### What Is a Mean-Variance Analysis?

Mean-variance analysis is the process of weighing risk, expressed as variance, against expected return. Investors use mean-variance analysis to make decisions about which financial instruments to invest in, based on how much risk they are willing to take on in exchange for different levels of reward. 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.

### Mean-Variance Analysis Explained

Mean-variance analysis is one part of modern portfolio theory, which assumes that investors will make rational decisions about investments if they have complete information. One assumption is that investors want low risk and high reward. There are two main parts of mean-variance analysis: variance and expected return. Variance is a number that represents how varied or spread out the numbers in a set are. For example, variance may tell how spread out the returns of a specific security are on a daily or weekly basis. The expected return is a probability expressing the estimated return of the investment in the security. If two different securities have the same expected return, but one has lower variance, the one with lower variance is the better pick. Similarly, if two different securities have approximately the same variance, the one with the higher return is the better pick.

In modern portfolio theory, an investor would choose different securities to invest in with different levels of variance and expected return.

### Sample Mean-Variance Analysis

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

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%.