The Capital Asset Pricing Model - CAPM: benchmark model …bertha.tepper.cmu.edu/telmerc/70391/annotations/09_capmB.pdfThe Capital Asset Pricing Model CAPM: benchmark model of the

70391 - Finance

The Capital Asset Pricing ModelCAPM: benchmark model of the cost of capital

70391 – Finance – Fall 2016Tepper School of BusinessCarnegie Mellon Universityc©2016 Chris Telmer. Some content from slides by Bryan Routledge. Used with permission.

11.07.2016 11:45

Market Cost of CapitalSource: Financial Times, February 27, 2015

CAPM 2

Market Cost of CapitalSource: Financial Times, February 27, 2015

CAPM 3

Question/Setup

Where did the 8.4% cost of capital come from?

Reminders:

:: It is the expected return that investors require, given the risk inherent inthe cash flows

:: It is the discount rate that we use to arrive at PV

:: We have used a shortcut: it reflects cash flow risk that is similar to theoverall stock market risk?

:: We have used less of a shortcut: it reflects the fraction of bonds andstocks that replicate the cash flow

A puzzle

:: Empirical fact: stock market cost of capital (now) is roughly 8%

:: So 8.4% for the retailers must mean similar risk ... right?

I Have a look at the stock returns (spreadsheet). Wow!

:: Solution: only systematic risk matters

CAPM 4

http://bertha.tepper.cmu.edu/telmerc/public_files/capm.xlsx

Question/Setup

Where did the 8.4% cost of capital come from?

Reminders:

:: It is the expected return that investors require, given the risk inherent inthe cash flows

:: It is the discount rate that we use to arrive at PV

:: We have used a shortcut: it reflects cash flow risk that is similar to theoverall stock market risk?

:: We have used less of a shortcut: it reflects the fraction of bonds andstocks that replicate the cash flow

A puzzle

:: Empirical fact: stock market cost of capital (now) is roughly 8%

:: So 8.4% for the retailers must mean similar risk ... right?

I Have a look at the stock returns (spreadsheet). Wow!

:: Solution: only systematic risk matters

CAPM 4


What is Risk?

CAPM 5

What IS Risk?

Alternatives:: $110, one year from now costs $100 today.

:: Risk-free rate is

10%.:: Expected return is (trivially) 10%.

:: A “lottery:” (a “gamble,” a “risky asset”): $120 w.p. 1/2,$100 w.p. 1/2.

:: If cost is $100, expected return is 10%:: We call this “risk neutrality.”

:: If cost is $95, expected return is 15.7%. Excess expectedreturn — the risk premium — is 5.7%.

:: We call this “risk aversion.”

:: What is risk? A positive risk premium. An expected returnthat is greater than the risk free interest rate.

:: Point: the risk in an asset is reflected in its expected return

CAPM 6

What IS Risk?


:: Risk-free rate is 10%.:: Expected return is (trivially) 10%.







CAPM 6

What IS Risk?




:: If cost is $100, expected return is

10%:: We call this “risk neutrality.”





CAPM 6

What IS Risk?




:: If cost is $100, expected return is 10%

:: We call this “risk neutrality.”





CAPM 6

What IS Risk?









CAPM 6

What IS Risk?





:: If cost is $95, expected return is

15.7%. Excess expectedreturn — the risk premium — is 5.7%.




CAPM 6

What IS Risk?





:: If cost is $95, expected return is 15.7%. Excess expectedreturn — the risk premium — is

5.7%.:: We call this “risk aversion.”



CAPM 6

What IS Risk?









CAPM 6

What IS Risk?









CAPM 6

What IS Risk?









CAPM 6

What IS Risk?








:: Point: the risk in an asset is reflected in its expected returnCAPM 6

Risk

Risk is manifest in the expected excess return:

Risk Premium = E(rit − rft

)

CAPM 7

Foundations of the CAPM

CAPM 8

Remember

A firm’s cost of capital *is* the expected return on its stock.1

1No leverage assumed here. Stay tuned.CAPM 9

Basics

Given:

:1: Investors prefer to hold well-diversified portfolios

:2: There is a limit to diversification:

We can increase the diversification benefit by adding

more risky investments

Volatility of an equally weighted portfolio vs. the number of stocks

std=40%, correlation=0.28

Point 2. says that it makes sense to write the following

CAPM 10

Slight Reformulation

:: Return on the market portfolio is rmt .I This is the market-cap weighted portfolio of all the assets.I It is also a well diversified portfolio: no idiosyncratic variation,

only common variation. It is the common variation.

