Risk and Return in Financial Markets

12

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Learning Objectives

•

Our goal in this chapter 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.

2

Expected and Unexpected Returns

•

The return on any stock traded in a financial market is composed of

two parts.

–

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Announcements and News

•

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significantly impact the profits of the firm.

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The impact of an announcement depends on how much of the

announcement represents new information.

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4

Systematic and Unsystematic Risk

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5

•

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

Systematic and Unsystematic

Components of Return

•

Recall:

  R – E(R)

= U

= Systematic portion

+ Unsystematic portion

= m + 



   R – E(R) = m + 



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.

•

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8

The Systematic Risk Principle

•

What determines the size of the risk premium on a risky

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9

Measuring Systematic Risk

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To be compensated for risk, the risk has to be special.

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Because assets with larger betas have greater systematic risks, they

will have greater expected returns.

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10

Betas with Respect

to the S&P 500 for

Individual Stocks

(based on monthly

data for 2004–2008)

11

Finding a Beta on the Web

12

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.

13

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?

12-14

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….

15

Efficient Frontier with Ten Stocks Versus

Three Stocks

16

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.

17

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:

18

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.

19

The Risk–Return Combinations  from Combining a Risk-

Free Investment and a Risky Portfolio

20

The Tangent or Efficient Portfolio

21

The Capital Market Line (CML)

22

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.

23

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

24

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

25

The Security Market Line

•

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–

According to the CAPM, if the expected return and

beta for individual securities are plotted, they should

all fall along the SML.

26

The Capital Market Line  and the Security

Market Line

27

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).

28

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.

29

Beta and the Risk Premium, I.

•

Consider a portfolio made up of asset A and a risk-free asset.

–

For asset A, 

E(R

A

)

 = 16% and 



A

 = 1.6

–

The risk-free rate 

R

f

 = 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%.

30

Beta and the Risk Premium, II.

31

Portfolio Expected Returns

and Betas for Asset A

32

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):

•

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33

Performance Evaluation Measures

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•

It is computed as a portfolio’s risk premium divided by

the standard deviation of the portfolio’s return.

34

Performance Evaluation Measures

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•

It is computed as a portfolio’s risk premium divided by

the portfolio’s beta coefficient.

35

The Basic Argument, I.

•

Recall that for asset A: 

E(R

A

)

 = 16% and 



A

 = 1.6

•

Suppose there is a second asset, asset B.

•

For asset B: 

E(R

B

)

 = 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

36

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.

12-37

Portfolio Expected Returns

and Betas for Asset B

38

Portfolio Expected Returns

and Betas for Both Assets

39

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:

40

The Fundamental Result, II.

I

n

g

e

n

e

r

a

l

…

•

T

h

e

r

e

w

a

r

d

-

t

o

-

r

i

s

k

r

a

t

i

o

m

u

s

t

b

e

t

h

e

s

a

m

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f

o

r

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l

l

a

s

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t

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n

a

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o

m

p

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t

i

t

i

v

e

f

i

n

a

n

c

i

a

l

m

a

r

k

e

t

.

•

If one asset has twice as much systematic risk as another

asset, its risk premium will simply be twice as large.

•

B

e

c

a

u

s

e

t

h

e

r

e

w

a

r

d

-

t

o

-

r

i

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,

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p

l

o

t

o

n

t

h

e

s

a

m

e

l

i

n

e

.

41

The Fundamental Result, III.

42

The Security Market Line (SML)

•

T

h

e

S

e

c

u

r

i

t

y

m

a

r

k

e

t

l

i

n

e

(

S

M

L

)

i

s

a

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r

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c

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l

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a

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p

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s

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m

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k

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e

x

p

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e

d

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t

u

r

n

i

n

f

i

n

a

n

c

i

a

l

m

a

r

k

e

t

s

.

•

For a market portfolio,

43

The Security Market Line, II.

•

T

h

e

t

e

r

m

E

(

R

M

)

–

R

f

i

s

o

f

t

e

n

c

a

l

l

e

d

t

h

e

m

a

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k

p

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m

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m

b

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c

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i

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k

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m

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m

o

n

a

m

a

r

k

e

t

p

o

r

t

f

o

l

i

o

.

•

For any asset 

i

 in the market:

•

S

e

t

t

i

n

g

t

h

e

r

e

w

a

r

d

-

t

o

-

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q

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k

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t

a

l

a

s

s

e

t

p

r

i

c

i

n

g

m

o

d

e

l

.

44

The Security Market Line, III.

•

T

h

e

C

a

p

i

t

a

l

A

s

s

e

t

P

r

i

c

i

n

g

M

o

d

e

l

(

C

A

P

M

)

i

s

a

t

h

e

o

r

y

o

f

r

i

s

k

a

n

d

r

e

t

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r

n

f

o

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c

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r

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p

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t

i

t

i

v

e

c

a

p

i

t

a

l

m

a

r

k

e

t

.

•

The CAPM shows that 

E(R

i

)

 depends on:

–

R

f

, the pure time value of money.

–

E(R

M

) – R

f

, the reward for bearing systematic risk.

–



i

, the amount of systematic risk.

45

The Security Market Line, IV.

46

The Security Market Line (SML)

47

Risk and Return Summary, I.

48

Risk and Return Summary, II.

