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 variables X and Y, where we can let:

• X = interest rate
• Y = inflation

The general formula used to calculate the covariance between two random variables, X and Y, is:

$$COV\ [X,\ Y] =E[(X-E[X]) (Y-E[Y])]$$

Where:

\(X\) = a random variable such as lending interest rates.

\(Y\) = another random variable, such as inflation rates.

\(E[X]\) = The expected value (mean) of X.

\(E[Y]\) = The expected value (mean) of Y.

The covariance between two random variables can be positive, negative, or zero.

• A positive number indicates co-movement (i.e., the variables tend to move in the same direction);
• A value of zero indicates no relationship; and
• A negative value shows that the variables move in opposite directions.

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:

$$ \begin{array}{c|ccc} \text{Asset} & A & B & C \\ \hline A & \text{Cov}(A, A) & \text{Cov}(A, B) & \text{Cov}(A, C) \\ B & \text{Cov}(B, A) & \text{Cov}(B, B) & \text{Cov}(B, C) \\ C & \text{Cov}(C, A) & \text{Cov}(C, B) & \text{Cov}(C, C) \\ \end{array} $$

• The off-diagonal terms represent variances since, for example, \(\text{Cov(C, C) = Var(C)}\)
• A two-asset portfolio would have a similar 2 × 2 matrix.
• A correlation matrix can also be created to represent the correlations between various assets in a large portfolio.

Example 1: Calculating the Covariance of a Portfolio of Two Assets

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.

Solution

Step 1: We calculate the weighted sum of each stock to get the expected return on Stock 1 and Stock 2

$$\begin{align} E(R_1)& = \frac{5\%+4.5\%+4.8\%+5.5\%+ 6\%}{5} = 5.2\% \\ E(R_2) &= \frac{6\%+ 6.2\%+ 5.7\%+ 6.1\%+ 6.5\%}{5} = 6.1\% \end{align}$$

Step 2: We subtract each year’s return from the expected return to obtain [R1-E(R1)] and [R2-E(R2)] as follows:

$$ \begin{array}{c|c|c|c|c} \textbf{Year} & R_1 & R_2 & R_1 – \mathbb{E}(R_1) & R_2 – \mathbb{E}(R_2) \\ \hline 1 & 5\% & 6\% & -0.2\% & -0.1\% \\ 2 & 4.5\% & 6.2\% & -0.7\% & 0.1\% \\ 3 & 4.8\% & 5.7\% & -0.4\% & -0.4\% \\ 4 & 5.5\% & 6.1\% & 0.3\% & 0\% \\ 5 & 6\% & 6.5\% & 0.8\% & 0.4\% \\ \end{array} $$

Step 3: We multiply the values obtained in Step 2 and we divide by the number of observations to get a mean observation.

$$\begin{align}
\text{Cov}(R_1, R_2) &= \mathbb{E}[(R_1 – \mathbb{E}[R_1])(R_2 – \mathbb{E}[R_2])] \\
&= \frac{
\begin{array}{l}
(-0.002 \times -0.001) + (-0.007 \times 0.001) + \\
(-0.004 \times -0.004) + (0.003 \times 0) + (0.008 \times 0.004)
\end{array}
}{5} \\
&= \frac{0.000002 + (-0.000007) + 0.000016 + 0 + 0.000032}{5} \\
&= \frac{0.000043}{5} = 0.0000086
\end{align}$$

Example 2: Covariance

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?

Solution

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

Correlation is the ratio of the covariance between two random variables and the product of their two standard deviations i.e.

$$\text{Correlation}\ (X_1,X_2\ )=\frac{Cov(X_1,X_2\ )}{\text{Standard deviation}\ (X_1\ )\times \text{Standard deviation}\ (X_2\ )}$$

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.

