How CAPM Evaluates Apple's Stock: Understanding Expected Returns

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 Capital Asset Pricing Model to estimate the expected return on the common stock of Apple (AAPL). 

Theoretically, the CAPM is expressed as:

E
(

R
i

)
=

R
f

+

β
i

 
×
 
[
E
(

R
M

)
−

R
f

]

(
1
)

E(R_i)=R_f+beta_i times [E(R_M)-R_f]qquadqquadqquadqquadsmall{(1)}

E(Ri​)=Rf​+βi​ × [E(RM​)−Rf​](1)

The model implies that the expected return of the asset E(R_i) equals the sum of the risk-free return and market risk premium multiplied by the beta, β_i, of the asset i. Beta of a particular asset reflects its systematic risk. The equation does not contain any unsystematic risk factor. β_i is the slope of the regression line of E(R_i) against the excess market return E(R_M)-R_f. Here is a step-by-step method to apply the CAPM to estimate the expected return of Apple.

Selecting a Market Portfolio Proxy for CAPM Analysis

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 $45 trillion, which is the sum of the market caps for all of the stocks in the index.

Calculating Apple’s Beta for CAPM

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

β
I

 
=
 

Cov
(
I
,
M
)

Var
(
M
)

 or 

β
I

 
=
 

ρ

I
,
M

σ
I

σ
M

 

(
2
)

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

βI​ = Var(M)Cov(I,M)​ or βI​ = σM​ρI,M​σI​​ (2)

Where Cov (I,M) is the covariance of Apple (I) and the market returns (S&P 500), Var (M) – variance of the market, ρ_I,M– correlation coefficient between the returns of S&P 500 and Apple shares, σI and σM 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 the beta of Apple as the slope of its regression line.

By calculating historicals with the help of equation (2), we get a historical beta of 1.26 (β_hist = 1.26). It is assumed that the beta of an asset has a 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:

β
adjusted

 
=
 
(
1
 
−
 
α
)
×

β
hist

+
α
 
×
 
1

(
3
)

beta_{text{adjusted}} = (1 – alpha)times beta_{text{hist}}+alpha times 1qquadqquadqquadquadsmall{(3)}

βadjusted​ = (1 − α)×βhist​+α × 1(3)

Where α is the speed at 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.

β
adjusted

 
=
 
0.67
 
×
 

β
hist

 
+
 
0.33
 
×
 
1
 
≈
 
1.18

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

βadjusted​ = 0.67 × βhist​ + 0.33 × 1 ≈ 1.18

Establishing Risk-Free Rate and Market Return for CAPM

Usually, the 10-year US government bond yield is used as a proxy for the 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 the 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 an annual 5.6% return.

Calculating Expected Return Using CAPM

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
 
×
 
(
5.6
%
 
−
 
2
%
)

6.25% = 2% + 1.18 times (5.6% – 2%)

6.25% = 2% + 1.18 × (5.6% − 2%)

Thus, based on CAPM, the expected annual return of Apple is 6.25%. In reality, Apple delivers much 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’s and the 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. The standard deviation of stock returns measures the degree of dispersion of returns around the mean, so the larger the standard deviation, the larger the 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 a 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 an annual expected return of about 6.25%.