031 Risk+and+Return031 Risk+and+Return031 Risk+and+Return031 Risk+and+Return031 Risk+and+Return031 Risk+and+Return031 Risk+and+Return031 Risk+and+Return031 Risk+and+Return031 Risk+and+Return031

7/28/2019 031 Risk+and+Return031 Risk+and+Return031 Risk+and+Return031 Risk+and+Return031 Risk+and+Return031 Risk+and+Return031 Risk+and+Return031 Risk+and+R

1/16

Risk and Return

2

The value of an investment of $1 in

$100,000

$10,000Common Stock 14,276

$1,000logs

cale)

T-Bills

$100llars

241

71

$10

D

$1

819

019

119

219

319

419

519

619

719

819

920

0

Start of Year

20

3

The real value of an investment of $1 in

Real Returns

$1,000

581

$100scale

)

qu t es

Bonds

Bills

ars(log

$10Dol

9.85

$1

.

8

1900

1909

1919

1929

1939

1949

1959

1969

1979

1989

1999

Start of Year

200

4


2/16


3/16

n es ma on o rm

percentage points below the historical average

History is unlikely to be repeated.

Histor ma overstate the risk remium

9

Avera e market risk remiums (nominalreturn on stocks minus nominal return on bills), 1900-2008

11

789 Risk premium, %

6 04 6.296.94 7.13

7.94 8.34 8.48.74 9.1

9.6110.21

456

4.29 4.695.05 . .

. . .

012

Denmar

Belgiu

itzerlan

Irelan

Spai

Norwa

Canad

U.K.

therland

Average

U.S.

Swede

Australi

thAfric

German

France

Japa

Ital

S Ne

So

Reason 1: Risk premium varies by country

If the inflation rate is i, then the real risk premium is (rm-rf)/(1+i)

The real risk premium may be significantly lower than nominal

premium for Italy.10

Dividend yields in the U.S.A. 1900

Reason 2: Mutual funds, pension

9.00

10.00

funds, and other financial

institutions make it easier for

7.00

8.00

investors to reduce risk

5.00

6.00

endYield(%)

3.00

4.00Divid

1.00

2.00

0.00

1900

1905

1910

1915

1920

1925

1930

1935

1940

1945

1950

1955

1960

1965

1970

1975

1980

1985

1990

1995

2000

2005

11

How to measure risk?

risk premium demanded.


4/16

Rates of return for U.S. common stocks:

Stock Market Index Returns

-

60.0

80.0

40.0

eturn

20.0

ntageR

-20.0

0.0

Perce

-40.0

-60.0

Year

Source: Ibbotson Associates13

Histogram of annual stock marketre urns -

21

24

24

#of Years

1713

20

11 1112

1 24 3

24

0

to-40

to-30

to-20

to-10

10to0

0to10

0to20

0to30

0to40

0to50

0to60

-5 -4 -3 -2- 1 2 3 4 5

Return %

14

Price changes vs. normal distribution:IBM - Da y % c ange 1988-2008

3.5

4.0

3.0

2.0

.

days

1.0

1.5

rtiono

0.0

0.5

Prop

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

15

Standard deviation vs. expected return:nves men

18

20

14ity

8

10

robabil

4

6%

0

2

-50 0 50

% return

16


5/16


18

14

lity

8

10

robabi

4

6%

0

2

-50 0 50

% return

17


18

14

lity

8

10

robabi

4

6%

0

2

-50 0 50

% return

18

easur ng r s

Variance - Avera e value of s uared deviations from mean A.

measure of volatility.

an ar ev a on - verage va ue o square ev a ons rom

mean. A measure of volatility. Coin toss game - calculating variance and standard deviation

19

n v ua secur es

interest are the:

Expected return

Covariance and correlation (to another security or index)

20


6/16

Expected return, variance, andcovar ance

.

There is a 1/3 chance of each state of the economy,

and the only assets are a stock fund and a bond fund.

