Mean, Variance and Covariance

What Are Mean, Variance and Covariance?

Mean, variance, and covariance are three of the most important statistical measures used in portfolio management. Together, they help investors evaluate expected returns, measure investment risk and understand how different assets move in relation to one another.

These concepts form the foundation of Modern Portfolio Theory and are essential for portfolio construction, diversification, and risk management. While the mean estimates an investment’s average return, variance measures the dispersion of returns around the average, and covariance shows whether two assets tend to move in the same or opposite directions.

In this study note, you’ll learn:

• What mean, variance, covariance and correlation measure.
• The difference between population and sample statistics.
• How covariance differs from correlation.
• Why these concepts are important in portfolio diversification.
• How these statistical measures are used in CFA Level I Portfolio Management.

Key Takeaways

• Mean measures the average return of an investment.
• Variance measures the dispersion of returns around the mean.
• Standard deviation is the square root of variance and represents investment risk.
• Covariance measures how two assets move together.
• Correlation standardizes covariance and ranges between -1 and +1.
• Diversification is most effective when assets have low or negative correlation.

The computation of mean, variance, and covariance statistics allows portfolio managers to compare the underlying securities’ return-risk characteristics and potential portfolio impact. These metrics are quantitatively determined and rely on historical price or return data. While we can compute the historical profile, this does not necessarily mean the relationship between assets or their return-risk profile will remain the same in the future.

Why Are Mean, Variance and Covariance Important?

Portfolio managers rarely evaluate investments based on expected returns alone. They also need to understand the level of uncertainty surrounding those returns and how different investments interact within a portfolio.

Mean, variance, and covariance provide this information.

• Mean estimates the average return an investor can expect.
• Variance measures how widely returns fluctuate around the average.
• Covariance shows whether two investments generally move together or in opposite directions.
• Correlation converts covariance into a standardized measure that is easier to interpret.

Together, these statistics allow investors to build diversified portfolios that balance expected return with acceptable levels of risk.

What Is Mean in Portfolio Management?

The mean of a set of values or measurements is the sum of all the measurements divided by the sum of all the measurements in the set:

$$ \text{Mean} = \frac{\sum_{i=1}^{n} x_{i}}{n} $$

If we compute the population’s mean, we call it the parametric or population mean, denoted by μ (read “mu”). If we get the mean of the sample, we call it the sample mean, denoted by the x bar.

What Is the Difference Between a Population and a Sample?

A population refers to the summation of all the elements of interest to the researcher.

• Examples: The number of people in a country, the number of hedge funds in the U.S., or even the total number of CFA candidates in a given year.

A sample is just a set of elements that represent the population as a whole. By analyzing sample data, we are able to make conclusions about the entire population.

• For example, if we sample the returns of 30 hedge funds spread across the U.S., we can use the results to make reasonable conclusions about the market as a whole (well over 10,000 hedge funds).

What Is Variance?

Variance is a measure of dispersion around the mean and is statistically defined as the average squared deviation from the mean. It is noted using the symbol σ².

$$ \sigma^2 = \frac{\sum_{i=1}^{N} (X_{i} – \mu)^2}{N} $$

Where μ is the population mean, and N is the population size.

The standard deviation, σ, is the square root of the variance and is commonly referred to as the volatility of the asset. Essentially, it is a measure of how far, on average, the observations are from the mean. A population’s variance is given by:

The population standard deviation equals the square root of the population variance. The sample variance is given by:

$$ S^2 = \frac{\sum_{i=1}^{N} (X_{i} – \bar{X})^2}{n-1} $$

Where X-bar is the sample mean, and n is the sample size.

Note that the sample standard deviation equals the square root of the sample variance.

What Is Covariance?

Covariance is a measure of how closely two assets move together. In covariance, we focus on the relationship between the deviations of some two variables rather than the deviation from the mean of one variable.

If the means of random variables \(X\) and \(Y\) are known, then the covariance between the two random variables can be determined as follows:

$$ { \hat { \sigma } }_{ xy }=\frac { 1 }{ n } \sum _{ i=1 }^{ n }{ \left( { x }_{ i }-{ \mu }_{ x } \right) } \left( { y }_{ i }-{ \mu }_{ y } \right) $$

If we do not know the means, then the equation changes to:

$$ { \hat { \sigma } }_{ xy }=\frac { 1 }{ n-1 } \sum _{ i=1 }^{ n }{ \left( { x }_{ i }-{ \hat { \mu } }_{ x } \right) } \left( { y }_{ i }-{ \hat { \mu } }_{ y } \right) $$

What Is Correlation?

