Measuring correlation between crypto assets

A correlation is computed on returns, not on prices, and the first decision is which return definition to use. A simple return is the percentage change from one observation to the next: the later price divided by the earlier price, minus one. A log return is the natural logarithm of that ratio. For small moves the two are almost identical; for large moves they diverge, and the divergence is systematic rather than random.

Log returns have two properties that make them the usual choice for this kind of work. They are additive over time, so a log return over a week is the sum of the daily log returns within it, which makes aggregation and annualisation straightforward. And they are symmetric in the sense that a log return of plus and minus the same magnitude represents the same proportional move up and down, whereas a simple return of minus fifty per cent requires a plus one hundred per cent move to recover. Simple returns are easier to explain and are what a reader intuitively expects, and they are the correct choice when the question is about the value of a holding rather than the behaviour of a series.

The choice changes the answer, and it changes it most in exactly the periods a crypto study cares about. In a window containing a very large single-day move, the simple-return series is bounded below at minus one hundred per cent and unbounded above, so the distribution is skewed and a handful of observations dominate the covariance. The log-return series compresses the same move and produces a different coefficient. Neither is wrong. The point is that the definition must be stated, because a reader comparing two studies that used different definitions is comparing two different measurements.

A correlation is always computed over a window, and the window is not a technical detail. A thirty-day window on daily returns contains thirty observations, which is enough to produce a number and not enough to produce a stable one: the standard error on a coefficient estimated from thirty observations is wide enough that two windows drawn from the same underlying process can differ substantially. A ninety-day window is more stable and less responsive. A one-year window is stable and can span a regime change, averaging a relationship that did not hold throughout.

The window also decides what the number is a statement about. A short window answers the question "how have these two assets moved together recently", which is what a trader wants and what a long-horizon reader should distrust. A long window answers "how have they moved together across the period", which is more stable and less useful for any decision about the present. Reporting a single coefficient without its window is therefore not a simplification; it is a different claim.

The honest presentation is a window and a sensitivity. State the window, then state what the coefficient does when the window is halved and doubled. If the three values are close, the relationship is reasonably stable over the period. If they are not, the single number was an artefact of the window chosen, and the reader is entitled to know that before they act on it.

Most of the machinery behind a correlation coefficient assumes the series being measured are stationary: that the process generating them has stable statistical properties over time, so that the covariance and the variances being estimated are not themselves moving. Crypto return series violate that assumption in ways that are not marginal. Volatility clusters, so the variance is not constant. The relationship between two assets changes with the market regime, so the covariance is not constant. And the assets themselves change: a network's liquidity, holder base and trading venues in 2017 are not the same as in 2024.

When the underlying relationship is not stable, the sample correlation is an average of a quantity that moved, weighted by where the variance happened to be. Because crypto volatility is concentrated in a small number of turbulent periods, the coefficient is often dominated by a handful of weeks. Two assets that were uncorrelated through a quiet year and moved together through one crash can produce a coefficient that describes the crash and is presented as a description of the assets.

There is no correction that makes this problem go away, but there are ways to be honest about it. Report the window. Report the sensitivity to the window. Report the coefficient computed with the largest few observations removed, so the reader can see how much of the result rests on them. And describe the regimes the window contains, because a coefficient spanning a bull market and a bear market is an average of two different relationships rather than a property of either.

A correlation between an altcoin and Bitcoin is a statement about co-movement. It is not evidence that Bitcoin moved the altcoin, and it is not evidence that the altcoin moved Bitcoin. Both are plausible in principle, and a coefficient cannot distinguish them. The most likely explanation in most windows is neither: both assets are exposed to the same market-wide factors, and the co-movement reflects a shared response to those factors rather than a transmission from one asset to the other.

Those common factors are worth naming, because they are what a reader should be thinking about instead of the coefficient. Broad risk appetite moves most speculative assets in the same direction at the same time. Dollar liquidity and the level of real interest rates change the discount applied to every long-duration asset. Leverage in derivatives markets is shared infrastructure: a liquidation cascade in one asset's perpetual futures pulls on the same collateral that supports positions in another. And the venues are shared, so a market-maker withdrawing from one order book withdraws from several at once.

Documented causation in this area is narrow and specific. It exists where there is a mechanism with a direction: a forced liquidation that propagates through shared collateral, an index rebalance that mechanically changes weights, a custody or exchange event that affects one asset's tradability directly. Those are claims about plumbing, and they can be checked against the plumbing. A correlation coefficient is not one of them, and no page on this site will present co-movement as though it were.