Chapter 7Principles of

Corporate FinanceTenth Edition

Introduction to Risk and Return

Slides by

Matthew Will

McGraw-Hill/Irwin Copyright © 2011 by the McGraw-Hill Companies, Inc. All rights reserved.

7-2

Topics Covered

Over a Century of Capital Market HistoryMeasuring Portfolio RiskCalculating Portfolio RiskHow Individual Securities Affect Portfolio

RiskDiversification & Value Additivity

7-3

$1

$10

$100

$1,000

$10,000

$100,000

Start of Year

Dol

lars

(log

sca

le)

Common Stock

US Govt Bonds

T-Bills

14,276

24171

2008

The Value of an Investment of $1 in 1900

7-4

$1

$10

$100

$1,000

Start of Year

Dol

lars

(log

sca

le)

Equities

Bonds

Bills

581

9.85

2.87

2008

Real Returns

The Value of an Investment of $1 in 1900

7-5

Average Market Risk Premia (by country)

4.29 4.69 5.05 5.43 5.5 5.61 5.67 6.04 6.29 6.94 7.137.94 8.34 8.4 8.74 9.1 9.61 10.21

0123456789

1011

Den

mar

k

Bel

giu

m

Sw

itze

rlan

d

Irel

and

Sp

ain

Nor

way

Can

ada

U.K

.

Net

her

lan

ds

Ave

rage

U.S

.

Sw

eden

Au

stra

lia

Sou

th A

fric

a

Ger

man

y

Fra

nce

Jap

an

Ital

y

Risk premium, %

Country

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Dividend Yield

Dividend yields in the U.S.A. 1900–2008

7-7

Rates of Return 1900-2008

Source: Ibbotson Associates Year

Per

cent

age

Ret

urn

Stock Market Index Returns

7-8

Measuring Risk

1 24

11 11

21

17

24

13

32

0

4

8

12

16

20

24

-50

to -

40

-40

to -

30

-30

to -

20

-20

to -

10

-10

to 0

0 to

10

10 t

o 20

20 t

o 30

30 t

o 40

40 t

o 50

50 t

o 60

Return %

# of Years

Histogram of Annual Stock Market ReturnsHistogram of Annual Stock Market Returns

(1900-2008)(1900-2008)

7-9

Measuring Risk

Variance - Average of squared deviations from mean.

Standard Deviation – Square root of the variance. A measure of volatility.

7-10

Measuring Risk

Coin Toss Game-calculating variance and standard deviation

7-11

Measuring Risk

Portfolio rate

of return=

fraction of portfolio

in first assetx

rate of return

on first asset

+fraction of portfolio

in second assetx

rate of return

on second asset

((

(())

))

7-12

Equity Market Risk (by country)

17.02 18.45 19.22 20.16 21.83 22.05 22.99 23.23 23.42 23.51 23.98 24.09 25.2828.32 29.57

33.93 34.3

0

5

10

15

20

25

30

35

40

Can

ada

Au

stra

lia

Sw

itze

rlan

d

U.S

.

U.K

.

Den

mar

k

Sp

ain

Net

her

lan

ds

Sou

th A

fric

a

Irel

and

Sw

eden

Bel

giu

m

Fra

nce

Nor

way

Jap

an

Ital

y

Ger

man

y

Sta

ndar

d D

evia

tion

of A

nnua

l Ret

urns

, %

Average Risk (1900-2008)

7-13

Dow Jones Risk

Annualized Standard Deviation of the DJIA over the preceding 52 weeks

(1900 – 2008)

Years

Sta

ndar

d D

evia

tion

(%

)

7-14

Measuring Risk

Diversification - Strategy designed to reduce risk by spreading the portfolio across many investments.

Unique Risk - Risk factors affecting only that firm. Also called “diversifiable risk.”

Market Risk - Economy-wide sources of risk that affect the overall stock market. Also called “systematic risk.”

7-15

Comparing Returns

7-16

Measuring Risk

05 10 15

Number of Securities

Po

rtfo

lio

sta

nd

ard

dev

iati

on

7-17

Measuring Risk

05 10 15

Number of Securities

Po

rtfo

lio

sta

nd

ard

dev

iati

on

Market risk

Uniquerisk

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Portfolio Risk

22

22

211221

1221

211221

122121

21

σxσσρxx

σxx2Stock

σσρxx

σxxσx1Stock

2Stock 1Stock

The variance of a two stock portfolio is the sum of these four boxes

7-19

Portfolio Risk

Example

Suppose you invest 60% of your portfolio in Exxon Mobil and 40% in Coca Cola. The expected dollar return on your Exxon Mobil stock is 10% and on Coca Cola is 15%. The expected return on your portfolio is:

%12)1540(.)1060(. ReturnExpected

7-20

Portfolio Risk

2222

22

211221

2112212221

21

)3.27()40(.σx3.272.181

60.40.σσρxxCola-Coca

3.272.181

60.40.σσρxx)2.18()60(.σxMobil-Exxon

Cola-CocaMobil-Exxon

Another Example

Suppose you invest 60% of your portfolio in Exxon Mobil and 40% in Coca Cola. The expected dollar return on your Exxon Mobil stock is 10% and on Coca Cola is 15%. The standard deviation of their annualized daily returns are 18.2% and 27.3%, respectively. Assume a correlation coefficient of 1.0 and calculate the portfolio variance.

