Capital asset pricing model (CAPM) is a classical method in finance and economy that states the relationship between the risk and expected return. It is based on the fact that every investment's expected pays-off depends on the level of risk taking on. More precisely, the return of a portfolio consists of two components of time value of money and an extra rate (usually called risk premium) to compensate exposing excess risk.

CAPM can be used for determining the performance of the portfolio and valuating risky securities.

# Different types of Risk

Every investment contains two types of risks; systematic risk and specific risk. We go through each kind of risk briefly and study their effects on the investing portfolio.

## Systematic Risk

Systematic risk is the uncertainty that arises from the entire or a large part of the market. For instance, it may represent the risk of certain market system such as US stock market. Inflation, interest rate shift and great recession in the market are examples of the causes for the systematic risk. It can not be diversified away even though one does not participate in any investing in the market. As an example, consider a person that holds cash and inflation in the market affects the value of the money.

## Specific Risk

This type of risk is associated to the risk exposure for the company of each underlying stock in an investing portfolio. It can be eliminated in a well-hedged portfolio (usually more than 25 different securities in a portfolio).

# Formula

To determine CAPM, one may consider the following well-known formula, which states the linear link between the expected return for a specific asset \(S\) over the risk-free rate and expected return of the market: $$ E(R_S) = R_f + \beta_S [E(R_M) -R_f], $$

where \(E(R_S)\), \(R_f \) and \(E(R_M)\) are expected return on security, risk free rate and expected return on market portfolio \(M\), respectively. Moreover, \( \beta_S \) is a coefficient that is related to non-diversifiable part of the security that can be determined by linear regression.

Knowing \(\beta_S \), one may use the CAPM formula to find the expected return for a certain security in a particular asset class.

# Beta Coefficient

\( \beta \) coefficient is a measure of systematic risk that depends on the chosen asset class and shows how risky a stock is compared to the whole market. In other words, beta is a measure of an asset's volatility compared to the entire market's volatility. A market benchmark (e.g., S&P 500) is used for this comparison to determine \( \beta\).

Mathematically, \( \beta \) can be calculated according to the following formula

$$ \beta_S= \frac{Cov(R_S , R_M)}{\sigma^2 (R_M)} = Corr(R_S, R_M) \frac{\sigma(R_s)}{\sigma(R_M)}. $$

In practice, we usually do not have these parameters. The clue is to estimate \(\beta\) using historical data of the company's share price for the specific stock which we aim to determine its beta coefficient. Moreover, historical data of the market benchmark (say S&P 500) are used for comparing the data with the market representative. The next step is to find daily returns for both of the stock and market index as follows:

$$\text{return}=\frac{\text{closing price}- \text{opening price}}{\text{opening price}}.$$

Sketching stock return versus market return and conducting linear regression, one may find \(\beta\) for the specific asset class. More precisely, beta coefficient is the slope of the linear regression line (usually called characteristic line) for the above mentioned returns.

To find \(\beta\) for a portfolio, one needs to find \(\beta\) for each component first and then multiply it by its weight in the portfolio. The portfolio's beta coefficient is then is calculated by adding up all the weighted betas.

Different \(\beta\) values correspond various security and market situations. We have the following cases:

- \(\beta < 0 \): It generally means that the security moves in opposite direction of the average market (e.g. gold stock).
- \(\beta = 0\): Expected return is like a risk-free rate or zero correlation with the market movements (such as cash or government bonds).
- \( 0 < \beta <1\): The portfolio is less risky than the average market.
- \(\beta = 1\): The same as the entire financial system.
- \(\beta > 1\): A riskier situation that the market risk.
- \(|\beta|>4\): It happens rarely, when the security is highly fluctuated in daily basis.

## SML

Security market line, SML, or characteristic line, depicts CAPM for different risks (beta). In other words, it is a chart with \(\beta\) as the \(x\)-axis against the expected return as \(y\)-axis. \(R_f\) is the \(y\)-intercept of the line and the point \( (1, R_M) \) is located on SML half-line, i.e., \(\beta_M=1\). Furthermore, the slope of SML is the market risk premium; \([E(R_M) -R_f]\).

## CML

Capital market line, CML, is another half-line that represents the relationship between expected return and the standard deviation of the total risk. It can be formulated as follows:

$$E(R_S)= R_f +\bigg(\frac{E(R_M)-R_f}{\sigma_M}\bigg)\sigma_S,$$

where \(\sigma_S \) indicates the stock \(S\)'s total risk. Moreover, the slope of the CML line is the the Sharpe ratio of the market portfolio.