averaging into euphoria. We benchmark against the sensible version of the alternative, not the dumbest one.
• Each investor is measured against their own money. When we compare drawdowns, each strategy’s loss is measured against the money that strategy had committed by that date -- the war-chest allocator against their war chest, the paycheck DCA-er against the paychecks they had fed in so far. Percent-of-own-money-in is the drawdown each of them actually lived.

10What a drawdown feels like: two lenses

“Maximum drawdown” sounds like one number, but there are three defensible ways to measure it for an accumulation strategy, and they answer different questions. Choosing the wrong one either terrifies or flatters. Consider an illustrative accumulator who has committed $100,000, of which Sigma has so far deployed $21,000 into positions -- and a bear market cuts those positions roughly in half:

• Deployed-slice vs its own cost (“how much are my positions down versus what I paid?”): about -50%. True -- but it describes a fifth of the account and ignores the $79,000 sitting safely in cash. This per-position diagnostic is what the research channel in Section 05 uses, because it is clean, per-asset, and significance-testable. It is not what the account holder experienced.
• Account peak-to-trough (“how far did my account fall from its high-water mark?”): the right lens for an active, take-profit account, which is exactly how the product reports the swing rotation. But for a long-term accumulator it has a perverse property: an account that runs up 100% and gives back a third while still far above every dollar contributed reads as a deep loss -- it counts surrendered gains as damage.
• Below money-in (“how far below the money I put in did my whole account -- positions plus dry powder -- ever sit?”): about -10% in the illustration. Floored at zero once the account is above its contributions. This is the lens the product reports for long-term accumulation, because it answers the only question a holder actually asks in a crash: am I below the money I put in, and by how much?

The product’s two accounts therefore report two different, deliberately-chosen risk numbers: money-in for accumulation (the hold-through-it experience) and peak-to-trough for the swing rotation (the trading experience, banked profit included). The research diagnostics in this paper additionally use the per-position cost-basis lens where per-asset statistical comparability matters. All three are computed from the same ledgers; they differ only in the question they answer -- and each surface in the product states which one it is using.

One consequence is worth stating because it cuts against Sigma: under the money-in lens the drawdown comparison with DCA becomes cycle-dependent. Aggressively buying into a crash can leave Sigma briefly deeper below its money-in than a slow DCA still holding most of its cash. The older cost-basis lens mechanically flattered Sigma here (averaging down always lowers your own average cost); the money-in lens does not, and we ship it anyway. The dry powder Sigma holds between discounts is what keeps the typical reading shallow -- often single-digit percentages at moments when a fully-invested holder sat 40-70% underwater.

11Robustness

A real edge should not be a knife-edge. We stress it three ways.

Parameters. The advantage over DCA is positive in the large majority of parameter cells -- 13 of 15 for Bitcoin, 13 of 15 for Apple, and 9 of 15 for Ethereum -- not at one fragile point. The negative cells are long-horizon presets that effectively never fired (zero buys, capital left idle). Because the shipped rule is a single fixed configuration chosen up front -- not selected after seeing the results -- the rule is non-fitted to the test data; it still embeds fixed choices (a deploy floor, confluence weights, a discount level), but none were tuned to the outcomes. Simplicity is the feature.

Costs. The advantage over DCA is essentially flat as round-trip transaction costs rise from a fraction of a percent to half a percent -- the edge is not an artifact of frictionless trading.

Sub-periods. Cutting each asset’s history into equal, non-overlapping windows, Sigma beats DCA in independent eras. The most recent slice is the thinnest -- a fair caution that the largest discounts came earlier in each asset’s life -- but the direction of the edge is consistent.

12Survivorship-honest view

Backtests that quietly drop dead assets flatter themselves. We include genuine collapses -- Terra/LUNA, TerraUSD, FTX’s FTT, Celsius, Serum, and others -- and report what each method actually did.

13Engine verification

Every number in this paper is produced by a backtest engine, and a backtest engine is code -- so before this edition, the engine was put through an adversarial line-by-line audit: an independent review pass instructed to try to break the methodology, not to defend it. The audit produced 58 findings, each triaged, fixed, and re-verified. The classes of issue it caught and closed are exactly the ones that quietly flatter backtests:

• Annualization. Risk and volatility figures now annualize by each price grid’s own bars-per-year rather than a hardcoded 365 -- a stock trading ~252 sessions a year was previously having its volatility and risk ratios inflated by roughly a fifth.
• Causality. The multi-day timeframes (3-Day / 5-Day / Weekly) are rebuilt from daily bars under automated tests asserting that no resampled bar can ever contain information from the future -- the classic look-ahead leak, guarded permanently.
• Benchmark symmetry. Fees, idle-cash yield, funding schedules, and deployment fractions are asserted identical across Sigma and every benchmark (Section 09).
• Valuation edges. Dead assets stop accruing contributions at their final bar rather than silently banking budget past their death; per-timeframe equity accounts mark to market on the daily grid rather than freezing between coarse bars; session-close conventions were aligned across mixed crypto/equity portfolios.

The full results set behind this paper was regenerated from scratch on the audited engine -- the significance nulls re-drawn, the robustness grids re-swept, the survivorship ledger re-run. Reproducibility is mechanical: every figure traces to a line of code via the calculation dictionary, every number is in the structured results file, and the build aborts if the rendered text violates the disclosure rules it ships with.

14Limitations & methodology

We list the caveats plainly; a method’s credibility is inseparable from the honesty of its disclosures.

