Mean, Variance and Covariance

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.

Mean

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.

Population vs. 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).

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.

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:

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

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.

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.