Covariance and correlation

How do you measure whether two returns move together?

Start with a picture. A scatter plot puts one variable on each axis, one dot per pair of observations. An upward drift means they tend to rise together, a downward drift means they move opposite ways, and a shapeless cloud means no clear pattern.

Covariance measures joint variability: it is the two-variable version of the variance. For each pair, multiply how far X is from its mean by how far Y is from its mean, then average.

$$\sigma_{X,Y} = \frac{\sum_{i=1}^{n} (X_i - \mu_X)(Y_i - \mu_Y)}{n}$$

With sample data, divide by n − 1 instead, for the same reason as the sample variance; that is what a spreadsheet’s COVARIANCE.S does (COVARIANCE.P divides by n). This is the sample covariance .

$$s_{X,Y} = \frac{\sum_{i=1}^{n} (X_i - \bar{X})(Y_i - \bar{Y})}{n - 1}$$

Covariance is positive when the variables tend to be above or below their means together, and negative when one is above while the other is below. Its size, though, depends on the units, so it cannot tell you how strong the link is.

In Excel and Google Sheets: COVARIANCE.P (divide by n), COVARIANCE.S (divide by n − 1) and CORREL.

How does correlation fix the size problem?

The correlation coefficient divides the covariance by the product of the two standard deviations. The units cancel, and the result lies between −1 and +1.

$$\rho_{X,Y} = \frac{\sigma_{X,Y}}{\sigma_X\,\sigma_Y}$$

Correlation is read straight from the number.

ρ	What the scatter plot looks like
+1	Points on an upward-sloping straight line: a perfect positive link
Close to +1	A tight upward pattern: a strong positive link
0	No straight-line pattern: no linear relationship
Close to −1	A tight downward pattern: a strong inverse link
−1	Points on a downward-sloping straight line: a perfect inverse link

The sign of the correlation is the sign of the covariance, because the standard deviations below it are always positive. It has nothing to do with whether the mean returns are positive or negative: two shares that both lose money on average can still be positively correlated.

Often the exam hands you the covariance and the standard deviations instead.

Drag a point and watch ρ and the covariance. Try the U-shape preset, then the One outlier preset.

Exam tip Rearranged, covariance = ρ × σ_X × σ_Y. Keep standard deviations as decimals (12% is 0.12) when the covariance is a decimal.

How closely does an asset track the market?

Know the idea

Take the asset’s excess returns (its return minus the risk-free rate) and the benchmark index’s excess returns, and correlate them. This is the excess-return correlation . A high positive correlation means the asset moves closely with the market once the risk-free return is set aside. If the risk-free rate barely changes, the answer is almost the same as for raw returns, because subtracting a constant does not change a correlation. The sensitivity behind this link becomes beta, estimated with a regression in Learning Module 10; correlation also drives portfolio risk in Learning Module 8.

What are the limits of correlation?

A correlation is a summary of a straight-line pattern. Before relying on it, check four things.

Correlation coefficient ρbetween −1 and +1, no units

• Reading ρ Memorize +1 upward line, −1 downward line, 0 no linear pattern; sign = sign of covariance
• Limitations Use it check these before trusting ρ
  • Linear only a strong curve can give ρ near 0
  • Outliers one point can create or hide correlation
  • Spurious correlation chance, a common third variable, or a shared divisor
  • Not causation association says nothing about cause

Limitation	What happens	Example
Nonlinear relation	ρ near 0 despite a strong relation	An option’s payoff against the share price around the strike
Outlier	One point creates or hides correlation	A single crash month
Spurious correlation	Chance, a hidden third variable, or a shared divisor	Ice-cream sales and sunburn cases, both driven by summer
Not causation	Association only	A high ρ never shows which variable moves the other

A spurious correlation is the appearance of a relationship that is not there, or is not direct. It arises three ways: by chance in a particular data set; because both variables respond to a third one; or because two unrelated series were both multiplied or divided by the same third series, especially one that is much more volatile than either. An example of the third: two small companies’ unrelated monthly outputs, each divided by the same highly volatile commodity price, will look correlated.

Do not build a strategy on a high correlation alone. Outliers can also create or hide a correlation; trimming or winsorizing the data can help, but deciding whether an extreme point carries information takes judgment.

Go deeper Why covariance has such awkward units

Covariance multiplies two deviations, so for two return series its unit is percent times percent, or %². That is why a covariance of 0.0066 tells you nothing until you set it against the two standard deviations. It is also why you should never compare covariances across asset pairs with different volatilities: a bigger covariance may only mean the assets are more volatile.

Go deeper Same summary, different pictures

Data sets can share the same means, standard deviations and correlation yet look completely different on a chart; a famous constructed example is Anscombe’s quartet. Summary statistics never replace a plot.

Go deeper Where this goes next

Correlation is the input that makes diversification work: the lower the correlation between assets, the more portfolio risk falls (Learning Module 8). Whether a sample correlation differs from zero is a hypothesis test (Learning Module 7), and the square of the correlation between two variables is the R² of a simple regression (Learning Module 10).