10
CAPM III A formal equation of the CAPM Applications, estimates of, and
determinants of ß What is 𝛼?
Key Assumptions of Portfolio Theory n We have been assuming that the investor knows all
individual assets’ n Mean returns n Variances n Covariances
n Reality: This information is unknown, and different people hold different beliefs; this may partially explain why people do not all hold the market portfolio as predicted
Example
Now that we have a market portfolio… n …we can measure the risk of every stock (or
asset) n It may seem natural to use the standard
deviation of that stock’s return; this makes sense for an investor who holds that stock in isolation
n But not for an investor who owns that stock as a component of the market portfolio…
Formally defining beta (ß) n ßi = si,M /sM2
n Ri is the return on asset i n RM is the return on the market portfolio n si,M is the covariance between Ri and RM n sM2 is the variance of the market portfolio
Rewriting beta (ß) n ßi = si,M /sM2 n Recall definition of correlation
coefficient: si,M = ρi,M si sM n ßi = (ρi,M si sM)/sM2 = ρi,M si /sM
n Assets whose returns are highly correlated with the market portfolio return have high betas
n Assets whose returns are risky have larger- magnitude betas
Example: Market Portfolio n Using ßi = si,M /sM2
ßM = sM,M /sM2 = sM2 /sM2 = 1
Also it can be shown that the average ßi across all stocks (weighting each stock by its share of the market portfolio) is 1
Interpreting Beta
n The right way to measure the riskiness of stock i’s return is not in isolation but instead relative to the return of the market portfolio n The reason is that investors optimally do
not hold asset i in isolation but rather as part of the market portfolio
Examples
Examples
Examples
Expected return on the market n In the market, we define the difference
between the risk-free return (RF) and the expected return on the market (RM) to be the risk premium n RM = RF + Risk premium
Intuition behind the CAPM equation n Just like the market as a whole
commands a premium for taking on risk, the same idea applies to a single security n The more risk the security adds to the
market portfolio, the higher its expected return
CAPM equation n The CAPM equation is
n Ri = RF + ßi (RM – RF) n In words, the expected return on a
security (Ri) is equal to the risk-free rate (RF) plus ßi times the risk premium (RM – RF)
Alternatively n The CAPM equation is
n Ri - RF = ßi (RM – RF) n The equity risk premium on a security is
equal to ßi times the market risk premium
Example n The risk-free return is 3% n The expected return on the market is 9% n A stock has a ß of 1.8 n What is the expected return of this stock?
n First, we have to calculate the risk premium: (9% – 3% = 6%)
n The expected return on the stock: 3% + 1.8 * 6% = 13.8%
Relationship between expected return and beta
ß of a security
Expected return of a security (R)
RF
1
RM = RF plus the risk premium
Security market line (SML)
0
Three Implications n ßi >1 Ri > RM : risky, growth stock
n 0< ßi < 1 RF < Ri < RM
n ßi < 0 Ri < RF : hedge
Cautionary note n In the CAPM equation, we find expected
return of a single security as a function of the beta of this security
n When finding the optimal investment portfolio, we find expected return of a portfolio as a function of the standard deviation of a portfolio’s return
n Make sure you do not confuse these two concepts
Note differences on the two graphs
Alpha n
Source: https://seekingalpha.com
Source: https://amazon.com
What is alpha? n Generalized CAPM:
n Ri - RF = 𝛼! + ßi (RM – RF) n 𝛼! is excess return on asset i not justified
by its risk (as properly measured), ßi n What does CAPM predict is 𝛼!?
n 𝛼! = 0 ! n Just because CAPM predicts no alpha does
not stop investors from searching for it
Estimating ß To estimate , run regression
Ri = ai + ßi (RM – RF) + ei, where ei is mean-zero error term
Formula from Statistics/Econometrics for ßi is
𝛽! = "#$ %!,%"'%# ()* %"'%#
= "#$ %!,%" ()* %"
= +!," +" % !
Estimating ß To estimate , run regression
Ri = ai + ßi (RM – RF) + ei, where ei is mean-zero error term
According to CAPM, estimated ai should be RF
Single-Factor Models Suppose we run regression
𝑅! = 𝑎! + 𝛽! 𝑅, − 𝑅- + 𝛿!𝑍! + 𝑒!, where ei is mean-zero error term and Zi is any other variable
According to CAPM, whatever is 𝑍! , 𝛿! = 0
The ”single factor” that affects Asset i’s return is the market risk premium
Multi-Factor Models Fama and French pioneered approach of adding more “factors” (Zi’s) to the regression equation to see whether they help to explain asset returns. They find that • High book-value firms outperform CAPM
prediction • Small-Cap firms outperform CAPM
predictions
Applications of CAPM n Let’s make some assumptions
n A new potential project by a firm has the same ß risk as for the firm as a whole
n 100% equity financing n I.e. no debt
n We can determine the appropriate discount rate for new projects n This is known as the cost of equity
Example n The risk-free return is 3% n The expected return on the
market is 9% n A stock has a ß of 1.8 n What is the expected return of
this stock? n First, we have to calculate the risk
premium: (9% – 3% = 6%)
n The expected return on the stock: 3% + 1.8 * 6% = 13.8%
n We return to our previous example
n In this case, the cost of equity (RS) is the same as the expected return on the stock
Example n Firms typically do not have all projects with
the same ß n Some projects are low-risk n Other projects are high-risk
n The discount rate for each project should depend on the risk for that project n Projects with low risk (and low ß) should have a
lower discount rate threshold than projects with high risk (and high ß)
Project-reliant values of ß
ß of a project
Discount rate of a project
RF
Firm’s overall cost of capital
SML (security market line)
We should accept this project (above SML)
We should reject this project (below SML)
How is ß determined? n Three factors help to determine ß
n Companies whose revenues go up and down more with the business cycle often have higher values of ß
n Firms with high fixed costs and low variable costs often have even higher fluctuations in profits based on sales è These firms often have high values of ß
n Firms with high fixed costs relative to variable costs are said to have high operating leverage
n Firms that rely more on debt than a comparable firm will have a higher ß
n Potential gains get divided among fewer shares of stock