Correlation Doesn’t Measure What You Think!

This article originally appears in my risk.net column in February, 2015, which you can find here.

Take a quick look at the two panels of Figure 1 and estimate the correlation for the two funds in both panels. Really, please do it now. What’s your gut feel of the correlation of each set? If you are like virtually everyone I asked, it is quite obvious that the two funds in the left panel are uncorrelated or possibly negatively correlated while those in the right panel are highly correlated with each other. Estimates for the left range from zero to -0.7, and estimates for the right panel are often above 0.7. In reality, though, the funds in the left pane have a return correlation of +0.95 and the correlation for the set on the right is -0.92. That’s right: the funds on the left are positively correlation, and quite highly, while the funds on the right are negatively correlated.

In the left panel, over this simulated three-year time period, Fund 1 shows a 49% total return corresponding to a 14% annualized return. Fund 2 suffers a 29.6% total loss, or an annualized loss of 11%. Now take another look at that left panel and estimate which of these two funds is more volatile. Some people interpret Fund 2 as more volatile because it suffers a loss while Fund 1 has stellar returns, equating “risk” or “loss” with volatility, but most people recognize that Fund 1’s volatility is at least somewhat greater than the volatility of Fund 2. In fact, the volatility of Fund 1 is 2.5 times greater than the volatility of Fund 2. Figure 1. Left Pane: two funds diverging in value. The blue fund’s total return is 86% while the red fund’s total loss is 22%. Right pane: Two funds both increasing in value.

Volatility

Oil’s annualized volatility in the second half of 2014 was 30%. What surprised me was the number of people who asked how it could have such a “low” volatility given that it had such large losses, revealing an expectation that losses imply volatility.

Let’s take a closer look at what volatility actually measures. Because many people colloquially equate highly volatile funds with risk, there is a natural tendency to associate volatility with loss, but the actual measurement of volatility – the standard deviation – ignores the direction of the fund’s movement and only cares about the consistency of the returns, or, rather, the lack thereof. Volatility measures the level of dispersion around the average, but the actual value of the average does not affect the measure of volatility at all. Fund 1 has an average monthly return of +1 while Fund 2 has an average monthly return of -1, and those average returns affect the gain/loss but not the volatility. Fund 1 has a monthly volatility of 2.5% while Fund 2 has a monthly volatility of 1%. The higher return fund has a much higher volatility than the losing fund. Rather than looking at a chart of the price over time as is shown in Figure 1, it is easier to see the relationship in a histogram of returns, as shown in Figure 2 below. Figure 2. Left panel: Fund 1’s return histogram with average monthly return of +1% and volatility (standard deviation) of 2.5%. Right panel: Fund 2’s return histogram with average monthly return -1% and volatility (standard deviation) of 1%. The two panels are drawn on the same scale for accurate visual comparison: the higher returning fund (left, blue) has a dispersion or volatility 2.5 times higher than the lower returning fund (right, red).

Here, it’s much easier to see that Fund 1 has a larger dispersion around its average than Fund 2. This example was constructed to explicitly show that a high-returning fund can very well have a much higher volatility than a fund with significant losses. Lesson: downward or, for that matter upward, trending funds do not necessarily have high volatility.

Correlation

The surprising result from the opening paragraph reveals a virtually universal misconception about correlations: correlations do not measure synchronization or co-movement of two time series. Rather, they measure the respective co-movement around each time series’ own individual averages. They are calculated in a way that normalizes this relative co-movement by their own volatilities so that the result is always between -1 and +1. In other words, “big” and “small” co-movements are defined by the funds’ own volatilities: a “big” move is big as compared with the volatility.

An added complexity in financial markets is that correlations (and volatilities) are always measured about the returns of the funds/securities rather than about their prices. No one considers the correlations of prices. So we analyze one set of data (returns) but we often graph another set: the prices.

For each time step in Figure 1, the correlation number measures how far away each of the fund’s returns is from its own average return and multiplies those numbers together to create a measure of co-movement. If the two funds move in the same direction from their own averages, then that time step’s contribution to the correlation is positive. If the two funds move in opposite directions from their own averages, then that time step’s contribution to the correlation is negative. The time steps’ contributions are added together to form the correlation over the entire timeframe. The farther the two funds move together – from their respective averages – the stronger the correlation. If one moves far from its average but the other moves only a little from its own average, then the correlation is smaller.

Although it may seem like nit-picking to place special emphasis on “from their own averages”, I do so because that is precisely the origin of the confusion in the example above. Correlation is a relative measure of dispersion, not an absolute measure of co-movement as many believe it to be. To be sure, there is a different and related absolute measure of co-movement between funds that is often confused with correlation. I explored the relationship between correlation and beta, the absolute measure for which it is often mistaken, in a previous article you can find here.

To better understand the nature of ‘correlation’, it is best viewed in scatter plot of the returns of one fund against another, as is shown in the two panels of Figure 3. Figure 3: Correlations between funds are best viewed as scatter plots of the returns of one fund against another. The left and right panes correspond to the same panes as in Figure 1: the left pain is highly correlated (+0.95) and the right pane is negatively correlated (-0.92).

The correlation between the funds is not measured by the steepness of the scatter plot — high correlation can be steep or flat. Rather, correlation is related to the shape of the scatter plot: high correlation is associated with cigar-shaped or pencil-shaped plots like those in Figure 3, while low correlation – near zero – is associated with more oval or circular plots. The closer the individual points in a scatter plot fall a straight line, the higher the correlation, with positive correlations sloping upward (left panel) and negative correlations sloping downward (right panel).

Recognizing that correlations measure two funds’ relative co-dispersion or co-movement of their returns from their own averages is the key to clarifying the confusion of why the funds in Figure 1 have counter-intuitive correlations. The growth of the funds’ values shown in Figure 1 is dominated by their average returns: +1% for the gaining fund and -1% for the losing fund in the left panel. The right panel of Figure 1 has both of those funds with average monthly returns of +1%. But correlation does not measure anything about those averages: like volatility for a single fund, it measures the dispersion around those averages, but in this case it measures the co-dispersion of two funds simultaneously. It just so happens that the funds shown in Figure 1 move around their average returns in a correlated way in the left pane and an anti-correlated way in the right pane. The shape of the fund’s value is governed mostly by the large monthly returns, not by their correlations. In other words, the correlation measures how ‘jiggly’ those returns are around that positive and negative trajectory.

Misconceptions about statistics – even about common terms such as volatility and correlation – abound in our industry, leading to incorrect assumptions and interpretations that can have a real impact on important trading and hedging decisions. Transparency into the calculations along with redundant and independent computations can help, but ultimately it is a matter of expertise to ensure proper interpretation.