Chapter

12 Return, Risk, and the Security Market Line

2

Learning Objectives

• Our goal in this ch2apter is to define risk more precisely, and discuss how to measure it.

• In addition, we will quantify the relation between risk and return in financial markets.

• More specifically, our goal is to understand

1. The difference between expected and unexpected returns.

2. The difference between systematic risk and unsystematic risk.

3. The security market line and the capital asset pricing model.

4. The importance of beta.

3

Expected and Unexpected Returns

• The return on any stock traded in a financial market is composed of two parts.– The normal, or expected, part of the return is the return that

investors predict or expect.– The uncertain, or risky, part of the return comes from unexpected

information revealed during the year.

E(R) -R U

Return Expected - Return Total Return Unexpected

Return Unexpected Return Expected Return Total

4

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.– When the situation is not as bad as previously thought, what seems

to be bad news is actually good news.– When the situation is not as good as previously thought, what

seems to be good news is actually bad news.

• News about the future is what really matters.– Market participants factor predictions about the future into the

expected part of the stock return.– Announcement = Expected News + Surprise News

5

Systematic and Unsystematic Risk

• Systematic risk is risk that influences a large number of assets. Also called market risk.

• Unsystematic risk is risk that influences a single company or a small group of companies. Also called unique risk or firm-specific risk.

Total risk = Systematic risk + Unsystematic risk

6

Pop Quiz:Systematic Risk or Unsystematic Risk?

• The government announces that inflation unexpectedly jumped by 2 percent last month.

Systematic Risk• One of Big Widget’s major suppliers goes bankruptcy.

Unsystematic Risk• The head of accounting department of Big Widget

announces that the company’s current ratio has been severely deteriorating.

Unsystematic Risk• Congress approves changes to the tax code that will

increase the top marginal corporate tax rate.

Systematic Risk

7

Systematic and Unsystematic Components of Return

• Recall:

R – E(R) = U

= Systematic portion+ Unsystematic portion

= m +

R – E(R) = m +

8

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 is also called diversifiable risk.

• Systematic risk is also called non-diversifiable risk.

9

The Systematic Risk Principle

• What determines the size of the risk premium on a risky asset?

• The systematic risk principle states:

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

• So, 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.

10

Measuring Systematic Risk

• To be compensated for risk, the risk has to be special.– Unsystematic risk is not special.– Systematic risk is special.

• The Beta coefficient () measures the relative systematic risk of an asset. – Assets with Betas larger than 1.0 have more systematic risk than

average.– Assets with Betas smaller than 1.0 have less systematic risk than

average.

• Because assets with larger betas have greater systematic risks, they will have greater expected returns.

Note that not all Betas are created equally.

11

Betas with Respect to the S&P 500 for Individual Stocks (based on monthly data for 2004–2008)

12

Finding a Beta on the Web

13

Portfolio Betas

• The total risk of a portfolio has no simple relation to the total risk of the assets in the portfolio.– Recall the variance of a portfolio equation– For two assets, you need two variances and the covariance.– For four assets, you need four variances, and six covariances.

• In contrast, a portfolio Beta can be calculated just like the expected return of a portfolio.– That is, you can multiply each asset’s Beta by its portfolio weight

and then add the results to get the portfolio’s Beta.

Example: Calculating a Portfolio Beta

• Using Value Line data from Table 12.1, we see – Beta for Southwest Airlines (LUV) is 1.05– Beta for General Motors (GM) 1.45

• You put half your money into LUV and half into GM.

• What is your portfolio Beta?

1.25

1.45.501.05.50

β.50β.50β GMLUVp

12-14

15

Capital Asset Pricing Model (CAPM)

• Now, we will go back to the Markowitz's efficient market frontier.

• In order to derive the CAPM, Capital Market Line (CML), Security Market Line (SML).

• Here, we will try to derive the security market line (SML), which is a linear line characterized with risk-free rate and the market portfolio.

• A series of graphs explain all….

16

Efficient Frontier with Ten Stocks Versus Three Stocks

17

Risk-Free Saving and Borrowing

• Risk can also be reduced by investing a portion of a portfolio in a risk-free investment, like T-Bills. However, doing so will likely reduce the expected return.

• On the other hand, an aggressive investor who is seeking high expected returns might decide to borrow money to invest even more in the stock market.

18

Investing in Risk-Free Securities

• Consider an arbitrary risky portfolio and the effect on risk and return of putting a fraction of the money in the portfolio, while leaving the remaining fraction in risk-free Treasury bills.

– The expected return would be:

[ ] (1 ) [ ]

( [ ] )

xP f P

f P f

E R x r xE R

r x E R r

19

Investing in Risk-Free Securities (cont'd)

• The standard deviation of the portfolio would be calculated as:

– Note: The standard deviation is only a fraction of the volatility of the risky portfolio, based on the amount invested in the risky portfolio.

