{"id":6313,"date":"2021-11-20T18:11:17","date_gmt":"2021-11-20T17:11:17","guid":{"rendered":"https:\/\/www.simtrade.fr\/blog_simtrade\/?p=6313"},"modified":"2024-01-30T21:56:07","modified_gmt":"2024-01-30T21:56:07","slug":"capital-market-line-cml","status":"publish","type":"post","link":"https:\/\/www.simtrade.fr\/blog_simtrade\/capital-market-line-cml\/","title":{"rendered":"Capital Market Line (CML)"},"content":{"rendered":"

\"Youssef_Louraoui\"<\/a><\/p>\n

In this article, Youssef LOURAOUI<\/a> (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022) presents the Capital Market Line (CML), a key concept in asset pricing derived from the Capital Asset Pricing Model (CAPM).<\/p>\n

This article is structured as follows: we first introduce the concept. We then illustrate how to estimate the capital market line (CML). We finish by presenting the mathematical foundations of the CML.<\/p>\n

Capital Market Line<\/h2>\n

An optimal portfolio is a set of assets that maximizes the trade-off between expected return and risk: for a given level of risk, the portfolio with the highest expected return, or for a given level of expected return, the portfolio with the lowest risk.<\/p>\n

Let us consider two cases: 1) when investors have access to risky assets only; 2) when investors have access to risky assets and a risk-free asset (earning a constant interest rate, 2% for example below).<\/p>\n

Risky assets<\/h3>\n

In the case of risky assets only, the efficient frontier (the set of optimal portfolios) is represented below in Figure 1.<\/p>\n

Figure 1. Efficient frontier with risky assets only.
\n\"img_Simtrade_CML_graph_1\"<\/a>
\nSource: Computation by the author.<\/p>\n

Risky assets and a risk-free asset<\/h3>\n

In the case of risky assets and a risk-free asset, the efficient frontier (the set of optimal portfolios) is represented below in Figure 2. In this case, the efficient frontier is a straight line called the Capital Market Line (CML).<\/p>\n

Figure 2. Efficient frontier with risky assets and a risk-free asset.
\n\"img_Simtrade_CML_graph_0\"<\/a>
\nSource: Computation by the author.<\/p>\n

The CML joins the risk-free asset and the tangency portfolio, which is the intersection with the efficient frontier with risky assets only. We can reasonably conclude from Figure 2 that, to increase expected return, an investor has to increase the amount of risk he or she takes to attain returns higher than the risk-free interest rate. As a result, the Sharpe ratio of the market portfolio equals the slope of the CML. If the Sharpe ratio is more than the CML, an investment strategy can be implemented, such as buying assets if the Sharpe ratio is greater than the CML and selling assets if the Sharpe ratio is less than the CML (Drake and Fabozzi, 2011).<\/p>\n

Investors who allocate their money between a riskless asset and the risky market portfolio M can expect a return equal to the risk-free rate plus compensation for the number of risk units \u03c3P<\/sub>) they accept. This result is in line with the underlying notion of all investment theory: investors perform two services in the capital markets for which they might expect to be compensated. First, they enable someone else to utilize their money in exchange for a risk-free interest rate. Second, they face the risk of not receiving the promised returns in exchange for their invested capital. The term E(rM<\/sub>)- Rf<\/sub>) \/ \u03c3M<\/sub> refers to the investor’s expected risk premium per unit of risk, which is also known as the expected compensation per unit of risk taken.<\/p>\n

Figure 3 represents the Capital Market Line which connect the risk-free asset to the efficient frontier line. The straight line in Figure 3 represents a combination of a risky portfolio and a riskless asset. Any combination of the risk-free asset and Portfolio A is similarly outperformed by some combination of the risk-free asset and Portfolio B. Continue drawing a line from Rf<\/sub> to the efficient frontier with increasing slopes until you reach Portfolio M’s point of tangency. All other possible portfolio combinations that investors could build are outperformed by the collection of portfolio possibilities along Line Rf<\/sub>-M, which is the CML. The CML, in this sense, represents a new efficient frontier that combines the Markowitz efficient frontier of risky assets with the ability to invest in risk-free securities. The CML’s slope is (E(rM<\/sub>)- Rf<\/sub>) \/ \u03c3(M<\/sub>), which is the highest risk premium compensation that investors can expect for each unit of risk they take on (Reilly and Brown, 2012) (Figure 3).<\/p>\n

If we fully invest our cash on the risk-free rate, we would be exactly on the y axis with an expected return of 2%. Each time we move along the curve that connects the risk-free rate to the optimum market portfolio, we allocate less weight to the risk-free rate, and we overweight more on riskier assets (Point A). Points M represents the optimal risky portfolio in the efficient frontier line, which minimizes the overall portfolio variance. It would have a weighting of 45% in stock A and a 55% in stock B, which would offer a 26.23% annualized return for a 17.27% annualized volatility. Point B represents a portfolio composition that is based on a leveraged position of 140% on the optimal risky portfolio and a short position on the risk-free asset of -40% (Figure 3).<\/p>\n

Figure 3. Efficient frontier with different points.
\n\"img_Simtrade_CML_graph_2\"<\/a>
\nSource: Computation by the author.<\/p>\n

Mathematical representation<\/h2>\n

We can define the CML as the line that is tangent to the efficient frontier which connects the risk-free asset with the market portfolio:<\/p>\n

\"img_SimTrade_CML_equations_0\"<\/a><\/p>\n

Where:<\/p>\n