Backward-Looking Floor Growth

The Bitcoin Floor Rate (BFR) is the Observatory’s primary planning metric for retirement tools, mortgage products, and floor bond pricing. It is forward-looking and model-dependent: it requires the power law to hold. We introduce two complementary backward-looking metrics that require no model at all.

First, the observed annual minimum price has grown at a trailing 5-year average of 34.3%, with 86% of complete years showing positive growth. Second, a scale-invariant metric — measuring minimum-price growth in trailing windows defined as a percentage of Bitcoin’s age — produces even stronger results: at a 15%-of-age window, 98% of observations exceed a 3.5% mortgage rate, with a worst case of −4.4%. At 20%-of-age, the record is 100% above 3.5%, worst case +10.7%.

The forward BFR (37.2%) and the trailing 5-year backward metric (34.3%) are consistent to within 3 percentage points — two different methodologies arriving at the same neighborhood. This paper is the companion to Paper 12. Paper 12 asks: “is our model correct?” This paper asks: “what if it is not?”

1. The Epistemic Gap

Every Observatory product depends on the Bitcoin Floor Rate. The BFR is computed as BFR(d) = ((d + 365) / d)^5.688 − 1, where d is days since genesis. It is a deterministic formula derived from the power law model. If the model is correct, the BFR is the floor’s growth rate. If the model is wrong, the BFR is meaningless.

This creates an epistemic gap. The Observatory’s retirement calculators, the Floor Bond coupon pricing, the Bitcoin Mortgage Stage 3 trigger, and the 1.6× loan safety rule all depend on the BFR. A skeptic who rejects the power law rejects all downstream products simultaneously.

1.1 What formal verification found

The formal verification paper (Paper 12) tested six core claims using two independent methods. Three findings are directly relevant:

Out-of-sample R² = 0.546. The model explains approximately half the variance in prices it has not seen. This is genuine predictive content but not the 0.956 headline number. The gap means approximately 45% of future price variance is unexplained.

Effective sample size = 24. The 5,735 daily observations carry the statistical information of approximately 24 independent draws. The HAC 95% confidence interval for beta spans [5.538, 5.850], translating to a BFR range of approximately 34–41%.

Floor multiplier is definition-dependent. Four floor definitions produce values from 0.314× to 0.480× trend, a range that translates to a €25,000 spread in today’s floor price.

1.2 The model-free alternative

This paper introduces a genuinely model-free alternative: use the observed data directly. How fast has the worst-case Bitcoin price actually grown? No formula. No parameters. No regression. Just the historical record. If this backward-looking metric produces a planning-useful growth rate, then the Observatory’s products can function even for users who reject the power law entirely.

2. The Backward-Looking Floor Growth Metric

For each calendar year from 2011 to 2026, we identify the minimum daily closing price. The year-over-year growth of this annual minimum is the backward-looking floor growth rate. It measures how fast the worst-case price in each year has grown compared to the worst-case price in the previous year.

This metric is:

Model-free. It requires no power law, no regression, no parameters. It is computed directly from observed prices.

Conservative. It uses the annual minimum, not the average or the close. Every year’s data point is the single worst day of that year.

Backward-looking. It measures what has already happened, not what a model predicts will happen.

Figure 1: Annual minimum price vs power law floor (0.432×) on log scale. The two series converge over time.

* 2026 is a partial year (January–March only). Summary statistics use 2012–2025 (14 complete years) as the primary sample.

Two complete years show negative annual minimum growth: 2015 (−43%) and 2022 (−46%). Both correspond to bear market bottoms (cycle 2 and cycle 4). Of 14 complete years, 12 (86%) show positive minimum growth, with a median of 110%.

Figure 2: Observed annual minimum YoY growth (bars) vs forward BFR (blue line). Green = positive, red = negative. 3.5% mortgage rate shown as dashed line.

3. Cycle-Bottom-to-Bottom Growth

The annual minimum YoY metric is noisy because it captures cycle timing effects. A more stable metric: the growth rate from one cycle’s absolute bottom to the next cycle’s absolute bottom.

