# The Bitcoin Decaying Sine Wave, the Pi Cycle High, and a PlanB S2F model revision

*Disclaimer: Nothing contained in this article should be considered as investment or trading advice.*

**Introduction**

This article shares hypotheses on intra-cycle price action and applies these to Plan B’s stock-to-flow model price predictions for the 2020 halving cycle. Also, if correct, the intra-cycle price action hypotheses suggest slight modifications to the method for calculating the stock-to-flow model merit investigation.

**Background**

Many bitcoin on-chain metrics take a continuous view of price development over time, for example moving averages or DMA multiples (see the golden ratio multiplier), and to a certain extent ignore the phase transitions that occur with a halving event. This is why Plan B’s recognition of the phase transitions (see Bitcoin Stock-to-Flow Cross Asset Model) is critical and allows us to review the data with a different lens.

If we restrict our view to intra-cycle price action as Bitcoin goes through price discovery immediately after halving to find its new equilibrium, it resembles the pattern of a decaying (co)sine wave with a series of waves reducing in amplitude as time progresses, stabilizing to a theoretic amplitude of zero. In practice this does not happen of course. The hypothesis is that the energy from the halving event slowly dissipates into price increases, but overshoots and corrects, much like a harmonic oscillator displaced from its equilibrium position. Rekt capital has a good description of this penomenon in a practical sense in the article ‘Bitcoin’s Four cycle: an extensive analysis’.

**Methodology and observations**

To approximate the equilibrium position of the cycle we take the average of the daily BTC prices at ‘market close’, for all days in the cycle starting with the day of halving. The data used in the research of this article was downloaded from coinmetrics.io. In this article we refer to this as the ‘cycle average BTC price’.

A number of constants were identified from the first 2 full cycles, the first beginning in 2012, the second in 2016:

1. Each cycle high is *Pi* (3.14) times the cycle average BTC price.

2. Each cycle low is a ~40% retracement from the average, and thus approximately ~.6 times the cycle average BTC price. This leads to a consistent ~80% retracement from top to bottom.

Granted, all these constants have 2 data points. One swallow does not a summer make. Neither do two. We’ll see whether this cycle invalidates the hypotheses.

**Application for price prediction**

One of the most well-known models to date is Plan B’s S2F cross-asset model, which predicts a price of $288K for this cycle. To err on the conservative side, for price prediction we will use the $288K as the cycle high and apply the constants to arrive at a cycle average btc price of $92K (288/3.14) and a low of $55K (92*.6).

There is another reason to use Plan B’s price prediction of $288K for this cycle as the high, and not the cycle average BTC price. In his S2F cross-asset model he uses the monthly data points and applies an algorithm (minimizing absolute distance) to quantify cycle clusters. If the hypothesis of the decaying sine wave is correct, in future research I invite Plan B to use the monthly data points from the latter half of the cycle when prices have stabilized around the cycle equilibrium. My suspicion, not founded in any form of mathematical modeling as yet, is that the revised S2F model will arrive at a lower market cap for Bitcoin for this cycle.

**Conclusion**

In this article I share the hypothesis that in each cycle the halving event causes Bitcoin to find its next equilibrium price in a pattern similar to a decaying sine wave. The cycle equilibrium price is approximated by averaging all daily prices in the cycle.

In both full halving cycles -starting in 2012 and 2016 — the cycle high was equal to *Pi* times the cycle average price, and the low was equal to ~.6 times the cycle average price.

Applying these observations to Plan B’s S2F price prediction hypothesizes that for the 2020 halving cycle the -very conservative- high will be $288K, the low $55K and the average $92K.

My apologies if I inadvertently shared observations of intra-cycle price dynamics that have already been published elsewhere. The internet is vast.

I invite you to like, comment, retweet, and heckle on twitter, and in particular to disprove my hypotheses.

C.B.Ferdinand — @CBFerdinand1

**References**

PlanB@100trillionUSD, *Modeling Bitcoin Value with Scarcity*, March 2019

PlanB@100trillionUSD, *Modeling Bitcoin Value with Scarcity*, April 2020

Philip Swift, The Golden Ratio Multiplier, June 2019

Rekt Capital, Bitcoin’s Four Cycle: an Extensive Analysis

Coinmetrics.io