Capital Asset Pricing Model (CAPM) is a model to estimate the expected return of an asset based solely on the systematic risk of the asset return. The logic behind why only systematic risk is priced in is that, in a perfectly efficient economic system, investors should be able to diversify their portfolio at no cost, so that it would allow them to eliminate unsystematic or firm-specific risk completely. Thus, if they can choose to invest in a diversified portfolio of assets rather than investing in a single asset, why should they demand a premium for single risk? One can easily argue that the financial world is far from being perfect and includes transaction costs, taxes, etc. Let's assume that it is possible to apply the CAPM to estimate the expected return on the common stock of Apple (AAPL).** **[For additional reading, refer to,

*Financial Concepts: Capital Asset Pricing Model (CAPM)*].

Theoretically, the CAPM is expressed as:

$E(R_i)=R_f+\beta_i\ \times\ [E(R_M)-R_f]\qquad\qquad\qquad\qquad\small{(1)}$

The model implies that the expected return of the asset ** E(R_{i})**, equals the sum of risk-free return and market risk premium multiplied by beta,

**β**

_{i,}of the asset

**. Beta of a particular asset reflects its systematic risk. The equation**

*i***does not contain any unsystematic risk factor.**

**β**is the slope of the regression line of

_{i }**against the excess market return**

*E(R*_{i})**. Here is a step-by-step method to apply the CAPM to estimate the expected return of Apple. (For related reading, refer to**

*E(R*_{M})-R_{f}*Beta: Know The Risk*).

#### Valuation Models: Apple’s Stock Analysis With CAPM

## 1. Choosing Proxy For Market Portfolio

The equity market portfolio is a portfolio that includes all the assets traded in the market. It would be too expensive and time consuming to construct such a portfolio; therefore, we can use an equity market index as a proxy for the market portfolio. The S&P 500 is a capitalization weighted index consisting of 500 leading large cap U.S. companies and covers approximately 80% of all traded equity market, with an approximate market capitalization of $25 trillion, which is the sum of the market caps for all of the stocks in the index.

## 2. Estimating Beta of Apple

We can estimate beta of the Apple stock by regressing Apple's returns against S&P 500's returns. The straightforward way of estimating beta is using the following formula:

$\beta_I\ =\ \frac{\text{Cov}(I,M)}{\text{Var}(M)}\text{ or }\beta_I\ =\ \frac{\rho_{I,M}\sigma_I}{\sigma_M}\qquad\qquad\qquad\qquad\qquad\ \small{(2)}$

where ** Cov (I,M)** is the covariance of Apple (I) and the market returns (S&P 500),

**– variance of the market,**

*Var (M)***– correlation coefficient between the returns of S&P 500 and Apple shares,**

*ρ*_{I,M}**and**

*σI***are standard deviations of Apple returns and the market returns, respectively. Our starting point in estimating the company beta is estimating its historical beta, based on the historical stock return data. For this, let's download the historical monthly Apple returns and S&P 500 returns (from January 2005–December 2014). The following plot of the Apple stock returns versus the S&P 500 returns helps illustrate beta of Apple as a slope of its regression line.**

*σM*
By calculating historicals with the help of equation (2), we get historical beta of 1.26 (**β _{hist }**= 1.26). It is assumed that beta of an asset has mean reverting property, which means it reverts to the market beta of 1 in the long run. Thus in practice, historical beta is adjusted to account for this nature of beta for ex-ante calculations. We adjust the historical beta of Apple through the following equation:

$\beta_{\text{adjusted}}\ =\ (1\ -\ \alpha)\times \beta_{\text{hist}}+\alpha\ \times\ 1\qquad\qquad\qquad\quad\small{(3)}$

where α is the speed of how fast long run beta approaches the market beta, which equals 1. So, the higher the α, the faster beta approaches 1. As a rule of thumb, α is taken as 0.33. Thus, we can calculate adjusted beta, *b _{adjusted}*

_{.}

_{$\beta_{\text{adjusted}}\ =\ 0.67\ \times\ \beta_{\text{hist}}\ +\ 0.33\ \times\ 1\ \approx\ 1.18$ }

## 3. Determining Risk Free Rate And Market Return

Usually the 10-year US government bond yield is used as a proxy for nominal risk free interest rate. As of February 11, 2015, the 10-year US Treasury bond yield was 2%. We can assume that the annual historical mean return of S&P 500 is a good proxy for the expected market return. Here, we calculate it as the arithmetic mean of monthly returns (again, based on the 10-year monthly data) and multiply it by 12, which yields approximately annual 5.6% return.

## 4. Estimating Expected Return

Now that we have all the relevant data, we can estimate the expected return on Apple, based on equation (1), assuming that Apple stockholders are compensated only for the systematic risk they bear.

$6.25\%\ =\ 2\%\ +\ 1.18\ \times\ (5.6\%\ -\ 2\%)$

Thus, based on CAPM, the expected annual return of Apple is 6.25%. In reality, Apple delivers quite larger actual returns. The CAPM does not capture the total risk of an asset. Standard deviation is a better estimator of the total risk. The following histograms compare the empirical distributions of Apple and S&P 500's returns (based on the same monthly returns from January 2005–December 2014).

The wide range of the annualized monthly stock returns of Apple implies how large the standard deviation of Apple's returns is. The high standard deviation explains why the actual returns may be significantly different from the expected return. Standard deviation of stock returns measures the degree of dispersion of returns around the mean, so the larger the standard deviation, the larger dispersion of returns around the mean.

The CAPM has several advantages and disadvantages. It is easier to apply and communicate the advantages of the model. However, the model does not reward investors for firm-specific risk. Not all investors can diversify their portfolio cheaply due to high transaction costs, and therefore bear significant unsystematic risk. Thus, if Apple stock carries high unsystematic risk, the CAPM will not capture it. Further, the model assumes that investors can lend and borrow at risk-free rate, which is a rare case in reality.

There are alternative models such as the Arbitrage Pricing Theory (APT) model and the French Fama model, which explain the expected return of the asset, adding more factors to reward.

## The Bottom Line

Although the CAPM (just as other models) has several drawbacks, it is an easier and effective starting point to estimate the expected return of an asset. It gives an overview of the level of return that investors should expect for bearing only systematic risk. Applying Apple, we get annual expected return of about 6.25%.