Transcript Standard Deviation - Amazon Web Services

Chapter 7
Principles of
Corporate Finance
Tenth Edition
Introduction to
Risk and Return
Slides by
Matthew Will
McGraw-Hill/Irwin
Copyright © 2011 by the McGraw-Hill Companies, Inc. All rights reserved.
Topics Covered
Over a Century of Capital Market History
Measuring Portfolio Risk
Calculating Portfolio Risk
How Individual Securities Affect Portfolio
Risk
Diversification & Value Additivity
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The Value of an Investment of $1 in 1900
$100,000
Common Stock
14,276
US Govt Bonds
T-Bills
$1,000
241
71
$100
$10
Start of Year
2008
$1
19
00
19
10
19
20
19
30
19
40
19
50
19
60
19
70
19
80
19
90
20
00
Dollars (log scale)
$10,000
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The Value of an Investment of $1 in 1900
Real Returns
$1,000
581
Equities
$100
Bills
$10
9.85
2.87
Start of Year
2008
$1
19
00
19
09
19
19
19
29
19
39
19
49
19
59
19
69
19
79
19
89
19
99
Dollars (log scale)
Bonds
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Average Market Risk Premia (by country)
Risk premium, %
Country
Italy
Japan
France
Germany
South Africa
Australia
9.61 10.21
8.74 9.1
8.34
8.4
7.94
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
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
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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
Percentage Return
80.0
60.0
40.0
20.0
0.0
-20.0
-40.0
-60.0
Source: Ibbotson Associates
Year
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Measuring Risk
Histogram of Annual Stock Market Returns
(1900-2008)
# of Years
24
24
21
20
17
16
11 11
12
8
1
2
-40 to -30
3
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
Measuring Risk
Variance - Average value of squared deviations
from mean. A measure of volatility.
Standard Deviation - Average value of squared
deviations from mean. A measure of volatility.
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Measuring Risk
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
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Average Risk (1900-2008)
40
35
30
25
20
15
Germany
Italy
Japan
28.32 29.57
Norway
France
Belgium
Sweden
Ireland
South Africa
Netherlands
Spain
Denmark
U.K.
U.S.
Switzerland
17.02
23.98 24.09 25.28
21.83 22.05 22.99 23.23 23.42 23.51
20.16
18.45 19.22
Australia
10
5
0
33.93 34.3
Canada
Standard Deviation of Annual Returns, %
Equity Market Risk (by country)
Dow Jones Risk
Annualized Standard Deviation of the DJIA over the preceding 52 weeks
Standard Deviation (%)
(1900 – 2008)
70
60
50
40
30
20
10
0
Years
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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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Comparing Returns
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Portfolio standard deviation
Measuring Risk
0
5
10
Number of Securities
15
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Portfolio standard deviation
Measuring Risk
Unique
risk
Market risk
0
5
10
Number of Securities
15
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Portfolio Risk
The variance of a two stock portfolio is the sum of these
four boxes
Stock1
Stock1
Stock 2
x 12σ 12
x 1x 2σ 12 
x 1x 2ρ 12σ 1σ 2
Stock 2
x 1x 2σ 12 
x 1x 2ρ 12σ 1σ 2
x 22σ 22
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
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%
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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.
Exxon- Mobil
Exxon- Mobil x12σ12  (.60) 2  (18.2) 2
Coca - Cola
x1x 2ρ12σ1σ 2  .40  .60
 1  18.2  27.3
Coca - Cola
x1x 2ρ12σ1σ 2  .40  .60
 1  18.2  27.3
x 22σ 22  (.40) 2  ( 27.3) 2
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%
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Portfolio Risk
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ExpectedPortfolioReturn (x1 r1 )  (x 2 r2 )
PortfolioVariance  x12σ 12  x 22σ 22  2(x1x 2ρ 12σ 1σ 2 )
Portfolio Risk
Example
Stocks
ABC Corp
Big Corp
s
28
42
Correlation Coefficient = .4
% of Portfolio
Avg Return
60%
15%
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
Stocks
ABC Corp
Big Corp
s
28
42
Correlation Coefficient = .4
% of Portfolio
Avg Return
60%
15%
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
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Portfolio Risk
Example
Stocks
Portfolio
New Corp
Correlation Coefficient = .3
s
% of Portfolio
28.1
50%
30
50%
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
Avg Return
17.4%
19%
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Portfolio Risk
The shaded boxes contain variance terms; the remainder
contain covariance terms.
1
2
3
STOCK
To calculate
portfolio
variance add
up the boxes
4
5
6
N
1
2
3
4
5 6
STOCK
N
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.
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Portfolio Risk
The return on Dell stock
changes on average
by 1.41% for each
additional 1% change in
the market return. Beta
is therefore 1.41.
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Portfolio Risk
The middle line shows that a well
Diversified portfolio of randomly
selected stocks ends up with 1
and a standard deviation equal to
the market’s—in this case 20%.
The upper line shows that a
well-diversified portfolio with 1.5
has a standard deviation of about
30%—1.5 times that of the market.
The lower line shows that a
well-diversified portfolio with .5
has a standard deviation of about
10%—half that of the market.
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Portfolio Risk
s im
Bi  2
sm
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Portfolio Risk
s im
Bi  2
sm
Covariance with the
market
Variance of the market
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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)
Deviation
Market
Anchovy Q from average
return
return
market return
-8%
-11%
-10%
4
8
2
12
19
10
-6
-13
-8
2
3
0
8
6
6
2
2
Variance = σm 2
(5)
(6)
Deviation
Squared
from average deviation
Anchovy Q from average
return
market return
-13%
100
6
4
17
100
-15
64
1
0
4
36
Total
304
= 304/6 = 50.67
Covariance = σim = 736/6 = 76
Beta (β) = σim /σm 2 = 76/50.67 = 1.5
(7)
Product of
deviations
from average
returns
(cols 4 x 5)
130
12
170
120
0
24
456
Web Resources
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