澳洲幸运5官方开奖结果体彩网:Capital Asset Pricing Model ꧃(CAPM) is a model to estimate the expected return of an asset based solely on the 澳洲幸运5官方开奖结果体彩网: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 澳洲幸运5官方开奖结果体彩网:portfolio at no cost, so that it would allow them to eliminate 澳洲幸运5官方开奖结果体彩网: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 澳洲幸运5官方开奖结果体彩网:single risk? One can easily argue that the financial world is far from being perfect and includes 澳洲幸运5官方开奖结果体彩网:transaction costs, taxes, etc. Let's assume that it is possible to apply the CAPM to estimate the expected return on the 澳洲幸运5官方开奖结果体彩网:common stock of Apple (AAPL). [For additional reading, refer to, Financial Concepts: Capital Asset Pricing Model (CAPM)].

Theoretically, the CAPM is expressed as:

 $E(R_i)=R_f+\beta_i\ \times\ [E(R_M)-R_f]\qquad\qquad\qquad\qquad\small{(1)}$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 澳洲幸运5官方开奖结果体彩网:risk-free return and 澳洲幸运5官方开奖结果体彩网:market risk premium multiplied by 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. (For related reading, refer to 澳洲幸运5官方开奖结果体彩网:Beta: Know The Risk).

1. Choosing Proxy For Market Portfolio

The equity 澳洲幸运5官方开奖结果体彩网: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 澳洲幸运5官方开奖结果体彩网:equity market index as a proxy for the market portfolio. The 澳洲幸运5官方开奖结果体彩网:S&P 500 is a 澳洲幸运5官方开奖结果体彩网:capitalization weighted index consisting of 500 leading 澳洲幸运5官方开奖结果体彩网:large cap U.S. companies and covers approximately 80% of all traded equity market, with an approximate 澳洲幸运5官方开奖结果体彩网: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 澳洲幸运5官方开奖结果体彩网:regressing Apple's returns against�♕� S&P 500's returns. The straightforward way of estimating beta is using the following formula:

$\beta_I\ =\ \frac{\text{Cov}(I,M)}{\text{Var}(M)}\text{ or }\beta_I\ =\ \frac{\rho_{I,M}\sigma_I}{\sigma_M}\qquad\qquad\qquad\qquad\qquad\ \small{(2)}$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– 澳洲幸运5官方开奖结果体彩网:correlation coefficient between the returns of S&P 500 and Apple shares, σI and σM are 澳洲幸运5官方开奖结果体彩网: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 slo💞pe of its regression line.

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 澳洲幸运5官方开奖结果体彩网: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 histor𒊎ical beta of Apple through the following equation:

$\beta_{\text{adjusted}}\ =\ (1\ -\ \alpha)\times \beta_{\text{hist}}+\alpha\ \times\ 1\qquad\qquad\qquad\quad\small{(3)}$βadjusted​ = (1 − α)×βhist​+α × 1(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 澳洲幸运5官方开奖结果体彩网:rule of thumb, α is taken as 0.33. Thus, we can calculate adjusted beta, b_adjusted.

_{$\beta_{\text{adjusted}}\ =\ 0.67\ \times\ \beta_{\text{hist}}\ +\ 0.33\ \times\ 1\ \approx\ 1.18$βadjusted​ = 0.67 × βhist​ + 0.33 × 1 ≈ 1.18}

3. Determining Risk Free Rate And Market Return

Usually the 澳洲幸运5官方开奖结果体彩网:10-year US government bond yield is used as a proxy for nominal risk free 澳洲幸运5官方开奖结果体彩网:interest rate. As of February 11, 2015, the 10-year US Treasury 澳洲幸运5官方开奖结果体彩网:bond yield was 2%. We can assume that the annual historical 澳洲幸运5官方开奖结果体彩网:mean return of S&P 500 is a good proxy for the expected market return. Here, we calculate it as the 澳洲幸运5官方开奖结果体彩网: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\%)$6.25% = 2% + 1.18 × (5.6% − 2%)

Thus, based on CAPM, the expected 澳洲幸运5官方开奖结果体彩网: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 澳洲幸运5官方开奖结果体彩网:histograms compare the empirical distributions of Apple 🌺and S&P 500's returns (based on the same monthly return꧋s 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 澳洲幸运5官方开奖结果体彩网:actual returns may be significantly different from the 澳洲幸运5官方开奖结果体彩网:expected return. Standard deviation of stock returns measures the degree of 澳洲幸运5官方开奖结果体彩网:dispersion of returns around the mean, so the larger the standard deviati🐻on, 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 po🐽rtfolio cheaply due to high transaction costs, and therefore bea🐽r 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 澳洲幸运5官方开奖结果体彩网:Arbitrage Pricing Theory (APT) model and the 澳洲幸运5官方开奖结果体彩网:French Fama model, which explain the expected return of the asset, adding more factors to rew🍨ard.

The Bottom Line

Although the CAPM (just as other models) has several d𓄧rawbacks, 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%.