# Does correlation of stocks represent magnitude?

When calculating the correlation between two stocks I get an 85% correlation. Does this indicate anything about the amount the stocks are going up (so if one goes up 10% so does the other) or just that when one goes up, so does the other, but at a totally different (positive) rate?

• Correlation is unitless. In order to get it in units, you need the (ordinary least squares) $\beta$, which is $\rho_{X,Y}\frac{\sigma_Y}{\sigma_X}$. This accounts for the volatilities of $X,Y$. Jan 3 at 18:13
• Percentages are also unitless. And I think "regression coefficient $\beta$" is more clear than "ordinary least squares $\beta$". Jan 4 at 6:19

Notice that linear correlation is just a standartized measure of variability for two variables around their mean values, loosely speaking.

In your concrete case of a linear correlation between stock returns, it won$$`$$t say anything about magnitude because the mean of each return series go into the computation. You can only say those stock returns have a strong positive linear relation.

• You can say a bit more than that. You can say that each moves at the same relative ratio in the same direction with each other most of the time which seems to be what the question is about, not absolute magnitudes. Jan 3 at 18:53

Should have been a comment as there are already brilliant answers, but posting as an answer only because it is a bit lengthy! Ignoring the sample/population nuances, here is a simple illustration that correlation is an indicator of the strength (and direction) of the linear relationship but not the 'magnitude' :

$$\text{Correl}(Y,X)= \frac{\text{Cov}(X,Y)}{\sqrt{\text{Var}(X) \, \text{Var}(Y)}}$$

$$\text{Slope}(Y,X)= \frac{\text{Cov}(X,Y)}{\text{Var}(X)}$$

Let's multiply Y by 10:

$$\text{Correl}(10 \times Y,X)= \frac{\text{Cov}(X,10 \times Y)}{\sqrt{\text{Var}(X) \, \text{Var}(10 \times Y)}}= \frac{10 \times \text{Cov}(X, Y)}{10 \times \sqrt{\text{Var}(X) \, \text{Var}( Y)}}=\text{Correl}(Y,X)$$

$$\text{Slope}(10 \times Y,X)= \frac{\text{Cov}(X,10 \times Y)}{\text{Var}(X)}= 10 \times \frac{\text{Cov}(X, Y)}{\text{Var}(X)}=10 \times \text{Slope}(Y,X)$$

Correlation is symmetric: the amount that $$X$$ is correlated with $$Y$$ is the same as the amount that $$Y$$ is correlated with $$X$$. What you're talking about is the regression coefficient, which is in a sense anti-symmetric: the coefficient of $$X$$ with respect to $$Y$$ is the reciprocal of coefficient of $$Y$$ with respect to $$X$$ (well, almost; you actually get slightly different lines depending on what you treat as the independent variable and which you treat as the dependent). If a small change in $$X$$ tends to correspond to a large change in $$Y$$, then clearly a large change in $$Y$$ tends to correspond to a small change in $$X$$.

If you draw a scatter plot of two variables and then find the line of best fit, the slope of that line is the regression coefficient. Correlation is how tightly spaced the points are around that line. If you swap $$X$$ and $$Y$$, the new slope will be (roughly) the reciprocal of the old one, but the correlation will be the same.

Correlation is about consistency, not magnitude. So, for instance, if you have

(100, 9)
(50, 6)
(200, 21)

You have high correlation, because the second number is consistently close to 10% of the first number. If you have

(100, 2000)
(50, 300)
(200, 3000)

You have lower correlation; the second number is somewhere around ten times the first number, but it's much less consistent.

High correlation between the prices of two stocks is an indication of relative percentage changes at around a certain ratio in the same direction most of the time, but not necessarily in similar percentages or magnitudes. If a stock’s price moves, for example, one-fifth of another stock’s price in percentage terms in the same direction most of the time (rises 1% when the other rises 5%, drops 0.3% when the other declines 1.5%, etc.), the two would still have high correlation.