capm lecture

Capital Asset Pricing ModelEcon 487

Outline

• CAPM Assumptions and Implications

• CAPM and the Market Model

• Testing the CAPM

• Conditional CAPM

CAPM Readings

• Zivot, Ch. 8 (pp. 185-191) (page #’s at top of page)

• Benninga, Ch. 10 (pp. 221-228)

• Perold (2004) (pp. 288-289)

What is the CAPM?

• Theory of asset price determination for firms

• Based on portfolio theory and Market Model

• The only thing that matters is Beta (co-movement with the market)

• Alternative to valuation theory for individual firms

CAPM Assumption #1

• Many investors who are all price takers

• I.e., financial markets are competitive

• Returns provide full summary of investment opportunities

CAPM Assumption #2

• All investors plan to invest over the same time horizon

• Abstracts from heterogeneity in investors (i.e., risk averse have different time preferences than the risk tolerant)

• Helps address any deviations from CER model

CAPM Assumption #3

• No distortionary taxes or transaction costs

• Clearly a false assumption (debt vs. equity)

CAPM Assumption #4

• All investors can borrow/lend at same risk-free rate

• Again, clearly false

• But we can consider Zero-Beta version of CAPM with short-sales

CAPM Assumption #5

• Preferences: Investors only care about expected return (like) and variance (dislike)

• Consistent with portfolio theory and CER model under Normality

• False if assets have different co-movements with state of the economy (i.e., recession vs. boom)

CAPM Assumption #6

• All investors have same information and beliefs about distribution of returns

• E.g., CER model, with same beliefs about parameters

CAPM Assumption #7

• Market portfolio that determines Beta consists of all publicly traded assets

Implication #1

• Investors will use Markowitz algorithm to determine same set of efficient portfolios

Implication #2

• Risk-averse investors will put most of their wealth in risk-free asset

• Risk-tolerant investors will put most of their wealth in risky assets

• In equilibrium, no net borrowing

=> risk-free rate and tangency portfolio

Implication #3

• Tangency portfolio = market portfolio

• Note: implies positive weights on all assets in tangency portfolio, even allowing for short sales

Implication #4

• Market portfolio is mean-variance efficient

• I.e., highest Sharpe Ratio

Implication #5

• “Security Market Line”(SML) pricing holds for all assets and portfolios

• I.e., expected return on asset i is fully determined by three things:

1. risk-free rate

2. market risk premium

3. Beta for asset i

Security Market Line (SML)

!

E[Rit ] = rf + "i,M (E[RMt ]# rf )

where i refers to an individual asset or portfolio Beta-Portfolio

!

E[Rpt ] = (1" xm )rf + xmE[RMt ]

= rf + xm (E[RMt ]" rf )

Set

!

xm

= "i,M

for a given asset i to compare

!

E[Rpt ] on Beta-Portfolio to

!

E[Rit].

Log-Linear Present Value Relationship

!

pt =c

1" #+ Et # j"1

(1" #)dt+ j

j=1

$

%&

' ( (

)

* + + " Et # j"1

rt+ j

j=1

$

%&

' ( (

)

* + +

Market Model

!

Rit

="i+ #

i,MRMt

+ $it

!

"it~ iidN(0,#"i

2)

!

cov(RMt,"it) = 0

Subtract

!

rf from both sides

!

Rit " rf =# i " rf + $ i,M RMt + %it

Add and subtract

!

"i,M rf from right-hand-side:

!

Rit " rf =# i " (1"$i,M )rf + $ i,M (RMt " rf ) + %it

Is the CAPM Useful?

The Capital Asset Pricing Model is an elegant theory with profound implicationsfor asset pricing and investor behavior. But how useful is the model given the idealizedworld that underlies its derivation? There are several ways to answer this question. First,we can examine whether real world asset prices and investor portfolios conform to thepredictions of the model, if not always in a strict quantitative sense, and least in a strongqualitative sense. Second, even if the model does not describe our current worldparticularly well, it might predict future investor behavior—for example, as a conse-quence of capital market frictions being lessened through financial innovation, im-proved regulation and increasing capital market integration. Third, the CAPM canserve as a benchmark for understanding the capital market phenomena that causeasset prices and investor behavior to deviate from the prescriptions of the model.

Suboptimal DiversificationConsider the CAPM prediction that investors all will hold the same (market)

portfolio of risky assets. One does not have to look far to realize that investors donot hold identical portfolios, which is not a surprise since taxes alone will causeidiosyncratic investor behavior. For example, optimal management of capital gainstaxes involves early realization of losses and deferral of capital gains, and so taxableinvestors might react very differently to changes in asset values depending on whenthey purchased the asset (Constantinides, 1983). Nevertheless, it will still be a positivesign for the model if most investors hold broadly diversified portfolios. But even herethe evidence is mixed. On one hand, popular index funds make it possible for investorsto obtain diversification at low cost. On the other hand, many workers hold concen-trated ownership of company stock in employee retirement savings plans and manyexecutives hold concentrated ownership of company stock options.

