**Cryptoasset Valuation: Introducing ***Beta* of Velocity

*Beta*of Velocity

*Note: This article is complementary to “**Considerations regarding “On Value, Velocity and Monetary Theory**.” The two pieces are best understood when read together.*

Written by Johnny Antos and Justin Shin

*Blockchain Advisory Group (BAG) provides technical and principled strategic advisory services to high-quality crypto core teams and traditional private and public corporate management teams.*

Since the inception of bitcoin ~10 years ago, many have considered cryptoassets as a form of money. The most popular formulations of “cryptoassets-as-money” use the Equation of Exchange or variations of it.

When limiting the scope of analysis strictly to the cryptoasset-as-money realm, by definition, MV=PQ. Many others have written about this, most notably Burniske, Evans, and Buterin. It’s helpful to be familiar with those pieces first before reading our thoughts below.

There has been much debate whether, at steady-state equilibrium, growth in cryptoasset Velocity (V) could outpace growth in cryptoasset GDP (PQ), which could reduce the Monetary Base (M), and cause a decline in token Price (P). However, we argue that it is incorrect to discuss “the velocity problem” in terms of magnitudes, and not in terms of rates of change. **Additionally, we believe that correlation is ineffective at capturing the relative magnitude of changes in PQ and V.**

Before we explore the issues associated with this way of thinking, it’s worth explicitly stating the underlying assumptions of using MV=PQ as a tool for contemplating future cryptoasset price.

**Table of Contents**

- Assumptions of MV=PQ in cryptoasset valuation
*Beta*of velocity: why is it useful?- Historical
*beta*of velocity - Appendix

**Assumptions of MV=PQ in cryptoasset valuation**

Cryptoasset valuation methods (unlike traditional monetarism) first focused on M as the dependent variable and PQ as the independent variable. Burniske’s initial work set V as a parameter (fixed value).

Because Evans uses average M (“Average VOLT Balance Held in U.S. Dollars”), not M at any given moment, the reasoning in his model is a bit different. The “VOLT Model” sets PQ as the independent variable, derives average M based on this PQ, and then derives V endogenously.

, where

Then,

, so

and

Evans acknowledges that it is **the general relationship between PQ and V** (relative to each other) that is informative when thinking about forecasting token price movement over time.

As it seems that most investors implicitly think in terms of the simple valuation methodology, MV=PQ (as per Burniske), and not in terms of Baumol-Tobin, the methodology we consider is:

- First set M aside completely, V = dependent variable, PQ = independent variable. Determine V based on PQ.
- Then bring in M: M = dependent variable, (PQ/V) = independent variable. This solves for the implied monetary base
*at a given moment in time*.

At a high level, the real question is, “Does this approach make sense?” If only thinking of cryptoassets as money, in a Medium of Exchange (MoE) capacity, this model has some merit.

- At any given moment in time, MV=PQ is true by definition. Thus, we are not as concerned with the specific magnitude of the variables. Instead, we are only interested in the interactions between changes in the variables themselves
- Do changes in PQ drive changes in V, or do changes in V drive changes in PQ? Which way does causality go?

**It is our view that changes in PQ drive changes in V**. Crypto core teams (and ultimately an ecosystem of stakeholders) determine what attributes and incentives support the specific use cases they want a token to provision. It would be nonsensical to arbitrarily “choose” a certain velocity, and then use that to determine PQ. In our view, crypto core teams *“choose”* PQ by determining which existing target market to disrupt (or in some cases, create innovative attributes and use cases that create entirely new markets). As an example, when the Filecoin team chose to attack the cloud storage market, they essentially chose a certain market size to go after, and consequently, the aggregate demand for the goods being provisioned, PQ.

**As a result, the magnitude of velocity is highly sensitive to the specific PQ chosen.** For example, for a Store of Value (SoV) use case, the magnitude of V is likely low, which is what many believe to be driving bitcoin price appreciation.

**As a corollary, changes in PQ (e.g., growing the network by adding more active users to the cryptoasset network, or existing users increasing their usage) are the driving factor behind how velocity changes over time**. If many people held bitcoin as a SoV in the early days, but 100% of new users exclusively use it for payments, it’s obvious that this specific increase in PQ causes a corresponding increase in V (the degree of this increase would be determined by the “**beta of velocity”**, which we define below). At the same time, complex interplay can exist between growth in PQ and prior existing PQ. (i.e., It is possible that the way new users are using a cryptoasset may influence the specific ways that existing users derive utility from the cryptoasset.)

