Bitcoin cycles from four observations

Six preregistered studies on bitcoin's price history since 2010 find that the four-year period is not separable from an autocorrelated null, that falling volatility accounts for shrinking drawdowns, that declining cycle multiples survive seven of eight denominators, and that peak projections differ by a factor of nine and a half across methods. Most of what four observations support concerns measurement rather than price.

Spectral periodicity

The dominant spectral period of detrended log price is 1,452 days (3.98 years), close enough to the halving interval that it is routinely read as confirming one. Against an AR(1) surrogate matched to the series' persistence the peak has p = 0.16; against a block-bootstrap surrogate, p = 0.73. Any trending, volatile, strongly autocorrelated series produces peaks of this size, and four cycles cannot separate a genuine oscillation from that background. The cycle templates below are therefore scenario arithmetic and carry no evidence of periodicity.

Drawdown depth and volatility

Claims that crashes are getting shallower usually rest on trough depths, which re-derived from price across the four cycles are −89.6%, −79.7%, −83.4% and −76.4%. The sequence is not monotone, since the 2017 trough was deeper than the 2013 trough; sequences that appear monotone typically omit the 2013 to 2015 cycle. A permutation test on the ordering gives p = 0.125 against an attainable floor of 0.042, so the declining trend is not established at conventional significance, which on four observations only a monotone ordering could have reached.

Annualised volatility across the same cycles fell 73%, from 2.46 to 0.66, while depth fell about 15%. Log drawdown per unit of volatility ran −0.92, −1.42, −2.12 and −2.18, more than doubling monotonically. Volatility therefore accounts for more than the entire shrinkage, and risk-adjusted drawdowns have deepened.

Cycle multiples under eight denominators

Successive cycle tops rose 35.06×, 17.19×, 3.47× and 1.85× over the prior top. The standard explanations of the decline, namely the growing base, the monetary environment and maturation, are claims about the correct denominator. Eight denominators were tested against a flattening criterion fixed in advance: a denominator explains the decline if it reduces the slope of the log-multiple sequence by at least half.

Denominator	Slope vs nominal	Explains?
Price per active address	15%	Yes
Price per stock-to-flow	83%	No
Price per unit hashrate	94%	No
Nominal, baseline	100%	—
Annual issuance value	100%	No
Consumer-price deflated	104%	No
Market value over broad money	115%	No
Market capitalisation	117%	No

Deflating by consumer prices moves the slope by 4%, which makes currency debasement a level effect rather than a cycle effect. Normalising by broad money or by market capitalisation steepens the decline, so neither liquidity nor base growth removes it. Only price per active on-chain address flattens the sequence, and active addresses have fallen since 2017, from roughly 958,000 to 595,000, over a period in which custody centralised and exchange-traded holdings generate no on-chain activity. A mechanically shrinking denominator flattens almost any numerator, so that result is at least as consistent with proxy degradation as with economics.

Flow-balance benchmark

Because each halving cuts issuance exactly in half, a flow-balance accounting benchmark with constant dollar demand implies a price response of 2.00×. Dividing the observed multiples by 2.00 gives the demand contribution net of supply mechanics: 17.53, 8.60, 1.73 and 0.92. The 2021 to 2025 cycle is the first whose price gain fell short of what the supply reduction alone implies. The dollar value of annual issuance peaked at $22.2 billion in 2021 and fell to $20.5 billion in 2025, so sell pressure from new supply has stopped growing.

Flow balance is an accounting identity rather than an economic law, and the stock-to-flow model built on it has failed out of sample. The arithmetic above uses the halving only as a fixed reference for measuring the residual and does not depend on that model.

Interest rates and inflation

Mean effective fed funds over the recent cycle was 3.88%, against 0.12%, 0.39% and 1.15% in the three prior cycles, and mean consumer inflation was 4.64%, against 2.16%, 1.29% and 2.32%, which invites the inference that tighter policy suppressed the return.

A direct rate adjustment does not support it. Net of the cash return forgone by compounding the policy rate daily across each cycle, the multiples are 34.96, 16.93, 3.31 and 1.59. Because the hurdle rose, the adjustment reduces the most recent multiple by 14.1% rather than rescuing it. Rank correlation between mean policy rate and log cycle multiple is −1.000, the strongest attainable and still uninformative: the Pearson critical value at four observations is 0.950, which the observed −0.863 does not reach, and the cycle index correlates +0.905 with the policy rate, so controlling for it flips the partial correlation positive.

At monthly frequency (190 observations), in regressions on forward three-month log returns with Newey-West standard errors, the policy rate level gives t = −1.05, the real policy rate t = +1.57 and the twelve-month change in policy rates t = −0.29. Realised consumer inflation is the one macroeconomic variable that clears the threshold, at t = −3.15, and its sign is negative, opposite to the hedging argument: higher inflation has been followed by lower bitcoin returns over the subsequent quarter. The result survives a Bonferroni correction across the four predictors tested at that horizon; the correction was not specified in advance and is disclosed rather than claimed.

Trough and peak projections

For the cycle trough, ten methods (arithmetic drawdown on closes and on lows, log drawdown, recovery multiple, a bounded logit transform, a volatility-adjusted estimate, an analytic estimate scaled by the measured fat-tail excess, and realised-price anchors) span 1.43× from lowest to highest. Drawdown is bounded below by total loss, and bounded quantities transform consistently.

For the following peak, eight methods span 9.5×, and 16.18× including the form disqualified below. Price has no upper bound, so a multi-year multiplicative extrapolation inherits whatever its functional form assumes about adoption, and those assumptions differ by an order of magnitude. A criterion set before the results were seen disqualified one form: fitting log multiples linearly implies they decline without limit, which projects the asset to roughly a thousand dollars within two further cycles. That form was the one in general use, and a form that produces an absurdity two steps out should not be trusted one step out.

A cycle trough can therefore be characterised and the following peak cannot, the peak range being an order of magnitude less informative. Neither range is a forecast.

Public valuation models

The public models use independent variables that are monotone in time, which guarantees in-sample fit against a monotone price series and nothing beyond it.

Stock-to-flow regresses log market value on the ratio of supply to annual issuance; it reports very high in-sample correlation, its explanatory power is confounded by the log-time trend, and it has little or no out-of-sample predictive ability. The power law regresses log price on log time since inception and currently implies a fair value near $378,000 against a market price near $60,000. Recent work characterises it as having weak structure and comparatively good forecasts, and Broido and Clauset found genuine power-law structure in roughly 4% of about a thousand real-world networks, so the prior on any particular series exhibiting one is low. Metcalfe-type models value the network by the square of its participants and face the measurement problem above, since the participant proxy has fallen while value rose. Network-value-to-transactions and market-value-to-realised-value are relative indicators rather than valuations; in our cross-correlation the latter lags price by roughly 240 days. Cost-of-production models are undermined by the direction of causality, since mining difficulty adjusts to price rather than anchoring it, and price per unit of hashrate has fallen in every cycle.

Academic asset-pricing work has largely not produced fair-value models, asking instead which factors price the cross-section and the time series, and has adopted none of the models above.

Model-free results

Annual issuance is now 0.86%, against United States broad money growth of 6.04% and consumer inflation of 3.73%, and it fell below money growth in April 2024. That is a statement about dilution and carries no implication about price, which demand determines.

Across the whole tradeable history, the maximum leverage at which a position entered at any point would have avoided liquidation is 1.076×, set by the decline that followed the June 2011 peak. It is a property of the worst path rather than of the average return, and it requires no model.