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# Minimum Variance Portfolio Definition, Quiz and Tutorial

Here we give a name to the one theoretical portfolio with the lowest expected risk.
1. Define - Define the Minimum Variance Portfolio.
2. Context - Use MVP in a sentence.
3. Video - See the video for the visuals in Excel.
4. Script - Follow along with the transcript below.
5. Quiz - Test yourself.
Updated: February 17, 2021
Modeling the lowest risk portfolio is practically a challenge because it requires using past observations to estimate the future.

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## Minimum Variance Portfolio definition and tutorial

Intermediate

Minimum Variance Portfolio is a term from portfolio theory that describes the one portfolio with the lowest risk, as measured by variance. It sits at the tip of the hyperbola containing all possible combinations of all risky assets. Portfolio return and risk combinations can be measured over three different time periods: historical, expected or forecast.

Synonyms: MVP, lowest risk portfolio

For context, recall that there are three different timeframes portfolio managers utilize when deciding on the weights to assign to assets in a portfolio at any given time. First is the past, second is the present and third is the future. Setting future expectations for the return and risk of assets is a tricky endeavor, but when all data is collected a Mean Variance Optimization software program can solve for the weights of one portfolio that provides the lowest risk, or variance.

This Minimum Variance Portfolio in recent times has been packaged into a product called a minimum variance ETF. Recall, there is no magic in this, and it isn't a panacea for investments. An ETF constructed this way is just one service provider's view on the expected return and expected risk of the assets held in a portfolio. It sure sounds appealing and smart though, and that's why smart beta investment shops have been able to raise billions of dollars in these mutual funds in the post-2000 timeframe.

### In a Sentence

Doc:  For investors who follow MPT, there is an ETF that runs MVO and manages a Minimum Variance Portfolio or MVP.
Leo:  OMG.

### Video

This video can be accessed in a new window or App here , at our YouTube Channel or within this page below.

Minimum Variance Portfolio definition for investment modeling (4:35)

### Video Script

The script includes two sections where we visualize and demonstrate the concept of the Minimum Variance Portfolio.

#### Visualize

We're sitting in Excel and this is a snippet from our boot camp course.

We cover all of the curves, lines and dots here in one 40-minute video there, but because most people can't sit still for that long, we have eleven separate 4-5 minute videos like this one. I'll provide a link to the boot camp video number 22 (see Quant 101 instead) at the end if you prefer one video.

Ok, let's keep it simple and focus on the Minimum Variance Portfolio, which is part of Modern Portfolio Theory, developed by Harry Markowitz in the 1950s.

First of all, we have a chart, with expected return on the y-axis and expected risk on the x-axis. Here we are using the expected timeframe which uses past observations as input, then after making adjustments, you have what is baked-in to market expectations. Of course there is a lot to setting expectations, and that is the focus of the boot camp.

Now the MVP is the white dot at the tip of the curve. And if you draw the whole curve you'll have a parabola. And the parabola contains all of the portfolios, invested in all risky assets around the world, plus all of the combinations of weights to all assets. So this is a theoretical concept.

For academic theories to work, scholars include a list of assumptions, meaning holding other variables constant. And here is a list of assumptions for MPT to review later.

So here sits a portfolio with weights that minimize portfolio risk.

#### Demonstrate

Let's now demonstrate what's going on here and make it more practical. Image simplifying this by narrowing our focus to just four large US stocks, Microsoft, eBay, Abbott Labs and Merck.

Let's say this dot corresponds with a portfolio constructed with 100% in eBay, and 0% in the other three stocks. Next, this dot could be 50% eBay and 50% Microsoft. This one could be 33% in each of eBay, Microsoft and Abbott Labs. And finally, this could be 25% in all four stocks.

We walk through the math elsewhere, but the takeaway should be that through diversification, a portfolio of non-correlated assets has lower risk than the sum of the component stocks. And here, the MVP is the one portfolio with the lowest expected variance.

### Quiz

The calculation of the Minimum Variance Portfolio requires a covariance matrix. | True or False

True

A challenge with creating a Minimum Variance Portfolio in practice is that estimates of risk are typically based on the past and the future may not be like the past. | True or False

True

Still unclear on the Minimum Variance Portfolio? Leave a question in the comments section on YouTube. Also, see a tutorial page and video called Ace Your Portfolio Theory Exam in the Quant 101 Course.

### Related Terms

Our trained humans found other terms in the category Modern Portfolio Theory you may find helpful.

## What's Next?

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• To see all terms in the Glossary, click Outline.
• To learn about Mean Variance Optimization, hit Back.
• Google is big enough, help the little guy, click Tip.
• For a discussion about the Normal Distribution, Click Next.

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