 ## 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

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 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

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*}

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