Download Portfolio Theory and the Capital Asset Pricing Model

Portfolio Theory and the Capital
Asset Pricing Model
723g28
Linköpings Universitet, IEI
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We have learned from last chapter risk and
return: (that for an individual investor)
Combining stocks into portfolios can reduce
standard deviation, below the level obtained
from a simple weighted average calculation.
Rational investors maximize the expected return
given risks. Or minimize risks given expected
return.
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Markowitz Portfolio Theory
• Efficient portfolio provides the highest return
for a given level of risk, or least risk for a given
level of return. The market portfolio is the one
that has the highest Sharpe ratio with the
return and risk.
• The Sharpe ratio is a measure of risk premium
per unit of risk in an investment asset or a
trading strategy
Sharpe Ratio 
rp  rf
p
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Effect of diversification on variance
Assuming the following:
• N independent assets, i.i.d. with covariance=0,
• σ= std of the return
• r= expected return
• Equally weighted portfolio,
Then, we have: the more the assets are in, the
lower the standard deviation σ.
σ
σ portfolio =
𝑛
𝜇𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 = 𝑟
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Markowitz Portfolio Theory
Price changes vs. Normal distribution
IBM - Daily % change 1988-2008
Proportion of Days
4,0
3,5
3,0
2,5
2,0
1,5
1,0
0,5
0,0
-7 -6
-5 -4
-3 -2
-1
0
1
2
3
4
5
6
7
8
Daily % Change
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Markowitz Portfolio Theory
Standard Deviation VS. Expected Return
Investment A
20
18
% probability
16
14
12
10
8
6
4
2
0
-50
0
50
% return
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Markowitz Portfolio Theory
Standard Deviation VS. Expected Return
Investment B
20
18
% probability
16
14
12
10
8
6
4
2
0
-50
0
50
% return
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Markowitz Portfolio Theory
Standard Deviation VS. Expected Return
Investment C
20
18
% probability
16
14
12
10
8
6
4
2
0
-50
0
50
% return
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Markowitz Portfolio Theory
Expected Returns and Standard Deviations vary given different
weighted combinations of the stocks
10
9
Boeing
Expected Return (%)
8
7
6
40% in Boeing
5
4
3
Campbell Soup
2
1
0
0.00
5.00
10.00
15.00
20.00
25.00
Standard Deviation
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A two asset portfolio constructed with
% of both assets, allow short selling of
one assets
0.027
Capital Market Line
0.025
0.023
0.021
Series1
Market
0.019
0.017
Market
0.015
0.05
0.07
0.09
0.11
0.13
0.15
0.17
0.19
0.21
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Efficient Frontier
TABLE 8.1 Examples of efficient portfolios chosen from 10 stocks.
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.
Efficient Portfolios – Percentages Allocated to Each Stock
Stock
Expected Return
Standard Deviation
Amazon.com
22.8%
50.9%
Ford
19.0
Dell
A
100
B
C
D
19.1
10.9
47.2
19.9
11.0
13.4
30.9
15.6
10.3
Starbucks
9.0
30.3
13.7
10.7
Boeing
9.5
23.7
9.2
10.5
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
Soup
3.1
15.8
8.4
33.9
3.6
Expected portfolio return
22.8
14.1
10.5
4.2
Portfolio standard deviation
50.9
22.0
16.0
8.8
Try graph the efficient frontier and find the market portfolio with the highest Sharpe
Ratio!
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Efficient Frontier
4 Efficient Portfolios all from the same 10 stocks
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Efficient Frontier
Lending or Borrowing at the risk free rate (rf) allows us to exist outside the
efficient frontier.
Expected Return (%)
S
rf
Minimum variance portfolio
T
Standard Deviation
The red line is the Capital Market Line, where you can hold a combination of
the risk free assets and the market portfolio and get any returns you like.
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Efficient Frontier
Another Example
Stocks

