The capital asset pricing model (or CAPM) is a widely-used asset pricing model. Its popularity arises from its simplicity and elegance. Analysts and investors use it to forecast returns or to estimate the cost of equity. In this lesson, we’ll explain this model together with its assumptions. And, you can find a handy CAPM calculator as well.

#### Contents

## Model and assumptions

In previous lessons, we covered the modern portfolio theory. According to this theory, individual investors engage in mean-variance optimization, such that they try to find the portfolios that yield the highest return for a given level of risk (as measured by return volatility). This leads to an efficient frontier on which there is a unique optimal risky portfolio when a risk-free asset is available.

CAPM model was developed in an attempt to answer the following question. If all investors act in compliance with the modern portfolio theory and hold the optimal risky portfolio, what would that mean for the market prices of risky assets such as stocks?

And, the answer to that question is as follows. If all investors hold the optimal risky portfolio, it becomes the market portfolio, which is a value-weighted portfolio of all risky assets. And, because the market portfolio is fully-diversified, it no longer contains any firm-specific risk. As a result, investors should only be rewarded for bearing systematic risk, which is still present in the market portfolio. It is this result that leads us to the CAPM equation.

### Equation

According to the **CAPM equation**, the expected return on a risky asset *E[R _{i}]* is linearly related to its beta

*β*, which is a measure of systematic risk, as follows:

_{i}where *R _{f}* is the risk-free rate of return and

*E[R*is the expected return on the market portfolio.

_{m}]We can also interpret this formula in a general sense as follows:

Expected return = Risk-free rate + Risk premium

where the risk premium* β_{i} (E[R_{m}] − R_{f})* is higher for stocks with higher betas. This is because beta is a measure of systematic risk, and high-beta stocks bear more systematic risk compared to low-beta ones. As mentioned above, there is no risk premium for firm-specific risk as it is diversified away (for free) when an investor holds the market portfolio.

It is worth noting that the CAPM equation applies to portfolios of assets (think of investment funds) as well as individual assets.

**Test your knowledge**

Daniel has identified a stock, which he wants to invest in. The stock has a beta of 1.3. At the moment, the risk-free rate in the economy is 5%. And, the market portfolio is expected to yield 8%. If CAPM holds, what level of return should Daniel expect if he invests in this stock?

Note: You can use the calculator below. The solution is provided at the bottom of this page.

### Assumptions

We can list the **CAPM assumptions** as follows:

- Investors act rationally and are mean-variance optimizers.
- They are assumed to agree on the expected return and variance of each asset and covariances between asset pairs. This means that they will all end up with the same optimal risky portfolio when they engage in mean-variance optimization.
- They have a single investment period (e.g., a year) in mind.
- All assets are tradeable and perfectly divisible.
- Investors are price takers (i.e., their trades do not influence prices).
- They can lend or borrow unlimited amounts at a common risk-free rate.
- Short selling is allowed without limits.
- There are no taxes and transaction costs (e.g., trading commissions).

Note that these are the assumptions for the standard version of the model. Extended versions of the model relax some of these assumptions. For example, one of the extensions allows for differential rates for borrowing and lending.

## CAPM calculator

You need the following inputs to use the **CAPM calculator**:

- The
**beta**of the asset. This can be estimated based on historical returns. - The
**risk-free rate of return**. This is often proxied by the yield on a selected government bond. **Expected return on the market portfolio**. Historical returns on a market index such as S&P500 can be a starting point, with the analyst adjusting this based on his beliefs about future performance.

## Video summary

##### What is next?

If you’ve enjoyed this lesson, we recommend you check out the following ones as well.

This lesson is part of our investments course.

**Next lesson**: We will focus on the security market line.**Previous lesson**: We explained the distinction between systematic risk and idiosyncratic risk.

Further reading:

Sharpe (1964) ‘Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk‘, The Journal of Finance, Vol. 19 (3), 425-442.

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**Solution for the “test your knowledge” exercise**

Using either the calculator above or the CAPM equation, we find that the stock’s expected return is:

5% + 1.3 (8% *−* 5%) = 8.9%

Therefore, Daniel can expect the stock to yield 8.9%. Of course, the realized return may end up being higher or lower than that as this is a risky investment.