# Capital Asset Pricing Model

Capital asset pricing model (CAPM) is a model which determines the minimum required return on a stock as equal to the risk-free rate plus the product of the stock’s beta coefficient and the equity risk premium. Where beta measures a stock’s exposure to systematic risk, the type of risk which can’t be diversified, and the equity risk premium is the additional return required on an average stock investment on top of the risk-free rate.

You might want to review the article on risk and return to obtain an understanding of the portfolio expected return, standard deviation and their mutual trade-off on an efficient frontier. The article on capital allocation line offers how a portfolio of risk-free asset and a portfolio of risky assets interact with each other.

Risk in stock investments is broadly classified into systematic risk (also called undiversifiable risk) and unsystematic risk (also called unique risk or diversifiable risk). Because unique risk can be reduced/eliminated through diversification, it need not be compensated through increased return. The basic logic behind the capital asset pricing model is that the required return on a stock should change in response to a change in systematic risk only.

## Formula

An asset must earn at least as much as the risk-free rate plus a premium on account of the additional systematic risk which should equal the premium that a (theoretical) market portfolio earns in general (i.e. the market risk premium) and the extent to which the stock’s returns vary with the market.

The market risk premium (also called equity risk premium) equals required return on the market (r_{m}) minus the risk-free rate (r_{f}) and the relationship between a stock’s risk and the market risk is given by the ratio of their mutual covariance to the ratio of variance of the market (σ_{m}^{2}). This can be expressed mathematically as follows:

$$ r_i=r_f+(r_m-r_f)\times\frac{Covar(r_i,r_m)}{{\sigma_m}^2} $$

The extent to which a stock’s return must change in response to a change in systematic risk is called the beta coefficient (β).

The above relationship can be expressed mathematically as follows:

$$ r_e=r_f+(r_m-r_f)\times\beta $$

It is the mathematical representation of the capital asset pricing model.

Risk-free rate (r_{f}) is the rate of an investment with zero risk. Return on a US Treasury securities with maturity that matches the maturity of the security for which required return is estimated is considered a good proxy for risk-free rate.

The market return is the required rate of return on the market portfolio. The return on some broad equity market index such as S&P 500 is treated as a proxy to the market return. The difference between the market return and risk-free rate is called the equity risk premium.

## CAPM: assumptions

Even though the capital asset pricing model (CAPM) is very popular, it is based on the following assumptions:

- Investors are rational and risk averse; all have the same investment horizon, they care only about expected return and risk and hold fully-diversified portfolios.
- The markets are efficient with no taxes and trading costs and all investors have the same information, i.e. there is strong-form market efficiency.
- Investors can borrow or lend at the risk-free rate.

## Example

The most popular use of the capital asset pricing model is to determine the appropriate required rate of return of a stock (i.e. cost of equity). Cost of equity is then used to find the intrinsic value of a stock or in calculating the weighted average cost of capital.

Let’s determine required rate of return on Amazon’s stock. The stock has a perpetual life so the rate on the long-term US Treasury securities is the appropriate proxy for risk-free rate. We can estimate the market risk premium based the difference between the broad market index such as S&P 500 and the relevant risk-free that has prevailed in the past. This approach is called the historical risk premium approach.

We can also obtain the risk-free rate, equity risk premium using financial databases such as Yahoo Finance, etc. and major financial services firms also offer their guidance. For example, in September 2017, Duff & Phelps communicated their expectation of the risk-free rate and US equity risk premium as 3.5% and 5% respectively. Amazon’s beta coefficient is 1.71 (as at Yahoo Finance).

Given this data, we can estimate Amazon’s required return on equity as follows:

$$ r_e=3.5\%+1.71\times5\%=12.05\% $$

This is the justified required return on Amazon’s stock given its systematic risk.