Category Archives: investing

Bitcoin Is Not Gold 2.0

It's not often we can use math to definitively shut down a claim being made to pitch an investment, but here we are.

The pitch involves the cryptocurrency Bitcoin (BTC). The claim, most famously made by the Winklevoss brothers in 2017, is that "Bitcoin is Gold 2.0". Here's a more recent clip of the brothers repeating their pitch on CNBC on 10 July 2019:

While they may be among the most prominent pitchmen for Bitcoin, they're far from alone in claiming Bitcoin has gold-like investing properties. Here's a selection of articles we turned up from 2013 through 2022 where analysts have made similar claims:

Let's address the elephant in the room. For Bitcoin to be Gold 2.0, it needs to share gold's top investing characteristic: it needs to provide an effective hedge against inflation by rising in value as inflation reduces real yields. Gold, or as BTC enthusiasts would describe it, Gold 1.0, does exactly that when real interest rates fall and become negative as the rate of inflation grows to exceed nominal interest rates. Here's the chart we featured in previous analysis showing the price of gold doing just that, rising in value as real interest rates decline in value and vice versa.

Gold Spot Price vs Inflation-Indexed Market Yield of 10-Year Constant Maturity U.S. Treasury, 2 January 2007 - 17 March 2022

Now, here's a chart that presents the value of Bitcoin with respect to the same data for the inflation-adjusted yields of 10-year Constant Maturity U.S. Treasuries over the period from 17 September 2014 through 21 April 2022, covering nearly the entire period where we can identify that the claim that "Bitcoin is Gold 2.0" has been prominently made. Spoiler alert: Bitcoin is not Gold 2.0!

Gold Spot Price vs Inflation-Indexed Market Yield of 10-Year Constant Maturity U.S. Treasury, 2 January 2007 - 17 March 2022

The chart of Bitcoin's "relationship" with real yields looks like something that could have been created on an Etch-a-Sketch. The value of BTC either moves sideways or up-and-down.

That observation aside, we see three main periods for Bitcoin's valuation history in this chart:

  1. 17 September 2014 to 7 October 2020. The price of Bitcoin in U.S. dollars has virtually no relationship with real interest rates, despite substantial changes in their value during this period. In math terms, the slope of the trendlines whenever real interest rates are changing in value is nearly equal to zero, because the value of Bitcoin isn't changing with them.
  2. 8 October 2020 - 6 January 2022. This is the period when the biggest changes in the valuation of Bitcoin has occurred, and we find that while Bitcoin's value has ranged all over the map from nearly $11,000 to a peak of $67,567 during this second period, it's matched against very little change in real interest rates. In math terms, the effectively vertical movement in Bitcoin's valuation is not defined with respect to changes in real interest rates. In practical terms, whatever moved Bitcoin prices during this time was unconnected to how inflation affected interest rates.
  3. 7 January 2022 - 21 April 2022. Shown as the green circles on the chart, we find once again there is little change in the value of Bitcoin even though real interest rates have substantially changed. Just like in the first period, the value of Bitcoin shows little to no, or dare we say, zero connection to changes in real interest rates.

For the record, we're just the latest to conclude that Bitcoin is not Gold 2.0, though perhaps the first to show it using tools available to middle and high school algebra students. Here is other analysis that finds Bitcoin lacks gold's most attractive investing properties for extra credit reading:

With respect to changes in inflation-adjusted interest rates, we've demonstrated the value of Bitcoin either moves sideways or up-and-down with little rhyme or reason, making it very different from how gold has performed during the period of their shared existence. Bitcoin is not Gold 2.0.

Previously on Political Calculations


Federal Reserve Economic Data. Market Yield on U.S. Treasury Securities at 10-Year Constant Maturity, Inflation-Indexed. [Online Database (Text File)]. Accessed 22 April 2022.

Yahoo! Finance. Bitcoin USD (BTC-USD), 14 September 2014 through 21 April 2022. [Online Database]. Accessed 22 April 2022.

How Gamestop’s Stock Price Was Squeezed

The story of how Gamestop (NYSE: GME) went from a value of $11.01 per share on 13 November 2020 to reach $325.00 per share on 29 January 2021 has become a stock market legend.

It has also become the subject of a fascinating academic study, where a new paper by Lorenzo Lucchini, Luca Maria Aiello, Laura Alessandretti, Gianmarco De Francisci Morales, Michele Starnini and Andrea Baronchelli investigated how social media contributed to the short squeeze that made GME into *the* prototype meme stock.

