January 2, 2020

Chapter 12

Return, Risk, and the Security Market

Line

Slides

12-1. Chapter 12

12-2. Return, Risk, and the Security Market Line

12-3. Learning Objectives

12-4. Return, Risk, and the Security Market Line

12-5. Expected and Unexpected Returns

12-6. Announcements and News

12-7. Systematic and Unsystematic Risk

12-8. Systematic and Unsystematic Components of Return

12-9. Diversification and Risk

12-10. The Systematic Risk Principle

12-11. Measuring Systematic Risk

12-12. Published Beta Coefficients

12-13. Finding a Beta on finance.yahoo.com: Southwest Airlines (LUV)

12-14. Portfolio Betas

12-15. Example: Calculating a Portfolio Beta

12-16. Beta and the Risk Premium, I.

12-17. Beta and the Risk Premium, II.

12-18. Portfolio Expected Returns and Betas for Asset A

12-19. The Reward-to-Risk Ratio

12-20. The Basic Argument, I.

12-21. The Basic Argument, II

12-22. Portfolio Expected Returns and Betas for Asset B

12-23. Portfolio Expected Returns and Betas for Both Assets

12-24. The Fundamental Result, I.

12-25. The Fundamental Result, II.

12-26. The Fundamental Result, III.

12-27. The Security Market Line (SML)

12-28. The Security Market Line, II.

12-29. The Security Market Line, III.

12-30. The Security Market Line, IV.

12-31. Risk and Return Summary, I.

12-32. Risk and Return Summary, II.

12-33. A Closer Look at Beta

12-34. Decomposition of Total Returns

12-35. Unexpected Returns and Beta

12-36. Where Do Betas Come From?

12-37. From Where Do Betas Come?

12-38. Using a Spreadsheet to Calculate Beta

12-39. Beta Measures Relative Movement

12-40. Another Way to Calculate Beta (The Characteristic Line)

12-41. Build a Beta

12-42. Why Do Betas Differ?

12-43. Extending CAPM

12-44. Important General Risk-Return Principles

12-45. The Fama-French Three-Factor Model

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Return, Risk, and the Security Market Line 12-2

12-46. Returns from 25 Portfolios Formed on Size and Book-to-Market, 1927-2014

12-47. Useful Internet Sites

12-48. Chapter Review, I.

12-49. Chapter Review, II.

Chapter Organization

12.1 Announcements, Surprises, and Expected Returns

A. Expected and Unexpected Returns

B. Announcements and News

12.2 Risk: Systematic and Unsystematic

A. Systematic and Unsystematic Risk

B. Systematic and Unsystematic Components of Return

12.3 Diversification, Systematic Risk, and Unsystematic Risk

A. Diversification and Unsystematic Risk

B. Diversification and Systematic Risk

12.4 Systematic Risk and Beta

A. The Systematic Risk Principle

B. Measuring Systematic Risk

C. Portfolio Betas

12.5 The Security Market Line

A. Beta and the Risk Premium

B. The Reward-to-Risk Ratio

C. The Basic Argument

D. The Fundamental Result

E. The Security Market Line

12.6 More on Beta

A. A Closer Look at Beta

B. Where Do Betas Come From?

C. Another Way to Calculate Beta

D. Why Do Betas Differ?

12.7 Extending CAPM

A. A (Very) Brief History of Testing CAPM

B. The Fama-French Three-Factor Model

12.8 Summary and Conclusions

Selected Web Sites

www.earningswhispers.com (earnings calendar)

biz.yahoo.com/z/extreme.html (earnings surprises)

www.portfolioscience.com (helps you analyze risk)

finance.yahoo.com (a source of betas)

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Return, Risk, and the Security Market Line 12-3

www.marketwatch.com (another source for betas)

www.fenews.co.uk (information on risk management)

www.moneychimp.com (for a CAPM calculator and other interesting tools)

mba.tuck.dartmouth.edu/pages/faculty/ken.french/ (source for data behind

the FAMA-French model)

www.morningstar.com (finding fund betas)

www.bloomberg.com (for current interest rates)

Annotated Chapter Outline

12.1 Announcements, Surprises, and Expected Returns

A. Expected and Unexpected Returns

The return on a security is represented by:

Total return - Expected return = Unexpected return

Or

R - E(R) = U

It is important to note that the average value of the unexpected return is zero

(otherwise, there would be a bias that investors could use to make profits).

