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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