12-1 Chapter 12 Return, Risk, and the Security Market Line

12-1

Chapter 12

Return, Risk, and the Security Market Line

12-2

Announcements and News• Firms make periodic announcements about

events that may significantly impact the profits of the firm.• Earnings• Product development• Personnel

• The impact of an announcement depends on how much of the announcement represents new information.

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Announcements & Surprises

Total return–Expected return = Unexpected return

R – E(R) = U

Announcement = Expected + Surprise

12-4

Announcements & Surprises

A company announces earnings $1.00 per share higher than the previous quarter.

Price IndicationDoesn't move No surpriseGoes up Market expected lower earningsGoes down Market expected higher earnings

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Systematic & Unsystematic Components of Return

(12.3) R – E(R)=Systematic +Unsystematic

(12.4) R – E(R) = U = m+ε

where m = market risk

ε = unsystematic risk

(12.5) Total risk=Systematic risk + Unsystematic risk

12-6

Systematic & Unsystematic Components of Return

(12.5) Total risk = Systematic risk + Unsystematic risk

= Market risk + firm-specific risk

If the latest Consumer Price Index numbers indicate that inflation has risen substantially. Will this impact systematic or unsystematic risk?

If a company announces that it will have to restate its financials for the last 3 years, which risk component will be affected?

12-7

Diversification and Risk• In a large portfolio:

• Some stocks will go up in value because of positive company-specific events, while

• Others will go down in value because of negative company-specific events.

• Unsystematic risk is essentially eliminated by diversification, so a portfolio with many assets has almost no unsystematic risk.

• Unsystematic risk = diversifiable risk.• Systematic risk = non-diversifiable risk.

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Systematic Risk“Unsystematic risk is essentially eliminated by diversification, so a

portfolio with many assets has almost no unsystematic risk.”

“The systematic risk principle states that the reward for bearing risk

depends only on the systematic risk of an investment.”

“The expected return on an asset depends only on its systematic risk.”

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The Systematic Risk Principle• The systematic risk principle states:

The expected return on an asset depends only on its systematic risk.

• No matter how much total risk an asset has, only the systematic portion is relevant in determining the expected return (and the risk premium) on that asset.

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Measuring Systematic Risk• The Beta coefficient ( ) measures the relative

systematic risk of an asset. • Beta > 1.0 more systematic risk than average.• Beta < 1.0 less systematic risk than average.

• Assets with larger betas = greater systematic risk

= greater expected returns.

Note that not all Betas are created equally.

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Risk & Beta

Std Dev BetaAsset 1 25% 1.30Asset 2 45% 0.80

• Asset 2 has more total risk because it has the greater standard deviation.

• Asset 1 has more systematic risk because it has the larger Beta.

• Asset 1 should have the higher expected return since it has the larger beta.

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Portfolio Beta

n

iiiP x

1

Xi = % of portfolio invested in asset i

βi = Beta of asset i

Wgt Beta Wgt*BAsset 1 40% 1.30 0.52Asset 2 60% 0.80 0.48

1.00

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Portfolio Beta

Security $ Invested E(R) BetaStock W $1,000 7.5% 0.85Stock X $2,500 9.0% 0.95Stock Y $3,500 12.0% 1.20Stock Z $1,000 10.5% 1.10

What is this portfolio’s expected return and Beta?

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Portfolio Return & Beta Example

Security $ Invested % (wgt)Stock W $1,000 12.50%Stock X $2,500 31.25%Stock Y $3,500 43.75%Stock Z $1,000 12.50%

$8,000 100.00%

Step 1: Calculate each stock’s % or weight in the portfolio

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Portfolio Return & Beta

Security % (wgt) E(R)Stock W 12.50% 7.5% 0.9%Stock X 31.25% 9.0% 2.8%Stock Y 43.75% 12.0% 5.3%Stock Z 12.50% 10.5% 1.3%

10.3%

Step 2: Apply the weights to each stock’s expected return to arrive at the portfolio’s expected return.

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Security % (wgt) BetaStock W 12.50% 0.85 0.106Stock X 31.25% 0.95 0.297Stock Y 43.75% 1.20 0.525Stock Z 12.50% 1.10 0.138

1.066

Step 3: Apply the weights to each stock’s beta to arrive at the portfolio’s beta.