Figure 3: Cycle-bottom-to-bottom CAGR is declining (397% → 95% → 54%) but remains well above the 3.5% mortgage rate and the forward BFR (37.2%).

The CAGRs are declining: 397%, 95%, 54%. This is consistent with the power law’s decelerating growth — each cycle’s floor grows by a smaller percentage than the previous. The most recent complete transition (C3 to C4) produced a 54% CAGR. Even if the next transition halves to 27%, it still exceeds a 3.5% mortgage rate by 7.7×.

The declining trend is the honest disclosure. n=3 cycle-bottom transitions does not support formal trend testing. The pattern is descriptive, not statistically established. The backward-looking metric is not a perpetual 54% or 34% — it is a rate that has been decelerating across cycles, consistent with the forward BFR’s structural deceleration.

4. Scale-Invariant Floor Growth

The calendar-year metric imposes an arbitrary human boundary onto a system that the Observatory models as scale-invariant. The halving-epoch metric uses protocol-defined boundaries but yields only n=3. A continuous generalization: measure the minimum price in a trailing window defined as a percentage of Bitcoin’s total age.

At any observation date d (days since genesis), the trailing X% window spans the most recent X% of Bitcoin’s entire history. As Bitcoin ages, the window grows in absolute terms but stays constant as a fraction of the asset’s lifetime. This respects the logarithmic time scale of the power law.

4.1 The metric

For each observation date, compute the minimum closing price in the trailing X%-of-age window. Compare it to the minimum in the immediately preceding X%-of-age window. Annualize the growth rate.

Figure 4: As window size increases, the worst case improves dramatically. At 20% of age, every observation exceeds the 3.5% mortgage rate.

The 15–20% range is the practical sweet spot for retirement and mortgage planning. It is long enough to smooth cycle noise (each window spans more than half a halving cycle) but short enough to be responsive to structural changes. At 15%, 98% of observations exceed a 3.5% mortgage rate. At 20%, the record is 100%.

Figure 5: Time series of scale-invariant floor growth at 10%, 15%, 20%, and 25% of Bitcoin’s age. Backward metric (green) oscillates around the forward BFR (red dashed).

4.2 Convergence with forward BFR

Figure 6: The 15%-of-age backward metric (green) and forward BFR (red dashed) occupy the same 30–80% range. The shaded area shows the spread between them.

Both metrics occupy the 30–80% range throughout 2021–2026. The backward metric is more volatile (it responds to recent price history) while the forward BFR decelerates smoothly. Their trajectories cross and recross depending on where we are in the cycle. What matters for planning is that both are comfortably above mortgage rates throughout.

5. Three Methods, One Neighborhood

Three independent methods produce floor growth estimates in the same range. None shares its methodology with the others. All three are above any realistic mortgage rate:

Figure 7: All backward metrics (green) exceed the forward BFR (red). The trailing 5-year average is the most conservative, differing by just 2.9pp from the model.

The forward BFR (37.2%) and the trailing 5-year backward metric (34.3%) differ by 2.9 percentage points. The 5-year window is highlighted because it is the most conservative multi-year average, incorporating two bear market bottoms (2022 and 2026 partial). It is the backward metric most suitable for conservative financial planning.

The proximity of the two methods is consistency, not confirmation. Both are derived from the same underlying price series. However, the backward metric uses only 15 data points from a 5,735-observation dataset. The overlap is minimal. A skeptic who rejects the power law formula but accepts historical price data arrives at approximately the same planning input (34% vs 37%).

6. Distribution of Annual Minimum Growth

The distribution of annual minimum growth rates (n=15 years, 2012–2026) is heavily right-skewed. The mean (333%) is much higher than the median (102%) because a few blow-off years produce extreme positive growth (2014: +2,233%, 2021: +489%). The left tail is bounded: the worst year was −46% (2022).

Figure 8: Histogram of annual minimum growth. Median 95%. Three negative years (left of dashed red line). Heavily right-skewed.

