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2.16

# Understanding CAPM: the Capital Asset Pricing Model

When considering the risk of a financial security, we may wish to ask the following questions:

1. What is an appropriate measure of its risk?

2. What is the appropriate rate of return which investors should require, given the risk of the security?

In step 2.11, you have seen that the risk of a security should not be measured in isolation, but rather with reference to a broader portfolio of securities: how does the security contribute to the overall risk of the portfolio? The capital asset pricing model, or CAPM, is a very useful tool which can give an answer to these questions. The CAPM says that:

1. The risk of a security is proportional to its covariance with the market portfolio, and is given by its beta;

2. Investors should require a risk premium on the security, which is proportional to its beta.

The beta values for companies are widely reported, and used by investors and risk managers. But what does beta mean exactly, and why is it important? In order to understand this, let’s look at the CAPM.

There is a simple formula for the CAPM. Let’s define the following symbols:

• $$R_i$$ is the rate of return on the security $$i$$;

• $$R_f$$ is a “safe”, or risk-free, rate of return: for instance, the rate of return on very short-term bonds issued by the government (Treasury bills);

• $$R_m$$ is the rate of return on the market: for the sake of simplicity, we can think of it as an average return in the market;

• $$\beta_i$$ is the beta of security $$i$$: this is proportional to the covariance of the return on security $$i$$ and the return on the market.

The CAPM says that the rate of return that investors require in order to be willing to hold the security is given by the formula:

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

where $$E (R_i)$$ is the required expected return on security $$i$$, and $$E (R_m)$$ is the expected return on the market.

### What does the formula for the CAPM tell us?

Let’s look at the components.

Look first at the component on the left of the equals sign: $$E (R_i)$$ is the rate of return that investors require to be willing to hold security $$i$$ in their portfolio.

Now consider the components to the right of the equals sign. If security $$i$$ carried no risk at all, it would have an expected return equal to the rate of return on a safe bond $$R_f$$. However, security $$i$$ is risky so investors require a risk premium: this is given by the product $$\beta_i \times [E (R_m) - R_f]$$.

The first term, $$\beta_i$$, is the beta of the security and measures the quantity of risk of security $$i$$. The second term, $$[E (R_m) - R_f]$$, is the difference between the expected return on the market and the safe return: this is the price of risk.

Hence, the CAPM tells us that:

1. Investors require a premium to hold a risky asset;

2. The appropriate measure of risk of security $$i$$ is its beta, $$\beta_i$$;

3. The risk premium that investors require is the product of the quantity of risk, $$\beta_i$$, times the price of risk.

The CAPM is a simple model, but it can be a very useful tool to measure the risk of financial securities and to assess whether the securities pay a high enough return, given the risk involved in holding them.