# Market Capitalization for Max- vs Unlimited-Supply Cryptocurrencies

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*This article is the first installment of a series exploring the intersection of blockchain and data. For this post, we will investigate the market capitalizations between tokens with and without supply maximums. In a future article, we will analyze historical, cryptocurrency market data (e.g. token age, price, rate of return, volatility, supply).*

# The Great Debate

Should a cryptocurrency have a maximum supply (e.g. BTC) or an unlimited supply (e.g. ETH)? There are good theoretical reasons for and against having a max supply. A currency with a max supply would be similar to a finite commodity (e.g. gold). Therefore, it would never depreciate from an increase in supply. However, it may not have as high of a circulation as that of an unlimited-supply currency. A currency whose supply grows depreciates (per unit a la inflation), so its users prefer to spend (or invest) than save, increasing its trade volume; a stablecoin such as Tether would be an example of an unlimited-supply currency since its supply adjusts to make sure each unit matches the price of a stable asset such USD. Then again, a fixed supply prevents minters (e.g. central banks) from increasing the currency supply to increase their assets at the expense of the rest of the network of currency holders. But a free market for currencies would effectively be an unlimited supply, and so on…

To shed light on the debate with real data, let us look at 1,375 cryptocurrencies from CoinMarketCap (as of July 19, 2018) to test the following hypothesis: **on average, a cryptocurrency with a maximum supply has a higher market capitalization than one without**. (Since this is an observational study, we cannot make causal inferences, but we might be able to make inferences to the two populations.)

# Comparing the Means

Let us look at the market cap data for 277 max-supply tokens and 1,098 unlimited-supply tokens. Here are some summary statistics:

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║ MARKET CAP ║ max ║ no max ║ difference ║

╠═════════════╬═════════════════╬════════════════╬════════════════╣

║ mean ║ 698,417,632 ║ 85,093,127 ║ 613,324,505 ║

║ median ║ 3,111,609 ║ 3,485,596 ║ -373,987 ║

║ std dev ║ 7,767,567,465 ║ 1,444,405,122 ║ 6,323,162,343 ║

║ skew ║ 16 ║ 32 ║ -16 ║

║ min ║ 2,423 ║ 494 ║ 1,929 ║

║ max ║ 126,913,779,660 ║ 47,251,527,471 ║ 79,662,252,189 ║

║ ex kurtosis ║ 252 ║ 1,035 ║ -783 ║

║ count ║ 277 ║ 1,098 ║ -821 ║

║ std err ║ 466,707,897 ║ 43,570,262 ║ 423,137,635 ║

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The means are $698M and $85M, respectively, with a difference of $613M. However, the standard errors are $467M and $44M, so the difference in means might not be statistically significant. (Note that the difference of the medians is only -$374k. We can estimate the standard errors of the medians through bootstrapping.)

## Testing the Hypothesis

In order to determine whether the greater max-supply mean is statistically significant, we can run a one-tailed, two-sample *z*-test. (The central limit theorem lets us approximate each sample mean distribution as normal.) You can refer to the tables below, and for more details, this Google Sheets file.

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║ Hypothesis ║ Description ║

╠═════════════╬═════════════════════╣

║ null ║ mu_max <= mu_no_max ║

║ alternative ║ mu_max > mu_no_max ║

╚═════════════╩═════════════════════╝

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║ Difference of means ║ Std err ║ z-value ║ p-value ║

╠═════════════════════╬═════════════╬═════════╬═════════╣

║ 613,324,505 ║ 468,737,271 ║ 1.308 ║ 0.095 ║

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The *p*-value is 9.5%, so at a significance level *α*** **of 10%, **the max-supply mean is statistically, significantly greater**. The max-supply outlier is Bitcoin with a market cap of $127B. Removing it decreases the difference of means to $156M. However, the *p*-value drops from 9.5% to 6.5%, so we still reject the null hypothesis that the max-supply mean is less than or equal to the unlimited-supply mean.

# Logging the Data

If the high skew of the distribution makes you uneasy, let us do the same mean analysis on the natural logarithm *ln* (*log* base *e*) of the market cap numbers.

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║ LOG MARKET CAP ║ max ║ no max ║ difference ║

╠════════════════╬═══════╬════════╬════════════╣

║ mean ║ 14.97 ║ 14.87 ║ 0.10 ║

║ median ║ 14.95 ║ 15.07 ║ -0.12 ║

║ std dev ║ 3.01 ║ 2.44 ║ 0.57 ║

║ skew ║ 0.30 ║ -0.05 ║ 0.35 ║

║ min ║ 7.79 ║ 6.20 ║ 1.59 ║

║ max ║ 25.57 ║ 24.58 ║ 0.99 ║

║ ex kurtosis ║ -2.64 ║ -2.95 ║ 0.31 ║

║ count ║ 277 ║ 1,098 ║ -821 ║

║ std err ║ 0.18 ║ 0.07 ║ 0.11 ║

╚════════════════╩═══════╩════════╩════════════╝

╔═════════════╦═════════════════════╗

║ Hypothesis ║ Description ║

╠═════════════╬═════════════════════╣

║ null ║ mu_max <= mu_no_max ║

║ alternative ║ mu_max > mu_no_max ║

╚═════════════╩═════════════════════╝

╔═════════════════════╦═════════╦═════════╦═════════╗

║ Difference of means ║ Std err ║ z-value ║ p-value ║

╠═════════════════════╬═════════╬═════════╬═════════╣

║ 0.101 ║ 0.195 ║ 0.517 ║ 0.303 ║

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The difference between the means estimates the log of the ratio of the max-supply market cap to the unlimited-supply market cap. This is because the log-transformed data have symmetric distributions and the log preserves ordering; therefore, mean[*log*[*X*]] = median[*log*[*X*]] = *log*[median[*X*]] where *X* is any random variable. According to the tables, the difference between the means is 0.1 with a *p*-value of 30%, which is greater than *α*(10%); we keep the null. Thus, **the estimated max-supply market cap median is not greater than that of the max-supply market cap**.

# Closing Remarks

The max-supply sample mean and median are greater than those for unlimited-supply, but only the former is statistically significant. (This conclusion may not hold for all future cryptocurrencies, say over the next decade.) **The data and statistical analyses show there is a difference in means, but they do not explain WHY there is difference.** Perhaps restricting the max-supply of a token makes it more scarce thus more valuable than a token with an unlimited supply. Maybe the market believes that fixing the currency supply prevents the minters (e.g. central banks) from lining their pockets at the expense of the token holders. Further research in the economics of max-supply currencies would be invaluable to token designers and users. Hopefully, this investigation made a small contribution.

Written by Lester Kim — Lester is a data scientist and the Director of Data at Littlstar.

*Note**: **Random sampling** tests were performed for the mean and median analyses with the same results. (See the last two sheets in the Google Sheet document above.)*

*Acknowledgements**: Thank you to **Vanessa Kincaid** for giving feedback on this article and **Eric Jiang** for sharing his thoughts on token design.*