## 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 investment decisions. Investors weigh 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.

### Key Takeaways:

- Mean-variance analysis is a tool used by investors to weigh investment decisions.
- The analysis helps investors determine the biggest reward at a given level of risk or the least risk at a given level of return.
- The variance shows 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 preferred.
- Similarly, if two different securities have approximately the same variance, the one with the higher return is preferred.

## Understanding Mean-Variance Analysis

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 seek low risk and high reward. There are two main components tof mean-variance analysis: variance and expected return. Variance is a number that represents 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. 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. The goal of this strategy is to differentiate investments, which reduces the risk of catastrophic loss in the event of rapidly changing market conditions.

## Example of 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%.