7 – L3 09 Capital Market Line V2

In the last video, you learned about the efficient frontier. You know that any point on the efficient frontier represents the portfolio that gives you the best risk return trade-off, and that these portfolios are known as market portfolios. But is that it? Is there a way to do better than the efficient frontier? In this video, we will be adding a risk-free asset into our portfolio and we’re going to introduce something called the capital market line. A risk-free asset is an investment instrument that entails absolutely no risk or uncertainty. In theory, if you invest in such an instrument, you receive a guaranteed rate of return called the risk-free rate and reality entirely risk-free assets don’t exist since all investments carry a certain level of risk. However, in practice people normally refer to the rate of return on a three month treasury bill as the risk-free rate. Now let’s add this risk-free asset into the picture. Let’s imagine we construct a new portfolio with any of the market portfolios that you prefer and the risk-free asset. Which market portfolio would you choose? Note that this new portfolio is a weighted sum of the risk-free asset and the market portfolio. For example, if you choose the green portfolio as your market portfolio and the red. as your risk-free asset, the line between them represents the potential portfolios that you can construct with the two assets. Now, which market portfolio would you choose so as to achieve the best return for a given level of risk? Well, you should be looking at the efficient frontier, the top edge of the set of possible portfolios. It turns out that you should choose the market portfolio that allows you to draw a straight line starting from the risk-free asset that just touches the top of the efficient frontier. To see why this is the best market portfolio for the job, let’s take a look at lines A, B, and C. Note that line A connects the red dot which represents the risk-free asset with the green dot, which represents the tangent market portfolio on the efficient frontier. Line A lies above the efficient frontier and intersects the tangent portfolio. All the portfolios represented by line A have higher returns than the ones on the efficient frontier given the same risk level. This is not the case for lines B and C. Portfolios on this tangent line are the best achievable. They even beat the efficient frontier which consists of purely risky assets. This line is called the capital market line. Now what is the expected return of the portfolio consisting of the risky portfolio and the risk-free asset? The formula for this quantity is also the formula for the capital market line because each point on this line gives the return as a function of risk of a possible combination of the risky portfolio and the risk-free asset. So let’s examine the graph to determine the equation. You have two points 0, rf which is the risk-free portfolio and sigma m, rm which is the market portfolio. The slope of a line simply equals rm minus rf divided by sigma m. This quantity the difference between the market return and the risk-free rate is called the market risk premium. The portfolio return is then just the risk-free rate of return as a baseline plus the slope times the portfolio volatility. The portfolio volatility will depend on what weights you choose to put on the risk-free asset and the market portfolio, i.e. where you choose to sit along the capital market line. The slope of the capital market line, which is of course the same for all points on the line is a special quantity known as the Sharpe Ratio of the market. We’re going to discuss that more in a future lesson. In fact, if you can borrow at the risk-free rate, you can achieve points on the capital market line to the right of the tangent market portfolio. This amounts to setting a negative weight on the risk-free asset. That is to say, shorting it. This is why professional investors care almost only about the Sharpe ratio because they can manufacture any level of risk or return with leverage.

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