Capital Asset Pricing Models

Market risk is the only risk left after diversification Return that investors get in the market is rewarded for market

risk only, not total risk Hence market risk is relevant risk, and specific risk is

irrelevant risk In the market, higher beta gets higher return, not higher std

gets higher return.

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std beta amount investedIBM 40% 0.95 2000AT&T 20% 1.10 40001. What is the beta of market portfolio2. Does IBM have more or less risk than the market3. Which stock has more total risk

Which stock has more systematic risk

Which stock is expected to have higher return in the market

4. In the boom market, which stock do you choose5. In the recession market, which stock do you choose6. How do investors know whether the return they get in the market is high

enough to reward for the level of risk taken

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It is the equilibrium model that underlies all modern financial theory.

Derived using principles of diversification with simplified assumptions.

Markowitz, Sharpe, Lintner and Mossin are researchers credited with its development.

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Investors care only about the mean-variance trade-off of their portfolios in the next period

All investors are price-takers. i.e., no investor is dominant such that her action alone will change prices – perfect competition assumption

Investors have homogeneous beliefs and equal investment opportunities

There is a risk-free asset and investors can borrow and lend at the same risk-free rate

Markets are frictionless, i.e., with no taxes and transaction costs. No limitation on the size of trading and short sales

All of investors’ wealth is in market traded assets

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All investors will hold the same portfolio for risky assets – market portfolio.

Market portfolio contains all securities and the proportion of each security is its market value as a percentage of total market value.

Risk premium on the the market depends on the average risk aversion of all market participants.

Risk premium on an individual security is a function of its covariance with the market.

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7

p

)E(R p

Rf

M

CML

Borrowing

Lending

The CML leads all investors to invest in the M portfolio. The only difference is the location on the CML depending on risk preferences◦ Risk averse investors will lend part of the portfolio at the risk-

free rate and invest the remainder in the market portfolio◦ Investors preferring more risk might borrow funds at Rf and

invest everything in the market portfolio◦ Two-fund separation theorem or “mutual fund theorem”

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Because Portfolio M lies at the point of tangency, it has the highest portfolio possibility line

Everybody will want to invest in Portfolio M and borrow or lend to be somewhere on the CML

Therefore this portfolio must include ALL RISKY ASSETS

Because the market is in equilibrium, all assets are included in this portfolio in proportion to their market value

Therefore, Portfolio M must be the market portfolio

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The tangency portfolio M is the market portfolio◦ All assets included in this portfolio are weighted in

proportion to their market value◦ Because portfolio M contains all risky assets, it is a

completely diversified portfolio. Only systematic risk remains in the market portfolio.

◦ Systematic risk may be measured by the standard deviation of returns on the market portfolio.

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Standard Deviation of Return

Number of Stocks in the Portfolio

Standard Deviation of the Market Portfolio (systematic risk)

Systematic Risk

Total Risk

Unsystematic (diversifiable) Risk

The risk premium on the market portfolio will be proportional to its risk and the degree of risk aversion of the investor: 2

2

( )

where is the variance of the market portolio and

is the average degree of risk aversion across investors

M f M

M

E r r A

A

The risk premium on individual securities is a function of the individual security’s contribution to the risk of the market portfolio

An individual security’s risk premium is a function of the covariance of returns with the assets that make up the market portfolio

Covariance of GE return with the market portfolio:

Therefore, the reward-to-risk ratio for investments in GE would be:

1 1

( , ) , ( , )n n

GE M GE k k k k GEk k

Cov r r Cov r w r w Cov r r

( ) ( )GE's contribution to risk premium

GE's contribution to variance ( , ) ( , )GE GE f GE f

GE GE M GE M

w E r r E r r

w Cov r r Cov r r

Reward-to-risk ratio for investment in market portfolio:

Reward-to-risk ratios of GE and the market portfolio:

And the risk premium for GE:

2

( )Market risk premium

Market varianceM f

M

E r r

2

( ) ( ( )

( , )GE f M f

GE M M

E r r E r r

Cov r r

2

( , )( ) ( )GE M

GE f M fM

Cov r rE r r E r r

CAPM holds for the overall portfolio because:

This also holds for the market portfolio:

P

( ) ( ) andP k kk

k kk

E r w E r

w

( ) ( )M f M M fE r r E r r

Assume: RFR = 6% RM = 12%

Implied market risk premium = 6%

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Stock Beta

A 0.70B 1.00C 1.15D 1.40E -0.30

RFR)-(RRFR)E(R Mi i

E(RA) = 0.06 + 0.70 (0.12-0.06) = 0.102 = 10.2%

E(RB) = 0.06 + 1.00 (0.12-0.06) = 0.120 = 12.0%

E(RC) = 0.06 + 1.15 (0.12-0.06) = 0.129 = 12.9%

E(RD) = 0.06 + 1.40 (0.12-0.06) = 0.144 = 14.4%

E(RE) = 0.06 + (-0.30) (0.12-0.06) = 0.042 = 4.2%

CAPM gives the relationship between risk and return. It gives the minimum return required by investors in order for

them to buy stock What is the meaning of E(R) calculated in the previous slide?

Remember earlier, we have

In CAPM, we have

What is the difference in meaning between the two expected return?

n

iiii rprE

1

ff rEr mii rrE

• When forecasted E(R) > required E(R), stock is undervalued or the price is too low

• When forecasted E(R) < required E(R), stock is undervalued or the price is too high

• In equilibrium, forecasted E(R) = required E(R)

Example: E(Rm) = 14%, Rf = 6%Stocks Beta E(R) (forecasted)IBM 1.2 17%ATT 1.5 14%

1. According to CAPM, what is the required E(R) for IBM and ATT

2. Which stock is undervalued, which stock is overvalued

Let alpha (α) be the difference between the actual (forecasted) E(R) and the required E(R)

In equilibrium, all assets and all portfolios of assets should plot on the SML ( i.e., α = 0)

Any security with an estimated return that plots above the SML is underpriced (α > 0 )

Any security with an estimated return that plots below the SML is overpriced ( α < 0 )

Previous example:◦ αIBM= 17 – 15.6 = 1.4 > 0. Actual E(R) is above the SML

◦ αATT= 14-18 = -4 > 0. Actual E(R) is below the SML

Expected return of a portfolio

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i

n

iip

Mi

n

ii

n

iMii

n

iiip

w

RfRwRf

RfRRfwREwRE

1

1

11

)()(

Q: Suppose that the risk premium on the market portfolio is estimated at 8% with a standard deviation of 22%. What is the risk premium on a portfolio invested 25% in GM and 75% in Ford, if they have betas of 1.10 and 1.25, respectively?

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Q: Two investment advisors: A: return=19%; beta=1.5B: return=16%; beta=1a) Who was better?b) If T-bill rate were 6% and market return were 14%, who would

be better?c) What if T-bill rate were 3% and market return were 15%?

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To move from expected to realized returns—use the index model in excess return form:

The index model beta coefficient turns out to be the same beta as that of the CAPM expected return-beta relationship

i i i M iR R e

εRRβRR

εRβaR

f,tM,tiif,ti,t

M,tiii,t

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where: Ri,t = rate of return for asset i during period tRM,t = rate of return for the market portfolio M during t

•Adjustments to Merrill-Lynch: Adjusted Beta = 2/3*(unadjusted. ) + 1/3

•Time interval problems•Different holding periods produce different beta•More pronounced for small company and illiquid stocks•Weekly and monthly returns better for estimation, not daily data.

error term random rate; riskfree ftR

Risk-free rate◦ Most use short-term Treasury bill returns◦ Notice that bill returns are variable, not truly risk-free.

Market risk premium◦ Historical data of excess return on market index◦ But expected return on market index may change over

time Proxies for market portfolio

◦ S&P (U.S. equity only)◦ World indices (ignore other assets, like real estates, etc.)

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effects. specfic firm measures which residual :

market theof premiumrisk or return exess:

marketon return :

regression theof slope andintercept are :,

iasset of premiumrisk or return exess :

iasset on return :

i

fm

m

ii

fi

i

ifmiifi

e

rr

r

rr

r

errrr

• alpha is the abnormal return = actual return – return predicted by CAPM

•According to CAPM, alpha should be = 0

•Beta is the systematic risk

All the points are actual values Line is the predicted relationship If there are a lot of specific risk, there will be a wide scatter of

points around the line. Hence, using market risk only in this case does not produce a precise estimate of expected return

If the points are close to the line, there is only small specific risk. Using market risk can explain most of the company return.