Rate of Return

Scenario Probabil i t Stock Fund Bond Fund

Recession 33.3% -7% 17%

Normal 33.3% 12% 7%

Boom 33.3% 28% -3%

21

xpec e re urn

Stock Fund Bond Fund

Rate of Squared Rate of Squared

Recession -7% 0.0324 17% 0.0100

Normal 12% 0.0001 7% 0.0000

Boom 28% 0.0289 -3% 0.0100

Expected return 11.00% 7.00%

ar ance . .

Standard Deviation 14.3% 8.2%

%)28(1%)12(1%)7(1)( SrE

%11)( SrE

22

ar ance



Scenario Return Deviation Return Deviation

Recession -7% 0.0324 17% 0.0100orma . .

Boom 28% 0.0289 -3% 0.0100

. .

Variance 0.0205 0.0067

. .

.

23

ar ance



cenar o Return Deviation Return Deviation

Recession -7% 0.0324 17% 0.0100orma . .

Boom 28% 0.0289 -3% 0.0100

. .

Variance 0.0205 0.0067


)0289.0001.0324(.0205.

24


7/16

an ar ev a on


e urn ev a on e urn ev a onRecession -7% 0.0324 17% 0.0100

. .

Boom 28% 0.0289 -3% 0.0100

. .

Variance 0.0205 0.0067


0205.0%3.14

25

The annual standard deviations andvar ances or t ree port o os, 1900-2008

Portfolio

deviation Variance

Treasury bills 2.8 7.7Government bonds 8.3 69.3

Common stocks 20.2 406.4

26

qu y mar e r s : y coun ryAverage Risk (1900-2008)

40

urns,

25

30

al

Ret

2 2 .9 9 2 3 .2 3 2 3 .4 2 2 3 .5 1 2 3 .9 8 2 4 .0 925.28

28.3229.57

33.93 34.3

15

20

ofAnn

17.0218.45 19.22

20.16. .

5

10

iation

a

nada

s

tralia

e

rland

U.S.

U.K.

mark

Spain

r

land

A

frica

r

eland

w

eden

lgium

ranc

o

rway

J

apan

Italy

many

rdDe

ASwit D

e

Neth

South I S B G

e

Stand

27

ow ones r sAnnualized Standard Deviation of the DJ IA over the recedin 52

70

weeks (1900 2008)

50

60

%)

40

iation(

30

ardDe

10Stan

0

Years

28


8/16

ompar ng re urns

29

ovar ance

Stock Bond

Scenario Deviation Devi ati on Product Wei ghted

Recession -18% 10% -0.0180 -0.0060

Normal 1% 0% 0.0000 0.0000

Boom 17% -10% -0.0170 -0.0057

Sum -0.0117

ovar ance - .

Deviation compares return in each state to the expected

return.

Weighted takes the product of the deviations multiplied by

t e pro a i ity o t at state.30

e re urn an r s or por o os


Recession -7% 0.0324 17% 0.0100

Normal 12% 0.0001 7% 0.0000

Boom 28% 0.0289 -3% 0.0100

Expected return 11.00% 7.00%

Variance 0.0205 0.0067


Note that stocks have a higher expected return than bonds and

-.portfolio that is 50% invested in bonds and 50% invested in

s oc s.31

or o os

Rate o Return

Scenario Stock fund Bond fund Portfolio squared deviation

Recession -7% 17% 5.0% 0.0016

Normal 12% 7% 9.5% 0.0000

Boom 28% -3% 12.5% 0.0012

Expected return 11.00% 7.00% 9.0%

. . .

Standard Deviation 14.31% 8.16% 3.08%

The rate of return on the portfolio is a weighted average of the

returns on the stocks and bonds in the portfolio:SSBBP rwrwr

32


9/16

or o os


Recession - . .

Normal 12% 7% 9.5% 0.0000

Boom 28% -3% 12.5% 0.0012


Variance 0.0205 0.0067 0.0010


The expectedrate of return on the portfolio is a weightedavera e of the ex ectedreturns on the securities in the

portfolio. )()()( SSBBP rEwrEwrE

33

or o os


Recession -7% 17% 5.0% 0.0016

Normal 12% 7% 9.5% 0.0000

Boom 28% -3% 12.5% 0.0012


ar ance . . .


