Ch 7

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Cha pter 7

Capital Asset Pricing and Arbitrage

Pricing Theory

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7-2

7.1 THE CAPITAL ASSET PRICING MODEL

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Capital Asset Pricing Model (CAPM)

• The CAPM is a centerpiece of modern financial

economics, which was proposed by William Sharpe, who was awarded the 1990 Nobel Prize for

economics

CAPM tells us 1) what is the price of risk?

(Market price of risk)

2) what is the risk of asset i?

(quantity of risk asset i)

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Capital Asset Pricing Model (CAPM)

• It is an “equilibrium” model derived using

principles of diversification and some simplified assumptions for the behavior of investors and the market condition

– The market equilibrium refers to a condition in which for all securities, market prices are established to

balance the demand of buyers and the supply of sellers. These prices are called equilibrium prices

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Capital Asset Pricing Model (CAPM)

• The CAPM is a model that relates the expected

required rate of return for any security to its risk as measured by beta

• Specifically:

Total risk = systematic risk + unsystematic risk CAPM says:

(1)Unsystematic risk can be diversified away. It can be avoided by diversifying at NO cost, the market will not reward the holder of unsystematic risk at all.

(2)Systematic risk cannot be diversified away without

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Capital Asset Pricing Model (CAPM

Diversification and Beta

Beta measures systematic risk

– Investors differ in the extent to which they will take risk, so they choose securities with different betas

• E.g., an aggressive investor could choose a portfolio with a beta of 2.0

• E.g., a conservative investor could choose a portfolio with a beta of 0.5

A measure of the sensitivity of a stock’s return to the returns on the market portfolio

β

_i

= Cov(R

_i

, R

_m

)/Var(R

_m

)

6

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Capital Asset Pricing Model (CAPM

So E(R_i)=R_f + β_i(E(R_m) – R_f) R_f + Units × Price.

• If we know the expected rate of return of a security, the theoretical

] R ]

) [E[R Var(R

) R , Cov(R R

]

E[R

_M _F

M M F i

i

  

] R ]

[E[R R

]

E[R

_i



_F

 

_i _M



_F

Number of units of

systematic risk () Market Risk Premium or the price per unit risk

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7-8

Assumptions for CAPM

• Single-period investment horizon

• Investors can invest in the universal set of publicly traded financial assets

• Investors can borrow or lend at the risk-free rate unlimitedly

• No taxes and transaction costs

• Information is costless and available to all investors

• Assumptions associated with investors

– Investors are price takers (there is no sufficiently wealthy investor such that his will or behavior can influence the whole market and thus security prices)

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Assumptions for CAPM

– All investors have the homogeneous expectations about the expected values, variances, and correlations of

security returns

– All investors attempt to construct efficient frontier portfolios, i.e., they are rational mean-variance

optimizers

(Investors are all very similar except their initial wealth and their degree of risk aversion)

Several assumptions are unrealistic

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Resulting Equilibrium Conditions

• Identical efficient frontier

– All investors are mean-variance optimizers and face the same universal set of securities, so they all derive the

identical efficient frontier and the same tangent portfolio (O) and the corresponding CAL given the current risk-free

• The market portfolio is the tangent portfolio

rate

O

– All investors will put part of their wealth on the same risky portfolio O and the rest on the risk-free asset

– The market portfolio is defined as the aggregation of the risky portfolios held by all investors

– Hence, the composition of the market portfolio must be identical to that of the tangent portfolio O, and thus E(r_M)

= E(r_O) and σ_M = σ_O

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Resulting Equilibrium Conditions (cont.)

• The capital market line (CML)

As a result, all investors will hold the same portfolio of risky assets–market portfolio,

which contains all publicly traded risky assets in the economy – The market portfolio is of course on the efficient frontier,

and the line from the risk-free rate through the market portfolio is called the capital market line (CML)

• We call this result the mutual fund theorem :Only one mutual fund of risky assets–the market

portfolio–is sufficient to satisfy the investment

demands of all investors

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The Efficient Frontier and the Capital Market Line

M = Market portfolio r_f = Risk free rate

E(r_M) - r_f = Market risk premium [E(r_M) – r_f] / σ_M = Slope of the CML

= Sharpe ratio for the market portfolio or for all combined portfolios on the CML

※ Note that the CML is on the E(r)-σ plane

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Evaluating the CAPM

theoretically the CAPM is untestable because Huge measurability problems because the market

portfolio is unobservable.