B. Announcements and News

An announcement can be broken into two parts, the anticipated or expected part,

and the surprise or innovation:

Announcement = Expected part + Surprise

The expected part is the part of the information that the market uses to form the

expectation, E(R), and the surprise is the news that influences the unanticipated

return on the stock, U. The current price reflects relevant publicly available

information, saying that markets are semistrong efficient. So when we speak of

news, we mean the surprise part of the announcement.

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Return, Risk, and the Security Market Line 12-4

12.2 Risk: Systematic and Unsystematic

The unanticipated part of the return, the portion resulting from surprises, is the

significant risk of any investment.

A. Systematic and Unsystematic Risk

Systematic Risk: Systematic risk influences a large number of assets.

This is also called market risk.

Unsystematic Risk: Unsystematic risk influences a single company or a

small group of companies. Unsystematic risk is also called unique or

asset-specific risk.

Systematic risk usually results from economic or macroeconomic factors,

whereas unsystematic risk results from company-specific events.

B. Systematic and Unsystematic Components of Return

The surprise component can be broken down into a systematic portion and an

unsystematic portion:

U = R - E(R) = Systematic portion + Unsystematic portion

or

R - E(R) = U = m +

12.3 Diversification, Systematic Risk, and Unsystematic Risk

A. Diversification and Unsystematic Risk

Unsystematic risk is essentially eliminated by diversification, so a portfolio with

many assets has almost no unsystematic risk.

B. Diversification and Systematic Risk

Total risk = Systematic risk + Unsystematic risk

Systematic risk is also called nondiversifiable risk or market risk. Unsystematic

risk is also called diversifiable risk, unique risk, or asset-specific risk. For a

diversified portfolio, unsystematic risk is negligible and almost all risk is

systematic.

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Return, Risk, and the Security Market Line 12-5

12.4 Systematic Risk and Beta

A. The Systematic Risk Principle

Systematic Risk Principle: 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. No matter

how much total risk an asset has, only the systematic portion is relevant in

determining the expected return (and risk premium) on that asset.

B. Measuring Systematic Risk

Beta Coefficient (): Measure of the relative systematic risk of an asset.

Assets with betas larger (smaller) than 1 have more (less) systematic risk

than average.

Beta is a measure of the systematic risk of an asset relative to an average asset.

The expected return and the risk premium on an asset depend only on its

systematic risk. Assets with larger betas have greater systematic risks and will

have greater expected returns.

C. Portfolio Betas

With a large number of assets in a portfolio, multiply each asset's beta by its

portfolio weight, and then sum the results to get the portfolio's beta:

i

N

1i

ip

βωβ

Note: Recall that variance is also a risk measure. However, the variance of the

return of a portfolio is not a simple calculation. If you have a 40 asset portfolio,

you need to calculate 40 variances, and 780 unique covariances. Forty times 40,

minus 40 is the number of covariances (1,560). Dividing by 2 gives the number of

unique covariance terms (780).

Unlike the variance of the return of the portfolio, the beta of a portfolio is a

weighted-average of the betas of the stocks in the portfolio.

12.5 The Security Market Line

A. Beta and the Risk Premium

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Return, Risk, and the Security Market Line 12-6

To calculate the expected return and beta when combining an asset with a risk-

free asset:

fAAAp

R)ω(1)E(RωRE

AAAAAp

βω0)ω(1βωβ

Figure 12.1 shows the result of combining a risky asset with a risk-free asset in

various proportions, from 0% to 150%. These combinations all fall on a straight

line.

B. The Reward-to-Risk Ratio

The slope of the line in Figure 12.1A is the reward-to-risk ratio and is calculated

as:

If an asset has a reward-to-risk ratio of 10.0%, the asset has a risk premium of

10.0% per unit of systematic risk.

C. The Basic Argument

Reviewing Figure 12.1 again, if an asset has a larger slope, or higher reward-to-

risk ratio, it offers a superior return for its level of risk, and it is preferred over an

asset with a lower reward-to-risk ratio.

D. The Fundamental Result

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

market. This must be so because otherwise investors could arbitrage a risk-free

profit. Figure 12.2 shows this result, as well as the fact that all securities should

plot on this line. If an asset plots above (below) this line, its price will rise (fall)

and its return will fall (rise) until it plots exactly on the line. This equilibrium will be

true for active financial markets.

E. The Security Market Line

Security Market Line (SML): Graphical representation of the linear

relationship between systematic risk and expected return in financial

markets.

