Transcript Chapter 7

Introduction to Risk and Return and
opportunity cost of capital
Chapter 11
BBM
08/29/12
1
Risk and Return
Risk and Return are related.
How?
We now focus on risk and return and their relationship
to the opportunity cost of capital.
11-2
Equity Rates of Return:
A Review
Percentage Return =
Capital Gain + Dividend
Initial Share Price
Dividend Yield =
Dividend
Initial Share Price
Capital Gain Yield =
11-3
Capital Gain
Initial Share Price
Real Rates of Return
Recall the relationship between real rates and nominal rates:
1  real rate of return =
1 + nominal rate of return
1 + inflation rate
Example: Suppose inflation from December 2009 to December 2010 was
1.5%. What was GE stock’s real rate of return, if its nominal rate of return was
23.93%?
11-4
Total Returns for Different Asset
Classes
The Value of an Investment of $1 in 1900
11-5
What Drives the Difference in Total
Returns?
Maturity Premium: Extra average return from investing in
longer term Treasury securities.
Risk Premium: Expected return in excess of risk-free return
as compensation for risk.
11-6
Risk Premium: Example
Expected Market Return =
11-7
Interest Rate on
Treasury Bills
+
Normal Risk
Premium
1981: 21.4%
=
14%
+
7.4%
2008:
=
2.2%
+
7.4%
9.6%
Returns and Risk
How are the expected returns and
the risk of a security related?
11-8
Measuring Risk
What is risk?
How can it be measured?
Variance: Average value of squared deviations from
mean. A measure of volatility.
Standard Deviation: Square root of variance.
measure of volatility.
11-9
Also a
Market Indexes
Dow Jones Industrial Average (The Dow)
Value of a portfolio holding one share in each of 30 large industrial firms.
Standard & Poor’s Composite Index (The S&P 500)
Value of a portfolio holding shares in 500 firms. Holdings are proportional to the
number of shares in the issues.
OMX SPI index
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OMX Stockholm PI
• OMX nordiska börs använder en gemensam indelning och
uppbyggnad av index för de nordiska marknaderna. En enhetlig
indexstandard ökar förståelsen för de nordiska indexen och
underlättar jämförelser mellan olika index.
• OMX Stockholm 30 (OMXS30) – PI
OMX Stockholm 30 är OMX Nordiska Börs Stockholms ledande
aktieindex. Indexet består av de 30 mest aktivt handlade aktierna
på den Nordiska Börsen i Stockholm.
• OMX Stockholm All-Share (OMXS) – PI
OMX Nordiska Börs Stockholms All-Share-index innefattar alla aktier
som är noterade på den Nordiska Börsen i Stockholm.
Basdatum för All-Share-index på OMX Nordiska Börs Stockholm är
den 31 december 1995, med basvärdet 100.
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Average Market Risk Premium (by country)
Risk premium, %
Italy
Japan
France
Germany
South Africa
Australia
9,61 10,21
9,1
8,74
7,94 8,34 8,4
Sweden
U.S.
6,94 7,13
Average
Netherlands
U.K.
Canada
Norway
Spain
Ireland
Switzerland
Belgium
6,04 6,29
5,05 5,43 5,5 5,61 5,67
4,69
4,29
Denmark
11
10
9
8
7
6
5
4
3
2
1
0
Country
Market risk premium = Market rate of return –
risk-free rate
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2005
2000
1995
1990
1985
1980
1975
1970
1965
1960
1955
1950
1945
1940
1935
1930
1925
1920
1915
1910
1905
1900
Dividend Yield (%)
Dividend Yield
Dividend yields in the U.S.A. 1900–2008
10.00
9.00
8.00
7.00
6.00
5.00
4.00
3.00
2.00
1.00
0.00
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Rates of Return 1900-2008
Stock Market Index Returns
80.0
Percentage Return
60.0
40.0
20.0
0.0
-20.0
-40.0
-60.0
Source: Ibbotson Associates
Year
14
Measuring Risk
Histogram of Annual Stock Market Returns
Ex. There has
been 24 years
with 20 to 30
% stock return.
