An Introduction to Correlation Coefficient

What is Correlation?

Most have a general idea that correlation is a measure of how closely the behavior of one variable mirrors the behavior of another. This is true, with some important qualifications we’ll discuss below.

Correlation is measured on a scale ranging from -1 to 1.

• -1: perfect negative correlation

• 0: no correlation

• 1: perfect positive correlation

Correlation is Linear!

Most often, correlation is measured with the Pearson product-moment correlation coefficient.

• The most important thing to remember about this measure is that is linear. It is a measure of linear dependence.

• The practical implication: multiplying either variable by a constant will not change the correlation.

`\text{Cor}(X, Y) = \text{Cor}(X, 5Y) = \text{Cor}(8X, 2Y) = \text{Cor}(7X, Y)`

• Where `Cor` denotes correlation and where we can, of course, replace the numbers 5, 8, 2 and 7 with any others.

Correlation, Beta, and Market Crashes

The above explains why, during a market crash, stock correlations approach 1 while many stocks have high (> 5) betas.

• Some stock returns effectively become linear multiples of the market return – they may decrease 6% for every 1% decline in the market.

This also helps explain the optimal hedge ratio (see the Cross Hedging and First Look at Risk presentations). Even if the assets are perfectly correlated, you will likely have a hedge ratio different from 1 given different volatilities.

Interactive App

In the app on the following slide, change the ‘Correlation’ slider to generate random numbers with a particular correlation. (Note: it may be a bit off because it is a random sample taken from a distribution with that correlation coefficient.)

• Changing the ‘Multiple’ slider will multiply the ‘x’ variable by that number. You’ll see it doesn’t change the correlation coefficient – only the slope of the graph. Linear transformations do not affect correlation.

• Then choose the radio button to square ‘z’. You’ll see this transformation will change the correlation. Nonlinear transformations do affect correlation.