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Beta and the Risk Premium, I.• Consider a portfolio made up of asset A and a

risk-free asset. • For asset A: E(RA) = 16% and A = 1.6

• The risk-free rate Rf = 4%. Note that for a risk-free asset, = 0 by definition.

• Varying the % invested in each asset will change the possible portfolio expected returns and betas.

• Note: if the investor borrows at the risk-free rate and invests the proceeds in asset A, the investment in asset A will exceed 100%.

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Beta and the Risk Premium

% in A % in Rf E( R ) Beta0% 100% 4.0% 0.0

25% 75% 7.0% 0.450% 50% 10.0% 0.875% 25% 13.0% 1.2100% 0% 16.0% 1.6125% -25% 19.0% 2.0150% -50% 22.0% 2.4

Portfolio

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Portfolio Expected Returns and Betas for Asset A

12-2012-20

Beta and the Risk Premium

% in B % in Rf E( R ) Beta0% 100% 4.0% 0.0

25% 75% 6.0% 0.350% 50% 8.0% 0.675% 25% 10.0% 0.9100% 0% 12.0% 1.2125% -25% 14.0% 1.5150% -50% 16.0% 1.8

Portfolio

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Portfolio Expected Returns and Betas for Asset B

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Portfolio Expected Returns and Betas for Assets A & B


0.0%

5.0%

10.0%

15.0%

20.0%

25.0%

0.0 0.3 0.6 0.9 1.2 1.5 1.8

Portfolio Beta

Po

rtfo

lio E

(R)

%67.6)(

B

fB RRE

%5.7)(

A

fA RRE

12-23

The Fundamental Relationship between Risk and Return

B

fB

A

fA RRERRE

)()(

(12.6)

“The reward-to-risk ratio must be the same for all assets in a competitive market.”

Reward-to-Risk (A) = 7.50%

Reward-to-Risk (B) = 6.67%

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The Fundamental Result• The situation we have described for assets A and B

cannot persist in a well-organized, active market• Investors will be attracted to asset A (and buy A shares)• Investors will shy away from asset B (and sell B shares)

• This buying and selling will make • The price of A shares increase• The price of B shares decrease

• This price adjustment continues until the two assets plot on exactly the same line.

• That is, until: B

fB

A

fA

β

RRE

β

RRE

12-25

The Fundamental Result

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Over- and Under-Valued

Security E(R) BetaStock W 7.5% 0.85Stock X 9.0% 0.95Stock Y 12.0% 1.20Stock Z 10.5% 1.10

If the Risk-free rate is 5%, are these stocks fairly valued?

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Security E(R) Beta RatioStock W 7.5% 0.85 2.94%Stock X 9.0% 0.95 4.21%Stock Y 12.0% 1.20 5.83%Stock Z 10.5% 1.10 5.00%


Stocks W and X offer insufficient returns for their level of risk compared to stocks Y and Z.

Stock Y offers the highest return for its level of risk.

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The Security Market Line (SML)

• The Security market line (SML) = a graphical representation of the linear relationship between systematic risk and expected return in financial markets.

• For a market portfolio,

fM

fM

M

fM

RRE

1

RRE

β

RRE

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The Security Market Line

• The term E(RM) – Rf = market risk premium because it is the risk premium on a market portfolio.

• For any asset i in the market:

• Setting the reward-to-risk ratio for all assets equal to the market risk premium results in an equation known as the capital asset pricing model.

fMi

fi RREβ

RRE

ifMfi βRRERRE

12-30

The Security Market Line• The Capital Asset Pricing Model (CAPM) is

a theory of risk and return for securities in a competitive capital market.

• The CAPM shows that E(Ri) depends on: • Rf, the pure time value of money.• E(RM) – Rf, the reward for bearing systematic risk. i, the amount of systematic risk.

ifMfi βRRERRE (12.7)

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ifMfi R)R(ER)R(E (12.7)

Market Risk Premium

Pure time value of money

Reward for bearing systematic risk

Amount of systematic risk

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CAPMReturn

Stock W 7.5% 0.85 9.25% OverStock X 9.0% 0.95 9.75% OverStock Y 12.0% 1.20 11.00% UnderStock Z 10.5% 1.10 10.50% Fair

Security E(R) Beta Value

ii

M

f

ifMfi

%)(%)R(E

%)R(E

%R

R)R(ER)R(E

55

10

5

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0.00%

2.00%

4.00%

6.00%

8.00%

10.00%

12.00%

14.00%

E(R

)

Beta

Security Market Line

RF = 5%

WX

Y

Z

12-35

Risk and Return Summary, I.