One of the most puzzling examples of suboptimal diversification is the so-

Figure 4The Securities Market Line (SML)

Beta of market ! 1.0 Beta

SML

Expe

cted

ret

urn Market

portfolio In equilibrium, allassets plot on the SML

EM " rf ! slope of SML

EM

rf

18 Journal of Economic Perspectives

Intuition for CAPM

• Investors should not be compensated for diversifiable risk

E.g., Beta=0

• If expected return > risk-free, borrow at risk-free to buy zero-beta asset

• If expected return < risk-free, sell zero-beta asset short and buy more risk-free

• Both cases imply higher portfolio return without higher risk (at the margin)

Equilibrium for Beta=0

• Risk-free rate and price of zero-beta asset adjust to equate expected return and risk-free rate

• E.g., if expected return < risk-free rate, price falls today to make future expected returns higher

• recall log-linear present-value relationship between price and expected returns


!



!



Set

!

xm

= "i,M


!


!

E[Rit].


!

pt =c

1" #+ Et # j"1

(1" #)dt+ j

j=1

$

%&

' ( (

)

* + + " Et # j"1

rt+ j

j=1

$

%&

' ( (

)

* + +

Market Model

!

Rit

="i+ #

i,MRMt

+ $it

!

"it~ iidN(0,#"i

2)

!

cov(RMt,"it) = 0

Subtract

!

rf from both sides

!


Add and subtract

!


!


E.g., Beta=1

• If expected return > market, sell other assets to buy high-return asset

• If expected return < market, sell asset to buy more of market portfolio

• Both cases imply higher portfolio return without higher risk (at the margin)

• Prices adjust to bring about equilibrium

In general

• Investors can choose mix of risk-free asset and market portfolio to achieve any desired expected return => “Beta Portfolio”

• Weight on market portfolio is Beta in SML

• If expected return on asset i is different than SML, prices will adjust as investors buy/sell Beta portfolio and asset i.


!



!



Set

!

xm

= "i,M


!


!

E[Rit].


!

pt =c

1" #+ Et # j"1

(1" #)dt+ j

j=1

$

%&

' ( (

)

* + + " Et # j"1

rt+ j

j=1

$

%&

' ( (

)

* + +

Market Model

!

Rit

="i+ #

i,MRMt

+ $it

!

"it~ iidN(0,#"i

2)

!

cov(RMt,"it) = 0

Subtract

!

rf from both sides

!


Add and subtract

!


!


Is the CAPM Useful?

The Capital Asset Pricing Model is an elegant theory with profound implicationsfor asset pricing and investor behavior. But how useful is the model given the idealizedworld that underlies its derivation? There are several ways to answer this question. First,we can examine whether real world asset prices and investor portfolios conform to thepredictions of the model, if not always in a strict quantitative sense, and least in a strongqualitative sense. Second, even if the model does not describe our current worldparticularly well, it might predict future investor behavior—for example, as a conse-quence of capital market frictions being lessened through financial innovation, im-proved regulation and increasing capital market integration. Third, the CAPM canserve as a benchmark for understanding the capital market phenomena that causeasset prices and investor behavior to deviate from the prescriptions of the model.

Suboptimal DiversificationConsider the CAPM prediction that investors all will hold the same (market)

portfolio of risky assets. One does not have to look far to realize that investors donot hold identical portfolios, which is not a surprise since taxes alone will causeidiosyncratic investor behavior. For example, optimal management of capital gainstaxes involves early realization of losses and deferral of capital gains, and so taxableinvestors might react very differently to changes in asset values depending on whenthey purchased the asset (Constantinides, 1983). Nevertheless, it will still be a positivesign for the model if most investors hold broadly diversified portfolios. But even herethe evidence is mixed. On one hand, popular index funds make it possible for investorsto obtain diversification at low cost. On the other hand, many workers hold concen-trated ownership of company stock in employee retirement savings plans and manyexecutives hold concentrated ownership of company stock options.

