MPT with Emperical testing with theories and literature

MPT with Emperical testing with theories and literature

1. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-1
Evolution of Modern Portfolio Theory

2. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-2
Evolution of Modern Portfolio Theory
–Efficient Frontier
–Single Index Model
–Capital Asset Pricing Model (CAPM)
–Arbitrage Pricing Theory (APT)

3. McGraw-Hill/Irwin
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10-3
Evolution of Modern Portfolio Theory
•Efficient Frontier
–Markowitz, H. M., “Portfolio Selection,” Journal of
Finance (December 1952).
•Rather than choose each security individually,
choose portfolios that maximize return for given
levels of risk (i.e., those that lie on the efficient
frontier). Problem: When managing large numbers
of securities, the number of statistical inputs
required to use the model is tremendous. The
correlation or covariance between every pair of
securities must be evaluated in order to estimate
portfolio risk.

4. McGraw-Hill/Irwin
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10-4
Evolution of Modern Portfolio Theory
(Continued)
•Single Index Model
–Sharpe, W. F., “A Simplified Model of Portfolio
Analysis,” Management Science (January 1963).
•Substantially reduced the number of required
inputs when estimating portfolio risk. Instead of
estimating the correlation between every pair of
securities, simply correlate each security with an
index of all of the securities included in the
analysis.

5. McGraw-Hill/Irwin
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10-5
Evolution of Modern Portfolio Theory
(Continued)
•Capital Asset Pricing Model (CAPM)
–Sharpe, W. F., “Capital Asset Prices: A Theory of Market
Equilibrium Under Conditions of Risk,” Journal of Finance
(September 1964).
•Instead of correlating each security with an index of
all securities included in the analysis, correlate each
security with the efficient market value weighted
portfolio of all risky securities in the universe (i.e.,
the market portfolio). Also, allow investors the option
of investing in a risk-free asset.

6. McGraw-Hill/Irwin
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10-6
Evolution of Modern Portfolio Theory
(Continued)
•Arbitrage Pricing Theory (APT)
–Ross, S. A., “The Arbitrage Theory of Capital Asset
Pricing,” Journal of Economic Theory (December 1976).
•Instead of correlating each security with only the
market portfolio (one factor), correlate each security
with an appropriate series of factors (e.g., inflation,
industrial production, interest rates, etc.).

7. McGraw-Hill/Irwin
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10-7
Markowitz’s Contribution
•Harry Markowitz’s “Portfolio Selection” Journal of
Finance article (1952) set the stage for modern
portfolio theory
–The first major publication indicating the important of
security return correlation in the construction of stock
portfolios
–Markowitz showed that for a given level of expected
return and for a given security universe, knowledge of the
covariance and correlation matrices are required
7

8. McGraw-Hill/Irwin
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10-8
Harry Markowitz Model
•Harry Max Markowitz (born August 24,
1927) is an American economist.
•He is best known for his pioneering work
in Modern Portfolio Theory.
•Harry Markowitz put forward this model
in 1952.
• Studied the effects of asset risk, return,
correlation and diversification on probable
investment portfolio returns
Harry
Markowitz
Essence of Markowitz Model
1.An investor has a certain amount of capital he wants to invest over a
single time horizon.
2.He can choose between different investment instruments, like stocks,
bonds, options, currency, or portfolio. The investment decision depends
on the future risk and return.
3.The decision also depends on if he or she wants to either maximize the
yield or minimize the risk

9. McGraw-Hill/Irwin
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10-9
Essence of Markowitz Model
1.Markowitz model assists in the selection of the most efficient by
analysing various possible portfolios of the given securities.
2.By choosing securities that do not 'move' exactly together, the HM
model shows investors how to reduce their risk.
3.The HM model is also called Mean-Variance Model due to the fact that
it is based on expected returns (mean) and the standard deviation
(variance) of the various portfolios.
 
Diversification and Portfolio Risk
P
o
r
t
f
o
l
i
o
R
i
s
k
Number of Shares
5101
5
20
Total
Risk
S
R
US
R
p 
p
deviation standard the
SR: Systematic Risk
USR: Unsystematic
Risk

10. McGraw-Hill/Irwin
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10-10
Assumptions
•An investor has a certain amount of capital he wants to
invest over a single time horizon.
 
•He can choose between different investment instruments,
like stocks, bonds, options, currency, or portfolio.
 
•The investment decision depends on the future risk and
return.
 
•The decision also depends on if he or she wants to either
maximize the yield or minimize the risk.
•The investor is only willing to accept a higher risk if he or
she gets a higher expected return.

