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Capital Asset Pricing Model

5. CAPMA. Ashta

For Université de Paris 6

Based largely on Bodie, Kane & Markus: Essentials of Investments, 5th Edition, McGraw Hill International, Chapter 7

& Shapiro & Balbirer: Modern Corporate Finance, Prentice Hall

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Capital Asset Pricing Model (CAPM)

! 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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Assumptions

! Individual investors are price takers

! Single-period investment horizon

! Investments are limited to traded financial assets

! No taxes and no transaction costs

! Investors are rational mean-variance optimizers

! Information is costless and available to all investors

! Homogeneous expectations

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Resulting Equilibrium Conditions

! All investors will hold the same portfolio for risky assets – market portfolio (which is optimum for each investor and gives the highest reward to risk ratio).

! Market portfolio contains all securities and is a value-weighted protfolio

! The proportion of each security is its market value as a percentage of total market value

! Risk premium on the market is proportional to its variance FM

2 and 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

– In fact it is directly proportional to its Beta.

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Capital Market Line (CML)(= CAL for Market portfolio)

E(r)

E(rM)

rf

MCML

!m

!Total Risk

Note: CML if for market portfolio. Therefore !M is the relevant risk

But each individual asset’s contribution to risk of market portfolio is only its covariance

(the rest can be diversified).

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Slope and Market Risk Premium

m = Market portfolio

i = Any portfolio rf = Risk free rate E(rM) - rf = Market risk premium

= Slope of CML

Note: Expected Premium / s.d. is higher for market porfolio than for any other combination of individual assets (tangency)

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Expected Return andSystematic Risk or Market Risk

– We know that systematic risk or market risk is the only important risk.

– Excess return (over risk free) has to vary with market risk or with covariance with the market

SML

Correlation

eliminated because

captured by

Covariance

Market risk

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From Covariance to Beta

! The same equation can be re-written in terms of Beta

SML

CAPM

SML

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CAPM : the equation

ri = rf + (rm – rf) x "

Required return of asset A

Risk free return

Market portfolio return

Risk unit, called Beta coefficient

or

index of non-diversifiable risk for asset A

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Same as SML Relationships

" = [COV(ri,rm)] _____________

!2 (rm)

Slope SML = E(rm) - rf

= market risk premium

SML = rf + [E(rm) - rf]"

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Sample Calculations for SML

E(rm) - rf = .08 rf = .03

If "x = 1.25

E(rx) = .03 + 1.25(.08) = .13 or 13%

If "y = .6

E(ry) =

12

E(r)

Rx=13%

SML

m

ß

ß1.0

Rm=11%

Ry=7.8%

3%

xß

1.25yß

.6

.08

Graph of Sample Calculations

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SML! SML provides a benchmark for evaluating expected

investment performance.

! It is a graphic representation of the relationship between expected return and Beta

! The upward slope reflects that investors want higher returns for higher risk. (Note: only systematic risk)

! If Beta = zero, the stock should give risk free return

! Properly valued assets plot exactly on the SML

! Individual assets may be mispriced because assumtions don’t always hold

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Q. 5 (try at home)

NPV = 15.64, IRR = 49.55%, Beta for this ra = 4.055

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! rf = 8%, rM = .18! Q. 15! Q. 16

! If Beta is 1, ! If Beta is zero, ! I would pay __________ too much

! Q. 17 (too easy, try later)– Beta = - .2

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E(r)

15%

SML

ß1.0

Rm=11%

rf=3%

1.25

Disequilibrium Example

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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%

! Underpriced: offering too high of a rate of return for its level of risk

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Reminder: Security

Excess Returns (i)

SCL

.

.

...

.

. .

. ..

. .

.

. .

. ..

.

..

. .

. ..

. ..

. .

. .

.

. ..

. .

.

. ... .

. .. .Excess returnson market index

Ri = # i + ßiRm + ei

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CAPM and SCL

! The CAPM implies that alphas (or intercept of the SCL) should be zero

! This intercept measures the excess reward on the security when market reward is zero.

! Note: The SCL is plotted using historical actuals: so the alpha is not always zero!

! Similarly, estimates of Beta based on past data are often adjusted to assess required future returns

! Beta is the slope of the SCL! So, using regression analysis we can estimate past

Betas

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Measuring Betas

Hewlett Packard Beta

Slope determined from 60 months of

prices and plotting the line of best

fit.

Price data - Jan 93 - Dec 97

Market return (%)

Hew

lett-Pack

ard retu

rn (%

)

R2 = .35

B = 1.69

SCL

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Questions 8 -14

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Possible/ Not possible?

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True or False (Shapiro Q. 3)

! c. A stock with a beta of .5 has a required return one half as high as the market.

Answer.

! e. If an asset lies above the security market line, it is overvalued.

Answer.

! g. An undiversified portfolio with a beta of 2 is twice as risky as the market portfolio.

Answer.

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Q. 6b: Comment

! Although our company has a very low beta, we feel it is misleading to our shareholders because our company is subject to very wide fluctuations in sales and profits.

! Answer.

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Shapiro Q. 7

! If the market portfolio actually yields a rate of return different from the expected return predicted by the CAPM, does this mean the CAPM is a bad model?

! Answer.

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Cost of Equity

! The CAPM is usually used to calculate the cost of equity

! If a company has no debt, the cost of equity and the cost of all capital of the company is the same.

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The company’s cost of capital can be

compared to the CAPM required return.

Required

return

Project Beta1.26

Company Cost of Capital

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5.5

0

SML

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Conclusion

! The return required by shareholders

= Return to be given to shareholders (deretmined by CAPM)

= Cost of capital of equity

! If company is completely financed by equity

Cost of equity = Cost of Capital for company

! If project risk = risk of company

Cost of capital for project = cost of capital for company

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Risk and DCFExample

All Equity Inc. has an expansion Project costing $ 200 million which is expected to produce CF = $100 million for each of three years. All Equity Inc. has no debt.

Given

a risk free rate of 6%,

a market premium of 8%,

and a beta of .75