6.1 Threshold analysis

For retirement and mortgage planning, the relevant question is not the average growth rate but how often the growth exceeds a given threshold:

86% of complete years (2012–2025) show annual minimum growth exceeding 3.5%. The two years that failed (2015 and 2022) are both bear market bottoms. No bull market or recovery year has failed. The caveat: failures cluster in bear markets, which is precisely when the mortgage holder needs the floor growth most. The failures are not random — they are systematically correlated with stress periods.

7. Product Implementation

The backward-looking metric becomes a toggle on every Observatory planning tool:

7.1 Three-mode retirement calculator

Model mode (forward BFR): Uses the power law formula. BFR = 37.2% today, decelerating. Primary mode for users who accept the power law. Projections extend 20–40 years.

Conservative mode (calendar backward): Uses the trailing 5-year average of annual minimum growth. Currently 34.3%. No model required. Updated annually. Projections limited to 5–10 years with explicit uncertainty.

Scale-invariant mode: Uses the 15%-of-age window annualized floor growth. Currently ~34%. Scale-invariant by construction. Recommended for users who want a model-free metric that respects Bitcoin’s logarithmic time structure. Worst observed value: −4.4%.

Bear market caveat: None of these backward metrics eliminate the need for a cash reserve during drawdowns. The annual minimum metric shows −46% in the worst single year. A mortgage holder or retiree must maintain a reserve fund covering 2 years of expenses, as specified in the Floor Bond and Bitcoin Mortgage papers.

7.2 Mortgage planning

The Stage 3 trigger (floor growth > mortgage interest) is currently computed from the forward BFR. In conservative mode, it uses the trailing 5-year average: 34.3% × floor value. At 1 BTC with a floor of €47,883, this produces €16,424 annual floor growth vs €12,250 mortgage interest — a 1.3× margin. Lower than the forward BFR margin (1.5×) but still positive.

7.3 Floor Bond pricing

The actuarial coupon derivation uses the forward BFR with a 30% model uncertainty haircut. The backward metric can replace or supplement this: trailing 5-year average (34.3%) with a 30% haircut produces 24.0%, versus 26.0% from the forward BFR. The senior coupon of 7.2% is 3.3× below the backward-derived haircut rate.

8. Limitations

Small sample size. 15 years of annual minimums produce 15 data points for the YoY metric and only 3 complete cycle-bottom transitions. The summary statistics are descriptive, not inferential.

Declining trend. The cycle-bottom CAGRs are declining (397%, 95%, 54%). The backward-looking metric is itself decelerating. Using the most recent value as a future input implicitly assumes the deceleration rate does not accelerate.

Bear market timing. The annual minimum metric is sensitive to where bear market bottoms fall relative to calendar year boundaries. The cycle-bottom CAGR is less sensitive to this effect.

Not a prediction. The backward-looking metric describes what has happened, not what will happen. It is a planning input, not a forecast. The observed 34% trailing growth could fall to zero or turn negative in any future year.

Survivorship. The metric only exists because Bitcoin has survived and appreciated. This is not circular reasoning but it is a limitation: the metric is conditioned on Bitcoin’s continued existence.

9. Conclusion

The Observatory’s forward-looking BFR (37.2%) and the backward-looking trailing 5-year average of annual minimum growth (34.3%) are consistent to within 3 percentage points. The two methods share minimal data overlap. The consistency does not validate either method. It means both point to the same planning neighborhood.

For institutional counterparties who do not accept the power law model, the backward-looking metric provides an alternative planning input that requires no model assumption. The observed annual minimum price has grown in 86% of complete years. The trailing 5-year average is 34.3%. The worst single year was −46%. Even the P25 (conservative quartile) is +5.5%, above the 3.5% mortgage rate.

The backward-looking metric does not replace the forward BFR. It complements it. The forward BFR is more precise (a deterministic formula). The backward metric is more defensible (no model required). For retirement calculators and mortgage products, offering both as a toggle — model mode and conservative mode — gives every user a planning input they can trust, regardless of their view on the power law.

This paper is the companion to Paper 12. Together they form the Observatory’s credibility package. Paper 12 says: we tested our own model, found three problems, and fixed them. This paper says: and even if you do not believe the model at all, the observed data produces approximately the same planning input. The model is useful. It is not required.