11. McGraw-Hill/Irwin
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10-11
10.1 Individual Securities
•The characteristics of individual securities that are of interest
are the:
–Expected Return
–Variance and Standard Deviation
–Covariance and Correlation

12. McGraw-Hill/Irwin
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10-12
10.2 Expected Return, Variance, and Covariance
Consider the following two risky asset world. 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
ScenarioProbabilityStock fundBond fund
Recession33.3% -7% 17%
Normal 33.3% 12% 7%
Boom 33.3% 28% -3%

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10-13
10.2 Expected Return, Variance, and Covariance
Stock fund Bond Fund
Rate of Squared Rate of Squared
Scenario Return Deviation Return Deviation
Recession -7% 3.24% 17% 1.00%
Normal 12% 0.01% 7% 0.00%
Boom 28% 2.89% -3% 1.00%
Expected return 11.00% 7.00%
Variance 0.0205 0.0067
Standard Deviation 14.3% 8.2%

14. McGraw-Hill/Irwin
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10-14
Stock fund Bond Fund
Rate of Squared Rate of Squared
Scenario Return Deviation Return Deviation
Recession -7% 3.24% 17% 1.00%
Normal 12% 0.01% 7% 0.00%
Boom 28% 2.89% -3% 1.00%
Expected return 11.00% 7.00%
Variance 0.0205 0.0067
Standard Deviation 14.3% 8.2%
10.2 Expected Return, Variance, and Covariance
%11)(
%)28(
3
1
%)12(
3
1
%)7(
3
1
)(


S
S
rE
rE

15. McGraw-Hill/Irwin
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10-15
Stock fund Bond Fund
Rate of Squared Rate of Squared
Scenario Return Deviation Return Deviation
Recession -7% 3.24% 17% 1.00%
Normal 12% 0.01% 7% 0.00%
Boom 28% 2.89% -3% 1.00%
Expected return 11.00% 7.00%
Variance 0.0205 0.0067
Standard Deviation 14.3% 8.2%
10.2 Expected Return, Variance, and Covariance
%7)(
%)3(
3
1
%)7(
3
1
%)17(
3
1
)(


B
B
rE
rE

16. McGraw-Hill/Irwin
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10-16
Stock fund Bond Fund
Rate of Squared Rate of Squared
Scenario Return Deviation Return Deviation
Recession -7% 3.24% 17% 1.00%
Normal 12% 0.01% 7% 0.00%
Boom 28% 2.89% -3% 1.00%
Expected return 11.00% 7.00%
Variance 0.0205 0.0067
Standard Deviation 14.3% 8.2%
10.2 Expected Return, Variance, and Covariance
%24.3%)7%11(
2


17. McGraw-Hill/Irwin
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10-17
Stock fund Bond Fund
Rate of Squared Rate of Squared
Scenario Return Deviation Return Deviation
Recession -7% 3.24% 17% 1.00%
Normal 12% 0.01% 7% 0.00%
Boom 28% 2.89% -3% 1.00%
Expected return 11.00% 7.00%
Variance 0.0205 0.0067
Standard Deviation 14.3% 8.2%
10.2 Expected Return, Variance, and Covariance
%01.%)12%11(
2


18. McGraw-Hill/Irwin
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10-18
Stock fund Bond Fund
Rate of Squared Rate of Squared
Scenario Return Deviation Return Deviation
Recession -7% 3.24% 17% 1.00%
Normal 12% 0.01% 7% 0.00%
Boom 28% 2.89% -3% 1.00%
Expected return 11.00% 7.00%
Variance 0.0205 0.0067
Standard Deviation 14.3% 8.2%
10.2 Expected Return, Variance, and Covariance
%89.2%)28%11(
2


19. McGraw-Hill/Irwin
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10-19
Stock fund Bond Fund
Rate of Squared Rate of Squared
Scenario Return Deviation Return Deviation
Recession -7% 3.24% 17% 1.00%
Normal 12% 0.01% 7% 0.00%
Boom 28% 2.89% -3% 1.00%
Expected return 11.00% 7.00%
Variance 0.0205 0.0067
Standard Deviation 14.3% 8.2%
10.2 Expected Return, Variance, and Covariance
%)89.2%01.0%24.3(
3
1
%05.2 

20. McGraw-Hill/Irwin
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10-20
Stock fund Bond Fund
Rate of Squared Rate of Squared
Scenario Return Deviation Return Deviation
Recession -7% 3.24% 17% 1.00%
Normal 12% 0.01% 7% 0.00%
Boom 28% 2.89% -3% 1.00%
Expected return 11.00% 7.00%
Variance 0.0205 0.0067
Standard Deviation 14.3% 8.2%
10.2 Expected Return, Variance, and Covariance
0205.0%3.14

21. McGraw-Hill/Irwin
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10-21
Stock fund Bond Fund
Rate of Squared Rate of Squared
Scenario Return Deviation Return Deviation
Recession -7% 3.24% 17% 1.00%
Normal 12% 0.01% 7% 0.00%
Boom 28% 2.89% -3% 1.00%
Expected return 11.00% 7.00%
Variance 0.0205 0.0067
Standard Deviation 14.3% 8.2%
10.3 The Return and Risk for Portfolios
Note that stocks have a higher expected return than bonds
and higher risk. Let us turn now to the risk-return tradeoff
of a portfolio that is 50% invested in bonds and 50%
invested in stocks.

