Bitcoin versus Gold update

June 2024 update and Kelly Criterion

This is not investment advice.

Gold has been the asset historically considered for protection against inflation. Bitcoin is the new contender that has outperformed gold greatly for the past decade and a half. It is worthwhile comparing Bitcoin to gold since the gold supply grows at less than 2% per year, whereas for the US dollar, the average growth over long time periods is about 7% for the M2 money supply. And fiat is debt-based, not an asset.

A little over two years ago I wrote this article comparing Bitcoin’s price denominated in gold: https://stephenperrenod.substack.com/p/bitcoin-vs-gold-update. It looked at performance from Bitcoin age of 4.25 years to an age of 13 years.

I found that the data could be equally well described by a power law of Bitcoin’s age

\(Price = A * t^k\)

with an index of k = 5.08 (t is time since January 2009) or by an exponential of the form 10^(0.28 * t) that corresponds to 90% compound growth.

Bitcoin vs. gold is a power law

In this update I have added data from years 2 to 4 at the front end as well as from years 13 to 15.25 at the recent end. The chart of a power law regression is shown in Figure 1. It is clearly following a power law, an exponential line would be straight on this log-linear chart. The power law fit has an R-squared (R^2) value of 0.94 with an index of 5.48. This is close to, but somewhat less than the slope of 5.7 found by Plan G in his models for Bitcoin vs. the dollar, and essentially equal to the power law slope of 5.4 that I find with a regression performed in Block calendar time, also against the US dollar.

Figure 1. Price of Bitcoin measured in gold ounces on a log-linear chart with time measured since Bitcoin began in January 2009. The green line is the best fit power law regression P ~ t ^5.48.

An exponential fit attempt is inferior, with a lower R^2 of 0.85 and an F-test of 301, less than half of the F-test 844 value found for a power law fit.

It is possible that the power law is flattening, for the regression performed on quarterly data since year 8 the power law index is a noticeably flatter 4.12. We need to monitor how prices develop to determine how real this is, it could reflect an S-curve such as the Weibull curves I have evaluated for the Bitcoin vs. dollar price trend. It is best to see how this new Epoch 5 unfolds in this regard over the next almost 4 years. A Weibull distribution looks like a power law until it approaches the knee of the curve.

Bitcoin appears to be on a path to supplanting gold as the ultimate reserve asset. We are interested in understanding when Bitcoin’s market cap, now $1.4 trillion, will reach that of all gold held by governments, close to $3 trillion and all gold, private and public, with a market cap of roughly $15 trillion.

Z-scores are positively skewed

Here is a chart of the Z-scores, the number of standard deviations of the residuals (actual price in gold ounces vs. model price in log terms, normalized by the standard deviation). The standard deviation is 0.33 in log10, or a multiplicative factor of 2.14. So if fair value from the regression is 37.2 ounces as now, the plus or minus one standard deviation (68% probability interval) range is [17.4, 79.6] ounces. The present value of 31.8 ounces has a Z of -0.07, slightly below the trend.

Figure 2. Z-scores for Bitcoin vs. gold best power law fit. Note that the Z score is “right tail” skewed, that is, the lowest values are at -1.5 but the highest values are well above that, nearly 10% of the data points exceed Z of +1.5. However the median is Z = -0.32.

The distribution of Z-scores is qualitatively similar to what is seen in power law regressions against the US dollar, with a skew to high values but a median below zero. A lot of time is spent in forming bases from which the major bubbles grow with an approximate four yearly cycle time.

The maximum value is 2.72, the minimum is -1.46, and while more points are found at Z < 0 there is a significant right tail with 9 % of the data points above 1.5.

One also sees that volatility seems to be coming down, the standard deviation for the data since year 8 is a lower 0.27 in log10 terms.

Two-asset Sharpe ratio maximization

We can construct a two asset portfolio of Bitcoin and gold starting from January 2015 until now, using the Sharpe ratio maximization tool at portfoliovisualizer.com, the highest Sharpe ratio observed is for a mix of 36% Bitcoin and 64% gold.

Figure 3. Equity curve for a Sharpe ratio maximized portfolio per portfoliovisualizer.com, with 36% Bitcoin and the remainder in gold (GLD ETF).

Over the interval the expect annual return of gold is 8% in dollar terms, and for Bitcoin it is 86%. Gold has underperformed the S&P over the interval but has basically covered the increase in the M2 money supply.

With this allocation of about 1/3 Bitcoin and  2/3 gold, one is trading off the higher return of Bitcoin with its higher volatility.

Kelly criteria

The Kelly criterion argues for a much larger allocation to Bitcoin in a two asset portfolio with gold. If we use annual rebalancing for all years since 2011, a full Kelly allocation is 77% Bitcoin.

The gambler’s version of the Kelly formula allows for complete losses and takes the form:

f = p - q/b

Here f is the optimal Kelly fraction to grow wealth over time, and it is a function of the win rate p (intervals when Bitcoin outperforms gold), q = 1- p or the percentage of time when gold outperforms, and b = w / l , the average win percentage during winning intervals for Bitcoin divided by the average loss percentage.

For the years 2011 through 2024 (partial) p = 0.786 and b = 15.36 which is extremely high because of the extra rapid growth of Bitcoin in early years, and this gives us a full Kelly allocation of f = 0.77. If we restrict ourselves to the past 5 years, p = 0.833 remains high, but b is a more reasonable 1.65, and the full Kelly fraction is still high at f = 0.73.

The investment formula version of Kelly allows for partial losses and is written as:

f = p/l - q/w

Here l and w are the same as in b = w/l  above. Applying this formula gives us f = 1.19 for all years and f = 1.13 for years since 2019. In other words, one would have a leveraged position that is fully long Bitcoin and also short gold against that, by around 13% to 19% of the portfolio.

A full Kelly allocation requires that one accept substantial drawdowns in return for maximizing the long-term growth of wealth. People often prefer to use a half-Kelly or 2/3 of a Kelly allocation to decrease drawdowns. Even then, for 2/3 of a Kelly one would have about 50% allocation to Bitcoin using the gambler’s formula and a 66% allocation with the investment formula version.

With shorter time horizons the allocation decreases. For example with monthly rebalancing between age 4.25 and age 15.42 years for Bitcoin, the gambler’s full Kelly allocation is 44% of the two asset portfolio.

The portfoliovisualizer.com site also has a Kelly criterion tool that takes yet a different form, conceptually similar to the investment formula version, but expressed as the expected return less the riskless rate (Treasury bill rate) divided by the observed variance, a measure of the volatility. For the years from 2015 to 2024, it indicates a 100% allocation to Bitcoin and 0% to gold as optimal. The drawdowns are substantial; there are two drawdowns of about 73% each during the 9 year interval.

And to close, here is the same chart as Figure 1, but inverted, showing how much Bitcoin is required to purchase one ounce of gold over the past 13 years. It has dropped from thousands down to about 1/30th of a Bitcoin to purchase one ounce of gold. The power law scaling has worked over 5 orders of magnitude, a range of 100,000.

Figure 4. Number of Bitcoin required to purchase one ounce of gold, the inverse of figure 1. It is of course also a power law. Gold/BTC ~ t^(-5.4). The required number has decreased from thousands of Bitcoin to 1/30th of a Bitcoin to buy an ounce of gold.

Feel free to share this article with Peter Schiff, and you must decide your own level of acceptable risk.