2 2

2

[ ] (1 ) ( ) ( ) 2(1 ) ( , )

( )

( )

xP f P f P

P

P

SD R x Var r x Var R x xCov r R

x Var R

xSD R0

20

The Risk–Return Combinations from Combining a Risk-Free Investment and a Risky Portfolio

21

The Tangent or Efficient Portfolio

22

The Capital Market Line (CML)

23

The Market Portfolio

• Because portfolio M lies at the point of tangency, it has the highest portfolio possibility line– So, everybody will want to invest in Portfolio M and borrow or

lend to be somewhere on the straight line. – This line is called the Capital Market Line (CML).

• Therefore this portfolio must include ALL RISKY ASSETS– not only U.S. common stocks but also all risky assets, such as

Non-U.S. stocks, options, real estate, coins, and art.

24

Continued

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

• Because it contains all risky assets, it is a completely diversified portfolio, which means that all the unique risk of individual assets (unsystematic risk) is diversified away

• All portfolios on the CML are perfectly positively correlated with each other and with the completely diversified market Portfolio M

• A completely diversified portfolio would have a correlation with the market portfolio of +1.00

25

The CML and the Separation Theorem

• Investors preferring more risk might borrow funds at the RFR and invest everything in the market portfolio

• The decision of both investors is to invest in portfolio M along the CML (the investment decision)

M

CML

RFR

B

A

( )portE R

port

26

The Security Market Line

• There is a linear relationship between a stock’s beta and its expected return (See figure on next slide). The security market line (SML) is graphed as the line through the risk-free investment and the market.

– According to the CAPM, if the expected return and beta for individual securities are plotted, they should all fall along the SML.

27

The Capital Market Line and the Security Market Line

28

The Capital Market Line and the Security Market Line

The CML depicts portfolios combining the risk-free investment and the efficient portfolio, and shows the highest expected return that we can attain for each level of volatility. According to the CAPM, the market portfolio is on the CML and all other stocks and portfolios contain diversifiable risk and lie to the right of the CML, as illustrated for Exxon Mobil (XOM).

29

The Capital Market Line and the Security Market Line

The SML shows the expected return for each security as a function of its beta with the market. According to the CAPM, the market portfolio is efficient, so all stocks and portfolios should lie on the SML.

30

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.

• We can calculate some different possible portfolio expected returns and betas by changing the percentages invested in these two assets.

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

31

Beta and the Risk Premium, II.

% of Portfolio in Asset A

Portfolio Expected Return

PortfolioBeta

0% 4 0.0

25 7 0.4

50 10 0.8

75 13 1.2

100 16 1.6

125 with borrowing 19 2.0

150 with borrowing 22 2.4

32

Portfolio Expected Returns and Betas for Asset A

33

The Reward-to-Risk Ratio

• Notice that all the combinations of portfolio expected returns and betas fall on a straight line.

• Slope (Rise over Run):

• What this tells us is that asset A offers a reward-to-risk ratio of 7.50%. In other words, asset A has a risk premium of 7.50% per “unit” of systematic risk.

7.50%

1.6

4%16%

β

RRE

A

fA

34

The Basic Argument, I.

• Recall that for asset A: E(RA) = 16% and A = 1.6

• Suppose there is a second asset, asset B.

• For asset B: E(RB) = 12% and A = 1.2

• Which investment is better, asset A or asset B?– Asset A has a higher expected return– Asset B has a lower systematic risk measure

The Basic Argument, II

• As before with Asset A, we can calculate some different possible portfolio expected returns and betas by changing the percentages invested in asset B and the risk-free rate.

% of Portfolio in Asset B

PortfolioExpected Return Portfolio Beta

0% 4 0.0

25 6 0.3

50 8 0.6

75 10 0.9

100 12 1.2

125 14 1.5

150 16 1.8

12-35

36

Portfolio Expected Returns and Betas for Asset B

37

Portfolio Expected Returns and Betas for Both Assets

38

The Fundamental Result, I.

• 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

39

The Fundamental Result, II.

In general …

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

• If one asset has twice as much systematic risk as another asset, its risk premium will simply be twice as large.

• Because the reward-to-risk ratio must be the same, all assets in the market must plot on the same line.

40

The Fundamental Result, III.

41

The Security Market Line (SML)

• The Security market line (SML) is 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

42

The Security Market Line, II.

• The term E(RM) – Rf is often called the 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

43

The Security Market Line, III.

• 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

44

The Security Market Line, IV.

45

The Security Market Line (SML)

46

Risk and Return Summary, I.

47

Risk and Return Summary, II.

48

A Closer Look at Beta

• R – E(R) = m + , where m is the systematic portion of the unexpected return.

• m = [RM – E(RM)]

• So, R – E(R) = [RM – E(RM)] +

• In other words:– A high-Beta security is simply one that is relatively sensitive to

overall market movements– A low-Beta security is one that is relatively insensitive to overall

market movements.

