Limited Time Offer: Save 10% on all 2021 and 2022 Premium Study Packages with promo code: BLOG10    Select your Premium Package »

Mean, Variance and Covariance

Mean, Variance and Covariance

The computation of mean, variance and covariance statistics allow portfolio managers to compare the return-risk characteristics and potential portfolio impact of underlying securities. These metrics are quantitatively determined and rely on historic price or return data. While we can compute the historic 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 mean of the population, 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 and it is denoted by 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 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 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 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, therefore, 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 is equal to 0, and move in perfectly opposite directions if their correlation is equal to -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.

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

Featured Study with Us
CFA® Exam and FRM® Exam Prep Platform offered by AnalystPrep

Study Platform

Learn with Us

    Subscribe to our newsletter and keep up with the latest and greatest tips for success
    Online Tutoring
    Our videos feature professional educators presenting in-depth explanations of all topics introduced in the curriculum.

    Video Lessons



    Sergio Torrico
    Sergio Torrico
    2021-07-23
    Excelente para el FRM 2 Escribo esta revisión en español para los hispanohablantes, soy de Bolivia, y utilicé AnalystPrep para dudas y consultas sobre mi preparación para el FRM nivel 2 (lo tomé una sola vez y aprobé muy bien), siempre tuve un soporte claro, directo y rápido, el material sale rápido cuando hay cambios en el temario de GARP, y los ejercicios y exámenes son muy útiles para practicar.
    diana
    diana
    2021-07-17
    So helpful. I have been using the videos to prepare for the CFA Level II exam. The videos signpost the reading contents, explain the concepts and provide additional context for specific concepts. The fun light-hearted analogies are also a welcome break to some very dry content. I usually watch the videos before going into more in-depth reading and they are a good way to avoid being overwhelmed by the sheer volume of content when you look at the readings.
    Kriti Dhawan
    Kriti Dhawan
    2021-07-16
    A great curriculum provider. James sir explains the concept so well that rather than memorising it, you tend to intuitively understand and absorb them. Thank you ! Grateful I saw this at the right time for my CFA prep.
    nikhil kumar
    nikhil kumar
    2021-06-28
    Very well explained and gives a great insight about topics in a very short time. Glad to have found Professor Forjan's lectures.
    Marwan
    Marwan
    2021-06-22
    Great support throughout the course by the team, did not feel neglected
    Benjamin anonymous
    Benjamin anonymous
    2021-05-10
    I loved using AnalystPrep for FRM. QBank is huge, videos are great. Would recommend to a friend
    Daniel Glyn
    Daniel Glyn
    2021-03-24
    I have finished my FRM1 thanks to AnalystPrep. And now using AnalystPrep for my FRM2 preparation. Professor Forjan is brilliant. He gives such good explanations and analogies. And more than anything makes learning fun. A big thank you to Analystprep and Professor Forjan. 5 stars all the way!
    michael walshe
    michael walshe
    2021-03-18
    Professor James' videos are excellent for understanding the underlying theories behind financial engineering / financial analysis. The AnalystPrep videos were better than any of the others that I searched through on YouTube for providing a clear explanation of some concepts, such as Portfolio theory, CAPM, and Arbitrage Pricing theory. Watching these cleared up many of the unclarities I had in my head. Highly recommended.