The variance of the rate of return on the two risky assets

portfolio is 222 w2 www

whereBS is the correlation coefficient between the returns

on t e stoc an on un s.34

or o os


Recession -7% 17% 5.0% 0.0016

Normal 12% 7% 9.5% 0.0000

Boom 28% -3% 12.5% 0.0012


Variance 0.0205 0.0067 0.0010


Observe the decrease in risk that diversification offers.

An equally weighted portfolio (50% in stocks and 50% in

bonds) has less risk than either stocks or bonds held in

isolation.

35

orre a on

baCov

ba

998.00117.

..

36


10/16

or o o r s

covariance terms.

o ca cu a e por o o var anceadd up the boxes.

1

2

The variance of the return on

a portfolio with3

4

many securities is

more dependent

5

6

on the covariances betweenthe individual securities than

on the variances of

the individual securities

N

1 2 3 4 5 6 N

STOCK37

% in stocks Risk Return

e e c en se or wo asse s

0% 8.2% 7.0%

5% 7.0% 7.2%

10% 5.9% 7.4%12.0%

or o o s an e urn om na ons

. .

20% 3.7% 7.8%

25% 2.6% 8.0%

30% 1.4% 8.2%100%

9.0%

10.0%

11.0%

etur

35% 0.4% 8.4%

40% 0.9% 8.6%

45% 2.0% 8.8% 100%7.0%

8.0%

folio

. . .

55% 4.2% 9.2%

60% 5.3% 9.4%

65% 6.4% 9.6%

on s

5.0%

.

0.0% 5.0% 10.0% 15.0% 20.0%Por

70% 7.6% 9.8%

75% 8.7% 10.0%80% 9.8% 10.2% We can consider other portfolio

ort o o s stan ar ev at on

. .

90% 12.1% 10.6%

95% 13.2% 10.8%

100% 14.3% 11.0%

weights besides 50% in stocks and

50% in bonds.

38

% in stocks Risk Return

e e c en se or wo asse s

Portfolio Risk and ReturnCombinations0% 8.2% 7.0%

5% 7.0% 7.2%

10% 5.9% 7.4% 12.0%turn

. .

20% 3.7% 7.8%

25% 2.6% 8.0%30% 1.4% 8.2%

10.0%

11.0%

io

R 100%stocks

35% 0.4% 8.4%

40% 0.9% 8.6%

45% 2.0% 8.8% 7.0%

8.0%

.

ortfo

100%

. .

55% 4.2% 9.2%

60% 5.3% 9.4%

65% 6.4% 9.6%

5.0%

6.0%

0.0% 5.0% 10.0% 15.0% 20.0%

P

70% 7.6% 9.8%

75% 8.7% 10.0%

80% 9.8% 10.2%

85% 10.9% 10.4%

Portfolio Risk (standard deviation)

90% 12.1% 10.6%

95% 13.2% 10.8%

100% 14.3% 11.0%

than others. They have higher returns

for the same level of risk or less.

39

or o os w var ous corre a ons

retur 100%

stocks = -1.0

Relationship depends on = 1.0correlation coefficient

-1.0 < < +1.0100%

bonds

= 0.2

If = +1.0, no risk reduction is possible

. ,

As long as


11/16

or o o

orre a on oe c en .

Stocks % of Portfolio Avg Return

Campbell 15.8 60% 3.1%

Boeing 23.7 40% 9.5%

= = .

Standard Deviation = Portfolio = 14.6

Return = we g te avg = Port o o = .7%

41

10

ar ow z or o o eory

8

9

Boein

7

)

5

640% in Boeing

turn(%

3

4

ectedr

2Campbell Soup

E

xp

0. . . . . .

Standard deviation42

ar ow z or o o eory

standard deviation, below the level obtained from a

simple weighted average calculation.

.

The efficient ortfolios offer the hi hest

expected return for any level of risk.

43

Expected StandardEfficient Portfolios Percentages Allocated to Each Stock

Stock Return Deviation A B C D

Amazon.com 22.8 50.9 100 19.1 10.9

Ford 19.0 47.2 19.9 11.0

Dell 13.4 30.9 15.6 10.3

Starbucks 9.0 30.3 13.7 10.7 3.6

. . . .