49

Decomposition of Total Returns

50

Unexpected Returns and Beta

51

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

52

Algebraically, beta can be expressed as follows:

ii iM

M

b

2

( , )i Mi

M

Cov r rb

53

Where Do Betas Come From?

54

Using a Spreadsheet to Calculate Beta

55

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.– To measure the overall market, the S&P 500 stock market index

is commonly used.– The calculated betas may be adjusted for various statistical

reasons.– And betas are time-sensitive (next slide)

56

Variation of CAPM Beta in Time

57

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.

58

Criticism toward CAPM and Conclusion

• The statistical tests have problems that make empirical verification virtually impossible.

• Investors seem to be concerned with both market risk and stand-alone risk. Therefore, the SML may not produce a correct estimate of the expected return.

• CAPM/SML concepts are based on expectations and future risk, yet betas are calculated using historical data. A company’s historical data may not reflect investors’ expectations about future riskiness.

• Other models are being developed that will one day replace the CAPM, but it still provides a good framework for thinking about risk and return.

59

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.

60

The Fama-French Three-Factor Model

• Professors Gene Fama and Ken French argue that two additional factors should be added.

• In addition to beta, two other factors appear to be useful in explaining the relationship between risk and return.– Size, as measured by market capitalization– The book value to market value ratio, i.e., B/M

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

61

The Size Effect: Being Small Means a Better Performance!

62

The Book-to-Market Ratio Effect

63

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.

64

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)

65

Chapter Review, I.

• Announcements, Surprises, and Expected Returns– Expected and unexpected returns– Announcements and news

• Risk: Systematic and Unsystematic– Systematic and unsystematic risk– Systematic and unsystematic components of return

• Diversification, Systematic Risk, and Unsystematic Risk– Diversification and unsystematic risk– Diversification and systematic risk

• Systematic Risk and Beta– The systematic risk principle– Measuring systematic risk– Portfolio Betas

66

Chapter Review, II.

• The Security Market Line– Beta and the risk premium– The reward-to-risk ratio– The basic argument– The fundamental result– The Security Market Line

• More on Beta– A closer look at Beta– Where do Betas come from?– Why do Betas differ?

• Extending CAPM– A (very) Brief History of Testing CAPM– The Fama-French three-factor model

67

• A risk that affects a limited number of securities is known as unsystematic risk.

A) True

B) False

68

• The time value of money is measured by the rate of return on the U.S. Treasury bill.

A) True

B) False

69

• An unexpected return is most apt to result from which one of these events?

A) A firm's price-earnings ratio was the same as the prior year's ratio.

B) A firm announced its quarterly dividend would continue to be paid.

C) Corporate income taxes rose according to the tax law approved two years ago.

D) A firm increased its annual dividend by the same percentage as in previous years.

E) The earnings of a firm declined due to an industrial accident.

70

• Which one of the following is most likely an unsystematic risk?

A) a change in the price of electricity

B) a market shortage of oil

C) the rapid decrease in the rate of GDP growth

D) a labor strike at a retail store

E) an increase in the cost of health insurance

71

• The reward for bearing risk depends on _____ risk.

A) unique

B) systematic

C) asset-specific

D) diversifiable

E) company

72

• What is the slope of the security market line if the U. S. Treasury bill has a return of 4.5 percent and the market has a return of 12.2 percent?

A) 5.45 percent

B) 6.00 percent

C) 7.28 percent

D) 7.70 percent

E) 7.85 percent

73

• Jensen and Sons common stock has an expected return of 14.8 percent. The risk-free rate is 2.8 percent and the market risk premium is 6.4 percent. What is the beta on Jensen and Sons stock?

A) 1.408

B) 1.750

C) 1.875

D) 1.932

E) 1.985

74

• You would like to invest $40,000 in a stock with a beta of 1.68. How much would you need to invest in U.S. Treasury bills to create a portfolio containing these two securities if you want the portfolio beta equal to the market beta?

A) $24,800

B) $27,200

C) $57,900

D) $64,800

E) $67,200

75

• A security which is overvalued will lie _____ and a security which is correctly valued will lie _____.

A) above the security market line (SML); on the horizontal axis

B) above the SML; on the SML

C) below the SML; on the horizontal axis

D) below the SML; on the SML

E) to the right of the SML; to the left of the SML

76

Performance Evaluation Measures

The Sharpe Ratio

• The Sharpe ratio is a reward-to-risk ratio that focuses on total risk.

• It is computed as a portfolio’s risk premium divided by the standard deviation of the portfolio’s return.

p

fp

σ

RRratio Sharpe

77

Performance Evaluation Measures

The Treynor Ratio

• The Treynor ratio is a reward-to-risk ratio that looks at systematic risk only.

• It is computed as a portfolio’s risk premium divided by the portfolio’s beta coefficient.

p

fp

β

RRratio Treynor