If you're not familiar with the story, here's the paper's summary of how the GME short squeeze was made, in which Reddit's r/wallstreetbets (WSB) plays a prominent role (we've added the bullet list formatting to make it easier to follow):

GameStop (GME) is a US video game retailer which was at the centre of the short squeeze in January 2021. The timeline of the events around the squeeze is summarized in table 1, and it unfolded as follows.

  • In 2019, Reddit user u/DeepFuckingValue entered a long position on GME, i.e. he bought shares of the GME stock, and started sharing regular updates in WSB.
  • On 27 October 2020, Reddit user u/Stonksflyingup shared a video explaining how a short position held by Melvin Capital, a hedge fund, could be used to trigger a short squeeze.
  • On 11 January 2021, GME announced a renewed Board of Directors, which included experts in e-commerce. This move was widely regarded as positive for the company, and sparked some initial chatter on WSB.
  • On 19 January, Citron Research (an investment website focused on shorting stocks) released a prediction that GME’s stock price would decrease rapidly.
  • On 22 January, users of WSB initiated the short squeeze.
  • By 26 January, the stock price increased more than 600%, and its trading was halted several times due to its high volatility. On that same date, business magnate Elon Musk tweeted ‘Gamestonk!!’ along with a link to WSB.
  • On 28 January, GME reached its all-time intra-day highest price, and more than 1 million of its shares were deemed failed-to-deliver, which sealed the success of the squeeze. A failure to deliver is the inability of a party to deliver a tradable asset, or meet a contractual obligation; a typical example is the failure to deliver shares as part of a short transaction.
  • On 28 January, the financial service company Robinhood, whose trading application was popular among WSB users, halted all the purchases of GME stocks.
  • On 1 and 2 February, the stock price declined substantially.

By the end of January 2021, Melvin Capital, which had heavily shorted GameStop, declared to have covered its short position (i.e. closed it by buying the underlying stock). As a result, it lost 30% of its value since the start of 2021, and suffered a loss of 53% of its investments, i.e. more than 4 billion USD.

While the paper's summary indicates a substantial decline in value, we'll point out that since it peaked at $325 per share, GME bottomed at $40.69 per share on 18 February 2022 before climbing back up to $300 per share on 8 June 2021, before dropping back toward the middle of that range. The following chart shows its stock price history:

GME Stock Price History: 12 October 2020 - 14 April 2022

That's all the "what happened", but it's the "why it happened" we find fascinating. The authors identify one key element in the postings of the WSB redditors that established their credibility, separating their postings from the ordinary run of the mill comments that dominate discussions on many other stock investing discussion sites, which they describe in the paper's introduction:

In this paper, we analyse discussions on WSB from 27 November 2020 to 3 February 2021 (table 1) and investigate how they translated into collective action before and during the squeeze that was initiated on 22 January and lasted until 2 February. Motivated by recent theoretical [10,11] and experimental [12] evidence that minorities of committed individuals may mobilize large fractions of a population [10,1315] even when they are extremely small [16], we investigate whether committed users on WSB had a role in triggering the collective action. To this aim, we operationalize the commitment of a user as an exhibited proof that the user has financial stakes in the asset.

We won't keep you in suspense. Here's the summary of what they found:

We show that a sustained commitment activity systematically pre-dates the increase of GameStop share returns, while simple measures of public attention towards the phenomenon cannot predict the share increase. Additionally, we also show that the success of the squeeze operation determines a growth of the social identity of WSB participants, despite the continuous flow of new users into the group. Finally, we find that users who committed early occupy a central position in the discussion network, as reconstructed by WSB posts and comments, during the weeks preceding the stock price surge, while more peripheral users show commitment only in the last phases of the saga.

The last part is to say that the early influencers who effectively established their credibility continued to be influential within the network. We think that continued influence is attibutable to their success, which had the short squeeze of GME's stock not occurred, would have led other WSB participants to discount the information value of their postings. People with opinions about a company's investment worthiness are a dime a dozen on a stock discussion board, but people who back up their talk with hard evidence of their bets that go on to pay off were granted credibility.

Much of the authors' study focuses on the dynamics between this "core" group of GME redditors and others who were on the periphery of the investing activity, which they describe as a "behavioural cascade" event. The core group attained a critical point of credibility, sweeping up peripheral redditors who transitioned from observers to participants as the short squeeze cascaded into a legendary event.


As for the hedge fund that lost billions on its attempted short of GME, the firm is considering returning what's left of the capital it controls to its remaining investors as its future with its current structure is in doubt. The fund lost 39% of its capital in 2021, with another 21% of losses to date in 2022. It's management would launch a new fund to replace it.


