The Risk of Individual Assets

1

© 2006 R.W.Parks/E. ZivotECON 422:CAPM1

Capital Market Equilibrium and the Capital Asset Pricing Model

Econ 422Investment, Capital & Finance

Summer 2006August 15, 2006


The Risk of Individual Assets

Investors require compensation for bearing risk.We have seen that the standard deviation of the rate of return is an appropriate measure of risk for one’s portfolio.Standard deviation is not the best measure of risk for individual assets when investors hold diversified portfolios.


The Risk of Individual Assets (continued)

For people holding a diversified portfolio it is the contribution of the individual asset to the portfolio’s standard deviation that matters.

[If your portfolio involved only one asset, e.g. young Bill Gates, the portfolio standard deviation would be the standard deviation of the single asset.]


The contribution of an individual asset to the portfolio’s standard

deviation: BetaBeta measures the sensitivity of an asset’s rate of return to variation in the market portfolio’s return.Beta for asset i can be computed as

β σσi

i m

m

im

m

r rV r

= =cov( , )

( ) 2

2


Beta as a Measure of Portfolio Risk: Class Example

Suppose you hold an equally weighted portfolio with 99 assetsWhat happens to the portfolio variance if a new asset, say IBM, is added to the portfolio?


Measuring BetasThe Market Model

Beta can be interpreted as the slope coefficient in a regression of the return on the ith security, ri, on the return for the market portfolio, rm.The interpretation rests on the market model:

rm represents “market risk” and εi represents “firm specific” risk independent of the market.See Spreadsheet example

it i i mt itr rα β ε= + +


Beta as a Measure of Risk for Individual Assets

Asset i’s Return ri

Market return rm

+10%

+10%

-10%

-10%

The intercept is given by α

The slope is given by β=cov(ri,rm)/V(rm)


The Capital Asset Pricing Model (CAPM)

CAPM describes the relationship between an asset’s beta risk and its expected return as follows:

E r r E r ri f i m f( ) ( )= + −β

= riskfree rate + beta x market risk premium.

3


The Security Market Line (SML)Describes the CAPM Relationship

β

E(ri) Security Market Line (SML)

rf Slope is the market risk premium = E(rm)-rf

1.0

E(rm)

E r r E r ri f i m f( ) ( )= + −β


The Capital Asset Pricing Model Example

If T-bill yield = 2%The beta for Microsoft = 1.61 (from Yahoo! See link for key statistics)The historical market risk premium =7.5%Then

[ ] ( [ ] )

2% 1.61*(7.5%) 14.075%msft f msft mt fE r r E R rβ= + −

= + =

Note: the return predicted from the CAPM is sometimes called the “risk-adjusted” return


The Market Model and the Measures of Risk

The total risk of an asset, held alone, i.e. not as part of a diversified portfolio, would be measured by its variance, V(ri).According to the market model

Total risk = systematic market risk + unique riskThe unique risk can be eliminated through diversification.

r rr V r V

it i i mt it

it i mt it

= + +

= +

α β ε

β ε

andV( ) ( ) ( )2


The Market Model and the Measures of Risk

R2 measures proportion of an asset’s total risk that is market risk:

2

2 2

22 2

( ) ( ) ( )1( ) ( ) ( )

1( ) ( ), 1

( ) ( )

it i mt it

it it it

i mt it

it it

Var r Var r VarVar r Var r Var r

R RVar r VarR R

Var r Var r

β ε

β ε

= = +

= + −

= − =

See spreadsheet for examples

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Portfolio Beta

The beta of a portfolio is a weighted average of the individual asset betas

1 1 2 2

portfolio share for asset i beta for asset i

p n n

i

i

x x x

x

β β β β

β

= + + +

==

See spreadsheet example


Testing the CAPM

Key CAPM prediction: stocks with high betas should have high average returns; stocks with low betas should have low average returnsSimple test: compute average returns and betas for a bunch of stocks and see if the high beta stocks have higher average returns than the low beta stocksCAPM predicts a straight line relationship between average return and beta


