Beta and CAPM

Beta

Beta is a measure of systematic risk, which refers to the risk inherent to the entire financial market. This is the risk that you cannot get rid of by diversifying across different securities.

A common misconception is that Beta is NOT the degree of correlation between security and the market; however, in the true sense, the Beta calculation uses the correlation between the security and the market.

The Beta formulae for company i is the following:

$$\beta_i=\frac{Cov(i,m)}{\sigma^2_m}=\frac{\sigma_{im}}{\sigma^2_m}$$

Where:

\(\sigma^2_m\) = the variance of the market index; and

\(\sigma_{im}\) = the covariance between the individual stock’s/asset’s return and that of the market;

Alternatively, by using the fact that we know that:

$${Cov(i,m)}=\frac{\rho_{im}\sigma_i\sigma_m}{\sigma_m^2}$$

We can write it as:

$${\beta_i}=\rho_{im}\times\frac{\sigma_i}{\sigma_m}$$

Where:

\(\rho_{im}\) = the correlation coefficient between returns of asset i and that of the market portfolio; and

\(\sigma_i\) = the standard deviation of asset i.

Interpreting Beta

A positive Beta indicates the asset moves in the same direction as the market, whereas a negative Beta would indicate the opposite.

The Beta of a risk-free asset is zero because the risk-free asset’s covariance and the market are zero. By definition, the Beta of the market is one, and most developed market stocks exhibit high positive betas.

Beta is essentially a multiplier

• A value of Beta above 1 indicates a stock/asset/portfolio that has, historically, amplified the return of the whole market (positive or negative).
• A beta close to zero would indicate a stock/asset/portfolio that provides a more stable return than the market as a whole.
• A negative beta would signify a stock/asset/portfolio whose performance is counter-cyclical, i.e., offsets the overall market experience.

Example 1: Calculating Beta

The correlation between an asset and the market is 0.6, the asset’s standard deviation is 18%, and the standard deviation of the market is 14%.

What is the Beta of the asset?

A. 0.77

B. 0.47

C. 0.99

Solution

The correct answer is A.

We know that:

$$\begin{align*}{\beta_i}=&\rho_{im}\times\frac{\sigma_i}{\sigma_m}\end{align*}$$

Thus,

$$\begin{align*}{\beta_i}&=\frac{0.6\times0.18}{0.14}\\&=0.77\end{align*}$$

The Capital Asset Pricing Model (CAPM)

The Capital Asset Pricing Model (CAPM) provides a linear relationship between the expected return for an asset and the Beta.

Assumptions of the CAPM model include:

• There are no transaction costs;
• There are no taxes;
• Assets are infinitely divisible;
• Unlimited short-selling is permissible;
• All assets are marketable/liquid;
• Investors are price takers whose individual buy and sell transactions do not affect the price;
• Investors’ utility functions are based solely on expected portfolio return and risk; and
• The only concern among investors is risk and return over a single period, and the single period is the same for all investors.

Application of CAPM

The application of the Capital Asset Pricing Model (CAPM) to compute the cost of equity is based on the following relationship:

$${E(R_i)}=R_f+\beta_i[E(R_m)-R_f]$$

Where:

\(E(R_i)\) = the cost of equity or the expected return on a stock;

\(R_f\) = the risk-free rate of interest (this may be estimated by the yield on a default-free government debt instrument);

\(\beta_i\) = the equity beta or return sensitivity of stock i to changes in the market return; and

\(E(R_m)\) = the expected market return.

Note: The expression \(E(R_m)-(R_f)\) is the expected market risk premium or equity risk premium

Example 2: CAPM Application

You have been provided the following:

• Risk-free rate = 5%
• Standard deviation of the security = 40%
• Security correlation with market = 0.80
• Standard deviation of the market = 20%
• Expected market return = 10%

Calculate the expected return for this security.

A. 12%

B. 13%

C. 21%

Solution

The correct answer is B.

First, find Beta:

$$\begin{align*}{\beta_i}=&\rho_{im}\times\frac{\sigma_i}{\sigma_m}\\=&\frac{0.80\times0.40}{0.20}\\=&1.6\end{align*}$$

Next, use the CAPM model to find the expected return:

$$\begin{align*}{E(R_i)}=&R_f+\beta_i[E(R_m)-R_f]\\=&{5\%}+{1.6(10\%-5\%)}\\=&13\%\end{align*}$$

FRM Part I & Part II Complete Course