Example: Calculating the Covariance

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%.

$$ \begin{align}
\text{cov}(\text{R}_{ \text{ABC}},\text{R}_{ \text{XYZ}}) & = 0.15(0.06 – 0.082)(0.04 – 0.04975) \\
& + 0.6(0.08 – 0.082)(0.05 – 0.04975) \\
& + 0.25(0.10 – 0.082)(0.055 – 0.04975) \\
& = 0.0000561 \\
\end{align} $$

Recall that,

$$\text{Corr}(\text{R}_{ \text{ABC}},\text{R}_{ \text{XYZ}})=\frac{\text{Cov}(\text{R}_{ \text{ABC}},\text{R}_{ \text{XYZ}}\ )}{(\text{Standard deviation}\ (\text{R}_{ \text{ABC}}\ )\times \text{Standard deviation} (\text{R}_{ \text{XYZ}}\ ))}$$

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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How does Correlation Impact Portfolio Risk?

As mentioned earlier, correlation ranges from -1 to +1

• +1 = returns are perfectly positively correlated.
• 0 = returns of two assets are not correlated.
• -1 = returns are perfectly negatively correlated.
• What happens to portfolio risk (in a portfolio of two risky assets) when the two assets are perfectly correlated?
• Risk is unaffected; no benefit of diversification

In conclusion, using negatively correlated investments to form a portfolio helps to reduce the overall volatility of the portfolio.

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FAQs on Covariance and Correlation

1. What is the covariance formula for CFA Level 1?

The formula for covariance between two random variables \( X \) and \( Y \) is:

\[ \text{Cov}(X, Y) = \mathbb{E}\left[(X – \mathbb{E}[X])(Y – \mathbb{E}[Y])\right] \]

Alternatively:

\[ \text{Cov}(X, Y) = \mathbb{E}[XY] – \mathbb{E}[X]\mathbb{E}[Y] \]

Where:

• \( X, Y \) = Random variables (e.g., returns, interest rates)
• \( \mathbb{E}[X], \mathbb{E}[Y] \) = Expected values (means) of \( X \) and \( Y \)
• \( \mathbb{E}[XY] \) = Expected value of the product \( XY \)

2. What’s the difference between covariance and correlation?

Covariance indicates the direction of co-movement between two variables but is affected by the units of measurement. Correlation standardizes this to a scale from -1 to +1.

\[ \rho(X, Y) = \frac{\text{Cov}(X, Y)}{\sigma_X \cdot \sigma_Y} \]

Where:

• \( \rho(X, Y) \): Correlation between \( X \) and \( Y \)
• \( \text{Cov}(X, Y) \): Covariance between \( X \) and \( Y \)
• \( \sigma_X, \sigma_Y \): Standard deviations of \( X \) and \( Y \), respectively

3. How do I convert covariance to correlation?

Use the correlation formula to convert covariance to a dimensionless metric:

\[ \text{Correlation} = \frac{\text{Cov}(X, Y)}{\sigma_X \cdot \sigma_Y} \]

Where:

• \( \text{Cov}(X, Y) \): Covariance between the two variables
• \( \sigma_X \): Standard deviation of variable \( X \)
• \( \sigma_Y \): Standard deviation of variable \( Y \)

4. Why is a correlation of zero not the same as independence?

A correlation of zero means there’s no linear relationship between variables, but they may still be dependent in other ways. Independence is a much stronger condition.

$$\begin{align}\rho(X, Y) = 0 &\Rightarrow \text{No linear relationship} \\ \text{Cov}(X, Y) = 0 & \Rightarrow \text{Independence} \end{align}$$

However, the converse is not always true: zero covariance does not imply independence unless the variables are jointly normally distributed.

5. How does correlation affect portfolio risk?

The correlation between assets determines the potential for diversification in a portfolio. The lower the correlation, the greater the risk-reduction benefit.

• \( \rho = +1 \): Perfect positive correlation – no diversification benefit
• \( \rho = 0 \): No linear relationship – some diversification benefit
• \( \rho = -1 \): Perfect negative correlation – maximum diversification benefit

In general, combining negatively or weakly correlated assets helps reduce portfolio volatility.