Testing the CAPM

Testing involves a number of measurement problems:» for expected returns» for beta» for the market portfolio


Some Approaches Used to Addressthe Measurement Problems

Early studies by Black Jensen & Scholes (1972) and by Famaand McBeth (1973)Measurement of betas: data from period 1 used to estimate individual stock betas. These betas used to construct “decile portfolios”: stocks with betas in the lowest 10% grouped as the first portfolio. Stocks with betas in the next lowest 10% grouped as the second portfolio etc.» Portfolio betas are more accurate than single asset betas

Data from period 2 used to re-estimate betas and average returns for the 10 portfoliosLook at the regression of average return on betas for the 10 portfolios

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Black, Jensen & Scholes CAPM Test (1972)Average Monthly Return

rf

Theoretical SML

Fitted SML

BetaHighBeta

Low Beta


Validity of the CAPMEvidence is mixed:» Long-run average returns are significantly related

to betas.» Beta is probably not a complete explanation.

– Low beta stocks have earned higher rates of return than predicted by the model.

» Recent results by Fama and French (1992)– Small company stocks stocks have earned

higher rates of return than predicted by the model. (Size Effect)

– Value company stocks have earned higher rates of return than predicted by the model. (Style Effect)


What is a Value Stock?

A stock that tends to trade at a lower price relative to it's fundamentals (i.e. dividends, earnings, sales, etc.) and thus considered undervalued by a value investor. Common characteristics of such stocks include a high dividend yield, low price-to-book ratio and/or low price-to-earnings ratio.


The CAPM Remains Attractive

It is simple and gives sensible answers.It distinguishes between diversifiable and non-diversifiable risk.

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The CAPM Remains Controversial

It is difficult to devise a definitive test» We don’t know how to define and measure the

market portfolio. If we use the wrong market index the resulting betas are mismeasured

Fama & French results cast doubt on the CAPM although their study is also subject to criticisms.

Multi-factor models have been developed to compete with the CAPM e.g. the Arbitrage Pricing Model. Ch 11


Applying the CAPM to Valuation

1 0 1

0

0 1 1

1 0 1

0

0

0

Recall the one-period holding period rate of return:

At time 0, is known, but and are not known; they are random variables.Take expectations:

( ) ( )( )

Solve for :

P P DrPt P P D

E P P E DE rP

P

P

− +=

=

− +=

= 1 1 1 1( ) ( ) ( ) ( )1 ( ) 1 [ ( ) ]

using the CAPM relation: ( ) [ ( ) ]f M f

f M f

E P E D E P E DE r r E r r

E r r E r r

β

β

+ +=

+ + + −

= + −


Applying the CAPM to Valuation(continued)

To value a future risky cash flow, discount the expected value of the cash flow to present value using the risk-adjusted expected return based on the CAPM.

1 10

( ) ( )1 [ ( ) ]f M f

E P E DPr E r rβ

+=

+ + −


Example: Stock valuation using CAPM

E[D1] = 5, g = 0.10, rf = 0.03β = 1.5, E[rm] – rf = 0.075

10

[ ] ( [ ] ) 0.03 1.5(0.075) 0.1425

[ ] 5 5Constant growth: 117.64( [ ] ) 0.1425 0.1 0.0425

f M fE r r E r r

E DPE r g

β= + − = + =

= = = =− −

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Using the CAPM to Determine the Discount Rate for Risky Projects

For risk-free projects you would use a risk-free interest rate for discounting a project’s cash flow.For risky projects you discount the expected cash flow by a risk adjusted interest rate, using the CAPM.


The Company Cost of Capital

The average rate of return for the entire company’s assets is called the company cost of capital.If the project under consideration has the same risk as these assets, then the company cost of capital is an appropriate discount rate.But the project may have a different risk. We would then use a discount rate matching the project’s risk.


