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The covariance is a measure of the degree of co-movement between two random variables. For instance, we could be interested in the degree of co-movement between the rate of interest and the rate of inflation.
The general formula used to calculate the covariance between two random variables, X and Y, is:
$$COV\ [X,\ Y] =E[(X-E[X]) (Y-E[Y])]$$
The covariance between two random variables can be positive, negative, or zero.
Moreover, if variables are independent, their covariance is zero, i.e.,
$$COV\left(X,Y\right)=E\left(XY\right)-E\left(X\right)E\left(Y\right)=0$$
Covariances can be represented in a tabular format in a covariance matrix as follows:
Asset |
A |
B |
C |
A |
Cov(A, A) |
Cov(A, B) |
Cov(A, C) |
B |
Cov(B, A) |
Cov(B, B) |
Cov(B, C) |
C |
Cov(C, A) |
Cov(C, B) |
Cov(C, C) |
A portfolio comprises two stocks – 1 and 2. The returns for the last 5 years are as follow:
Stock 1: 5%; 4.5%; 4.8%; 5.5%; 6%.
Stock 2: 6%; 6.2%; 5.7%; 6.1%; 6.5%.
Compute the covariance.
Step 1: We calculate the weighted sum of each stock to get the expected return on Stock 1 and Stock 2
$$ E(R_1) = \frac{5\%+4.5\%+4.8\%+5.5\%+ 6\%}{5} = 5.2\% $$
$$ E(R_2) = \frac{6\%+ 6.2\%+ 5.7\%+ 6.1\%+ 6.5\%}{5} = 6.1\% $$
Step 2: We subtract each year’s return from the expected return to obtain [R1-E(R1)] and [R2-E(R2)] as follows:
Year |
R_{1} |
R_{2} |
[R_{1}-E(R_{1})] |
[R_{2}-E(R_{2})] |
1 |
5% |
6% |
-0.2% |
-0.1% |
2 |
4.5% |
6.2% |
-0.7% |
0.1% |
3 |
4.8% |
5.7% |
-0.4% |
-0.4% |
4 |
5.5% |
6.1% |
0.3% |
0% |
5 |
6% |
6.5% |
0.8% |
0.4% |
Step 3: We multiply the values obtained in Step 2 and we divide by the number of observations to get a mean observation.
$$Cov\left(R_1,R_2\right)=E\left[R_1-E\left(R_1\right)\right]\left[R_2-E\left(R_2\right)\right]$$
$$=\frac{(-0.2\%\times-0.1\%)+(-0.7\%\times0.1\%)+(-0.4\%\times-0.4\%)+(0.3\%\times0\%)+(0.8\%\times0.4\%)}{5}$$
$$=\frac{0.000043}{5}=0.0000086$$
Consider a set of a well-diversified portfolios X and Y. Suppose that the mean returns for X and Y are 6.12 and 7.04, respectively, what is the covariance between these portfolios?
Recall,
\(COV\left(X,Y\right)=E\left(XY\right)-E\left(X\right)E\left(Y\right)\), for independent variables,
\(E\left(XY\right)=E\left(X\right)E\left(Y\right)=6.12\times7.04=43.0848\)
Which implies that,
\(COV\left(X,Y\right)=E\left(XY\right)-E\left(X\right)E\left(Y\right)=43.0848-6.12\left(7.04\right)=0\)
Correlation is the ratio of the covariance between two random variables and the product of their two standard deviations i.e.
$$\text{Correlation}\ (X_1,X_2\ )=\frac{Cov(X_1,X_2\ )}{Standard\ deviation\ (X_1\ )\times Standard\ deviation\ (X_2\ )}$$
Correlation measures the strength of the linear relationship between two variables. While the covariance can take on any value between negative infinity and positive infinity, the correlation is always a value between -1 and +1.
A correlation of -1 indicates a perfect inverse relationship (i.e. a unit change in one means that the other will have a unit change in the opposite direction). Secondly, a correlation of +1 indicates a perfect linear relationship (i.e. the two variables move in the same direction with the unit changes being equal). Finally, a correlation of zero implies that there is no linear relationship between the variables.
We anticipate that there is a 15% chance that next year’s stock returns for ABC Corp will be 6%, a 60% probability that they will be 8% and a 25% probability that they will be 10%. We already know the expected value of returns is 8.2% and the standard deviation is 1.249%.
We also anticipate that the same probabilities and states are associated with a 4% return for XYZ Corp, a 5% return, and a 5.5% return. The expected value of returns is then 4.975 and the standard deviation is 0.46%.
$$ \begin{align*}
\text{Covariance}, \text{cov}(\text R_{ \text{ABC}},\text R_{ \text{XYZ}}) & = 0.15(0.06 – 0.082)(0.04 – 0.04975) \\
& + 0.6(0.08 – 0.082)(0.05 – 0.04975) \\
& + 0.25(0.10 – 0.082)(0.055 – 0.04975) \\
& = 0.0000561 \\
\end{align*} $$
$$\text{Correlation}\ (R_i,R_j\ )=\frac{Cov(R_i,R_j\ )}{(Standard\ deviation\ (R_i\ )\times Standard\ deviation\ (R_j\ ))}$$
Thus,
$$\text{Correlation}=0.0000561\left(0.01249\times0.0046\right)=0.976$$
Interpretation: The correlation between the returns of the two companies is very strong (almost +1) and the returns move linearly in the same direction.
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As mentioned earlier, correlation ranges from -1 to +1
In conclusion, using negatively correlated investments to form a portfolio helps to reduce the overall volatility of the portfolio.
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