12-36

Risk and Return Summary, II.

12-37

More on Beta

(12.8 and 12.4) R - E(R)= m + ε

(12.9) m = [RM –E(RM)] x β

(12.10) R-E(R)= [RM –E(RM)] x β + ε

Systematic portion of the unexpected return on the market M

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Decomposition of Total Return

• Suppose the expected return on the market is 10% and the risk-free rate is 5%.

• Asset A has a beta of 0.80.• The expected return on Asset A is 9%.

• One year the return on the market is actually 8% and the return on Asset A in the same year is 6.5%. Decompose these returns.

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Decomposition of Total Return

• E(RM) = 10% Actual RM = 8%

• RF = 5% βA = 0.80

• E(RA) = 9% Actual RA =6.5%

• UNE(RM)= [RM-E(RM)] = 8% - 10% = -2%

• UNE(RA) = [RA-E(RA)] = 6.5% - 9% = -2.5%

• Systematic UNE = [RM-E(RM)] x β = -2% x .80 = -1.6%

• Unsystematic portion = [RA-E(RA)] - [RM-E(RM)] x β

= (-2.5%) – (-1.6%) = -0.90%

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Decomposition of Total Returns

12-41

Unexpected Returns and Beta

12-42

Where Do Betas Come From?• A security’s Beta depends on:

• How closely correlated the security’s return is with the overall market’s return, and

• How volatile the security is relative to the market.

• A security’s Beta is equal to the correlation multiplied by the ratio of the standard deviations.

m

iMii σ

σR,RCorrβ

12-43

Where Do Betas Come From?

12-44

Using a Spreadsheet to Calculate Beta

12-45

Calculating Beta

2

),(

M

im

M

iim

MiMii RRCorr

Product ofYear Asset H Market Asset H Market Asset H Market Deviations

(1) (2) (3) (4) (5) (6) (7)2004 15.00% 10.00% 8.40% 5.20% 0.007056 0.002704 0.0043680 2005 11.50% 8.00% 4.90% 3.20% 0.002401 0.001024 0.0015680 2006 -4.00% -1.00% -10.60% -5.80% 0.011236 0.003364 0.0061480 2007 12.50% 7.00% 5.90% 2.20% 0.003481 0.000484 0.0012980 2008 -2.00% 0.00% -8.60% -4.80% 0.007396 0.002304 0.0041280

Totals 33.00% 24.00% 0.00% 0.00% 0.031570 0.009880 0.0175100

Asset H 33% / 5 = 6.60% .031570 / 4 = 0.007893 √.007893 = 8.88%Market 24% / 5 = 4.80% .009880 / 4 = 0.002470 √.002470 = 4.97%

Covariance = .01751 / 4 = 0.0044Correlation= 0044/(.088 x .0497) = 0.9914Beta = .9914 x (.0888/.0497) = 1.77

Returns Return Deviations Squared Deviations

Average Returns Variances Standard Deviations

(12.11)

Beta = .0044/.00247=1.77

12-46

Calculating Beta

Step 1: Compute average returns for Asset H and the Market

Year Asset H Market(1) (2)

2004 15.00% 10.00%2005 11.50% 8.00%2006 -4.00% -1.00%2007 12.50% 7.00%2008 -2.00% 0.00%

Totals 33.00% 24.00%

Asset H 33% / 5 = 6.60%Market 24% / 5 = 4.80%

Returns

Average Returns

12-47

Calculating Beta

Step 2: Compute variance and standard deviations for Asset H and the Market

Year Asset H Market Asset H Market Asset H Market(1) (2) (3) (4) (5) (6)

2004 15.00% 10.00% 8.40% 5.20% 0.007056 0.0027042005 11.50% 8.00% 4.90% 3.20% 0.002401 0.0010242006 -4.00% -1.00% -10.60% -5.80% 0.011236 0.0033642007 12.50% 7.00% 5.90% 2.20% 0.003481 0.0004842008 -2.00% 0.00% -8.60% -4.80% 0.007396 0.002304

Totals 33.00% 24.00% 0.00% 0.00% 0.031570 0.009880

Asset H 33% / 5 = 6.60% .031570 / 4 = 0.007893 √.007893 = 8.88%Market 24% / 5 = 4.80% .009880 / 4 = 0.002470 √.002470 = 4.97%

Returns Return Deviations Squared Deviations

Average Returns Variances Standard Deviations

12-48

Calculating BetaStep 3: Compute the covariance between the Market and

Asset H.

