A Surface Level Look at the Distributions of Daily Percentage Returns of Bitcoin (BTC) and Gold (GLD)

Brian Koralewski is the founder of Austere Capital Advisory LLC, a digital asset derivative consulting firm headquartered in New York City.

Bitcoin vs. GLD Daily Percentage Returns

Right away, we observe that the Kurtosis is considerably higher than a normal distribution would suggest. This indicates that the majority of returns do not hover around the mean, and instead there is greater concentration in the tails, which implies a mean that is “peaky” or sharp (since less observations than normal occur around the average).

This also gives credibility that the “tail wags the dog” — that extreme deviations outside the average returns play a larger role than a normal distribution would infer.

All the above is quite obvious — all asset prices as well as their daily returns are too erratic to be quaintly normally distributed, and thus a lognormal distribution is generally used when inferring price and return volatility. For simplicity purposes we use a normal distribution here to highlight the extent of activity outside of 3 standard deviations. To look at the tails of a lognormal distribution would be a tail risk analysis in and of itself and will be the subject of a future paper.

We also see that BTC’s daily returns are slightly negatively skewed — which indicates that the magnitude of the “down” days are greater than the “up” days, even though overall BTC has more positive return days than negative. Let’s now take a look at a frequency distribution to further isolate BTC daily returns and their respective probabilities:

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