Expected Value, Variance, Standard Deviation, Covariances, and Correlations of Portfolio Returns

A portfolio is a collection of investments a company, mutual fund, or individual investor holds. A portfolio consists of assets such as stocks, bonds, or cash equivalents. Financial professionals usually manage a portfolio.

Portfolio Expected Return

Portfolio expected return is the sum of each individual asset’s expected returns multiplied by its associated weight. Therefore:

$$ E(R_p) = \sum {W_i R_i} \text{ where i = 1,2,3 … n} $$

Where:

\(W_i\) = Weights (market value) attached to each asset \(i\).

\(R_i\) = Returns expected by each each asset \(i\).

Example: Portfolio Expected Return

Assume that we have a simple portfolio of two mutual funds, one invested in bonds and the other in stocks. Let us further assume that we expect a stock return of 8% and a bond return of 6%, and our allocation is equal in both funds. Then:

$$ \begin{align*} E(R_p)& = 0.5 × 0.08 + 0.5 × 0.06 \\ & = 0.07 \text{ or } 7\% \\ \end{align*} $$

Portfolio Variance

The variance of a portfolio’s return is always a function of the individual assets as well as the covariance between each of them. If we have two assets, A and B, then:

$$ \text{Portfolio variance} = { W }_{ A }^{ 2 } { \sigma }^{ 2 }\left( { R }_{ A } \right) +{ W }_{ B }^{ 2 }{ \sigma }^{ 2 }\left( { R }_{ B } \right) +2 \left( { W }_{ A } \right)  \left( { W }_{ B } \right)  Cov\left( { R }_{ A },{ R }_{ B } \right) $$

Portfolio variance is a measure of risk. More variance translates to more risk. Investors usually reduce the portfolio variance by choosing assets that have low or negative covariance, e.g., stocks and bonds.

Portfolio Standard Deviation

This is simply the square root of the portfolio variance. It is a measure of the riskiness of a portfolio. Thus:

$$ \text{Standard deviation}=\sqrt{ { W }_{ A }^{ 2 }  { \sigma }^{ 2 }\left( { R }_{ A } \right) +{ W }_{ B }^{ 2 }  { \sigma }^{ 2 }\left( { R }_{ B } \right) +2  \left( { W }_{ A } \right)  \left( { W }_{ B } \right)  \text{cov} \left( { R }_{ A },{ R }_{ B } \right)  } $$

Where:

\(\text{cov} ( { R }_{ A },{ R }_{ B })\) = Directional relationship between the returns on assets A and B.

Covariance

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 interest rate and the inflation rate. The general formula used to calculate the covariance between two random variables, \(X\) and \(Y\), is:

$$ \text{cov}[X,Y ] = E [(X – E[X ])(Y – E[Y ])] $$

While the abovementioned covariance formula is correct, we use a slightly modified formula to calculate the covariance of returns from a joint probability model. It is based on the probability-weighted average of the cross-products of the random variables’ deviations from their expected values for each possible outcome. Therefore, if we have two assets, \(i\) and \(j\), with returns \(R_i\) and \(R_j\) respectively, then:

$$ { \sigma }_{ { R }_{ i },{ R }_{ j } }=\sum _{ i=1 }^{ n }{ P\left( { R }_{ i } \right) \left[ { R }_{ i }-E\left( { R }_{ i } \right) \right] \left[ { R }_{ j }-E\left( { R }_{ j } \right) \right] } $$

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.
• A negative value shows that the variables move in opposite directions.

Correlation

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

$$ { \text{Correlation} }\left( { R }_{ i },{ R }_{ j } \right) =\frac { \text{Covariance}\left( { R }_{ i },{ R }_{ j } \right) }{ \text{Standard deviation}\left( { R }_{ i } \right) × \text{Standard deviation}\left( { R }_{ j } \right) } $$

Correlation measures the strength of the linear relationship between two variables. While covariance can take on any value between negative infinity and positive infinity, correlation is always a value between -1 and +1.

• +1 indicates a perfect linear relationship (i.e., the two variables move in the same direction with equal unit changes).
• Zero indicates no linear relationship at all.
• -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.

Example: Calculating the Correlation Coefficient #1

We anticipate 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 of a 10% return. We already know that 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%.

Suppose we wish to calculate the covariance and the correlation between ABC and XYZ returns, then:

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

$$ { \text{Correlation} }\left( { R }_{ i },{ R }_{ j } \right) =\frac { \text{Covariance}\left( { R }_{ ABC },{ R }_{ XYZ } \right) }{ \text{Standard deviation}\left( { R }_{ ABC } \right) × \text{Standard deviation}\left( { R }_{ XYZ } \right) } $$

Therefore:

$$ \begin{align*} \text{Correlation} & =\cfrac {0.0000561}{(0.01249 × 0.0046)} \\ & = 0.976 \\ \end{align*} $$

Interpretation: The correlation between the returns of the two companies is very strong (almost +1), and the returns move linearly in the same direction.

Example: Calculating Correlation Coefficient #2

An analyst is analyzing the impact of changes in interest rates introduced by the Central Bank on the country’s inflation rate. He analyzes historical data for five years. The covariance between the interest rate and the inflation rate is -0.00075, while the standard deviation of the interest rate is 5.5%, and the inflation rate is 12%. Calculate and interpret the correlation between interest rate and inflation rate.

Solution

$$\begin{align}\text{Correlation}_{\text{(Interest rate, Inflation)}}&=\frac{\text{Covariance (Interest rate, Inflation)}}{\text{Standard deviation of interest rate}\times \text{Standard deviation of inflation}}\\ &=\frac{-0.00075}{0.055\times 0.12}=-0.11364\end{align}$$

Interpretation: A correlation of -0.11364 indicates a negative correlation between the interest rate and the inflation rate.

Question

Assume that we have equally invested in two different companies; ABC and XYZ. We anticipate 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%. In addition, we already know that the expected value of returns is 8.2%, and the standard deviation is 1.249%.

Besides, we anticipate that the same probabilities 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%.

The portfolio standard deviation is closest to:

A. 0.0000561.

B. 0.00007234.

C. 0.00851.

The correct answer is C.

Actual calculation:

$$ \text{Portfolio variance} = { W }_{ A }^{ 2 } × { \sigma }^{ 2 }\left( { R }_{ A } \right) +{ W }_{ B }^{ 2 } × { \sigma }^{ 2 }\left( { R }_{ B } \right) +2× \left( { W }_{ A } \right) × \left( { W }_{ B } \right) × Cov\left( { R }_{ A },{ R }_{ B } \right) $$

First, we must calculate the covariance between the two stocks:

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

Since we already have the weight and the standard deviation of each asset, we can proceed and calculate the portfolio variance:

$$ \begin{align*} \text{Portfolio variance} & = 0.5^2× 0.01249^2+ 0.5^2× 0.0046^2+ 2 × 0.5 × 0.5× 0.0000561 \\ & = 0.00007234 \\ \end{align*} $$

Therefore, the standard deviation is \(\sqrt{0.00007234} = 0.00851\).