The **market portfolio** is the market value-weighted portfolio of all risky assets in an economy. In this post, we show how the market portfolio is equivalent to the optimal risky portfolio when all investors behave according to the modern portfolio theory.

Learning objectives

- Understand the link between the optimal risky portfolio and the market portfolio.
- Evaluate how investors can hold the market portfolio in practice.

## Moving from the optimal risky portfolio to the market portfolio

According to the modern portfolio theory, investors engage in mean-variance optimization to identify efficient portfolios that lie on the efficient frontier. These portfolios are optimal in the sense that they offer the best risk-return tradeoff in the market. Furthermore, when there is a risk-free asset, investors can split their funds between the risk-free asset and efficient portfolios by constructing capital allocation lines (CAL). The optimal risky portfolio is the efficient portfolio that yields the maximum Sharpe ratio (essentially a risk/reward ratio) such that the CAL is tangent to the efficient frontier. Therefore, modern portfolio theory advises investors to locate the optimal risky portfolio and divide their funds between that portfolio and the risk-free asset according to their degree of risk aversion.

### Investing in the optimal risky portfolio

What happens if all investors heed this advice and indeed invest in the optimal risky portfolio? 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 will invest $1,000 in the optimal risky portfolio, and Steve will invest $800. In order to hold the optimal risky portfolio, Linda should invest $500 in A, $300 in B, and $200 in C. Steve should invest $400 in A, $240 in B, and $160 in C. This information is summarized in Table 1 below.

Assets | Optimal risky portfolio | Linda | Steve | Market value |

Asset A | 50% | $500 | $400 | $900 |

Asset B | 30% | $300 | $240 | $540 |

Asset C | 20% | $200 | $160 | $360 |

Total | 100% | $1,000 | $800 | $1,800 |

### The composition of the market portfolio

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.

Now, the market portfolio is a 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%. So, the composition of the market portfolio is as shown in Table 2.

Assets | Market portfolio |

Asset A | 50% |

Asset B | 30% |

Asset C | 20% |

Total | 100% |

You are right, these figures do look familiar. They are the exact same weights of these 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. You can compute betas for portfolios of assets as well as individual assets. How about the beta of the market portfolio? What would that be? The answer is easy: It is equal to 1. In fact, we can easily prove this by using the mathematical definition of beta:

*β _{i} = σ_{im} / σ_{m}^{2}*

where *β _{i}* is the beta of asset

*i*,

*σ*is the covariance between the returns of asset

_{im}*i*and the market portfolio, and

*σ*is the variance of market portfolio returns. If we let asset

_{m}^{2}*i*be the market portfolio, then we have:

*β _{m} = σ_{mm} / σ_{m}^{2}*

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

*β _{m} = σ_{mm} / σ_{m}^{2}* =

*σ*

_{m}^{2}*/ σ*= 1

_{m}^{2}You can interpret this as the market portfolio carrying 1 unit of market risk. This sets a benchmark against which we can compare other assets. For example, a stock with beta equal to 2 carries twice as much market risk than the market portfolio. A corporate bond with beta equal to 0.5 bears half as much market risk as the market portfolio, and so on.

## Investing in index funds to gain market exposure

In theory, the market portfolio should contain all the risky assets in a market. This presents lots of challenges. First, there are tens of thousands of risky assets such as stocks, bonds (corporate, government), derivatives (futures, options, …), alternative investments, etc. in developed capital markets around the world. Second, there are many risky assets that do not trade on organized exchanges, which makes it very difficult to buy or sell them.

Given these challenges, is there a practical solution available to investors? Fortunately, yes. There are investment products available to help investors invest in proxies of the market portfolio. For example, there are mutual funds and exchange-traded funds that provide investors with broad exposure to capital markets. The cost of investing in index funds that track broad market indices is generally low as these funds do not require active management. Therefore, investors do not need to pay much to gain exposure to a broad range of assets. If you are interested in investing ETFs in the US market, some of the popular investment companies that offer such products are (in alphabetical order):

Note that this is not meant to be an exhaustive list, and there are many other reputable providers in the market.

### Wrapping up

In this post, we have shown that the market value-weighted portfolio of all risky assets in a market is equivalent to the optimal risky portfolio when all investors seek to invest in the latter. Do investors in reality do that? No, they don’t. Evidence suggests that there are many investors out there who concentrate their investments on a small number of assets, such as a few stocks. This is generally a bad idea because such portfolios are expected to be inefficient. That is, investors can obtain a higher level of return for the same level of risk.

The theory suggests that investors would be better off by investing in a proxy of the market portfolio than holding a concentrated portfolio. The former is well diversified and is expected to be efficient. There are many index funds that track broad market movements. Fees associated with such funds are low as they can be managed passively. As a result, investors do not need to worry about constructing the market portfolio themselves. They can gain exposure to the market movements indirectly and at a low cost via index funds.

##### What is next?

This post is part of our free course on investments. In the previous post, we introduced the concept of the optimal risky portfolio when a risk-free asset is available. Next, we will make a distinction between systematic risk and unsystematic risk and will explain how the latter can be eliminated via portfolio diversification.

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