Product ofYear Asset H Market Asset H Market Deviations

(1) (2) (3) (4) (7)2004 15.00% 10.00% 8.40% 5.20% 0.0043680 2005 11.50% 8.00% 4.90% 3.20% 0.0015680 2006 -4.00% -1.00% -10.60% -5.80% 0.0061480 2007 12.50% 7.00% 5.90% 2.20% 0.0012980 2008 -2.00% 0.00% -8.60% -4.80% 0.0041280

Totals 33.00% 24.00% 0.00% 0.00% 0.0175100

Asset H 33% / 5 = 6.60% .031570 / 4 = 0.007893Market 24% / 5 = 4.80% .009880 / 4 = 0.002470

Covariance = .01751 / 4 = 0.0044

Returns Return Deviations

Average Returns Variances

12-49

Calculating Beta

Asset H 33% / 5 = 6.60% .031570 / 4 = 0.007893Market 24% / 5 = 4.80% .009880 / 4 = 0.002470

Covariance = .01751 / 4 = 0.0044Correlation= 0044/(.088 x .0497) = 0.9914Beta = .9914 x (.0888/.0497) = 1.77

Average Returns Variances

Step 4: Compute the correlation coefficient and the Beta for Asset H.

MiMii )R,R(Corr (12.11)

12-50

Why Do Betas Differ?

• Betas are estimated from actual data. Different sources estimate differently, possibly using different data.• For data, the most common choices are three to

five years of monthly data, or a single year of weekly data.

• The S&P 500 index is commonly used as a proxy for the market portfolio.

• Calculated betas may be adjusted for various statistical and fundamental reasons.

12-51

Beta - β

Beta (β) measures a specific asset’s market risk relative to an average asset.

Exxon Mobil 0.90 0.75General Motors 1.45 1.21IBM 1.10 1.62Microsoft 1.00 0.94Wal-Mart 0.80 0.54

CompanyValue Line

S&P

Market Beta = 1.00

Risk-free asset beta = 0.00

12-52

Finding a Beta on the Web

12-53

Beta and Return General Motors GM 1.21Wal-Mart WMT 0.80Market M 1.00

CompanyValue Line

0%

5%

10%

15%

20%

25%

1 2 3 4 5

GM WMT MARKET

12-54

Extending CAPM• The CAPM has a stunning implication:

• What you earn on your portfolio depends only on the level of systematic risk that you bear

• As a diversified investor, you do not need to worry about total risk, only systematic risk.

• But, does expected return depend only on Beta? Or, do other factors come into play?

• The above bullet point is a hotly debated question.

12-55

Important General Risk-Return Principles

• Investing has two dimensions: risk and return.• It is inappropriate to look at the total risk of an

individual security.• It is appropriate to look at how an individual

security contributes to the risk of the overall portfolio

• Risk can be decomposed into nonsystematic and systematic risk.

• Investors will be compensated only for systematic risk.

12-56

The Fama-French Three-Factor Model• FF propose two additional factors as useful in

explaining the relationship between risk and return.• Size, as measured by market capitalization• The book value to market value ratio

• Whether these two additional factors are truly sources of systematic risk is still being debated.

12-57

Returns from 25 Portfolios Formed on Size and Book-to-Market

• Note that the portfolio containing the smallest cap and the highest book-to-market have had the highest returns.

12-58

Useful Internet Sites• earnings.nasdaq.com (to see recent earnings surprises)• www.portfolioscience.com (helps you analyze risk)• money.cnn.com (a source for betas)• finance.yahoo.com (a terrific source of financial information)• www.smartmoney.com (another fine source of financial

information)• www.moneychimp.com (for a CAPM calculator)• http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/

(source for data behind the FAMA-French model)

Documents

12-1 Chapter 12 Return, Risk, and the Security Market Line