One of the most puzzling examples of suboptimal diversification is the so-

Figure 4The Securities Market Line (SML)

Beta of market ! 1.0 Beta

SML

Expe

cted

ret

urn Market

portfolio In equilibrium, allassets plot on the SML

EM " rf ! slope of SML

EM

rf

18 Journal of Economic Perspectives

Key Point

• Even if there is an implicit present-value model of stock price determination, there is no need to forecast future dividends for firm i

• Given SML, all that matters for pricing firm i is Beta (i.e., responsiveness to market return)

• Easier to estimate Beta than to forecast future dividends

CAPM and the Market Model

• Market Model is a statistical model

• CAPM is a theory that places parameter restrictions on Market Model

• Consider “excess return” version of Market Model


!



!



Set

!

xm

= "i,M


!


!

E[Rit].


!

pt =c

1" #+ Et # j"1

(1" #)dt+ j

j=1

$

%&

' ( (

)

* + + " Et # j"1

rt+ j

j=1

$

%&

' ( (

)

* + +

Market Model

!

Rit

="i+ #

i,MRMt

+ $it

!

"it~ iidN(0,#"i

2)

!

cov(RMt,"it) = 0

Subtract

!

rf from both sides

!


Add and subtract

!


!


Excess-Return Market Model

!

Rit " rf =# i

$+ % i,M (RMt " rf ) + &it ,

where

!

" i

#=" i $ (1$% i,M )rf .

Take expectations of both sides and take

!

rf to right-hand-side:

!

E[Rit ] = rf +" i

#+ $i,M (E[RMt ]% rf )

The SML for CAPM implies

!

"i

#= 0 or, equivalently,

!

" i = (1#$ i,M )rf .

Testing the CAPM1. Test H0: α*=0 for excess-return Market

Model (t-test for one asset or F-test for a joint test for a set of assets)

2. Check if market portfolio is efficient and equal to tangency portfolio for assets in market portfolio

3. Check predictions for expected returns based on Beta and SML

Why might tests reject CAPM in practice? (I)

• CAPM does not hold

• restrictions on short sales

• heterogeneous probability assessments

Why might tests reject CAPM in practice? (II)

• Maybe CAPM only holds for portfolios, not individual assets

• liquidity issues for individual assets

Why might tests reject CAPM in practice? (III)• Set of assets in market portfolio proxy

incomplete

• foreign assets, bonds, real estate

• explains if market portfolio appears less efficient than tangency portfolio

• Roll critique (only test is if true market portfolio is mean-variance efficient)

Why might tests reject CAPM in practice? (IV)

• Non-tradable assets matter

• human capital

• co-movement of traded assets with business cycle (multiple factor model)

• given same Beta, prefer less correlation with labour income

Early Studies

• Black, Jensen, Scholes (72), Fama and MacBeth (73), Blume and Friend (73) support mean-variance efficiency of market portfolio

• But estimated mean return on zero-beta portfolio > risk-free rate

Anomalies Literature

• Portfolios based on firm characteristics appear to have higher ex-post Sharpe ratios than market proxy

• Portfolios emphasizing smaller size, lower P/E ratios, and higher book/market ratios appear to be better

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Y

Counter-Attack

• Little theory behind additional factors

• data snooping? (alphabet experiment)

• firms with long data on BM ratios have survivorship bias

• large econometric effects of even small measurement error for true market portfolio

Statistical Issues

• We will consider validity of CER model assumptions

• For example, Beta not stable over time

• Conditional CAPM?

Excess-Return Market Model

!

Rit " rf =# i

$+ % i,M (RMt " rf ) + &it ,

where

!

" i

#=" i $ (1$% i,M )rf .

Take expectations of both sides and take

!

rf to right-hand-side:

!

E[Rit ] = rf +" i

#+ $i,M (E[RMt ]% rf )

The SML for CAPM implies

!

"i

#= 0 or, equivalently,

!

" i = (1#$ i,M )rf .

Conditional CAPM

!

Et[Rit+1] = rft + " it (Et[RMt+1]# rft )

where

!

"it

=cov

t(R

it+1,RMt+1)

vart(R

Mt+1)

In practice

!

vart(R

Mt+1) changes a lot over time.

Unconditional Expectations

!

E[Rit+1] = rft + " i(E[RMt+1]# rft ) + cov(" it ,Et[RMt+1])

I.e.,

!

"i

#= cov($

it,E

t[R

Mt+1]) if conditional CAPM holds

Conditional CAPM

• Lewellaen and Nagel (06) find that short-horizon estimates of time-varying betas and the market risk premium do not explain α*’s for B/M or momentum portfolios

Conclusions (I)

• CAPM does not hold in simple form

• Yet it can explain a large portion of variation in expected returns

• Normative model (investors should not needlessly expose themselves to diversifiable risk)

Conclusions (II)

• Modern finance still mostly based on factor approach instead of individual firm valuation

• Need to address statistical problems with Market Model

• For conditional CAPM, need good estimates of time-varying betas and market risk premium

capm lecture

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