Project Discount Rate and the Company Cost of Capital

Required return

Rf

Company Cost of Capital

Project Beta

Security marketline, CAPM


Determining Project RiskE[r] = 2% + β*(7.5%)

10%1.06Cost improvement

15% (company cost of capital)

1.73 = estimated beta from data

Expansion of existing business

20%2.4New Products

30%3.73Speculative ventures

Discount RateProject BetaCategory

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Estimating the Cost of Capitalfor an All-Equity Firm

For an all-equity firm the riskiness of the stock is the same as the riskiness of the company’s assets.Stock betas are easy to obtain or estimate.Use the CAPM to estimate the company cost of capital: This is the discount rate for project with the same riskiness as the firm’s current assets.

( ) [ ( ) ]A f E M fE r r E r rβ= + −


Estimating the Cost of Capitalfor an All-Equity Firm: Example

For Microsoft MSFT I found the following information: (From Yahoo Finance, key statistics)Microsoft is an all equity firm, i.e. it has essentially no debt, debt/equity ratio is 0.Its stock beta is 1.61rf= 1.75%=current T-Bill yield[E(rM) - rf]=expected market risk premium=7.5%,E(re)=rf + β[E(rM) - rf]

=1.75%+1.61[7.5%]=13.825%


A Firm’s Capital Structure: Relative Amounts of Debt and Equity

Assets Liabilities

Cash Debt D

Receivables

Buildings

Equipment Equity E

Patents

V= Total Assets Total Liabilities =V=D+E


The Risk of A Company’s Assets is Shared by Its Owners

A

A

A

If the risk of the company's current assets is then

Note that since , 1.

If we apply the CAPM to the three components, it must be the case that

E(r ) ( ) (

D E D E

D

D E D ED E D E V V

D ED E VV V

D EE r E rV V

β

β β β β β= + = ++ +

+ = + =

= + ).E

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Examples of Asset Betas

0 .1 0 .9A D Eβ β β= ⋅ + ⋅

All equity firm: D/V = 0, E/V =1

Firm with small debt: D/V = 0.1, E/V =0.9

0 1A D E Eβ β β β= ⋅ + ⋅ =

Note: Equity beta, βE, is estimated from stock returns (easy); Debt beta, βD, may be estimated from returns on corporate debt (not so easy)


The Riskiness of the Equity Owners Rises When There is Debt

Consider the special case when the debt is risk-free:

A

A A A A

If 0,

D E

D

E

D EV V

V E D DE E E

β β β

β

β β β β β

= +

=+

= = = +


The Riskiness of the Equity Owners Rises When There is Debt (continued)

D/E

BA

BE

Debt is risk free

A AEDE

β β β= +


The Discount Rate for a Project in a Leveraged Firm

Equity beta are easy to determine. Asset betas would not be easy to estimate directly.Use the equity beta, and undo the effect of the leverage.Estimate or look up the equity beta, estimate the debt beta if it is not zero. Then use the beta relationship to get the asset beta:

A

Then use the CAPM to determine the corresponding expected return.( ) [ ( ) ]

Use this rate as the discount when project has same risk as company's existing assets.

D E

A f A M f

D ED E D E

E r r E r r

β β β

β

= ++ +

= + −

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The Discount Rate for a Project in a Leveraged Firm: Example

Sea Star Inc. has a debt/equity ratio of 0.1, an equity beta of 1.21 and a debt beta of 0.11. The current risk-free interest rate is 4%. The expected market risk premium is 8.5%. Find the discount rate for a project with the same risk as the company’s current assets.



Note:

0.1 0.1

0.1 0.1 10.1 1.1 11

1 101 111 11

D D EE

D E ED E E E E

E DD E D E

= ⇒ = ×

× ×= = =

+ × + ×

= − = − =+ +



Therefore,

1 10 1 10.11 1.21 .01 1.10 1.1111 11 11 11

Then use the CAPM:

( ) ( ) 4% 1.11*8.5% 13.435%

A D E

A f A M fE r r E r r

β β β

β

= + = + = + =

⎡ ⎤= + − = + =⎣ ⎦

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The Risk of Individual Assets