•However, practically the CAPM is testable and could still be a useful predictor of expected returns. Empirical testing shows that the CAPM works reasonably well

Betas are not as useful at predicting returns as other measurable factors may be.

• More advanced versions of the CAPM that do a better job at estimating the market portfolio are useful at predicting stock returns.

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CAPM and the Real World

the principles we learn from the CAPM are still entirely valid

– Investors should diversify (invest in the market portfolio)

– Differences in risk tolerances can be handled by changing the asset allocation decisions in the complete portfolio

– Systematic risk is the only risk that matters (thus we

have the relationship between the expected return and the beta of each security)

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Expected Returns On Individual Securities

The risk premium, defined as the expected return in excess of r

_f

, reflect the compensation for

securities holders

In the equilibrium, the ratio of risk premium to beta should be the same for any two securities or

portfolios (including the market portfolio)

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Expected Returns On Individual Securities

– Therefore, for all securities,

※The competition among investors for pursuing the securities with higher risk premiums and smaller betas will result in the above equality

– Rearranging gives us the CAPM’s expected return-beta relationship

or

( ) ( ) ( )

1

M f M f i f

M i

E r r E r r E r r

 

  

 

( )_i _f _i[ ( )_M _f ]

E r  r  E r  r E r( )_i  r_f _i[ ( )E r_M  r_f ]

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Expected Returns On Portfolios

• Since the expected return-beta relationship

according to the CAPM is linear and holds not only for ALL INDIVIDUAL ASSETS but also for ANY

PORTFOLIO, the beta of a portfolio is simply the weighted average of the betas of the assets in the portfolio

β

_P

=  W

_i

β

_i

If you put half your money in a stock with a beta of 1.5 and

30% of your money in a stock with a beta of 0.9 and the

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Security Market Line (SML) Relationships

E(r

_i

) = r

_f

+ β

_i

[E(r

_M

) – r

_f

]

β

_i

= cov(R

_i

,R

_M

) / var(R

_M

)

E(r

_M

) – r

_f

=

market risk premium

For example: E(r

_M

) – r

_f

= 8% and r

_f

= 3%

β

_x

= 1.25  E(r

_x

) = 3% + 1.25 × (8%) = 13%

β

_y

= 0.6  E(r

_y

) = 3% + 0.6 × (8%) = 7.8%

※ SML: graphical representation of the expected return-beta relationship of the CAPM (on the E(r)-beta plane)

※ For the stock with a higher beta, since it is with higher systematic risk, it needs to offer a higher expected return to attract investors

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E(r)

E(r_x)=13%

SML

1 β

E(r_M)=11%

E(r_y)=7.8%

3%

1.25 0.6

slope is 0.08, which is the market risk premium

Graph of Security Market Line

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Applications of the CAPM

• In reality, not all securities lie on the SML in the economy

• Underpriced (overpriced) stocks plot above

(below) the SML: Given their betas, their expected rates of return are higher (lower) than the

predication by the CAPM and thus the securities are underpriced (overpriced)

• The difference between actually rate of return on a

security and the expected and is the abnormal rate

of return on this security, which is often denoted

as alpha (α)

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Disequilibrium Example

E(r)

15%

SML

R_m=11%

r_f=3%

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Disequilibrium Example

• Suppose a security with a  of 1.25 is offering expected return of 15%.

• According to SML, it should be 13%.

• Under-priced: offering too high of a rate of return for its level of risk.

• This kind of security is a more attractive

investment target

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Problem 1

b . (c on t in u e d ) S h a r p e R a t ios ii. H e ld Alon e :

Sharpe Ratio X = Sharpe Ratio Y =

Sharpe Ratio Index =

(0.14 – 0.05)/0.36 = 0.25 (0.17 – 0.05)/0.25 = 0.48

(0.14 – 0.05)/0.15 = 0.60 Bette

r

σ r Ratio E(r)

Sharpe  ^f



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Problem 2

E(r

_P

) = r

_f

+  [E(r

_M

) – r

_f

]

20% = 5% +  (15% – 5%)

 = 15/10 = 1.5

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Problems 5 & 6

5.

Not possible. Portfolio A has a higher beta than Portfolio B, but the expected return for Portfolio A is lower.

Possible.

Portfolio A's lower expected rate of return can be paired with a higher standard deviation, as long as Portfolio A's beta is lower than that of Portfolio B.

5.

6.