Correlation is a concept that is closely related to covariance in the following way:

$$ { \rho }_{ xy }=\frac { { \sigma }_{ xy } }{ { \sigma }_{ x }{ \sigma }_{ y } } $$

Correlation ranges between +1 and -1 and is much easier to interpret than covariance. Two variables are perfectly correlated if their correlation is equal to +1. Note that they are uncorrelated if their correlation equals 0 and move in perfectly opposite directions if their correlation equals -1.

What Is the Difference Between Covariance and Correlation?

Covariance and correlation both measure the relationship between two variables, but they differ in how the relationship is expressed.

Covariance indicates whether two assets generally move in the same direction or opposite directions. However, because its value depends on the units of measurement, it is difficult to compare across different asset pairs.

Correlation standardizes covariance by dividing it by the product of each asset’s standard deviation. This produces a value between -1 and +1, making relationships much easier to interpret.

For this reason, portfolio managers often rely on correlation when evaluating diversification opportunities.

Mean vs. Variance vs. Covariance vs. Correlation

Although covariance indicates whether assets move together, correlation makes those relationships much easier to compare because it always falls between -1 and +1.

Real-World Example of Mean, Variance and Covariance

Suppose an investor is comparing two stocks.

Both Stock A and Stock B have an average annual return of 8%.

Although their expected returns are identical, Stock A experiences much larger fluctuations in price throughout the year. As a result, Stock A has a higher variance and is considered riskier.

Now suppose Stock A and Stock C tend to move in opposite directions during market fluctuations. Their covariance is negative, meaning that losses in one investment are often offset by gains in the other.

By combining assets with lower or negative covariance, investors can reduce overall portfolio risk without necessarily sacrificing expected returns.

CFA Exam Tip

Candidates frequently confuse covariance with correlation.

Remember:

• Covariance indicates the direction of the relationship but is not standardized.
• Correlation indicates both the direction and strength of the relationship using values between -1 and +1.
• Perfect positive correlation = +1.
• Perfect negative correlation = -1.
• Zero correlation indicates no linear relationship.

Understanding these distinctions is essential when answering portfolio diversification questions on the CFA exam.

Glossary

Mean — The average value of a dataset or investment returns.

Variance — A statistical measure of the dispersion of observations around the mean.

Standard Deviation — The square root of variance and the most common measure of investment risk.

Covariance — A measure of how two variables move relative to one another.

Correlation — A standardized measure of the relationship between two variables ranging from -1 to +1.

Diversification — The process of reducing portfolio risk by combining investments with different return patterns.

Frequently Asked Questions

What is mean in portfolio management?

Mean represents the average return generated by an investment over a specified period.

What is variance?

Variance measures how widely investment returns are dispersed around their average value. Higher variance generally indicates greater investment risk.

What is covariance?

Covariance measures whether two investments tend to move in the same direction or opposite directions.

What is the difference between covariance and correlation?

Covariance measures joint movement but is not standardized. Correlation standardizes covariance, making relationships easier to compare using values between -1 and +1.

Why is covariance important?

Covariance helps investors identify diversification opportunities by showing how investments move relative to one another.

What does negative covariance mean?

Negative covariance indicates that two investments generally move in opposite directions.

Why is correlation preferred over covariance?

Correlation provides a standardized measure that allows easier comparison between different asset pairs.

Summary of Mean, Variance and Covariance

These statistical measures provide the foundation for portfolio risk analysis.

Understanding how these measures work together enables portfolio managers to evaluate risk, estimate expected returns, and build diversified investment portfolios.

Question

In a two-asset portfolio, which combination of assets would result in the most diversified portfolio?

A. Correlation coefficient = 0.75.

B. Correlation coefficient = -0.2.

C. Correlation coefficient = 0.

Solution

The correct answer is B.

A diversified portfolio is produced, and portfolio risk is lowered within a two-asset portfolio by combining negatively correlated assets.