Lastly, **M is determined by the interplay between the relative growth rates of PQ and V.**

In this conception, M isn’t chosen by a core team. Core teams choose the specific attributes and use cases that their token provisions. This effectively determines the PQ of the network (i.e., capturing a certain percent of the market, following a particular adoption curve). In turn, that “chosen” PQ determines the velocity because specific user holding or usage behavior is inseparably embedded in the chosen PQ. Thus, any change in velocity is determined either by existing users changing their usage behavior, or by new users with different behavior patterns, and more than likely, some combination of both of these effects. Through this process, M, being the required size of the monetary base, constantly fluctuates at every moment depending on the relative changes in the growth rates of PQ and V, or the quantity (PQ/V).

This seems to be the most robust justification for using the basic formulation of MV=PQ in cryptoasset valuation.

**We believe that reducing cryptoasset ecosystems to simple monetary economies obscures many other new possibilities that are enabled by cryptoassets**. We are currently exploring frameworks beyond the simple “cryptoassets-as-money” formulation.

**Beta of velocity: why is it useful?**

In his exploration of the velocity thesis, Evans writes:

“The real question is how

changes in velocity correlate with changes in PQ.Strong positive correlations approaching 1 effectively decouple token value from network transaction growth (note that while this is a drag on the upside, it is protective of value on the downside). If the two are uncorrelated, then token utility value grows (and declines) linearly with demand for the underlying utility (this is what happens in the INET model).Therefore, we need to compare the growth rates of velocity and PQ to formalize the velocity thesis: The thesis states that if velocity grows faster than PQ, token utility value declines.”(emphasis our own)

Evans acknowledges that what matters is the rate of growth in PQ relative to the rate of growth in V. This is intuitive in the “instantaneous” MV=PQ formulation:

- If PQ grows faster than V grows, then M increases (and token value in USD increases, all else equal)
- If V grows faster than PQ grows, then M decreases (and token value in USD decreases, all else equal)

**However, ***correlation *alone is insufficient to describe the relationship between V and PQ.

*correlation*alone is insufficient to describe the relationship between V and PQ.

To observe the mechanics clearly, consider two hypothetical economies, both with the same GDP growth over time.

The left-hand table shows an economy in which velocity is growing at the same rate as GDP. As a result, the monetary base stays constant over time. The correlation between PQ and V is perfect at 1.

The right-hand table shows an economy with the same GDP growth but with velocity growing at double the GDP growth rate — 10%. As velocity outpaces GDP growth, the monetary base declines over time, as does token value. One can easily see that two economies with the same GDP profile can have different monetary bases, depending on the growth rate of velocity relative to the growth rate of GDP.

Notice that the correlation between PQ and V are both exactly 1 (or ~1). Correlation will always be extremely high for a series of values that share the same directional movement.

This is because correlation only measures whether increasing values in one variable tend to correspond to increasing values in the other variable. In other words, **it fails to capture the magnitude of the corresponding changes**.

We argue that it is neither the ** correlation **between the magnitudes of GDP and velocity nor the

*correlation**of their rates of change that matters when evaluating the impact on token utility value. Knowing only the correlation between PQ and V is inadequate because it fails to provide useful information about the magnitude of the relative, simultaneous volatility between PQ and V. Covariance alone also fails to provide adequate insight, as it does not take into account the variance of the independent variable.*

We can borrow a simple concept from classic portfolio construction (and basic linear regression) to make our point more clear: ** beta**.

*Note: While the formula is analogous to the traditional beta of a stock, this does not refer to the traditional definition of **beta**, which compares the volatility of a stock’s return to the return of the market portfolio.*

In the realm of cryptoassets-as-money (MV=PQ), changes in M are determined by:

Where:

In Burniske’s INET model, setting V as a fixed parameter is equivalent to setting *beta* of velocity = 0. That is, for a given increase or decrease in PQ, there is no change in V.

Evans’ VOLT model is different than the popular conception of MV=PQ, due to its different underlying assumptions, as noted above. In the VOLT model:

For Burniske and for our “instantaneous” conception, M is the total monetary base required to support PQ and V at any given moment, and we are interested in how relative changes in PQ and V drive changes in M from one point in time to another, so we are interested in how MV=PQ shifts from one point in time to the next.