22. McGraw-Hill/Irwin
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10-22
10.3 The Return and Risk for Portfolios
Rate of Return
Scenario Stock fundBond fundPortfoliosquared deviation
Recession -7% 17% 5.0% 0.160%
Normal 12% 7% 9.5% 0.003%
Boom 28% -3% 12.5% 0.123%
Expected return 11.00% 7.00% 9.0%
Variance 0.0205 0.00670.0010
Standard Deviation14.31% 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 
%)17(%50%)7(%50%5 

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10-23
Rate of Return
Scenario Stock fundBond fundPortfoliosquared deviation
Recession -7% 17% 5.0% 0.160%
Normal 12% 7% 9.5% 0.003%
Boom 28% -3% 12.5% 0.123%
Expected return 11.00% 7.00% 9.0%
Variance 0.0205 0.00670.0010
Standard Deviation14.31% 8.16% 3.08%
10.3 The Return and Risk for Portfolios
The rate of return on the portfolio is a weighted average of
the returns on the stocks and bonds in the portfolio:
%)7(%50%)12(%50%5.9 
SSBBP
rwrwr 

24. McGraw-Hill/Irwin
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10-24
Rate of Return
Scenario Stock fundBond fundPortfoliosquared deviation
Recession -7% 17% 5.0% 0.160%
Normal 12% 7% 9.5% 0.003%
Boom 28% -3% 12.5% 0.123%
Expected return 11.00% 7.00% 9.0%
Variance 0.0205 0.00670.0010
Standard Deviation14.31% 8.16% 3.08%
10.3 The Return and Risk for Portfolios
The rate of return on the portfolio is a weighted average of
the returns on the stocks and bonds in the portfolio:
%)3(%50%)28(%50%5.12 
SSBBP
rwrwr 

25. McGraw-Hill/Irwin
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10-25
Rate of Return
Scenario Stock fundBond fundPortfoliosquared deviation
Recession -7% 17% 5.0% 0.160%
Normal 12% 7% 9.5% 0.003%
Boom 28% -3% 12.5% 0.123%
Expected return 11.00% 7.00% 9.0%
Variance 0.0205 0.00670.0010
Standard Deviation14.31% 8.16% 3.08%
10.3 The Return and Risk for Portfolios
The expected rate of return on the portfolio is a weighted
average of the expected returns on the securities in the
portfolio.
%)7(%50%)11(%50%9 
)()()(
SSBBP rEwrEwrE 

26. McGraw-Hill/Irwin
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10-26
Rate of Return
Scenario Stock fundBond fundPortfoliosquared deviation
Recession -7% 17% 5.0% 0.160%
Normal 12% 7% 9.5% 0.003%
Boom 28% -3% 12.5% 0.123%
Expected return 11.00% 7.00% 9.0%
Variance 0.0205 0.00670.0010
Standard Deviation14.31% 8.16% 3.08%
10.3 The Return and Risk for Portfolios
The variance of the rate of return on the two risky assets
portfolio is
BSSSBB
2
SS
2
BB
2
P
)ρσ)(wσ2(w)σ(w)σ(wσ 
where 
BS
is the correlation coefficient between the returns
on the stock and bond funds.

27. McGraw-Hill/Irwin
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10-27
Rate of Return
Scenario Stock fundBond fundPortfoliosquared deviation
Recession -7% 17% 5.0% 0.160%
Normal 12% 7% 9.5% 0.003%
Boom 28% -3% 12.5% 0.123%
Expected return 11.00% 7.00% 9.0%
Variance 0.0205 0.00670.0010
Standard Deviation14.31% 8.16% 3.08%
10.3 The Return and Risk for Portfolios
Observe the decrease in risk that diversification offers.
An equally weighted portfolio (50% in stocks and 50%
in bonds) has less risk than stocks or bonds held in
isolation.

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10-28
Efficient Frontier
 Construct a risk/return plot of all possible
portfolios
Those portfolios that are not dominated
constitute the efficient frontier
28

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10-29
Efficient Frontier (cont’d)
29
Standard Deviation
Expected Return
100% investment in security
with highest E(R)
100% investment in minimum
variance portfolio
Points below the efficient
frontier are dominated
No points plot above
the line
All portfolios
on the line
are efficient

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10-30
Efficient Frontier (cont’d)
•When a risk-free investment is available, the shape of the
efficient frontier changes
•The expected return and variance of a risk-free rate/stock
return combination are simply a weighted average of the two
expected returns and variance
•The risk-free rate has a variance of zero
30

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10-31
Efficient Frontier (cont’d)
31
Standard Deviation
Expected Return
R
f
A
B
C

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10-32
Efficient Frontier (cont’d)
•The efficient frontier with a risk-free rate: Extends from the
risk-free rate to point B
•The line is tangent to the risky securities efficient frontier
Follows the curve from point B to point C
32

33. McGraw-Hill/Irwin
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10-33
Portfolo Risk and Return Combinations
5.0%
6.0%
7.0%
8.0%
9.0%
10.0%
11.0%
12.0%
0.0%2.0%4.0%6.0%8.0%10.0%12.0%14.0%16.0%
Portfolio Risk (standard deviation)
P
o
r
tfo
lio
R
e
tu
r
n
% in stocks Risk Return
0% 8.2% 7.0%
5% 7.0% 7.2%
10% 5.9% 7.4%
15% 4.8% 7.6%
20% 3.7% 7.8%
25% 2.6% 8.0%
30% 1.4% 8.2%
35% 0.4% 8.4%
40% 0.9% 8.6%
45% 2.0% 8.8%
50.00% 3.08% 9.00%
55% 4.2% 9.2%
60% 5.3% 9.4%
65% 6.4% 9.6%
70% 7.6% 9.8%
75% 8.7% 10.0%
80% 9.8% 10.2%
85% 10.9% 10.4%
90% 12.1% 10.6%
95% 13.2% 10.8%
100% 14.3% 11.0%
10.4 The Efficient Set for Two Assets
We can consider other
portfolio weights besides
50% in stocks and 50% in
bonds …
100%
bonds
100%
stocks

