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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
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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.
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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
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Standard deviation vs. expected return:nves men
18
14
lity
8
10
robabi
4
6%
0
2
-50 0 50
% return
17
Standard deviation vs. expected return:nves men
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
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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
Stock Fund Bond Fund
Rate of Squared Rate of Squared
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
Stock Fund Bond Fund
Rate of Squared Rate of Squared
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
Standard Deviation 14.3% 8.2%
)0289.0001.0324(.0205.
24
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an ar ev a on
Rate of Squared Rate of Squared
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
Standard Deviation 14.3% 8.2%
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
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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
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%
Variance 0.0205 0.0067
Standard Deviation 14.3% 8.2%
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
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or o os
Scenario Stock fund Bond fund Portfolio squared deviation
Recession - . .
Normal 12% 7% 9.5% 0.0000
Boom 28% -3% 12.5% 0.0012
Expected return 11.00% 7.00% 9.0%
Variance 0.0205 0.0067 0.0010
Standard Deviation 14.31% 8.16% 3.08%
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
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%
ar ance . . .
Standard Deviation 14.31% 8.16% 3.08%
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
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%
Variance 0.0205 0.0067 0.0010
Standard Deviation 14.31% 8.16% 3.08%
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
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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
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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.
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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
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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
54 Campbell et al. (2001) JF
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
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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.
58 Campbell et al. (2001) JF
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
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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