Lorenzo Lucchini, Luca Maria Aiello, Laura Alessandretti, Gianmarco De Francisci Morales, Michele Starnini and Andrea Baronchelli. From Reddit to Wall Street: the role of committed minorities in financial collective action. Royal Society Open Science. Volume 9, Number 4. 6 April 2022. DOI: 10.1098/rsos.211488.

Real Yields and the Price of Gold

Not long ago, we were asked about how inflation affects the price of gold. It's something of an occupational hazard, where a lot of people assume that because we know quite a lot about how stock prices work, that knowledge directly carries over to things like commodities.

For most commodities, like copper, or oil, or turkeys, we would default to supply and demand analysis or dig into the factors affecting their cost of production to explain their changes in prices over time.

Gold falls into a different category that doesn't fit well into that basic mold. Aside from its use in jewelry, it's not used in great quantity to support any industrial applications like nearly all other metals are. Instead, it's used in financial applications, often as a hedge for inflation. Which is to say that when inflation runs hot, the price of gold will rise. But not always. Something else affects it as well.

That something else would appear to be interest rates, where the recent history for gold prices points to their having an inverse relationship with inflation-indexed interest rates. By inverse relationship, that means that as real interest rates fall, the price of gold tends to rise in response. The following chart shows the relationship we found between the daily closing spot price for gold and the daily inflation-indexed market yield of 10-year constant maturity U.S. Treasuries over the past 15 years.

Gold Spot Price vs Inflation-Indexed Market Yield of 10-Year Constant Maturity U.S. Treasury, 2 January 2007 - 17 March 2022

While the chart picks up the action at the start of January 2007, the pattern it shows began taking hold in 2006, coinciding with the deflation phase of the U.S. housing bubble. The inverse relationship we've identified only holds for the current period.

The most recent data point shown in the chart, for 17 March 2022, has the inflation-indexed market yield on 10-Year constant maturity U.S. Treasury securities at -0.71%, which is paired with gold's closing spot price for the day of $1,944.05. For reference, the nominal market yield for a 10-Year constant maturity U.S. Treasury was 2.20%. The inflation-indexed yield is negative because expected inflation rate over the 10 year period of the security is greater than the non-inflation indexed yield. That situation is consistent with a high price for gold.

After finding this relationship, we went searching for insight from other analysts who were already well familiar with it. Here's how Longtermtrends describes how gold prices work:

According to Erb and Harvey the correlation between real interest rates and the price of gold is -0.82. In other words, when real yields go down gold goes up. This correlation explains why inflation is gold's best friend while rate hikes are its worst enemy.

Here is a possible explanation for this relationship. Rising interest rates also mean rising opportunity costs of holding gold. Gold neither pays dividends nor interest. Thus, it is relatively expensive to hold it in the portfolio when real interest rates are high. On the other hand, when real yields are negative, holders of cash and bonds are losing wealth. In such a scenario, they are more prone to buy gold.

The Erb and Harvey study Lontermtrends references is from 2013, but it's good to see that the relationship we found is still very much in the same ballpark as the correlation they found. Just so, the price of gold is still governed by supply and demand. It's just that the demand for gold works quite a bit differently than it does for other commodities given its primary use as a hedge for negative real yields in today's financial applications.


Claude B. Erb and Campbell R. Harvey. The Golden Dilemma. Financial Analysts Journal. Vol. 69. No. 4. July/August 2013, pp 10-42. DOI: 10.2139/ssrn.2078535.

Federal Reserve Economic Data. Market Yield on U.S. Treasury Securities at 10-Year Constant Maturity, Inflation-Indexed. Online Database (Text File)]. Accessed 19 March 2022.

USAGold. Daily Gold Price History. [Online Database]. Accessed 19 March 2022.

Heavy Tails and Taylor’s Law

Power laws show up in a lot of different places. That includes things like stock prices and income distributions, which is why developments in maths related to the topic of power laws catch our attention.

That brings us to a story from earlier this year that we've recently caught up with. We came across it via a university's PR news release with the clickbait title "Financial crashes, pandemics, Texas snow: How math could predict 'black swan' events".

We don't think much of that type of PR sensationalism, so we set it aside to review later. After all, we've seen the PR departments at universities function much like those scam scientific journals that will literally publish gibberish if you pay them. Usually after we get to reviewing such pieces, we end up sweeping the stories into the garbage. Where most of them belong.

This one turned out to have some meat behind it. It relates to when and how Taylor's Law may be successfully used to estimate the variance of a population based on the mean of smaller samples measuring it.