ABC Corp
28
Big Corp
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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Efficient Frontier
Return
B
A
Risk
(measured as
)
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Efficient Frontier
Return
B
AB
A
Risk
16
Efficient Frontier
Return
B
AB
N
A
Risk
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Efficient Frontier
Return
B
ABN
AB
N
A
Risk
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Efficient Frontier
Goal is to move up and
left.
Return
WHY?
B
ABN
AB
N
A
Risk
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Efficient Frontier
The ratio of the risk premium to the standard
deviation is the Sharpe ratio.
In a competitive market, the expected risk
premium varies in proportion to portfolio standard
deviation. P denotes portfolio. Along the Capital
Market Line one holds the risky assets and a risk
free loan.
Sharpe Ratio 
rp  rf
rp  rf
p
p

rm  rf
m
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Capital Asset Pricing Model
ri  rf  i (rm  rf )
 im
i  2
 m
CAPM
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Security Market Line
Stock Return
ri
.
r
Market Return = m
Market Portfolio
Risk Free Return =
rf
(Treasury bills)
1.0
2,0
BETA
risk
𝑟𝑖 = 2 𝑟𝑚 − 𝑟𝑓
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Efficient Frontier
Return
Low Risk
High Risk
High Return
High Return
Low Risk
High Risk
Low Return
Low Return
Risk
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Capital Market Line
Return
Tangent portfolio
Market Return = rm
.
Market Portfolio
Risk Free Return =
(Treasury bills)
rf
Risk
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Security Market Line
Return
.
r
Market Return = m
Market Portfolio
Risk Free Return =
rf
(Treasury bills)
1.0
BETA
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Market Risk Premium: Example
14
Example:
Let,
rf  4%
rm  12%
Market Risk Premium = 8%
Expected Return (%)
12
market risk premium  8%
10
Market Portfolio
(market return = 12%)
8
6
4
rf  4%
2
0
0
0,2
0,6
0,4
0,8
Beta
According to CAPM, the expected return on the asset is
r  rf    (rm  rf )  4%  1.2  (8%)  13.6%
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Security Market Line: depicts the
Return
CAPM
SML
Security Market Line
rf
1.0
BETA
SML Equation = rf + β( rm - rf )
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Expected Returns
These estimates of the returns expected by investors in
February 2009 were based on the capital asset pricing model.
We assumed 0.2% for the interest rate r f and 7 % for the
expected risk premium r m − r f .
Stock
Beta (β)
Amazon
Ford
Dell
Starbucks
Boeing
Disney
Newmont
ExxonMobil
Johnson & Johnson
Soup
2.16
1.75
1.41
1.16
1.14
.96
.63
.55
.50
.30
Expected Return
[rf + β(rm – rf)]
15.4
12.6
10.2
8.4
8.3
7.0
4.7
4.2
3.8
2.4
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SML Equilibrium
• In equilibrium no stock can lie below the security market line. For
example, instead of buying stock A, investors would prefer to lend
part of their money and put the balance in the market portfolio. And
instead of buying stock B, they would prefer to borrow and invest in
the market portfolio. (lend=save, borrow is leveraging.) risk free
assets and the market portfolio can span the whole Security market
line)
Higher risk
lower return
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Testing the CAPM
Beta vs. Average Risk Premium: low beta
portfolio fared better than high beta
Average Risk Premium
portfolio 1931-2008
1931-2008
20
SML
Investors
12
Market
Portfolio
0
1.0
Portfolio Beta
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Testing the CAPM
Beta vs. Average Risk Premium
Average Risk Premium
1966-2008
12
8
Investors
SML
4
Market
Portfolio
0
1.0
Portfolio Beta
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Testing the CAPM: Return vs. Book-toMarket
Dollars
(log scale)
Cumulated difference of Small minus big firm stocks
Cumulated difference of High minus low book-to-market firm stocks
100
High-minus low book-to-market
2008
10
2006
1996
1986
1976
1966
1956
1946
1936
1
1926
Small minus big
0,1
http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html
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