What is the optimal risky portfolio?

What is the optimal risky portfolio?

In the previous post, we explained that when there is no risk-free asset in an economy, investors should invest in one of the efficient portfolios that lie on the efficient frontier based on their risk tolerance. We added that, if a risk-free asset exists, then there is a unique efficient portfolio that all investors should invest in. This unique efficient portfolio is known as the optimal risky portfolio, which we examine in detail in this post.

Learning objectives

  • Understand what a capital allocation line is.
  • Locate the optimal risky portfolio on the efficient frontier.

Capital allocation line

When a risk-free asset exists in an economy, investors can add that asset into their portfolios if they wish so. In the risk-return space, the combination of the risk-free asset and any risky portfolio is a straight line. This line is called the capital allocation line as it shows how an investor’s capital is allocated between the risk-free asset and the risky portfolio. We discuss below an example where an investor splits his wealth between the riskless asset and an efficient portfolio.

Suppose that an investor has $1,000 to invest. She decides to split her funds between the risk-free asset Rf and an efficient portfolio A. In Figure 1, this is represented as the capital allocation line that goes through Rf and A.

For example, if the investor invests $500 in Rf and $500 in A, her portfolio lies at point B, which is exactly halfway between Rf and A. If she invests $200 in Rf and $800 in A, her portfolio is still in between Rf and A but is closer to the latter as her portfolio is now 80% invested in A. Finally, if the investor borrows, say, $600 at the risk-free rate to invest $1,000 + $600 = $1,600 in A, her portfolio lies at point C, which is located to the right of A.

capital allocation line
Figure 1: Capital allocation line

Combining the risk-free asset with efficient portfolios

There is an infinite number of efficient portfolios that lie on the efficient frontier. Figure 1 showed the case where the investor split her funds between Rf and a specific efficient portfolio A. Is this the best she can do? The answer is, no. She can do better by investing in another efficient portfolio. To see this, we let the investor consider investing in another efficient portfolio B.

As shown in Figure 2, B is further up on the efficient frontier as compared to A. As a result, the capital allocation line that connects Rf and B lies above the one that connects Rf and A.

This has an important implication. Look at portfolio A’. It is not only a combination of the risk-free asset and B but also lies directly above A. So, A and A’ have the same level of risk, but the latter offers a higher return. Thus, the investor would prefer A’ over A. In fact, for any combination of Rf and A, we can find a combination of Rf and B with the same risk but more return. As a result, the investor is better off by splitting her wealth between Rf and B instead of Rf and A.

combining the risk-free asset with efficient portfolios
Figure 2: Combining the risk-free asset with efficient portfolios

Locating the optimal risky portfolio

Can the investor still do better than investing in Rf and B? Yes, she can! The investor has to seek the efficient portfolio that is the furthest along the efficient frontier and that can still be combined with the Rf. This is equivalent to maximizing the slope of the capital allocation line that goes through the risk-free asset and the efficient frontier.

We illustrate this in Figure 3. As we discussed earlier, B is better than A, since the capital allocation line involving the former has a higher slope. By the same logic, C is better than B, because it leads to a better capital allocation line. Furthermore, the capital allocation line that connects Rf and C is tangent to the efficient frontier. This means that this line just touches the efficient frontier. It also means that it is impossible to combine Rf with any other efficient portfolio that lies further along the efficient frontier than C. Thus, this is the point where the slope of the capital allocation line is maximized. And, C is the optimal risky portfolio!

optimal risky portfolio
Figure 3: Optimal risky portfolio

Using Excel to find the optimal risky portfolio

It is easy to find the optimal risky portfolio using Excel or other software. If you’re using Excel, you will need to use the Solver add-in. If you haven’t used that before, you can enable it by going to File > Options > Add-ins. Once you are at the Add-ins tab, click the Go button where it says “Manage Excel Add-ins”. In the menu that pops up, simply check the “Solver Add-in” and hit the OK button. Now, you can go to the “Data” menu. And, you will find the Solver icon there.

To find the optimal portfolio, you need:

  1. The return and variance of each security you are examining.
  2. The covariance (or correlation) between each pair of securities.

Maximizing the Sharpe ratio using Solver

If we look at Figure 3, the capital allocation line that goes through the optimal risky portfolio C has the steepest slope compared to other efficient portfolios such as A or B. In fact, the slope of that capital allocation line can be interpreted as the Sharpe ratio of C:

( RC − Rf ) / σC

where RC is the return of the optimal risky portfolio C and σC is the return volatility of the same portfolio, which is a measure of portfolio risk. So, you need to use the Solver to maximize this Sharpe ratio. If you click the Solver button, pay attention to the following fields in the menu that pops up:

  • Set objective: Assign this to the cell where you compute the Sharpe ratio using the equation above.
  • By Changing Variable Cells: These are the cells that will contain the investment weights for each security.
  • Subject to the Constraints: You should create a cell where you sum up the investment weights. Then, you should add a constraint that sets the value of this cell equal to 1.

In summary, your objective is to maximize the Sharpe ratio by changing the investment weights of the assets you are examining with the constraint that investment weights add up to 100%.

Wrapping up

According to the modern portfolio theory, investors engage in mean-variance optimization to find the efficient frontier. The portfolios that lie on the efficient frontier offer the best risk-return tradeoff in the market. In this post, we have shown that when there is a risk-free asset in the market, there is a unique optimal risky portfolio. Then, investors’ remaining task is to simply decide how to split their funds between the risk-free asset and the optimal risky portfolio. Their decision will be driven by their risk preferences such that more (less) risk-averse investors would invest a lower (higher) proportion of their money in the optimal risky portfolio.

What is next?

This post is part of our free course on investments. We will introduce the concept of the market portfolio in our next post. In the previous post, we discussed the efficient frontier, distinguishing between efficient portfolios and inefficient ones.

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