:: Slight reformulation:

rit − rft = αi + βi(rmt − rft

)+ εit

:: Take expected value

E(rit − rft

)= αi + βiE

(rmt − rft

):: CAPM follows from αi = 0

E(rit)

= rft + βiE(rmt − rft

)

CAPM 11






)+ εit


E(rit − rft

)= αi + βiE

(rmt − rft


E(rit)


)

CAPM 11






)+ εit


E(rit − rft

)= αi + βiE

(rmt − rft

)

:: CAPM follows from αi = 0

E(rit)


)

CAPM 11






)+ εit


E(rit − rft

)= αi + βiE

(rmt − rft


E(rit)


)CAPM 11

Graphically

CAPM 12

Capital Market Line

35

So far

Why do mean-variance optimization?Computations involving: one riskless and

one risky asset and two risky assetsThe Sharpe ratioDiversification and correlationMany assets

Source: Burton Hollifield

CAPM 13

CAPM

E(rit)


):: Beta is all that matters for one firm’s cost of capital relative

to another’s.

:: More sophisticated versions of this?

I Multi-factor models (more than one beta)

CAPM 14

CAPM

E(rit)


):: Beta is all that matters for one firm’s cost of capital relative

to another’s.

:: More sophisticated versions of this?

I Multi-factor models (more than one beta)

CAPM 14

Beta is the OLS Regression Coefficient

Mathematically it must be the case that

βi =Cov

(ri , rm

)Var(rm)

=Cov

(ri , rm

)σ2m

σiσi

=Cov

(ri , rm

)σm σi

σiσm

= Corr(ri , rm

) σiσm

= ϕi ,mσiσm

CAPM 15

CAPMQuantity and price of risk in asset i

E(ri − rf

)= βiE

(rM − rf

)

Security Market Line (SML)

CAPM 16

Applying the CAPM

CAPM 17

Value Weighting Works for Betas

Recall, expected returns for portfolios:

µp = γ1µ1 + γ2µ2

Sub-in CAPM, recall that γ1 + γ2 = 1, and then simplify:

rf + βp E(rm − rf

)= γ1

[rf + β1 E

(rm − rf

)]+ γ2

[rf + β2 E

(rm − rf

)]βp E

(rm − rf

)=

[γ1β1 + γ2β2

]E(rm − rf

)

Cancel the market risk premium:

βp = γ1 β1 + γ2 β2

Portfolio betas are value-weighted averages of the betasof the things in the portfolio

CAPM 18



µp = γ1µ1 + γ2µ2



)= γ1

[rf + β1 E

(rm − rf

)]+ γ2

[rf + β2 E

(rm − rf

)]βp E

(rm − rf

)=

[γ1β1 + γ2β2

]E(rm − rf

)


βp = γ1 β1 + γ2 β2


CAPM 18



µp = γ1µ1 + γ2µ2



)= γ1

[rf + β1 E

(rm − rf

)]+ γ2

[rf + β2 E

(rm − rf

)]βp E

(rm − rf

)=

[γ1β1 + γ2β2

]E(rm − rf

)


βp = γ1 β1 + γ2 β2


CAPM 18



µp = γ1µ1 + γ2µ2



)= γ1

[rf + β1 E

(rm − rf

)]+ γ2

[rf + β2 E

(rm − rf

)]βp E

(rm − rf

)=

[γ1β1 + γ2β2

]E(rm − rf

)


βp = γ1 β1 + γ2 β2


CAPM 18

Value-Weighting

rp =N∑i=1

γi ri

µp =N∑i=1

γiµi

βp =N∑i=1

γiβi

CAPM 19

Expected Returns on the Retailers

The date is January 2002. You have 1,200 shares of AEO and2,300 shares of GPS.

I What is the expected return over the next year on each of the stocks?Use the betas that appear below. Calculate expected returns based onpair of betas.

I What is the annual expected return on your portfolio? How much moneydo you expect your portfolio to be worth in one year?

I What is your portfolio beta?

I What is the expected return on your portfolio, based on its beta?

CAPM 20

Exercise

Spreadsheet-based exercise

CAPM 21

http://bertha.tepper.cmu.edu/telmerc/public_files/capmInvestorProblem.xlsx

A Capital Budgeting Problem

American Eagle is considering a new marketing campaign.

:: Cost = 400

:: Payoffs = 220 in one year and 225 in two years

:: Compute NPVI Based on sample betasI Based on industry betas

CAPM 22

Exercise

Spreadsheet-based exercise

CAPM 23

http://bertha.tepper.cmu.edu/telmerc/public_files/capmFirmProblem.xlsx

American Eagle Diversifies

Suppose that American Eagle buys 20% of First Energy’s (FE)regulated utility business. What is the new firm’s cost of capital?