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10-34
Portfolo Risk and Return Combinations
5.0%
6.0%
7.0%
8.0%
9.0%
10.0%
11.0%
12.0%
0.0%2.0%4.0%6.0%8.0%10.0%12.0%14.0%16.0%
Portfolio Risk (standard deviation)
P
o
r
tfo
lio
R
e
tu
r
n
10.4 The Efficient Set for Two Assets
We can consider other
portfolio weights besides
50% in stocks and 50% in
bonds …
100%
bonds
100%
stocks
% in stocks Risk Return
0% 8.2% 7.0%
5% 7.0% 7.2%
10% 5.9% 7.4%
15% 4.8% 7.6%
20% 3.7% 7.8%
25% 2.6% 8.0%
30% 1.4% 8.2%
35% 0.4% 8.4%
40% 0.9% 8.6%
45% 2.0% 8.8%
50% 3.1% 9.0%
55% 4.2% 9.2%
60% 5.3% 9.4%
65% 6.4% 9.6%
70% 7.6% 9.8%
75% 8.7% 10.0%
80% 9.8% 10.2%
85% 10.9% 10.4%
90% 12.1% 10.6%
95% 13.2% 10.8%
100% 14.3% 11.0%
% in stocks Risk Return
0% 8.2% 7.0%
5% 7.0% 7.2%
10% 5.9% 7.4%
15% 4.8% 7.6%
20% 3.7% 7.8%
25% 2.6% 8.0%
30% 1.4% 8.2%
35% 0.4% 8.4%
40% 0.9% 8.6%
45% 2.0% 8.8%
50% 3.1% 9.0%
55% 4.2% 9.2%
60% 5.3% 9.4%
65% 6.4% 9.6%
70% 7.6% 9.8%
75% 8.7% 10.0%
80% 9.8% 10.2%
85% 10.9% 10.4%
90% 12.1% 10.6%
95% 13.2% 10.8%
100% 14.3% 11.0%
% in stocks Risk Return
0% 8.2% 7.0%
5% 7.0% 7.2%
10% 5.9% 7.4%
15% 4.8% 7.6%
20% 3.7% 7.8%
25% 2.6% 8.0%
30% 1.4% 8.2%
35% 0.4% 8.4%
40% 0.9% 8.6%
45% 2.0% 8.8%
50% 3.1% 9.0%
55% 4.2% 9.2%
60% 5.3% 9.4%
65% 6.4% 9.6%
70% 7.6% 9.8%
75% 8.7% 10.0%
80% 9.8% 10.2%
85% 10.9% 10.4%
90% 12.1% 10.6%
95% 13.2% 10.8%
100% 14.3% 11.0%
% in stocks Risk Return
0% 8.2% 7.0%
5% 7.0% 7.2%
10% 5.9% 7.4%
15% 4.8% 7.6%
20% 3.7% 7.8%
25% 2.6% 8.0%
30% 1.4% 8.2%
35% 0.4% 8.4%
40% 0.9% 8.6%
45% 2.0% 8.8%
50% 3.1% 9.0%
55% 4.2% 9.2%
60% 5.3% 9.4%
65% 6.4% 9.6%
70% 7.6% 9.8%
75% 8.7% 10.0%
80% 9.8% 10.2%
85% 10.9% 10.4%
90% 12.1% 10.6%
95% 13.2% 10.8%
100% 14.3% 11.0%
% in stocks Risk Return
0% 8.2% 7.0%
5% 7.0% 7.2%
10% 5.9% 7.4%
15% 4.8% 7.6%
20% 3.7% 7.8%
25% 2.6% 8.0%
30% 1.4% 8.2%
35% 0.4% 8.4%
40% 0.9% 8.6%
45% 2.0% 8.8%
50% 3.1% 9.0%
55% 4.2% 9.2%
60% 5.3% 9.4%
65% 6.4% 9.6%
70% 7.6% 9.8%
75% 8.7% 10.0%
80% 9.8% 10.2%
85% 10.9% 10.4%
90% 12.1% 10.6%
95% 13.2% 10.8%
100% 14.3% 11.0%
% in stocks Risk Return
0% 8.2% 7.0%
5% 7.0% 7.2%
10% 5.9% 7.4%
15% 4.8% 7.6%
20% 3.7% 7.8%
25% 2.6% 8.0%
30% 1.4% 8.2%
35% 0.4% 8.4%
40% 0.9% 8.6%
45% 2.0% 8.8%
50% 3.1% 9.0%
55% 4.2% 9.2%
60% 5.3% 9.4%
65% 6.4% 9.6%
70% 7.6% 9.8%
75% 8.7% 10.0%
80% 9.8% 10.2%
85% 10.9% 10.4%
90% 12.1% 10.6%
95% 13.2% 10.8%
100% 14.3% 11.0%
% in stocks Risk Return
0% 8.2% 7.0%
5% 7.0% 7.2%
10% 5.9% 7.4%
15% 4.8% 7.6%
20% 3.7% 7.8%
25% 2.6% 8.0%
30% 1.4% 8.2%
35% 0.4% 8.4%
40% 0.9% 8.6%
45% 2.0% 8.8%
50% 3.1% 9.0%
55% 4.2% 9.2%
60% 5.3% 9.4%
65% 6.4% 9.6%
70% 7.6% 9.8%
75% 8.7% 10.0%
80% 9.8% 10.2%
85% 10.9% 10.4%
90% 12.1% 10.6%
95% 13.2% 10.8%
100% 14.3% 11.0%
% in stocks Risk Return
0% 8.2% 7.0%
5% 7.0% 7.2%
10% 5.9% 7.4%
15% 4.8% 7.6%
20% 3.7% 7.8%
25% 2.6% 8.0%
30% 1.4% 8.2%
35% 0.4% 8.4%
40% 0.9% 8.6%
45% 2.0% 8.8%
50% 3.1% 9.0%
55% 4.2% 9.2%
60% 5.3% 9.4%
65% 6.4% 9.6%
70% 7.6% 9.8%
75% 8.7% 10.0%
80% 9.8% 10.2%
85% 10.9% 10.4%
90% 12.1% 10.6%
95% 13.2% 10.8%
100% 14.3% 11.0%
% in stocks Risk Return
0% 8.2% 7.0%
5% 7.0% 7.2%
10% 5.9% 7.4%
15% 4.8% 7.6%
20% 3.7% 7.8%
25% 2.6% 8.0%
30% 1.4% 8.2%
35% 0.4% 8.4%
40% 0.9% 8.6%
45% 2.0% 8.8%
50% 3.1% 9.0%
55% 4.2% 9.2%
60% 5.3% 9.4%
65% 6.4% 9.6%
70% 7.6% 9.8%
75% 8.7% 10.0%
80% 9.8% 10.2%
85% 10.9% 10.4%
90% 12.1% 10.6%
95% 13.2% 10.8%
100% 14.3% 11.0%
% in stocks Risk Return
0% 8.2% 7.0%
5% 7.0% 7.2%
10% 5.9% 7.4%
15% 4.8% 7.6%
20% 3.7% 7.8%
25% 2.6% 8.0%
30% 1.4% 8.2%
35% 0.4% 8.4%
40% 0.9% 8.6%
45% 2.0% 8.8%
50% 3.1% 9.0%
55% 4.2% 9.2%
60% 5.3% 9.4%
65% 6.4% 9.6%
70% 7.6% 9.8%
75% 8.7% 10.0%
80% 9.8% 10.2%
85% 10.9% 10.4%
90% 12.1% 10.6%
