Matakuliah
Tahun
: F0892 - Analisis Kuantitatif
: 2009
COVARIANCE DAN CORRELATION
PORTFOLIO
Pertemuan 11
COVARIANCE
• 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.
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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.
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In the formula above, the covariance between X and Y is shown as sigma-sub-XY
• 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().
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CORRELATION
• The correlation coefficient (often denoted by Greek
rho=ρ) 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).
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The Covariance Between Two Rates of Return
• The covariance is a (statistical) measure of how two random
variables (in this case, the returns of two securities) “move
together;”
• A positive covariance between the returns of two securities
indicates that the returns of the two securities tend to move in
the same direction, that is, better-than-expected returns for one
security are likely to occur when better-than-expected returns
occur for the other security;
• A negative covariance between the returns of two securities
indicates that the returns of the two securities tend to move in
opposite directions, that is, better-than-expected returns for one
security are likely to occur when worse-than-expected returns
occur for the other security;
• A relatively small or zero covariance between the
returns of two securities indicates that there is little or
no relationship between the returns of the two
securities;
• We denote the covariance between the return of
security i and the return of security j by  ij (the Greek
letter sigma);
• Note that
; ij ji
• You may use the Excel function “COVAR” to compute
the covariance between the returns of two securities.
The Correlation Coefficient
• The correlation coefficient is a statistical measure closely
related with the covariance;
• The interpretation of the correlation coefficient is that of a
“normalized covariance;”
• We denote the correlation coefficient between the return of
security i and the return of security j by ij (the Greek letter
pho);
• The relation between covariance and correlation is given by
the following equation:  ij ij  i ; j
• The correlation coefficient between the return of security i and
the return of security j lies between -1 and 1;
.
• If the correlation coefficient between the returns of two
securities is positive, then the returns of the two securities tend
to move in the same direction, that is, better-than-expected
returns for one security are likely to occur when better-thanexpected returns occur for the other security;
• If it is negative, then the returns of the two securities tend to
move in opposite directions, that is, better-than-expected
returns for one security are likely to occur when worse-thanexpected returns occur for the other security;
• If it is close to 0, then there is little or no relationship between
the returns of the two securities;
• You may use the Excel function “CORREL” to compute the
correlation coefficient between the returns of two securities.
70%
50%
30%
-40%
10%
10%
-10%
-30%
Weekly returns on stock A (y axis) versus Stock B(x axis)
in the period October XX-January XY. A & B an internet company
60%
Weakly Return
80%
70%
60%
50%
40%
30%
20%
10%
0%
-10%
-20%
-30%
Weekly returns on two internet stocks (A=red and B=blue) in the period October XX-January XY
12%
8%
4%
-30%
-10%
10%
0%
30%
-4%
Weekly returns on stock C(y axis) and Stock B (x axis) in the
period October XX-January XY
50%
70%
Weakly Return
80%
70%
60%
50%
40%
30%
20%
10%
0%
-10%
-20%
-30%
Weekly returns on stock B (blue) and stock C (red) in the period October XX-January XY
• Using the Excel formula “CORREL” the correlation
coefficient between the weekly returns of Stock A and
Stock B in the period of October XX-January XY
turned out to be 0.48;
• The correlation coefficient between the weekly returns
of Stock C and Stock B in that period was 0.16.