7-21

Portfolio Risk

Another Example

Suppose you invest 60% of your portfolio in Exxon Mobil and 40% in Coca Cola. The expected dollar return on your Exxon Mobil stock is 10% and on Coca Cola is 15%. The standard deviation of their annualized daily returns are 18.2% and 27.3%, respectively. Assume a correlation coefficient of 1.0 and calculate the portfolio variance.

% 21.8 0.477 DeviationStandard

0.47718.2x27.3)2(.40x.60x

]x(27.3)[(.40)

]x(18.2)[(.60) Variance Portfolio22

22

7-22

Portfolio Risk

)rx()r(x Return PortfolioExpected 2211

)σσρxx(2σxσxVariance Portfolio 21122122

22

21

21

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Portfolio Risk

Example Correlation Coefficient = .4

Stocks % of Portfolio Avg Return

ABC Corp 28 60% 15%

Big Corp 42 40% 21%

Standard Deviation = weighted avg = 33.6

Standard Deviation = Portfolio = 28.1

Real Standard Deviation:

= (282)(.62) + (422)(.42) + 2(.4)(.6)(28)(42)(.4)

= 28.1 CORRECT

Return : r = (15%)(.60) + (21%)(.4) = 17.4%

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Portfolio Risk

Example Correlation Coefficient = .4

Stocks % of Portfolio Avg Return

ABC Corp 28 60% 15%

Big Corp 42 40% 21%

Standard Deviation = weighted avg = 33.6

Standard Deviation = Portfolio = 28.1

Return = weighted avg = Portfolio = 17.4%

Let’s Add stock New Corp to the portfolio

7-25

Portfolio Risk

Example Correlation Coefficient = .3

Stocks % of Portfolio Avg Return

Portfolio 28.1 50% 17.4%

New Corp 30 50% 19%

NEW Standard Deviation = weighted avg = 31.80

NEW Standard Deviation = Portfolio = 23.43

NEW Return = weighted avg = Portfolio = 18.20%

NOTE: Higher return & Lower risk

How did we do that? DIVERSIFICATION

7-26

Portfolio Risk

The shaded boxes contain variance terms; the remainder contain covariance terms.

1

2

3

4

5

6

N

1 2 3 4 5 6 N

STOCK

STOCKTo calculate portfolio variance add up the boxes

7-27

Portfolio Risk

Market Portfolio - Portfolio of all assets in the economy. In practice a broad stock market index, such as the S&P Composite, is used to represent the market.

Beta - Sensitivity of a stock’s return to the return on the market portfolio.

7-28

Portfolio Risk

The return on Dell stockchanges on averageby 1.41% for eachadditional 1% change inthe market return. Betais therefore 1.41.

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Portfolio Risk

The middle line shows that a wellDiversified portfolio of randomlyselected stocks ends up with 1and a standard deviation equal tothe market’s—in this case 20%.The upper line shows that awell-diversified portfolio with 1.5has a standard deviation of about30%—1.5 times that of the market.The lower line shows that awell-diversified portfolio with .5has a standard deviation of about10%—half that of the market.

7-30

Portfolio Risk

2m

imiB

7-31

Portfolio Risk

2m

imiB

Covariance with the market

Variance of the market

7-32

Beta

(1) (2) (3) (4) (5) (6) (7)Product of

Deviation Squared deviationsDeviation from average deviation from average

Market Anchovy Q from average Anchovy Q from average returnsMonth return return market return return market return (cols 4 x 5)

1 -8% -11% -10% -13% 100 1302 4 8 2 6 4 123 12 19 10 17 100 1704 -6 -13 -8 -15 64 1205 2 3 0 1 0 06 8 6 6 4 36 24

Average 2 2 Total 304 456

Variance = σm2 = 304/6 = 50.67

Covariance = σim = 736/6 = 76

Beta (β) = σim/σm2 = 76/50.67 = 1.5

Calculating the variance of the market returns and the covariance between the returns on the market and those of Anchovy Queen. Beta is the ratio of

the variance to the covariance (i.e., β = σim/σm2)