R-square: 0.2866 ANOVA table

Total risk = systematic risk + unsystematic risk

7449.17 = 2224.696 + 5224.45

(100%) = 29.87% + 70.13%

Alpha = 0.8890 > 0 (positive alpha, undervalued or overvalued?) During the period Jan 99-Dec03: the risk-adjusted or abnormal return of

GM = 0.8990% or actual return is higher than CAPM predicted Is this value statistically different from 0? Is this still consistent with

CAPM 95% confidence interval (-1.5690 to 3.3470) Beta = 1.2384 Is beta statistically different from 0?

CAPM is a benchmark about the fair (required) expected return on a risk asset. Investors calculate the return they actually earn based on their input and compare with the return they get from the CAPM

Compare the performance of the mutual fund: we use alpha or risk-adjusted return rather than regular return

Compute the cost of equity for capital budgeting

Typical Tests◦ Time-series test (Black-Jensen-Scholes)

◦ Cross-sectional tests (Fama-MacBeth)

◦ Most tests are done in portfolios

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.0 if see Test to

)(

i

itftMtiiftit RRRR

.0 , ,0 if see Test to

residual) ( ,ˆ

~~'

~

210

210

RFRR

CHARR

RR

M

iiiii

itMtiiit

High beta portfolios do not necessarily generate high returns◦ Controversial results: low betas have positive alpha, high

betas have negative alpha. Size and book-to-market value ratio seem to have

explanatory power for returns ◦ Fama and French

Momentum in returns◦ Relative strength

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Is the CAPM wrong? ◦Problems with the proxy for market portfolio◦Possible missing risk factors -> Multi-factor

models◦Relaxing assumptions

Important intuitions from the CAPM◦Diversification◦Only covariance with systematic risks matters

CAPM is powerful at the conceptual level. It is a useful way to think about risk and return

Empirical data does not support CAPM fully but it is simple, logical, easy to use, so use CAPM with caution

Zero-Beta Model◦ Helps to explain positive alphas on low beta

stocks and negative alphas on high beta stocks

Consideration of labor income and non-traded assets

Merton’s Multiperiod Model and hedge portfolios◦ Incorporation of the effects of changes in

the real rate of interest and inflation

CAPM is limited (true market portfolio is unobservable), nice idea though!

Other factors also matter, e.g., Fama-French book-to-market and size factors

Arbitrage Pricing Theory (APT): no free lunch (for diversified portfolio)!

FIN 2802, Spring 09 - TangChapter 9: Asset Pricing Theories

CAPM is a single factor model. The market risk premium is the only factor

In CAPM, all the news, uncertainties affect the market, then the market affect the stock individually

In APT, there are n factors that can influence stock return so there will be n-sources of risk or n-channels of uncertainties

Empirical evidence support APT (more than 1 factor affect stock return), but unable to identify these factors.

So if the purpose is to get cost of capital only, then APT is appropriate

If we want to know sources of risk then APT is not useful

Use of more than a single factor Requires formation of factor portfolios What factors?

◦Factors that are important to performance of the general economy

◦Fama-French Three Factor Model

Work of Chen, Roll, and Ross◦Chose a set of factors based on the

ability of the factors to paint a broad picture of the macro-economy

IP = % change in industrial production EI = % change in expected inflation UI = % change in unanticipated inflation CG = excess return of long-term corporate

bond over long-term government bond GB = excess return of long-term

government bond over T-bills

Fama and French propose three factors:◦ The excess market return, rM-rRF.

◦ the return on, S, a portfolio of small firms (where size is based on the market value of equity) minus the return on B, a portfolio of big firms. This return is called rSMB, for S minus B.

◦ the return on, H, a portfolio of firms with high book-to-market ratios (using market equity and book equity) minus the return on L, a portfolio of firms with low book-to-market ratios. This return is called rHML, for H minus L.

The factors chosen are variables that on past evidence seem to predict average returns well and may capture the risk premiums

Where:◦ SMB = Small Minus Big, i.e., the return of a portfolio of small

stocks in excess of the return on a portfolio of large stocks◦ HML = High Minus Low, i.e., the return of a portfolio of stocks

with a high book to-market ratio in excess of the return on a portfolio of stocks with a low book-to-market ratio

it i iM Mt iSMB t iHML t itr R SMB HML e

A multi-index CAPM will inherit its risk factors from sources of risk that a broad group of investors deem important enough to hedge

The APT is largely silent on where to look for priced sources of risk