• The per-cycle multiples are hindsight (this paper’s biggest caveat). The per-cycle results we lead with (Sections 04 and 06 — the cycle tables and the annualized XIRR view) value each cycle at its top, located with a 2-year-forward top detector — i.e. they assume you sold near the peak. They are an upper-bound “if-you’d-sold-near-the-peak” benchmark, not a live result, and we label them as such wherever they appear. The full-history cost-basis, significance, drawdown, and realized-cash figures (Sections 05, 07, and 10), by contrast, are causal: they use only information available at the time, are held to the most recent bar, and use no forward look.
• Per-cycle win-rate is conditional, not universal. Sigma accumulation beat DCA in 21 of 30 cycles — dominant where genuine deep discounts exist (BTC, ETH, XRP, AAPL), mixed where they do not (NVDA, S&P). In the losing cycles Sigma paid a higher average cost than DCA because the cheapest window was an un-flaggable launch or steady grind.
• It needs well-timed discounts to work. On a parabolic asset whose deepest discount readings only register after the cheapest early window has passed, Sigma stays fully deployed but buys at a higher average cost than steady DCA and underperforms. The method is a discount-capture strategy, not a momentum-chasing one.
• A launch-era asset was tested and excluded from the headline set. Solana’s listed history is essentially one un-flaggable 2020 launch plus a single cycle; on it Sigma deployed fully but bought at a higher average cost than DCA (worse than ~99.5% of random timing, p = 0.995). We exclude it from the headline set because a one-launch history distorts every cross-asset chart, and we disclose the exclusion here precisely so it cannot read as survivorship curation: the full Solana run remains in the structured results file, and the boundary condition it demonstrates — no genuine discounts, no edge — is stated wherever the edge is claimed.
• No protection against total collapse. Accumulate-and-hold goes to roughly -100% on a wipeout; the rotation can but does not always pull cash out first.
• Single price feed. Results use a single historical price source (yfinance). Near a borderline discount reading, a different feed (such as a live composite index) can flip individual signals, producing small differences from a live deployment.
• Minor annualization conventions. Post-audit, the product’s tearsheets annualize by each price grid’s own bars-per-year; this paper’s drawdown-adjusted return (Calmar) keeps a calendar-day convention for consistency with the money-weighted figures.² Footnote-level; it does not move the cost-basis or significance conclusions.
• Research metrics vs product metrics. The drawdown channel in this paper’s alpha decomposition uses the per-position underwater-vs-cost diagnostic (chosen for per-asset statistical comparability); the product reports account-level lenses -- money-in for accumulation, peak-to-trough for the swing rotation (Section 10). The product numbers therefore read shallower than the research diagnostic. This is a difference of question, not of ledger -- all are computed from the same trades.
• Largest discounts came earlier. The deepest discounts in each asset’s history tend to sit in its early, more volatile years; the most recent sub-periods show a thinner edge. Future opportunity depends on future volatility offering genuine discounts.
• Liquidity and capacity are not modelled at scale. The transaction-cost sweep covers 5-50 bps on BTC, ETH, and AAPL only. The survivorship / dead-coin rotation assumes fills at the daily close on small-cap, impaired names (SRM, MIR, ANC, CEL) that could not in practice be transacted at scale -- real spreads and slippage would erode the realized cash the rotation shows on those names. Treat the rotation’s collapse-era cash figures as an upper bound, not an executable result.
• Symmetric idle-cash yield can push lifetime deployment past 100%. Idle cash earns a money-market-style yield while it waits, and that yield is reinvested, so cumulative dollars deployed can slightly exceed the starting budget. This is applied identically to DCA, so it does not advantage Sigma -- both methods earn the same yield on the same idle cash.
• Per-asset budgets differ. Budgets are $100k for BTC / AAPL / NVDA / S&P, $50k for ETH, and $10k for XRP. Every comparison holds capital fixed within each asset; terminal multiples are scale-invariant, so cross-asset multiples remain comparable.

15Conclusion

A fixed, simple rule that deploys capital into deep discounts — and banks gains into premiums — produces a real, conditional edge whose engine is a lower average cost basis. Cycle by cycle, Sigma beat dollar-cost averaging in 21 of 30 market cycles across the headline set, valued at each cycle’s top (a hindsight benchmark). The mechanism is clean and not inflated: where a genuine deep discount exists, Sigma pays less per unit, buys more units, and earns a higher multiple and a higher annualized return. That cost-basis edge clears a 1,000-run random-timing null with p ≤ 0.002 on 6 of 6 assets, sitting at the cost-efficiency ceiling, and it survives parameters, costs, and sub-period. It is not a free lunch: it needs real, well-timed discounts (it loses where the cheapest window is an un-flaggable launch or secular grind — launch-era coins, and early NVIDIA / S&P cycles), it offers no shield against total collapse, and its drawdown benefit is asset-dependent. The rule ships as two accounts with two honest yardsticks — a long-term accumulator measured against the money put in, and a swing rotation measured against its own high-water mark — every benchmark handicap-free by construction, and every figure regenerated on an adversarially audited engine. Stated precisely and without overreach — and without leaning on inflated whole-chart terminal numbers — the result is a credible, repeatable allocation edge: buying the discount, and getting paid to wait.

16Appendix & references

Supporting materials, kept separate from this paper’s causal headline figures:

• Exhibit report -- per-asset accumulation and rotation charts, including the retrospective cycle-top framing: product_report/bitcoin_report.html.
• Calculation dictionary -- every reported figure traced to the line of code that computes it: CALCULATION_DICTIONARY.md.
• Structured results -- the full machine-readable result set behind every number and chart in this paper: results.json.