Again, remember that for Evans, average M = “average VOLT balance held in USD at equilibrium. Notice that in the VOLT model, **M is not the money supply required to support PQ and V at any given moment**. Instead, the M that Evans refers to is the **average** monetary base over each 12-month period. Taking into account simultaneous, instantaneous demand is something that Sall attempts to do, though we do not believe that queuing theory is the best fit for assets like bitcoin, where human decision-making transactions drive PQ.

Both conceptions of M (average and instantaneous) are useful when thinking about MV=PQ, as both provide different types of information. However, if we are more interested in considering the causal effects of a changing PQ on a changing V (and the net effect on M at any given moment), the instantaneous conception better allows us to think about the dynamic effects between discrete points in time.

Here is a basic sign chart that might help with interpretation. For example, starting in the far right column *beta* > 1, we see that when PQ increases, V increases and the net effect on M is a decrease. This is true by definition, as *beta* > 1.

*Historical beta of velocity for bitcoin*

*Historical beta of velocity for bitcoin*

We fully acknowledge that the V in this historical dataset is a calculated metric (simply equal to observed PQ / observed M). Therefore, the result of mapping historical changes of PQ against changes of V with *beta *is simply a different representation (perfectly following the movements) of the changes in bitcoin’s historical market cap over time.

Because we are using historical M (market cap of bitcoin) and historical PQ (estimated USD transaction value), we are calculating historical V. Thus, the purpose of showing the historical chart is to demonstrate that *beta* provides us with the relevant information for forecasting changes in M. It is useful to remember that when trying to make future projections, the math is opposite of the historical charts: M is predicted (not observed), and velocity and PQ are outputs.

**We believe beta serves the purpose of bringing some clarity to the discourse about “the velocity problem.” Instead of making vague claims about potential future “high / low velocity” or “GDP outpacing velocity”, the conversation should be in terms of whether a specific cryptoasset’s velocity beta might be <1, =1, or >1 and how that beta might evolve over time.**

It seems more useful to think in terms of *beta* and to consider future *beta* as opposed to speaking in terms of static magnitudes, because what investors usually care about most is the expected change in token price, not static values. *Beta* directly addresses the relative, simultaneous changes of PQ and V that drive the increase or decrease of token prices. Our hope is that this type of thinking allows core teams, and their advisors, to perform honest reflection on what type of velocity *beta* their specific use case (target market) might have.

More specific language allows those analyzing crypto ecosystems to standardize how we discuss and experimentally test the “velocity thesis,” and hopefully accelerates the development of even more robust metrics for modeling this new class of cryptoassets.

*Blockchain Advisory Group (BAG) provides technical and principled strategic advisory services to high-quality crypto core teams and traditional private and public corporate management teams across the organizational lifecycle, from initial concept to liquidity events.*

*Appendix*

*Appendix*

The usual caveats around the empirical historical data apply: Historical velocity *beta* over some time period is not predictive of changes in future velocity, especially when the specific utility / use-case of a cryptoasset shifts over time. In fact, until solutions are found to better measure PQ and M with a higher degree of accuracy, using past velocity *beta* is akin to momentum trading based on bitcoin market cap / price movements.

Others have written about the many methodological issues of accurately measuring historical velocity, namely:

**For PQ (a cryptoasset’s GDP)**:

- Should you include all transactions, or exclude trading / exchange volume?
- If removing certain transactions due to the nature of those transactions, how confident are we that we have adjusted appropriately?

**For M (a cryptoasset’s monetary base)**:

- How should market capitalization be adjusted to ensure that PQ and M are on an apples-to-apples basis? If removing certain transactions from PQ, then how could you determine which coins to remove from M?
**Note:**For our graphs, we use “on-chain transaction volume” for PQ, but total market cap for M. This is in-line with the approach of Woo and other approaches, even with its noted flaws.

The following works mention some existing problems with the measurement of these variables:

- A Fistful of Bitcoins: Characterizing Payments Among

Men With No Names, Meiklejohn et al. - On The Difficulty of Estimating On-chain Transaction Volume, Coinmetrics
- BlockSci: Design and Applications of a Blockchain Analysis Platform, Kalodner et al.
- https://blog.chainalysis.com/reports/money-supply

Lastly, for reference, you can see how the *beta* of velocity for bitcoin is smoothed when it is calculated over longer rolling timeframes.