Market Risk Premium: The risk premium on a market portfolio; i.e., a

portfolio made of all assets in the market

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Slope=E(RA)−Rf

βA

Return, Risk, and the Security Market Line 12-7

Capital Asset Pricing Model (CAPM): A theory of risk and return for

securities in a competitive capital market.

The line that results when the expected returns and beta coefficients are plotted

is called the Security Market Line (SML). A portfolio made up of all assets in the

market is called the market portfolio. The risk premium on a market portfolio is

also the slope of the SML, and it is called the market risk premium:

The Capital Asset Pricing Model (CAPM) shows that the expected return for an

asset depends on:

The pure time value of money (as measured by the risk-free rate)

The reward for bearing systematic risk (as measured by E(RM) - Rf)

The amount of systematic risk (as measured by beta)

It is important to emphasize that the CAPM is an “equilibrium” model. As an

equilibrium example, use supply and demand curves. When those are drawn,

they represent an equilibrium process. If we are jostled away from where the

demand and supply curves intersect, market forces will act to move the market

back toward equilibrium.

For the CAPM, here is a simple example. Suppose we have all solved the

efficient portfolio problem (using the same information about expected returns

and variances). We all have our share holdings in the risk-free rate and in the

assets of the tangency portfolio. But, somebody, for whatever reason, thinks that

they need to increase their weight in a particular stock, say IBM. But, they have

to bid these shares away from other investors. This will drive up the price of IBM,

which lowers the expected return for IBM. Then, we all have to solve the efficient

portfolio problem all over again!

The equation for the CAPM is:

This relationship is shown in Figure 12.3. The CAPM is relevant for portfolios of

assets, as well as individual assets. The risk and return relationship is

summarized in Table 12.2.

12.6 More on Beta

A. A Closer Look at Beta

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SML Slope=E(RM)−Rf

βM

=E(RM)−Rf

1. 0 =E(RM)−Rf

E(Rj)=Rf+βj

(

E(RM)−Rf

)

Return, Risk, and the Security Market Line 12-8

A high-beta security is one that is relatively sensitive to overall market

movements, and a low-beta security is relatively insensitive to market

movements.

B. Where Do Betas come from?

Covariance: A measure of how two variables (like security returns) tend to

move or vary together.

Lecture Tip:

Sensitivity is how closely correlated the security’s return is with the market’s

return and how volatile the security is relative to the market. The beta for a

security is equal to the correlation multiplied by the ratio of the standard

deviations.

To calculate the covariance, sum the products of the return deviations, and then

divide the total by n – 1 (as shown in Table 12.4). Covariance measures the

tendency of the returns of two assets to move, or vary, together. It is similar to the

correlation coefficient, although the covariance numbers are harder to interpret.

C. Another Way to Calculate Beta

Beta is an estimate of the relationship of movement in a security’s return given a

movement in the market (or benchmark). Thus, this is really “rise over run,” which

is the slope coefficient of a regression line of the security (y) on the market (x).

D. Why Do Betas Differ?

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Running the regression:

Ri=α+βiRM+εi

results in an estimate slope, βi, that equals:

βi=COV(Ri,RM)

σM

2

Because,

COV(Ri,R M)=Corr

(

Ri,RM

)

×σi×σm

βi can be also written as:βi=Corr

(

Ri,RM

)

xσi

σM

Return, Risk, and the Security Market Line 12-9

It is important to note that different sources provide different betas. These

sources may use different data. Differences may be due to:

Using daily, weekly, monthly, quarterly, or annual returns.

Estimating betas over short periods (a few weeks) versus long periods (5-

10 years).

Choice of the market index (S&P 500 index versus all risky assets).

Some sources adjust betas for statistical and fundamental reasons (such

as Value Line).

12.7 Extending CAPM

For investors, the CAPM has a stunning implication: What investors earn on their

portfolio depends only on the level of systematic risk they bear. The corollary is

equally striking: As a diversified investor, you do not need to be concerned with

the total risk or volatility of any individual asset in your portfolio—it is simply

irrelevant.

Of course, you should note that the CAPM is a theory, and, as with any theory,

whether it is correct is a question for the data. The students will most likely ask,

“So, does the CAPM work or not?” You can answer by rephrasing the question

to: “You mean, does expected return depend on beta, and beta alone, or do other

factors come into play?”

A. A (Very) Brief History of Testing CAPM

The CAPM was introduced in the mid-1960s (but, perhaps surprisingly, tests of

this model began to appear only in the early 1970s). When researchers test the

CAPM, they essentially look to see whether average returns are linearly related

to beta. That is, they want to know if asset returns and beta line up as shown in

Figure 12.3.

