Capital Asset Pricing Model and Security Market Line
A lot of the financial literature devotes exclusive discussions to the Capital Asset Pricing Model (CAPM). In this section, we will explore key concepts that highlight the importance of linearity in finance.
In the famous CAPM, the relationship between risk and rates of return in a security is described as follows:
For a security, i, its returns are defined as Ri and its beta as βi. The CAPM defines the return of the security as the sum of the risk-free rate, Rf, and the multiplication of its beta with the risk premium. The risk premium can be thought of as the market portfolio’s excess returns exclusive of the risk-free rate. The following is a visual representation of the CAPM:
Beta is a measure of the systematic risk of a stock — a risk that cannot be diversified away. In essence, it describes the sensitivity of stock returns with respect to movements in the market. For example, a stock with a beta of zero produces no excess returns regardless of the direction the market moves in. It can only grow at a risk-free rate. A stock with a beta of 1 indicates that the stock moves perfectly with the market.
The beta is mathematically derived by dividing the covariance of returns between the stock and the market by the variance of the market returns.
The CAPM model measures the relationship between risk and stock returns for every stock in the portfolio basket. By outlining the sum of this relationship, we obtain combinations or weights of risky securities that produce the lowest portfolio risk for every level of portfolio return.
Beta is a measure of the systematic risk of a stock — a risk that cannot be diversified away. In essence, it describes the sensitivity of stock returns with respect to movements in the market. For example, a stock with a beta of zero produces no excess returns regardless of the direction the market moves in. It can only grow at a risk-free rate. A stock with a beta of 1 indicates that the stock moves perfectly with the market.
The beta is mathematically derived by dividing the covariance of returns between the stock and the market by the variance of the market returns.
An investor who wishes to receive a particular return would own one such combination of an optimal portfolio that provides the least risk possible. Combinations of optimal portfolios lie along a line called the efficient frontier.
Along the efficient frontier, there exists a tangent point that denotes the best optimal portfolio available and gives the highest rate of return in exchange for the lowest risk possible. This optimal portfolio at the tangent point is known as the market portfolio.
There exists a straight line drawn from the market portfolio to the risk-free rate. This line is called the Capital Market Line (CML). The CML can be thought of as the highest Sharpe ratio available among all the other Sharpe ratios of optimal portfolios. The Sharpe ratio is a risk-adjusted performance measure defined as the portfolio’s excess returns over the risk-free rate per unit of its risk in standard deviations. Investors are particularly interested in holding combinations of assets along the CML line. The following diagram illustrates the efficient frontier, the market portfolio, and the CML:
Another line of interest in CAPM studies is the Security Market Line (SML). The SML plots the asset’s expected returns against its beta. For a security with a beta value of 1, its returns perfectly match the market’s returns. Any security priced above the SML is deemed to be undervalued since investors expect a higher return given the same amount of risk. Conversely, any security priced below the SML is deemed to be overvalued, as follows:
Suppose we are interested in finding the beta, βi, of a security. We can regress the company’s stock returns, Ri, against the market’s returns, RM, along with an intercept, α, in the form of the Ri=α+βRM equation.
Consider the following set of stock return and market return data measured over five time periods:
The scipty.stats.linregress function returns five values: the slope of the regression line, the intercept of the regression line, the correlation coefficient, the p-value for a hypothesis test with a null hypothesis of a zero slope, and the standard error of the estimate. We are interested in finding the slope and intercept of the line by printing the values of beta and alpha, respectively:
The beta of the stock is 0.5077 and the alpha is nearly zero.
The equation that describes the SML can be written as follows:
The term E[RM]−Rf is the market risk premium, and E[RM] is the expected return on the market portfolio. Rf is the return on the risk-free rate, E[Ri] is the expected return on asset, i, and βi is the beta of the asset.
Suppose the risk-free rate is 5% and the market risk premium is 8.5%. What is the expected return of the stock? Based on the CAPM, an equity with a beta of 0.5077 would have a risk premium of 0.5077×8.5%, or 4.3%. The risk-free rate is 5%, so the expected return on the equity is 9.3%.
If the security is observed in the same time period to have a higher return (for example, 10.5%) than the expected stock return, the security can be said to be undervalued, since the investor can expect a greater return for the same amount of risk.
Conversely, should the return of the security be observed to have a lower return (for example, 7%) than the expected return as implied by the SML, the security can be said to be overvalued. The investor receives a reduced return while assuming the same amount of risk.
The CAPM suffers from several limitations, such as the use of a mean-variance framework and the fact that returns are captured by one risk factor — the market risk factor. In a well-diversified portfolio, the unsystematic risk of various stocks cancels out and is essentially eliminated.
Resources:
Udemy: Python for Finance: Investment Fundamentals & Data Analytics
Yves Hilpisch Python for Finance: Analyze Big Financial Data