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Problem 7

7 . Calculate Sharpe ratios for both portfolios:

Not possible. The reward-to-variability ratio for Portfolio A is better than that of the market, which is not possible according to the CAPM, since the CAPM predicts that the market portfolio is the portfolio with the highest return per unit of risk.

.12 0.5 .10 Sharpe_A .16   .24 0.33

.10 Sharpe_M .18  

σ r Ratio E(r)

Sharpe  ^f



7.

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Problem 9

9 . Given the data, the SML is:

E(r) = 10% + (18% – 10%)

A portfolio with beta of 1.5 should have an expected return of:

E(r) = 10% + 1.5(18% – 10%) = 22%

Not Possible: The expected return for Portfolio A is 16% so that

Portfolio A plots below the SML (i.e., has an  = –6%), and hence is an overpriced portfolio. This is inconsistent with the CAPM.

9.

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Problem 11

1 1 .

Sharpe A = Sharpe M =

Possible: Portfolio A's ratio of risk premium to standard

deviation is less attractive than the market's. This situation is consistent with the CAPM. The market portfolio should provide the highest reward-to-variability ratio.

(16% - 10%) / 22% = .27 (18% - 10%) / 24% = .33

11.

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Problem 13

b.

r₁ = 1 9 % ; r₂ = 1 6 % ; ₁ = 1 .5 ; ₂ = 1 .0

We can’t tell which adviser did the better job selecting stocks because we can’t calculate either the alpha or the return per unit of risk.

r₁ = 1 9 % ; r₂ = 1 6 % ; ₁ = 1 .5 ; ₂ = 1 .0 , r f = 6 % ; r_M = 1 4 %

₁ =

₂ =

Th e s e c on d a d vis e r d id t h e b e t t e r job s e le c t in g s t oc k s (b ig g e r + a lp h a )

19% – 16% –

19% – 18% = 1%

16% – 14% = 2%

CAPM: r_i = 6% + β(14%-6%)

Part c?

[6% + 1.5(14% – 6%)] = [6% + 1.0(14% – 6%)] =

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Problem 13

c .

r₁ = 19%; r₂ = 16%; ₁ = 1.5; ₂ = 1.0, rf = 3%; r_M = 15%

₁ =

2 =

Here, not only does the second investment adviser appear to be a 19% – [3% + 1.5(15% – 3%)] =

16% – [3%+ 1.0(15% – 3%)] =

19% – 21% = –2%

16% – 15% = 1%

CAPM: r_i = 3% + β(15%-3%)

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7.4 MULTIFACTOR MODELS AND THE

CAPM

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Fama French Three-Factor Model

• In reality, the systematic risk is not from one source

• It is obvious that developing models that allow for

several systematic risks can provide better descriptions of security returns

• In addition to the market risk premium , Fama and French propose the size premium and the book-to- market premium

– The size premium is constructed as the difference in returns between small and large firms and is denoted by SMB (“small minus big”)

– The book-to-market premium is calculated as the difference in

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Fama French Three-Factor Model

• The Fama and French three-factor model is

– r_SMB is the return of a portfolio consisting of a long

position of $1 in a small-size-firm portfolio and a short position of $1 in a large-size-firm portfolio

– r_HML is the return of a portfolio consisting of a long

position of $1 in a higher-B/M (value stock) portfolio and a short position of $1 in a lower-B/M (growth stock)

portfolio

– The roles of r_SMB and r_HML are to identify the average

reward compensating holders of the security i exposed to the sources of risk for which they proxy

– Note that it is not necessary to calculate the excess return for r_SMB and r_HML

HML HML SMB SMB

( )_i _f _iM [ ( )_M _f )] _i ( ) _i ( ) E r  r  E r  r   E r   E r

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Fama-French (FF) 3 factor Model

r_GM – rf =α_GM + β_M(r_M – rf ) + β_HMLr_HML + β_SMBr_SMB + e_GM

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Arbitrage Pricing Theory

• Arbitrage –Creation of riskless profits by trading relative mispricing among securities

1. Constructing a zero-investment portfolio today and earn a profit for certain in the future

2. 2. Or if there is a security priced differently in two

markets, a long position in the cheaper market financed by a short position in the more expensive market will

lead to a profit as long as the position can be offset each other in the future

• Since there is no risk for arbitrage, an investor will

create arbitrarily large positions to obtain large levels of profit

– No arbitrage argument: in efficient markets, profitable arbitrage opportunities will quickly disappear