# Market portfolio

We have so far learned how to calculate the risk and return of portfolios and how to trace an efficient frontier through mean-variance optimization. It is now time to introduce a special portfolio that will play a significant role when we discuss the CAPM: The market portfolio.

## What is the market portfolio?

The market portfolio is the value-weighted portfolio of all risky assets in an economy. Value-weighting (as opposed to equal-weighting) means that assets with high market values (e.g., large-cap stocks) would have a higher weighting in the portfolio than those with low market values (e.g., small-cap stocks). In practice, broad-based market indices such as the S&P500 serve as a proxy of this portfolio.

According to the capital asset pricing model, the market portfolio is equivalent to the optimal risky portfolio when all investors behave according to the modern portfolio theory by engaging in mean-variance optimization.

Let’s illustrate this with an example. Say, there are only three risky assets in a market A, B, and C. And, let’s suppose the composition of the optimal risky portfolio is:

50% A, 30% B, and 20% C.

Let’s also assume there are only two investors in this market: Linda and Steve. Linda has \$1,000 to invest in the market, and Steve has \$800. In order to “hold” the optimal risky portfolio, Linda should invest \$500 in A, \$300 in B, and \$200 in C. Why? This ensures that \$500/\$1,000 = 50% of her portfolio is A, \$300/\$1,000 = 30% is B, and \$200/\$1,000 = 20% is C, which reflects the composition given above. Following the same logic, Steve should invest \$400 in A, \$240 in B, and \$160 in C. This information is summarized in Table 1 below.

If Linda and Steve are the only investors in the market and their total investments in risky assets are \$1,000 and \$800, respectively, then the total market value of all risky assets (i.e., A, B, and C) is \$1,000 + \$800 = \$1,800.

We can now find the composition of the market portfolio, which is the value-weighted portfolio of assets A, B, and C. What is the overall weight of asset A in the market? \$900 / \$1,800 = 50%. How about B and C? \$540 / \$1,800 = 30% and \$360 / \$1,800 = 20%, respectively. This is summarized in Table 2.

Do these investment weights look familiar? They should! They exactly match the weights of these three assets in the optimal risky portfolio. Therefore, when all investors attempt to hold the optimal risky portfolio, it becomes the market portfolio!

## The beta of the market portfolio

The beta of an asset quantifies that asset’s exposure to market risk. The same is true for portfolios of assets as well as individual assets. If that is the case, what should be the beta of the market portfolio? The answer is easy: It should be equal to 1. In fact, we can easily prove this using the mathematical definition of beta:

where βi is the beta of asset iσim is the covariance between asset i and the market, and σm2 is the variance of market returns. If we let asset i be the market portfolio, then we have:

As we have discussed before an asset’s covariance with itself is equal to its variance, so we have:

This completes the proof.

##### Summary

In this lesson, we have explained what is meant by the market portfolio. We have defined it as the value-weighted portfolio of all risky assets in a particular market. Moreover, we have shown that this portfolio has a beta of 1 and that it is typically proxied by a market index. We will further examine this portfolio within the context of CAPM in the following lessons.