Variances and covariances
Michael Wood
Printed March 6, 2016
This handout explains some of the mathematical properties of variances and covariances.
This may be helpful for your understanding of some theories in finance, but you will not
be expected to be familiar with this material in the Decision and Risk Analysis exam
(although it might enhance your answers to some questions). Read this in conjunction
with the example on the spreadsheet VarianceCovariance.xls .
The variance is the square of the standard deviation. In symbols this is written
VAR(X) =(1/n) Σ (X−XBar)2
(Equation 1)
Where n is the sample size, and XBar is the mean of all the X’s. (Σ means “the sum of the
bit to the right worked out for each individual X”. Sometimes—if there is a possibility of
ambiguity—these individual X’s are called Xi.)
The covariance is a bit like the variance except that it involves another variable, Y:
COVAR(X,Y)=(1/n) Σ (Xi−XBar)(Yi−YBar)
(Equation 2)
To see how the covariance works imagine first that there is a strong positive correlation
between X and Y. This means that if Xi is above the mean of the X’s (Xbar), then Yi is
also likely to be above the mean of the Y’s (Ybar). On the other hand if Xi is below the
mean of the X’s (Xbar), then Yi is also likely to be below the mean of the Y’s (Ybar). In
either case multiplying them together will give a positive result, so the covariance will be
positive.
Now imagine X and Y are negatively correlated. This means if one is above the mean, the
other is likely to be below, so the signs are likely to be different and result is negative.
Adding up lots of negative terms, of course, is likely to give a negative covariance.
Obviously, of the correlation is around zero, some terms will be positive, others negative,
and the result is likely to be a covariance that is around zero.
This means the covariance is a bit like a correlation coefficient. The only problem is that
its magnitude will depend on size of X’s and Y’s. For this reason we divide by the
standard deviation of the X’s and of the Y’s (as it’s the deviations from the mean that
feed into the covariance, not the absolute size of the X’s and Y’s) if we want a correlation
coefficient. This gives Pearson’s correlation coefficient, r:
r = COVAR(X,Y)/(sx sy)
(Equation 3)
There is also a formula for the slope of a regression line in terms of the covariance and
variances. If you are using a single variable, X, to predict Y, the slope of the least squares
prediction line is
Slope = COVAR(X,Y)/VAR(X)
(Equation 4)
This is easy to prove if you know a bit of calculus (see a book on mathematical statistics,
or ask me if you are interested).
The variance and covariance are particularly useful because they enable us to analyse
sums and averages of a series of variables. The following formula is always true for
any two variables X and Y:
VAR(X+Y) = VAR(X) + VAR(Y) + 2COVAR(X,Y)
(Equation 5)
This is proved below—but don’t worry about this unless you are interested.
This formula is particularly useful when X and Y are uncorrelated, because then
COVAR(X,Y)=0 and the formula becomes
VAR(X+Y) = VAR(X) + VAR(Y)
(Equation 6)
With any particular set of data, this will only be exactly true if the correlation (and
covariance) is exactly zero. However, it is a useful approximation for two variables
whose correlation is approximately zero.
Equations 5 and 6 can be extended to cover more than two variables. Equation 5 is a bit
complicated to extend, but Equation 6 is easier:
VAR(X+Y+Z+…) = VAR(X) + VAR(Y) + VAR(Z) + …
(Equation 7)
provided that the variables are not correlated (i.e. the correlations between each pair of
variables is zero).
In ordinary language, Equation 7 says that if variables are uncorrelated, the variance of
the sum is the same as the sum of the variances.
Equation 7 can be used to work out the predicted standard deviation of the mean of a
random sample. If the sample is X1, X2, … Xn we can write the mean as
(1/n)X1+ (1/n)X2, + … (1/n)Xn . The individual members of the sample are obviously
uncorrelated, so the variance of this sum is just the sum of the variances. It should be
obvious from Equation 1 that
VAR{(1/n)X} = (1/n)2 VAR(X)
so if all the X’s are from the same population
VAR(sample mean) = (1/n)2 VAR(X) + … + (1/n)2 VAR(X)
or (remembering the sample size is n)
VAR(sample mean) = (1/n) VAR(X)
So
Standard deviation of the sample mean = (standard deviation of X)/√n (Equation 8)
This is often called the standard error of the mean.
Proof of Equation 5
VAR(X+Y) = (1/n)∑{(X+Y) – (XBar-YBar)}2
= (1/n)∑{(X-XBar) – (Y-YBar)}2
= (1/n)∑{(X-XBar)2 + (Y-YBar)2 + 2(X-XBar)(Y-YBar)}
= VAR(X) + VAR(Y) + 2COVAR(X,Y)
which is Equation 4.