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# Mean Variance Optimization Definition and Quiz

Since portfolio theory postulates that all investors balance expected returns of a portfolio against expected risk, shouldn't we at least know how that works?
1. Define - Define Mean Variance Optimization for portfolios.
2. Context - Use MVO in a sentence.
3. Quiz - Test yourself.
Updated: February 17, 2021
This aspect of portfolio theory can be anything but practical, yet MVO still offers structure for a well-thought-out portfolio management process.

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## Mean Variance Optimization definition

Intermediate

Mean Variance Optimization (MVO) is the framework for maximizing the reward-to-risk ratio of a portfolo. Mean in the numerator refers to the mean expected return and variance in the denominator refers to a measure of expected variability. MVO is a key takeaway from Modern Portfolio Theory as it is assumed that all investors maximize returns relative to risk. A software program called an MVO optimizer generates weights to securities while maximizing the reward-to-risk ratio.

Synonyms: Mean-Variance, MVO, Mean-Variance Optimization

For context, recall that nearly every individual views the world differently. And there are about as many investment processes as there are people. Yet, in academia, scholars have settled on the idea that investors find an expected return on all risky assets and the correlation among those assets. From there, investors create a covariance matrix and utilize mean variance optimization to find the weights of all holdings.

The objective function in the optimizer is to maximize the expected return, or mean in MVO, while minimizing expected risk using the measure variance.

• Sharpe Ratio = Expected Return (mean) / Standard Deviation (variance)

In a plain vanilla optimization case some investors instead use the square root of variance, which is standard deviation. In this case, the MVO optimizer maximizes the ratio called the Sharpe Ratio. In the end, the mean variance optimizer returns the weights to each security. Constrants can be added to fine-tune the resulting portfolio so as to not short stocks, to limit the number positions or to restrict the weights allocated to sectors.

To visualize on a scatterplot, risk is typically charted on the x-axis and return is charted on the y-axis, so portfolios up and to the left, or in the northwest quadrant are preferred over portfolios in any other quadrant.

### In a Sentence

Doc:  One MPT assumption is that all investors are MVO aware.
Doc:  In theory, right? My stockbroker says that almost no one follows MPT.

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### Quiz

Click box for answer.

Since standard deviation is the square root of variance, you would get similar results if you maximize the mean-to-standard-deviation ratio? | True or False

True

The primary purpose of a mean variance optimization is to select the weights assigned to each asset in a portfolio that results in the highest ratio of expected risk to expected return? | True or False

False, only the last part is reversed.

#### Comments or Questions?

Still unclear on Mean Variance Optimization? Check out our Excel Series called Quant 101 with 27 videos on portfolio return and risk measurement.

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