49

A Closer Look at Beta

•

R – E(R) = m

 + 



, where 

m

 is the systematic portion of

the unexpected return.

•

m

 = 





 [

R

M

–

E(R

M

)

]

•

So, 

R – E(R) = 





 [

R

M

–

E(R

M

)

] + 



•

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.

50

Decomposition of Total Returns

51

Unexpected Returns and Beta

52

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.

53

Where Do Betas Come From?

55

Using a Spreadsheet to Calculate Beta

56

Beta Measures Relative Movement

•

A security with a beta of 1 is expected to move, on average, with the

market.

–

In this case, the characteristic line would have  a 45 degree angle, or a slope of 1.

–

This slope means that if the market goes up, say, 3%, we expect the stock to go up 3%.

–

In mathematics, we measure slope as “the rise over the run.”

–

So, if the market return is the “x data” and the stock return is the “y data,” the slope is the

beta of the security.

•

Apply this method to the scatter plot on the next slide. What do you

think the beta is?

–

When the market has a 10% return, the security return is slightly less than 10%.

–

The characteristic line suggests that the beta for this security is less than 1. Why?

•

You might recall from your statistics class an alternative method for

estimating a “line of best fit,” a simple linear regression.

–

We  regress the returns of the security (y data) on the returns of the market (x data).

–

Excel has a built-in regression function.

•

The output has an estimate of the “Market Returns” coefficient, i.e., the slope.

•

The highlighted “Market Returns” coefficient is the beta estimate.

57

Another Way to Calculate Beta

(The Characteristic Line)

58

Build a Beta

59

Why Do Betas Differ?

•

Betas are estimated from actual data. Different sources

estimate differently, possibly using different data.

–

F

o

r

d

a

t

a

,

t

h

e

m

o

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w

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k

l

y

d

a

t

a

.

–

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

commonly used.

–

The calculated betas can be adjusted for various statistical reasons.

–

And betas are time-sensitive (next slide)

•

T

h

e

b

o

t

t

o

m

-

l

i

n

e

l

e

s

s

o

n

:

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e

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,

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g

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s

t

o

r

i

c

a

l

d

a

t

a

.

•

Anytime we use the past to predict the future, there is the

danger of a poor estimate.

60

Variation of Beta in Time

61

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.

62

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.

•

B

u

t

,

d

o

e

s

e

x

p

e

c

t

e

d

r

e

t

u

r

n

d

e

p

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n

d

o

n

l

y

o

n

B

e

t

a

?

O

r

,

d

o

o

t

h

e

r

f

a

c

t

o

r

s

c

o

m

e

i

n

t

o

p

l

a

y

?

•

The above bullet point is a hotly debated question.

63

•

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.

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.

65

The Size Effect:

Being Small Means a Better Performance!

66

The Book-to-Market Ratio Effect

67

Returns from 25 Portfolios Formed

on Size and Book-to-Market

1926-2006, 1927-2014

•

Note that the portfolio containing the smallest cap and

the highest book-to-market have had the highest returns.

68

•

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

unsystematic risk.

A

)

T

r

u

e

B

)

F

a

l

s

e

•

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

Treasury bill.

A

)

T

r

u

e

B

)

F

a

l

s

e

•

The reward for bearing risk depends on _____ risk.

A

)

u

n

i

q

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e

B

)

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D

)

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r

s

i

f

i

a

b

l

e

E

)

c

o

m

p

a

n

y

•

An unexpected return is most apt to result from which

one of these events?

A

)

A

f

i

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m

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.

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.

C

)

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.

D

)

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.

E

)

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a

c

c

i

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n

t

.

•

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

A

)

a

c

h

a

n

g

e

i

n

t

h

e

p

r

i

c

e

o

f

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t

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i

t

y

B

)

a

m

a

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k

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a

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f

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i

l

C

)

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G

D

P

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D

)

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E

)

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o

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t

o

f

h

e

a

l

t

h

i

n

s

u

r

a

n

c

e

•

A security which is overvalued will lie _____ and a security

which is correctly valued will lie _____.

A

)

a

b

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v

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t

h

e

s

e

c

u

r

i

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y

m

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(

S

M

L

)

;

o

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t

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h

o

r

i

z

o

n

t

a

l

a

x

i

s

B

)

a

b

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e

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S

M

L

;

o

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t

h

e

S

M

L

C

)

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S

M

L

;

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t

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r

i

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o

n

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a

l

a

x

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s

D

)

b

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l

o

w

t

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S

M

L

;

o

n

t

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e

S

M

L

E

)

t

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r

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h

t

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f

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S

M

L

;

t

o

t

h

e

l

e

f

t

o

f

t

h

e

S

M

L

•

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

.

4

5

p

e

r

c

e

n

t

B

)

6

.

0

0

p

e

r

c

e

n

t

C

)

7

.

2

8

p

e

r

c

e

n

t

D

)

7

.

7

0

p

e

r

c

e

n

t

E

)

7

.

8

5

p

e

r

c

e

n

t

•

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

.

4

0

8

B

)

1

.

7

5

0

C

)

1

.

8

7

5

D

)

1

.

9

3

2

E

)

1

.

9

8

5

•

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

)

$

2

4

,

8

0

0

B

)

$

2

7

,

2

0

0

C

)

$

5

7

,

9

0

0

D

)

$

6

4

,

8

0

0

E

)

$

6

7

,

2

0

0