# Capital asset pricing model (CAPM)

The capital asset pricing model (or the CAPM) is probably one of the most commonly taught topics in finance. This is not because it is a perfect model. In fact, it is far from it. However, the CAPM is built upon some of the most fundamental concepts in finance. Furthermore, CAPM is widely used in practice despite its drawbacks. Not least because, it is relatively easy to understand and implement. Finally, it is a milestone on the path to more sophisticated asset pricing models. In this post, we explain this popular model and offer an easy-to-use CAPM calculator.

Learning objectives

• Understand the logic behind the CAPM equation.
• Evaluate the model’s assumptions and its practical uses.

## What is the capital asset pricing model?

In the previous posts, we explained that when all investors seek to hold the optimal risky portfolio, that portfolio ends up becoming the market portfolio. Because the market portfolio contains all risky assets, it is a well-diversified portfolio. As we have discussed in the previous lesson, investors can get rid of unsystematic risk via diversification. As a result, the market portfolio contains systematic risk only.

CAPM argues that because investors can get rid of unsystematic risk freely by investing in the market portfolio, they do not need to be compensated for bearing this type of risk. In other words, they should expect a risk premium on systematic risk only as this type of risk is non-diversifiable.

### CAPM formula

The capital asset pricing model posits that a risky asset’s risk premium will depend on that asset’s beta, which is a measure of systematic risk. This leads to the famous CAPM formula as follows:

Expected return on a risky asset = Risk-free rate of return + Risk premium

E[Ri] = Rf + βi (E[Rm] − Rf)

where E[Ri] is the expected return on asset i, Rf is the risk-free rate of return, βi is the beta of asset i, and E[Rm] is the expected return on the market portfolio. Note that this formula applies to portfolios of assets as well as individual assets.

According to the formula, if asset i has a higher expected return than asset j (E[Ri] > E[Rj]), this is because βi > βj. That is, high-beta assets command higher expected returns than low-beta assets. This is because high-beta assets carry a larger amount of systematic risk than low-beta assets, and investors demand a higher risk premium on the former type of assets as a result of that.

We can also verify that:

I. The beta of the market portfolio is equal to 1, since:

E[Rm] = Rf + βm (E[Rm] − Rf)

E[Rm] Rf = βm (E[Rm] − Rf)

βm = (E[Rm] − Rf) / (E[Rm] − Rf) = 1

II. And, the beta of the risk-free asset is equal to 0, since

Rf = Rf + βf (E[Rm] − Rf)

βf = (Rf Rf) / (E[Rm] − Rf) = 0

## Security market line

The CAPM posits a linear relationship between an asset’s expected return E[Ri] and its beta βi. Therefore, the CAPM equation, which we provide again below, is the equation of a line. We call this the security market line (SML). In particular, the risk-free rate of return Rf is the intercept of SML, and the market risk premium E[Rm] − Rf is the slope of SML. This is illustrated in Figure 1.

E[Ri] = Rf + βi (E[Rm] − Rf)

According to the capital asset pricing model, all risky assets should lie on the SML. Such assets include individual assets as well as portfolios. The reason is that if an asset lies above or below the SML, this creates an arbitrage opportunity. However, we expect arbitrage opportunities to disappear quickly in competitive markets. This means that assets that lie off the SML would revert back to it as investors exploit arbitrage opportunities. We show this in Figure 2 below.

Assets A and B are both mispriced, whereas asset C is fairly (or correctly) priced according to the CAPM. Asset A is undervalued as it is offering a return higher than an asset with the same beta but that lies on the SML. So, investors would exert buying pressure on A. This would increase its price and lower its expected return, pulling it down to the SML.

In contrast, asset B is overvalued as its return is higher than an asset on the SML with the same beta. In this case, the demand for B would fall, creating selling pressure. This would lower B’s price and boost its expected return, pushing it up to the SML.

## The assumptions of CAPM

Like all models, CAPM is a simplification of reality. As such, it is based on a number of simplifying assumptions.

First, it assumes that investors act rationally and are mean-variance optimizers.

Second, investors 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.

Third, investors have a single investment period (e.g., a year) in mind.

Fourth, all assets are tradeable and perfectly divisible.

Fifth, investors are price takers (i.e., their trades do not influence prices).

Sixth, they can lend or borrow unlimited amounts at a common risk-free rate.

Seventh, short selling is allowed without limits.

Eight, there are no taxes and transaction costs (e.g., trading commissions).

## Cost of equity

It is common for stock analysts to determine the intrinsic value of a stock by conducting discounted cash flows (DCF) analysis. One of the key inputs to the DCF analysis is the cost of equity. Analysts often rely on the CAPM to estimate the cost of equity. In particular, they would regress past stock returns on the past market returns to obtain an estimate of beta. Then, they would use yields on treasury bills (or bonds) as the risk-free rate. Finally, they would come up with the expected market return factoring in past market performance and their own judgment about future performance.

A simple example would be as follows. Suppose, we estimate a stock’s beta as 1.2. Furthermore, assume that the yield on Treasury bills is 2%. Finally, you expect the market to yield 7%. Then, the stock’s expected return is:

E[Ri] = Rf + βi (E[Rm] − Rf) = 2% + 1.2 ( 7% 2% ) = 8%

This can be taken as an estimate of the stock’s cost of equity.

## The CAPM calculator

We provide a CAPM calculator for you to easily calculate the expected return on an asset using the CAPM equation. There are three inputs you need to use our CAPM calculator:

• The beta of the asset,
• The risk-free rate of return,
• And the expected return on the market portfolio.
##### Summary

In this post, we offer a detailed discussion of the capital asset pricing model and provide an online CAPM calculator as well. A key insight of the model is that expected returns on risky assets should be positively linked to their exposure to market risk, which is captured by beta. If you are interested in finding out more about the origins of this model, you can find a link to Sharpe’s seminal paper below.