(1900-2008)
# of Years
24
24
21
20
17
16
11 11
12
8
1
2
-40 to -30
3
Probability
distribution of
stock returns
2
50 to 60
40 to 50
30 to 40
20 to 30
10 to 20
0 to 10
-10 to 0
-20 to -10
Return %
-30 to -20
0
4
-50 to -40
4
13
15
Thinking About Risk
• Message 1
– Some Risks Look Big and Dangerous but Really Are
Diversifiable
• Message 2
– Market Risks Are Macro Risks
• Message 3
– Risk Can Be Measured
Historical Risk
(1900-2010)
11-17
Risk and Diversification
Diversification
Strategy designed to reduce risk by spreading a 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.”
11-18
Measuring Risk
Variance - Average value of squared deviations
from mean. A measure of volatility.
Standard Deviation (STD) – square root of variance.
A measure of volatility.
 ~
- 
 ~ 
Var  rm  = E  rm - r 






 ~ 
STD = Var  rm 


2
Expected value: average value with equal weight.
~
_
Mean    E ri  r
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Measuring Risk
Example from page: 320. Coin Toss Game-calculating
variance and standard deviation
(1)
(2)
(3)
Percent Rate of Return Deviation from Mean Squared Deviation
+ 40
+ 30
900
+ 10
0
0
+ 10
0
0
- 20
- 30
900
Variance = average of squared deviations = 1800 / 4 = 450
Standard deviation = square of root variance =
450 = 21.2%
Measuring Risk
You start with 100 kr. Toss two coins at a time. Head up you
gain 20%, tails up you lose 10%. There are all together 4
outcomes. (HH) (HT) (TH) (TT). Coin Toss Game-calculating
variance and standard deviation
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Measuring Risk
(
(
)(
)(
Portfolio rate
fraction of portfolio
=
x
of return
in first asset
rate of return
on first asset
)
)
fraction of portfolio
rate of return
+
x
in second asset
on second asset
Expected return is just a weighted
average of individual stock returns.
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Dow Jones Risk
Annualized Standard Deviation of the DJIA over the preceding 52 weeks
(1900 – 2008)
70
Standard Deviation (%)
60
50
40
30
20
10
0
Years
23
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.”
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Market Index
Obs: step 1, get the Mean 8,5%, step 2, get the deviations from
the mean and step 3, square it to get the variance. Since we have
6 years’ observations, divide it with 6.
Risk and Diversification
11-26
Portfolio Risk
The variance of a two stock portfolio is the
sum of these four boxes
Stock1
Stock1
Stock 2
x12 σ12
x1x 2 σ12 
x1x 2ρ12 σ1σ 2
Stock 2
x1x 2 σ12 
x1x 2ρ12 σ1σ 2
x 22 σ 22
PortfolioVariance  x12σ 12  x 22σ 22  2(x1x 2ρ 12σ 1σ 2 )
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Portfolio Risk
Example
Suppose you invest 60% of your portfolio in
Campbell Soup and 40% in Boeing. The expected
dollar return on your Campbell Soup stock is 3.1%
and on Boeing is 9.5%. The expected return on
your portfolio is:
ExpectedReturn  (.60 3.1)  (.40 9.5)  5.7%
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Portfolio Risk
Example
Suppose you invest 60% of your portfolio in Campbell Soup and
40% in Boeing. The expected dollar return on your Campbell
Soup stock is 3.1% and on Boeing is 9.5%. The standard
deviation of their annualized daily returns are 15.8% and
23.7%, respectively. Assume a correlation coefficient of 1.0
and calculate the portfolio variance.
CampbellSoup
CampbellSoup
Boeing
x12 σ12  (.60) 2  (15.8) 2
x1x 2ρ12 σ1σ 2  .40 .60
 1 15.8  23.7
Boeing
x1x 2ρ12 σ1σ 2  .40 .60
 1 15.8  23.7
x 22 σ 22  (.40) 2  (23.7) 2
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Do stock prices move together?
What effect does diversification have on a
portfolio’s total risk, unique risk and market risk?
11-30
Portfolio Risk
Example
Suppose you invest 60% of your portfolio in Campbell Soup and 40% in
Boeing. The expected dollar return on your Campbell Soup stock is
3.1% and on Boeing is 9.5%. The standard deviation of their
annualized daily returns are 15.8% and 23.7%, respectively. Assume a
correlation coefficient of 1.0 and calculate the portfolio variance.
