Apr 12

Covariance and Correlation

by David Harper, CFA, FRM, CIPM

FRM | Quant |

Covariance measures the linear relationship between two variables. For example, how do the asset returns of a high yield hedge fund strategy relate (or compare) to the asset returns of a market neutral strategy. Positive covariance says the variables tend to increase or decrease together; negative covariance says they tend to move in opposite directions. If they negatively covary, that tells us that one asset can be a hedge for the other. If the variables are totally independent, the covariance is zero.

Why covariance matters?

Portfolio diversification benefits are realized when assets in the portfolio do not perfectly covary. Or we could also say, when assets are imperfectly correlated. Correlation is the unitless version of covariance: it is covariance translated into a standardized measure, from -1.0 (perfect negative correlation) to 1.0 (perfectly correlated). Covariance is the linear relationship expressed in situation-specific units; correlation is the same but without units.

Given a portfolio, we can generally reduce portfolio variance (and standard deviation) by adding uncorrelated assets.

If the variables are (X) and (Y), covariance is equal to the expected product of the variables, E(XY) minus the product of their mean values:

 

In the formula above, the covariance between X and Y is shown as sigma-sub-XY. We could also show this as cov(x,y). In Excel, covariance is given by the function =COVAR().

The correlation coefficient (often denoted by Greek rho) is given by the following important equation:

In  words, the correlation coefficient is the covariance divided by the product of the standard deviations. If we rearrange this formula, we get the relationship: covariance is equal to the product of (correlation coefficient)(standard deviation of first variable)(standard deviation of second variable).

An example with hedge fund indices

I pulled ten years of hedge fund index data courtesy of Hedge Fund Research, Inc.. Monthly returns are given for three strategies, equity hedge (EH), high yield (HY), and market neutral (MN):

 

This gives us three series, 120 data points in each series (10 years x 12 months/year). In Excel, we can calculate the standard deviation =STDEVP(), the correlation =CORREL(), and the covariance =COVAR():

 

For example, the correlation between equity hedge and high-yield is about 38.6%. The covariance between equity hedge and high-yield is about 0.91.

The covariance is not intuitively meaningful but the correlation coefficient is intuitive. Note above that we used Excel to calculate the correlation coefficient between EH and MN to be 38.6%. Let's try to calculate correlation the "long way," by taking the covariance and dividing it by the product of the standard deviations:

...and we get the same result, 38.6%.

Comments

  1. John C. Vesey 19 Apr 2007

    Dear David,

    As always you do a superb job. I haven’t contacted you about the CFP exam as my wife has been rather ill disposed. She underwent surgery last week and is now in the recovery phase. One good think about aging is that one’s sense of humor seems to improve. Everything is a double entendre!! In any event, I shall get my thoughts to you by next week and shall continue to access your web site. By the way, is it time for me to renew my subscription. Please let me know as I shall comply tuite suite. John

  2. David Harper, CFA, FRM 19 Apr 2007

    Hi John,

    I am sorry to hear that but I wish you both well, sincerely. Thank you for offering a ‘silver lining’ to the aging dilemma ...i am not so funny i so i shall need to find some other recompense.

    Re the subscription, John, you were my first customer. Your support goes way back I am a grateful. Your money is not so good here...Good luck with the important matters at hand

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