In this article, Youssef Louraoui (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022). presents the Capital Market Line (CML), a pilar concept derived from the CAPM.

This article is structured as follows: we present an introduction of the concept. We then illustrate how to estimate the capital market line. We finish by presenting the mathematical foundations of this concept.

## Capital Market Line

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.

## Estimation of the Capital Market Line

You can download an Excel file with data to estimate the Capital Market Line.

Let us consider two cases: 1) investors have access to risky assets only; 2) investors have access to risky assets and a riskless asset (earning a risk-free interest rate).

We download a time series of 2 years’ worth of monthly data for two stocks (Apple and CML Microsystems Plc) to create an investment portfolio. We also assume that the risk-free rate is equal to 2%.

In the first case, the efficient frontier which represents the set of optimal portfolios is represented below in Figure 1.

Figure 1. Efficient frontier with risky assets only.

Source: Computation by the author. Data source: Thomson Reuters.

In the second case, the capital market line which connects the risk-free rate asset with the optimal risky portfolio is represented below:

Figure 2. Efficient frontier with risky assets and a riskless asset.

Source: Computation by the author. Data source: Thomson Reuters.

In this case, the efficient frontier is a straight line called the Capital Market Line (CML). The CML joins the riskless 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 returns, an investor has to increase the amount of risk he or she takes to attain returns higher than the risk-free asset. 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).

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 σ_{P}) they accept (Reilly and Brown, 2012). 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[r_{M}– R_{f}) / σ_{M} 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.

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 R_{f} 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 R_{f}-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(R_{m} -RFR) / σ_(_{M}), 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).

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% Apple stock and a 55% S&P500 index, 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).

Figure 3. Efficient frontier with different points.

Source: Computation by the author. Data source: Thomson Reuters.

## Mathematical representation

Mathematically, we can deconstruct the Capital Market Line as:

Where

- E(R
_{P}) represents the expected return of the portfolio - R
_{f}is the risk-free interest rate - σ
_{P}represents the volatility of the portfolio - E(R
_{M}) represents the expected return of the market - σ
_{M}represents the volatility of the market - E[r
_{M}– R_{f}] represents the market risk premium.

## Why should I be interested in this post?

If you are a business school or university undergraduate or graduate student, this content will help you in broadening your knowledge of finance.

## Related posts on the SimTrade blog

▶ Louraoui Y. Portfolio

▶ Louraoui Y. Systematic and unsystematic risk

▶ Louraoui Y. Alpha

▶ Louraoui Y. Factor Investing

▶ Louraoui Y. Origin of factor investing

▶ Louraoui Y. Markowitz Modern Portfolio Theory

▶ Walia J.Capital Asset Pricing Model (CAPM)

## Useful resources

### Academic research

Pamela, D. and Fabozzi, F., 2010. The Basics of Finance: An Introduction to Financial Markets, Business Finance, and Portfolio Management. *John Wiley and Sons Edition.*

Lintner, J. 1965a. The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets. The Review of Economics and Statistics 47(1): 13-37.

Lintner, J. 1965b. Security Prices, Risk and Maximal Gains from Diversification. *The Journal of Finance*, 20(4): 587-615.

Mossin, J. 1966. Equilibrium in a Capital Asset Market. *Econometrica*, 34(4): 768-783.

Reilly, R. K., Brown C. K., 2012. Investment Analysis & Portfolio Management, Tenth Edition.

Sharpe, W.F. 1963. A Simplified Model for Portfolio Analysis. *Management Science*, 9(2): 277-293.

Sharpe, W.F. 1964. Capital Asset Prices: A theory of Market Equilibrium under Conditions of Risk. *The Journal of Finance*, 19(3): 425-442.

## About the author

The article was written in November 2021 by Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022).