PortfolioVariance [(.60)2 x(15.8)2 ]
 [(.40)2 x(23.7)2 ]
 2(.40x.60x
15.8x23.7) 359.5
Standard Deviation 359.5  19.0%
Obs: Since the correlation coefficient is 1, there is no portfolio risk
reduction at all! The average standard deviation for the two stocks
is the same 18,96%=15,8*0,6+23,7*0,4
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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 expected return on your
portfolio is:
ExpectedReturn  (.60  10)  (.40  15)  12%
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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.
PortfolioVariance  [(.60)2 x(18.2)2 ]
 [(.40)2 x(27.3)2 ]
 2(.40x.60x
18.2x27.3) 477.0
Standard Deviation 477.0  21.8%
Again the correlation coefficient= 1, no gain on diversification! But it
sure lowered the risk and the return by averaging.
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Portfolio Risk
ExpectedPortfolioReturn (x1 r1 )  (x 2 r2 )
PortfolioVariance  x12σ 12  x 22σ 22  2(x1x 2ρ 12σ 1σ 2 )
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Portfolio Risk
Example
Stocks
ABC Corp
Big Corp
Correlation Coefficient = .4
s%
% of Portfolio
28%
60%
42%
40%
Avg Return
15%
21%
Standard Deviation = weighted avg. = 33.6 (this is an average of the std)
Standard Deviation = Portfolio = 28.1
Real Standard Deviation:
Portfolio Variance = (282)(.62) + (422)(.42) + 2(.4)(.6)(28)(42)(.4)
STD=sqrt (Variance) = 28.1 CORRECT
Mean: r = (15%)(.60) + (21%)(.4) = 17.4%
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Portfolio Risk
Example
Stocks
ABC Corp
Big Corp
Correlation Coefficient = .4
s
% of Portfolio
28%
60%
42%
40%
Avg Return
15%
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
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Portfolio Risk
Example
Stocks
Portfolio
New Corp
Correlation Coefficient = .3
s
% of Portfolio
28.1
50%
30
50%
Avg Return
17.4%
19%
NEW Standard Deviation = weighted avg. std= 31.80
NEW Standard Deviation = Portfolio = 23.43
NEW Mean = weighted avg = Portfolio = 18.20%
NOTE: Higher return & Lower risk
How did we do that? DIVERSIFICATION
37
The Variance Covariance Matrix
The shaded boxes contain variance terms; the remainder
contain covariance terms. Adding them up you get the portfolio
variance.
1
2
3
STOCK
To calculate
portfolio variance
add up the boxes
4
5
6
N
1
2
3
4
5 6
STOCK
N
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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.
39
The Security Market Line
The return on Dell
stock changes on
average by 1.41%
for each additional
1% change in
the market return.
Beta = 1.41.
E (r )  rf 
E (rM )  r f
sm
s im
 r f   i E (rM )  r f 
The model is the famous CAPM: Capital asset pricing model, you get the r--required rate of return or cost of capital from this!
40
Portfolio Risk
s im
i  2
sm
σim = Covariance with the market
σm^ 2 = Variance of the market
If beta is 1,5, market
volatility is 20%,
Then, Covariance of the
portfolio with the
market return is
=(20%)^2*1,5=0,06
41
Portfolio Risk
s im
Bi  2
sm
Covariance with the
market
Variance of the market
42
Portfolio Risk
The middle line shows that a well
Diversified portfolio of randomly
selected stocks ends up with β= 1
and a standard deviation equal to
the market’s—in this case 20%.
The upper line shows that a
well-diversified portfolio with β=
1.5 has a standard deviation of
about 30%, i.e. 1.5 times that of
the market.
The lower line shows that a
well-diversified portfolio with β =
.5 has a standard deviation of
about 10%—half that of the
market.
43
Beta
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 = σ im/σm2)
(1)
Month
1
2
3
4
5
6
Average
(2)
(3)
(4)
(5)
(6)
(7)
Product of
Deviation
Squared
deviations
Deviation
from average deviation
from average
Market Anchovy Q from average Anchovy Q from average returns
return
return
market return return
market return (cols 4 x 5)
-8
-11
-10
-13
100
130
4
8
2
6
4
12
12
19
10
17
100
170
-6
-13
-8
-15
64
120
2
3
0
1
0
0
8
6
6
4
36
24
2
2
Total
304
456
2
Variance = σm = 304/6 = 50.67
Check the figure
see if it is right!
Find out how to
use excel to
calculate variance
and mean of a
portfolio.
Covariance = σim = 456/6 = 76
Beta (β) = σim/σm2 = 76/50.67 = 1.5
44