Named after ecologist Lionel Roy Taylor, who developed it after realizing the total variance of the animal population in a region (such as moths) is proportional to a power of the mean of the sampled population of animals counted at the same time at several different sites within it (such as the counts of moths captured overnight in traps spread over the region).

Here's the basic math formula for Taylor's Law:

Variance = A*(Mean)B

In this power law equation, A and B are constant values, the mean is that for the different samples of the population, and the resulting variance applies for the total population. This expression becomes a linear equation when using logarithms:

log(Variance) = log(Constant A) + (Constant B)*log(Mean)

It had been assumed that Taylor's Law would only work well when dealing with a population whose variance in either space or time is described by a normal Gaussian distribution, the kind represented by a standard bell-curve in statistics. Joel E. Cohen, Richard A. Davis, and Gennady Samorodnitsky however have demonstrated in a 2020 paper that Taylor's Law holds for heavy-tailed distributions.

The difference in the variance between a normal-tailed distribution and a heavy-tailed distribution in statistics is that more extreme variances are more likely to be seen in heavy-tailed distributions. The variation of stock prices is a good example, because big changes in them are observed much more frequently than are predicted by normal variance distribution statistics. That has real world meaning because the failure to recognize that fact has resulted in some of the investing world's biggest failures.

Adem Tumerkan: Normal vs Heavy-Tailed Distribution

Cohen, Davis, and Samorodnitsky see how their work might be extended beyond its native field of ecology:

Until now, Taylor's Law was thought to have no place in these heavy-tailed systems. It helped plot our paths along the normal circumstances of daily life, but when it came to extreme occurrences like the present pandemic, Taylor's Law seemed irrelevant.

But a few years ago, Cohen and colleagues at Columbia University made a striking discovery—a way of looking at heavy-tailed variables that yields surprisingly orderly connections between the mean and the variance. "It was as if we took all of the pieces of a car, put it in a box, and the car still ran," Cohen says. "This combination of variables gave us the same result regardless of how they were connected."

A collaboration of excited mathematicians culminated in this new study, which collects many more examples of the phenomenon and concludes with mathematical proof that extreme, heavy-tailed events are indeed well described by Taylor's Law.

This does not mean that any individual extreme event can be predicted with a simple mean-to-variance formula. But the research effectively breaks Taylor's Law out of its shell, giving scientists good reason to test whether market fluctuations and natural disasters obey the same Taylor's Law that governs insect populations and the progression of cancerous growths.

Cohen hopes that this work will stimulate further basic research on the mathematics of heavy-tailed distributions and that scientists will use it to better understand the extreme events wherever heavy-tailed distributions lurk. "Advances like these are the mathematical analogue of bioimaging," he says.

"They make it possible to see what was previously invisible."

Potentially. It will be exciting to see.


Joel E. Cohen et al. Heavy-tailed distributions, correlations, kurtosis and Taylor's Law of fluctuation scaling, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences (2020). DOI: 10.1098/rspa.2020.0610. [Ungated PDF Document].

J.N. Perry. Taylor's Power Law for Dependence of Variance on Mean in Animal Populations. Journal of the Royal Statistical Society. Series C (Applied Statistics). Vol. 30, No. 3 (1981), pp. 254-263. DOI: 10.2307/2346349.

Adam Tumerkan. Hunting and Profiting from 'Fat Tails' - Be Long HighVol Assets. Seeking Alpha. [Online Article]. 20 July 2017.

Previously on Political Calculations

Image Credit: Adem Tumerkan.

The S&P 500 and Campbell’s Tomato Soup

Longtime readers know our two favorite data series are the price histories for an iconic No. 1 can of Campbell's Condensed Tomato Soup and for the S&P 500 stock market index. We're tapping both of them today to answer a burning question we had: How many cans of Campbell's tomato soup could a hypothetical investor in the S&P 500 buy if they cashed in the equivalent of one share of the index.

We're presenting the graphical answer to that question below, where we find that as of the index' all-time record highs in early September 2021, our hypothetical index investor could buy 4,752 cans of tomato soup!

Number of Cans of Campbell's Condensed Tomato Soup per Value of One Share of the S&P 500, January 1898 - September 2021

Expressed a little differently, that's the same as 198 cases of 24 cans each.

Compare that quantity to January 1898, when Campbell's condensed tomato soup first began being shipped across the country. We find the equivalent value of a share of the S&P 500 (actually an an index made up of its predecessor individual stock components) would have bought 46 cans, or rather, just under 2 cases worth of soup.

Now, which would you rather have in your investment portfolio or pantry today: one equivalent share of the S&P 500 or 198 cases of Campbell's condensed tomato soup?

Cases of Campbell's Condensed Tomato Soup