I Utility beta = 0.3, retailer beta = 1.03.I FE EV = $36b, so a 20% piece is worth $7.2b. AEO EV =

$3.1b.I AEO is now 30% retailer, 70% investor-owned utility.

CAPM 24

Where the Numbers Came From

Open spreadsheet capm.xlsx What to watch for:

:: Calculation and summary statistics for total returns

:: Portfolio analysisI How well did your portfolio from above workout over the 13.5

year time period?

:: Calculation of betas and implied expected returns

CAPM 25


Further Applications

CAPM 26

CAPM: Estimating β on Industry PortfoliosCAPM: Estimating β on Industry Portfolios

−.3

−.2

−.1

0.1

.2D

aily

Ret

urn

−.2 −.1 0 .1Market all NYSE, AMEX, and NASDAQ [FF]

Gold Industry Portfolio Excess Return fit

FindIndustryBeta.do

Daily data: 1980 : 2012

Industry: Gold Beta=0.44

Expected Returns 27

CAPM 27


−.2

−.1

0.1

.2D

aily

Ret

urn


Guns Industry Portfolio Excess Return fit

FindIndustryBeta.do

Daily data: 1980 : 2012

Industry: Guns Beta=0.70

Expected Returns 27

CAPM 28


−.2

−.1

0.1

Dai

ly R

etur

n


Rtail Industry Portfolio Excess Return fit

FindIndustryBeta.do

Daily data: 1980 : 2012

Industry: Rtail Beta=0.95

Expected Returns 27

CAPM 29


−.2

−.1

0.1

.2D

aily

Ret

urn


Aero Industry Portfolio Excess Return fit

FindIndustryBeta.do

Daily data: 1980 : 2012

Industry: Aero Beta=0.99

Expected Returns 27

CAPM 30


−.2

−.1

0.1

.2D

aily

Ret

urn


Fin Industry Portfolio Excess Return fit

FindIndustryBeta.do

Daily data: 1980 : 2012

Industry: Fin Beta=1.28

Expected Returns 27

CAPM 31

Industry Betas

Aswath Damodaran’s Website

CAPM 32

http://pages.stern.nyu.edu/~adamodar/New_Home_Page/datafile/Betas.html

Time Varying Risk Free Rate

:: It’s important to use today’s risk-free rate

:: Cost of capital must change with level of interest rates

See spreadsheet

CAPM 33

http://bertha.tepper.cmu.edu/telmerc/public_files/cost_of_capital_as_riskfree_rate_changes.xlsx

Empirical Performance of CAPM

:: We’ll leave this for more advanced classes

:: Good discussion in text book.

:: CAPM is foundational. More sophisticated models areimportant in practice, but knowing/understanding/usingCAPM is a pre-requisite.

:: Optional exercise with Fama-French factors: (basic,with simulation

CAPM 34

http://bertha.tepper.cmu.edu/telmerc/public_files/fama_french_sol.xlsx

http://bertha.tepper.cmu.edu/telmerc/public_files/fama_french_with_simulations_optional.xlsm

Takeaways

CAPM 35

Takeaways

:: CAPM: cost of capital depends only on beta:

E(ri)

= rf + βi E(rm − rf

):: Portfolio Calculations:

I Portfolio expected returns and betas are value-weightedaverages of the expected returns and betas inside the portfolio

:: Coming in Finance II:I Portfolio choiceI More rigorous treatment of CAPMI More state-of-the-art versions of CAPM (e.g., Fama-French

3-factor model)

CAPM 36

Where Did the 8.4% Come From?Opportunity Cost of Capital

Expected Returns 28

CAPM 37

Implications

The return on Microsoft has

:: A higher standard deviation that the return on the S&P500

:: A beta close to 1.0

Is Microsoft riskier than the market?

CAPM 38

ImplicationsSecurity Market Line

Consider assets A, B and C :

:: Which asset would you hold?

:: Would you hold any of asset C?

:: What are the betas: βA, βB , βC?

CAPM 39

Jargon

Capital Market Line (µ against σ)

Security Market Line (µ against β)

CAPM 40

Documents

The Capital Asset Pricing Model - CAPM: benchmark model …bertha.tepper.cmu.edu/telmerc/70391/annotations/09_capmB.pdfThe Capital Asset Pricing Model CAPM: benchmark model of the