The earliest tests of the CAPM suggested that return and risk (as measured by

beta) showed a reasonable relationship. However, the relationship was not so

strong that financial researchers were content to move on and test other theories.

To summarize years of testing, the relationship between returns and beta

appeared to vary depending on the time period that was studied. Over some

periods, the relationship was strong. In others, it was apparent but not strong. In

others, it was seemingly nonexistent.

Over the years, researchers have refined their techniques to measure betas. In

addition, the question was raised whether researchers could calculate betas at

all. The basic argument was that betas could not be calculated relative to the

overall market portfolio because we cannot observe the true market portfolio.

Nonetheless, despite this insightful critique, researchers continue to test CAPM

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Return, Risk, and the Security Market Line 12-10

and debate the findings of CAPM research to this day.

Students are comforted, however, when you disclose that few researchers

question these four general principles:

1. Investing has two dimensions: risk and return.

2. It is inappropriate to look at the risk of an individual security. What is

appropriate is how the individual security contributes to the risk of a

diversified portfolio.

3. Risk can be decomposed into systematic risk and nonsystematic risk.

4. Investors will be compensated only for taking on systematic risk.

B. The Fama-French Three-Factor Model

Table 12.5 illustrates an important finding from years of research into stock

market returns. As shown, two groups of stocks have tended to do noticeably

better than the market as a whole: (1) stocks with a small-market capitalization

(small-cap stocks), and (2) stocks that have a higher than average ratio of book

(or accounting) value to market value (so-called value stocks).

Table 12.5 is formed as follows. First, for each year of historic data, a large set of

stocks are ranked on the basis of their market cap, or size. The smallest 20

percent of the stocks are placed into the market cap quintile number 1, the next

smallest 20 percent are placed into market cap quintile number 2, and so on.

Then, the same stocks are ranked on the basis of their book/market (B/M) ratio.

The smallest 20 percent are placed into B/M quintile number 1, the next smallest

20 percent are placed into B/M quintile number 2, and so on. A high B/M ratio is

indicative of a “value” stock, while a low B/M would be affiliated with a “growth”

stock.

Lecture Tip: Students will be a bit confused with this dual sorting. (The best way

to understand it is to have them replicate Table 12.5.) Basically, you will have to

show them that two sets of quintiles means that any particular stock could wind

up in one of twenty five buckets (i.e., those shown in Table 12.5).

Three things should jump out at you in Table 12.5. (1) Cell 1-5, which contains

stocks with the smallest cap and highest B/M has had the highest returns. (2)

Looking down each column, you can see that in three columns the highest return

belongs to the smallest cap quintile (i.e., in columns 3, 4, and 5). (3) Looking

across each row, you can see that in five of five rows, the highest return belongs

to the highest B/M quintile.

12.8 Summary and Conclusions

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Return, Risk, and the Security Market Line 12-11

Lecture Tip: A student project will help reinforce the concepts presented in this

chapter. This project is best completed using a spreadsheet. Have students

collect two to four years of monthly stock prices for a firm of their choice and for a

market index, such as the S&P 500. This data is easily available at

finance.yahoo.com. Calculate monthly returns from the price data for the security

and the index. Calculate the correlation and covariance between the security and

the index, and then calculate the beta for the security. To find the slope (beta) of

the characteristic line, you regress the security returns (dependant variable)

against the index returns (independent variable).

Another useful student project is to have the students gather six years of monthly

data. Calculate a beta using the first five years of data, and calculate another

beta using the second five years of data (i.e., drop the first year of the data and

add the last year of data). Students will be surprised to see that betas may not be

very stable!

Return data for these portfolios, or for index ETFs, can be used to show that

there is more non-systematic risk in a single security than for a portfolio, and that

betas for portfolios are most likely more “stable” than betas for individual

securities. We say “most likely” because it is always possible to select a single

security randomly that might exhibit a very stable beta.

In addition, using 5 years of monthly data, students could gather data for 30

firms. Then, they could calculate 30 betas and 30 average returns. Students

could then plot (or even regress) the average returns against beta. They will be

surprised to see that the betas “do not line up.” Using the average return on the

S&P 500 and the current T-bill rate as the risk-free rate, a historical SML can be

over-laid into the plot. This will reinforce the notion that the SML is for expected,

not realized, returns.

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