Disney 7.7 19.6 8.8 11.2

Newmont 7.0 36.1 9.9 10.2

ExxonMobil 4.7 19.1 9.7 18.4

Johnson & Johnson 3.8 12.6 7.4 33.9

Campbell Soup 3.1 15.8 8.4 33.9Expected portfolio return 22.8 14.1 10.5 4.2

Portfolio standard deviation 50.9 22.0 16.0 8.8

44 Note: Standard deviations and the correlations between stock returns were estimated from monthly returns

January 2004-December 2008. Efficient portfolios are calculated assuming that short sales are prohibited.


12/16


13/16

c en ron erGoal is to move

Return up and left.

WHY?

B

NABN

A

RiskThe ratio of the risk remium to

fp rr RatioSharpethe standard deviation is called

49

p

easur ng r s

-

spreading the portfolio across many investments.

Unique Risk - Risk factors affecting only that firm., ,

unsystematic risk, or residual risk.

Market Risk - Economy-wide sources of risk that

.

undiversifiable risk or systematic risk.

50

s : ys ema c an unsys ema c

e can rea own e o a r s o o ng a s oc

into two components: systematic risk and

2

unsystematic risk:URR o a r s

becomes

where

Nonsystematic Risk: risksystematictheism

Systematic Risk: m

n51

Excess standard deviation against timean num er o s oc s

. ,

attains a large fraction of the total benefits of

vers cat on.

In the first two subsam les a ortfolio of20 stocksreduced annualized excess standard deviation to about five

, ,

excess standard deviation required almost 50 stocks.

The increase in idiosyncratic volatility over time has

increased the number of randoml selected stocks neededto achieve relatively complete portfolio

.52 Campbell et al. (2001) JF

E t d d d i ti i t ti


14/16

Excess standard deviation against timean num er o s oc s

The annualized excess standard deviation against

the number of stocks in the portfolio, for

sam le eriods 1963 to 1973 solid line , 1974

to 1985 (bottom dashed line), and 1986 to 1997

53 Campbell et al. (2001) JF

xcess s an ar ev a on aga ns me


xcess s an ar ev a on aga ns me

year, calculated from daily data during the year, of

equa y weig te port o ios containing 2, 5, 20, an 50

stocks over the standard deviation of an e uallweighted index.

e gure s ows a mo es ncrease n e excess

standard deviation of a typical 50-stock portfolio, but a

much more dramatic increase in the excess standard

deviation of a t ical 2-stock ortfolio from about 25percent in the early 1960s to a peak of 50 percent in the

ear y s.55 Campbell et al. (2001) JF

Average correlations among individuals oc s

56

Average R2 statistics of market model


15/16

Average R2 statistics of market modelor n v ua s oc s

57

Campbell, Lettau, Malkiel, and Xu (2001)

explanatory power of the market model for a typical

stock have declined, whereas the number of stocks

increased.


s w en o ng e mar e por o o

-

economy. In practice a broad stock market index,

such as the S&P Composite Index.

-

the market portfolio.A stocks contribution to

portfolio risk. Beta measures the responsiveness of a

. .,

systematic risk).

2

, Mi

i

59

Beta of Anchovy Queen restaurantc a n

456/6 =76

60


16/16

or o o r s

The return on Dell stock

chan es on avera e

by 1.41% for each

the market return. Beta

s t ere ore . .

61

s ma ng w regress on

Security Returns

Slope = i

Return on

R = + R +

62

or o o r s

63

ys ema c r s an e as

FFFRR kk ...2211

Where

is specific to a particular stock anduncorre ate w t t e term or ot er stoc s.

Use an index of stock market returnslike the

S&P500, as the single factor:

RR PSPS )( 500&500&

MM

64

Documents

031 Risk+and+Return031 Risk+and+Return031 Risk+and+Return031 Risk+and+Return031 Risk+and+Return031 Risk+and+Return031 Risk+and+Return031 Risk+and+Return031 Risk+and+Return031 Risk+and+Return031