*Basic

Mean-variance analysis is the process of weighing risk, expressed as variance, against expected return.[1] Investors use mean-variance analysis to make investment decisions. Investors weigh how much risk they are willing to take on in exchange for different levels of reward.[2] Mean variance analysis allows investors to find the biggest reward at a given level of risk or the least risk at a given level of return.[3]

**Intermediate

The expected return is a probability expressing the estimated return on the investment in the security.[4] If two securities have the same expected return, but one has lower variance, the one with lower variance is preferred.[5] Similarly, if two different securities have approximately the same variance, the one with the higher return is preferred.[6] Mean variance analysis is one part of modern portfolio theory, which assumes that investors will make rational choices about investments if they have complete information.[7] One assumption is that investors seek low risk and high reward.[8] There are two main components of mean-variance analysis: variance and expected return.[9] Variance is a number that says how varied or spread out the numbers are in a set. For example, variance may tell how spread out the returns of a specific security are on a daily or weekly basis.[10] The expected return is the probability expressing the estimated return of the investment in the security.[11]

***Advanced

It is possible to calucate which investments have the greatest variance and expected return. Assume the following investments are in an investor's portfolio:1[12]

Investment A: Amount = $100,000 and expected return of 5%

Investment B: Amount = $300,000 and expected return of 10%

In a total portfolio value of $400,000, the weight of each asset is:

Investment A weight = $100,000 / $400,000 = 25%

Investment B weight = $300,000 / $400,000 = 75%

Therefore, the total expected return of the portfolio is the weight of the asset in the portfolio multiplied by the expected return:

Portfolio expected return = (25% x 5%) + (75% x 10%) = 8.75%. Portfolio variance is more complicated to calculate because it is not a simple weighted average of the investments' variances. The correlation between the two investments is 0.65. The standard deviation, or square root of variance, for Investment A is 7%, and the standard deviation for Investment B is 14%. 

In this example, the portfolio variance is:

Portfolio variance = (25% ^ 2 x 7% ^ 2) + (75% ^ 2 x 14% ^ 2) + (2 x 25% x 75% x 7% x 14% x 0.65) = 0.0137

The portfolio standard deviation is the square root of the answer: 11.71%.

Sources

[1] Editors (2023). Mean-Variance Analysis: Overview, Components, Example. Corporatefinanceinstitute.com
[2] Editors. Modern portfolio theory. Wikipedia. en.wikipedia.org
[3] Editors (2022). What is mean variance analysis? Indeed.com
[4] Editors (2022). Mean-Variance Analysis – Explained. Thebusinessprofessor.com
[5] Editors. Mean-Variance Optimization. Columbia.edu
[6] Editors. Mean-Variance Portfolio Theory. Analystprep.com
[7] Editors (2023). How mean-variance optimization works in investing. Smartasset.com
[8] Levy, H (2004). Prospect Theory and Mean-Variance Analysis. Jstor.org
[9] Brunnermeier, M. K. Mean-Variance Analysis & Capital Asset Pricing Model (CAPM). Princeton.edu
[10] Editors. Mean Variance Portfolio Theory. Kent.ac.uk
[11] Campbell, J. Y. Long-Horizon Mean-Variance Analysis: A User Guide. Scholar.harvard.edu
[12] Editors. Mean-Variance Analysis. Gpennacc.web.illinois.edu