95% 13.2% 10.8%
100% 14.3% 11.0%
% in stocks Risk Return
0% 8.2% 7.0%
5% 7.0% 7.2%
10% 5.9% 7.4%
15% 4.8% 7.6%
20% 3.7% 7.8%
25% 2.6% 8.0%
30% 1.4% 8.2%
35% 0.4% 8.4%
40% 0.9% 8.6%
45% 2.0% 8.8%
50% 3.1% 9.0%
55% 4.2% 9.2%
60% 5.3% 9.4%
65% 6.4% 9.6%
70% 7.6% 9.8%
75% 8.7% 10.0%
80% 9.8% 10.2%
85% 10.9% 10.4%
90% 12.1% 10.6%
95% 13.2% 10.8%
100% 14.3% 11.0%
% in stocks Risk Return
0% 8.2% 7.0%
5% 7.0% 7.2%
10% 5.9% 7.4%
15% 4.8% 7.6%
20% 3.7% 7.8%
25% 2.6% 8.0%
30% 1.4% 8.2%
35% 0.4% 8.4%
40% 0.9% 8.6%
45% 2.0% 8.8%
50% 3.1% 9.0%
55% 4.2% 9.2%
60% 5.3% 9.4%
65% 6.4% 9.6%
70% 7.6% 9.8%
75% 8.7% 10.0%
80% 9.8% 10.2%
85% 10.9% 10.4%
90% 12.1% 10.6%
95% 13.2% 10.8%
100% 14.3% 11.0%
% in stocks Risk Return
0% 8.2% 7.0%
5% 7.0% 7.2%
10% 5.9% 7.4%
15% 4.8% 7.6%
20% 3.7% 7.8%
25% 2.6% 8.0%
30% 1.4% 8.2%
35% 0.4% 8.4%
40% 0.9% 8.6%
45% 2.0% 8.8%
50% 3.1% 9.0%
55% 4.2% 9.2%
60% 5.3% 9.4%
65% 6.4% 9.6%
70% 7.6% 9.8%
75% 8.7% 10.0%
80% 9.8% 10.2%
85% 10.9% 10.4%
90% 12.1% 10.6%
95% 13.2% 10.8%
100% 14.3% 11.0%
% in stocks Risk Return
0% 8.2% 7.0%
5% 7.0% 7.2%
10% 5.9% 7.4%
15% 4.8% 7.6%
20% 3.7% 7.8%
25% 2.6% 8.0%
30% 1.4% 8.2%
35% 0.4% 8.4%
40% 0.9% 8.6%
45% 2.0% 8.8%
50% 3.1% 9.0%
55% 4.2% 9.2%
60% 5.3% 9.4%
65% 6.4% 9.6%
70% 7.6% 9.8%
75% 8.7% 10.0%
80% 9.8% 10.2%
85% 10.9% 10.4%
90% 12.1% 10.6%
95% 13.2% 10.8%
100% 14.3% 11.0%
% in stocks Risk Return
0% 8.2% 7.0%
5% 7.0% 7.2%
10% 5.9% 7.4%
15% 4.8% 7.6%
20% 3.7% 7.8%
25% 2.6% 8.0%
30% 1.4% 8.2%
35% 0.4% 8.4%
40% 0.9% 8.6%
45% 2.0% 8.8%
50% 3.1% 9.0%
55% 4.2% 9.2%
60% 5.3% 9.4%
65% 6.4% 9.6%
70% 7.6% 9.8%
75% 8.7% 10.0%
80% 9.8% 10.2%
85% 10.9% 10.4%
90% 12.1% 10.6%
95% 13.2% 10.8%
100% 14.3% 11.0%
% in stocks Risk Return
0% 8.2% 7.0%
5% 7.0% 7.2%
10% 5.9% 7.4%
15% 4.8% 7.6%
20% 3.7% 7.8%
25% 2.6% 8.0%
30% 1.4% 8.2%
35% 0.4% 8.4%
40% 0.9% 8.6%
45% 2.0% 8.8%
50% 3.1% 9.0%
55% 4.2% 9.2%
60% 5.3% 9.4%
65% 6.4% 9.6%
70% 7.6% 9.8%
75% 8.7% 10.0%
80% 9.8% 10.2%
85% 10.9% 10.4%
90% 12.1% 10.6%
95% 13.2% 10.8%
100% 14.3% 11.0%
% in stocks Risk Return
0% 8.2% 7.0%
5% 7.0% 7.2%
10% 5.9% 7.4%
15% 4.8% 7.6%
20% 3.7% 7.8%
25% 2.6% 8.0%
30% 1.4% 8.2%
35% 0.4% 8.4%
40% 0.9% 8.6%
45% 2.0% 8.8%
50% 3.1% 9.0%
55% 4.2% 9.2%
60% 5.3% 9.4%
65% 6.4% 9.6%
70% 7.6% 9.8%
75% 8.7% 10.0%
80% 9.8% 10.2%
85% 10.9% 10.4%
90% 12.1% 10.6%
95% 13.2% 10.8%
100% 14.3% 11.0%
% in stocks Risk Return
0% 8.2% 7.0%
5% 7.0% 7.2%
10% 5.9% 7.4%
15% 4.8% 7.6%
20% 3.7% 7.8%
25% 2.6% 8.0%
30% 1.4% 8.2%
35% 0.4% 8.4%
40% 0.9% 8.6%
45% 2.0% 8.8%
50% 3.1% 9.0%
55% 4.2% 9.2%
60% 5.3% 9.4%
65% 6.4% 9.6%
70% 7.6% 9.8%
75% 8.7% 10.0%
80% 9.8% 10.2%
85% 10.9% 10.4%
90% 12.1% 10.6%
95% 13.2% 10.8%
100% 14.3% 11.0%
% in stocks Risk Return
0% 8.2% 7.0%
5% 7.0% 7.2%
10% 5.9% 7.4%
15% 4.8% 7.6%
20% 3.7% 7.8%
25% 2.6% 8.0%
30% 1.4% 8.2%
35% 0.4% 8.4%
40% 0.9% 8.6%
45% 2.0% 8.8%
50% 3.1% 9.0%
55% 4.2% 9.2%
60% 5.3% 9.4%
65% 6.4% 9.6%
70% 7.6% 9.8%
75% 8.7% 10.0%
80% 9.8% 10.2%
85% 10.9% 10.4%
90% 12.1% 10.6%
95% 13.2% 10.8%
100% 14.3% 11.0%
% in stocks Risk Return
0% 8.2% 7.0%
5% 7.0% 7.2%
10% 5.9% 7.4%
15% 4.8% 7.6%
20% 3.7% 7.8%
25% 2.6% 8.0%
30% 1.4% 8.2%
35% 0.4% 8.4%
40% 0.9% 8.6%
45% 2.0% 8.8%
50% 3.1% 9.0%
55% 4.2% 9.2%
60% 5.3% 9.4%
65% 6.4% 9.6%
70% 7.6% 9.8%
75% 8.7% 10.0%
80% 9.8% 10.2%
85% 10.9% 10.4%
90% 12.1% 10.6%
95% 13.2% 10.8%
100% 14.3% 11.0%
% in stocks Risk Return
0% 8.2% 7.0%
5% 7.0% 7.2%
10% 5.9% 7.4%
15% 4.8% 7.6%
20% 3.7% 7.8%
25% 2.6% 8.0%
30% 1.4% 8.2%
35% 0.4% 8.4%
40% 0.9% 8.6%
45% 2.0% 8.8%
50% 3.1% 9.0%
55% 4.2% 9.2%
60% 5.3% 9.4%
65% 6.4% 9.6%
70% 7.6% 9.8%
75% 8.7% 10.0%
80% 9.8% 10.2%
85% 10.9% 10.4%
90% 12.1% 10.6%
95% 13.2% 10.8%
100% 14.3% 11.0%

35. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-35
Portfolo Risk and Return Combinations
5.0%
6.0%
7.0%
8.0%
9.0%
10.0%
11.0%
12.0%
0.0%2.0%4.0%6.0%8.0%10.0%12.0%14.0%16.0%
Portfolio Risk (standard deviation)
P
o
r
tfo
lio
R
e
tu
r
n
% in stocks Risk Return
0% 8.2% 7.0%
5% 7.0% 7.2%
10% 5.9% 7.4%
15% 4.8% 7.6%
20% 3.7% 7.8%
25% 2.6% 8.0%
30% 1.4% 8.2%
35% 0.4% 8.4%
40% 0.9% 8.6%
45% 2.0% 8.8%
50% 3.1% 9.0%
55% 4.2% 9.2%
60% 5.3% 9.4%
65% 6.4% 9.6%
70% 7.6% 9.8%
75% 8.7% 10.0%
80% 9.8% 10.2%
85% 10.9% 10.4%
90% 12.1% 10.6%
95% 13.2% 10.8%
100% 14.3% 11.0%
10.4 The Efficient Set for Two Assets
100%
stocks
100%
bonds
Note that some portfolios are
“better” than others. They have
higher returns for the same level of
risk or less. These compromise the
efficient frontier.

36. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-36Portfolio Risk/Return Two Securities:
Correlation Effects
•Relationship depends on correlation coefficient
•-1.0 <  < +1.0
•The smaller the correlation, the greater the risk reduction
potential
•If= +1.0, no risk reduction is possible

37. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-37
Two-Security Portfolios with Various
Correlations
100%
bonds
r
e
t
u
r
n

100%
stocks
 = 0.2
 = 1.0
 = -1.0

38. The Sharpe Index Model

39. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-39
Need for Sharpe Model
In Markowitz model a number of co-variances have to
be estimated.
If a financial institution buys 150 stocks, it has to
estimate 11,175 i.e., (N
2
– N)/2 correlation
co-efficients.
Sharpe assumed that the return of a security is linearly
related to a single index like the market index.

40. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-40
Single Index Model
•Casual observation of the stock prices over a period of time
reveals that most of the stock prices move with the market
index.
•When the KSE-100index increases, stock prices also tend to
increase and vice – versa.
•This indicates that some underlying factors affect the market
index as well as the stock prices.

41. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-41
Stock prices are related to the market index and this
relationship could be used to estimate the return of stock.
R
i = 
i + 
i R
m + e
i
where R
i — expected return on security i

i — intercept of the straight line or alpha co-
efficient

i
— slope of straight line or beta co-efficient
R
m — the rate of return on market index
e
i — error term

42. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-42
Definition of Risk When Investors Hold the
Market Portfolio
•Researchers have shown that the best measure of the risk of
a security in a large portfolio is the beta ()of the security.
•Beta measures the responsiveness of a security to
movements in the market portfolio.
)(
)(
2
,
M
Mi
i
R
RRCov



43. Beta
“What Is Beta and How Is It Calculated?”

44. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-44
Beta
•A “coefficient measuring a stock’s relative volatility”
•Beta measures a stock’s sensitivity to overall market
movements
Source:UBS Warburg Dictionary of Finance and Investment
Terms

45. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-45
•In practice, Beta is measured by comparing changes in a
stock price to changes in the value of the 1 index over a
given time period
•The S&P 500 index has a beta of 1

46. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-46
A Generic Example
•Stock XYZ has a beta of 2
•The S&P 500 index increases in value by 10%
•The price of XYZ is expected to increase 20% over the same
time period

47. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-47
Beta can be Negative
•Stock XYZ has a beta of –2
•The S&P 500 index INCREASES in value by 10%
•The price of XYZ is expected to DECREASE 20% over the
same time period

48. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-48
•If the beta of XYZ is 1.5 …
•And the S&P increases in value by 10%
•The price of XYZ is expected to increase 15%

49. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-49
•A beta of 0 indicates that changes in the market index cannot
be used to predict changes in the price of the stock
•The company’s stock price has no correlation to movments
in the market index

50. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-50
•If beta is a measure of risk, then investors who hold stocks
with higher betas should expect a higher return for taking on
that risk
•What does this remind you of?

51. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-51
Beta and Risk
•Beta is a measure of volatility
•Volatility is associated with risk

52. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-52
How to Calculate Beta
Beta = Covariance(stock price, market index)
Variance(market index)
**When calculating, you must compare the percent change in
the stock price to the percent change in the market index**

53. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-53
Risk
•Systematic risk= 
i
2
× variance of market index
= 
i
2

m
2
Unsystematic risk= Total variance – Systematic risk
e
i
2
= 
i
2
– Systematic risk
Thus the total risk= Systematic risk + Unsystematic risk
= 
i
2

m
2
+ e
i
2

54. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-54
Portfolio Variance
2
N N
2 2 2 2
p i i m i i
i=1 i=1
σ = x β + x e
 
   
 
  
     
 
where
σ
2
p
= variance of portfolio
σ
2
m
= expected variance of market index

e
2
i
= Unsystematic risk
x
i = the portion of stock i in the portfolio

55. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-55
Example
•The following details are given for x and y companies’
stocks and the Sensex for a period of one year. Calculate the
systematic and unsystematic risk for the companies stock. If
equal amount of money is allocated for the stocks , then
what would be the portfolio risk ?
• X stock Y stock Index
•Average return 0.15 0.25 0.06
•Variance of return 6.30 5.86 2.25
•Βeta 0.71 0.27

56. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-56
Company X
•Systematic risk= 
i
2
× variance of market index
= 
i
2

m
2
= ( 0.71)
2
x 2.25 = 1.134
Unsystematic risk= Total variance – Systematic risk
e
i
2
= 
i
2
– Systematic risk = 6.3 – 1.134 =5.166
Total risk= Systematic risk + Unsystematic risk
= 
i
2

m
2
+ e
i
2
= 1.134 + 5.166 = 6.3

57. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-57
Company Y
•Systematic risk= 
i
2
× variance of market index
= 
i
2

m
2
= ( 0.27)
2
x 2.25 = 0.1640
Unsystematic risk= Total variance – Systematic risk
e
i
2
= 
i
2
– Systematic risk = 5.86 – 1.134 =5.166

58. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-58
•σ
2
p = [ ( .5 x .71 + .5 x .27)
2
2.25 ] + [ ( .5)
2
(5.166) + (.5 )
2
( 5.696) ]
• = [ ( .355 + .135 )2 2.25 ] + [ ( 1.292 + 1.424 ) ]
• = 0.540 + 2.716
• = 3.256

59. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-59
Sharpe’s optimal portfolio
i f
i
R R
β

The selection of any stock is directly related to its
excess return to beta ratio.
where R
i = the expected return on stock i

R
f
= the return on a risk less asset


i = Systematic risk

60. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-60
Capital Asset Pricing Model (CAPM)
•Introduction
•Systematic and unsystematic risk
•Fundamental risk/return relationship revisited
60

61. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-61
Introduction
•The Capital Asset Pricing Model (CAPM) is a theoretical
description of the way in which the market prices investment
assets
•The CAPM is a positive theory
61

62. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-62
Portfolio Risk as a Function of the Number
of Stocks in the Portfolio
Nondiversifiable risk;
Systematic Risk;
Market Risk
Diversifiable Risk;
Nonsystematic Risk;
Firm Specific Risk;
Unique Risk
n

In a large portfolio the variance terms are effectively
diversified away, but the covariance terms are not.
Thus diversification can eliminate some, but not all of the
risk of individual securities.
Portfolio risk

63. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-63Systematic and
Unsystematic Risk
•Unsystematic risk can be diversified and is irrelevant
•Systematic risk cannot be diversified and is relevant
Measured by beta
Beta determines the level of expected return on a security
or portfolio (SML)
63

64. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-64
10.9 Relationship between Risk and Expected
Return (CAPM)
•Expected Return on the Market:
•Expected return on an individual security:
PremiumRisk Market 
F
MRR
)(β
F
M
iF
i RRRR 
Market Risk Premium
This applies to individual securities held within well-diversified
portfolios.

65. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-65
Expected Return on an Individual Security
•This formula is called the Capital Asset Pricing
Model (CAPM)
)(β
F
M
iF
i RRRR 
•Assume 
i
= 0, then the expected return is R
F
.
•Assume 
i = 1, then MiRR
Expected
return on
a security
=
Risk-
free rate
+
Beta of the
security
×
Market risk
premium

66. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-66
Relationship Between Risk & Expected Return
E
x
p
e
c
t
e
d

r
e
t
u
r
n

)(β
F
M
iF
i RRRR 
F
R
1.0
M
R
)(β
F
M
iF
i RRRR 

67. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-67
Relationship Between Risk & Expected Return
E
x
p
e
c
t
e
d

r
e
t
u
r
n

%3
FR
%3
1.5
%5.13
5.1β
i %10MR
%5.13%)3%10(5.1%3 iR

68. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-68
CAPM (cont’d)
•The CAPM deals with expectations about the future
•Excess returns on a particular stock are directly related to:
–The beta of the stock
–The expected excess return on the market
68

69. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-69
CAPM (cont’d)
CAPM assumptions:
–Variance of return and mean return are all investors care
about
–Investors are price takers, they cannot influence the
market individually
–All investors have equal and costless access to
information
–There are no taxes or commission costs
69

70. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-70
SML and CAPM
If you show the security market line with excess
returns on the vertical axis, the equation of the SML
is the CAPM
–The intercept is zero
–The slope of the line is beta

71. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-71
Note on the CAPM Assumptions
Several assumptions are unrealistic:
–People pay taxes and commissions
–Many people look ahead more than one period
–Not all investors forecast the same distribution
Theory is useful to the extent that it helps us learn
more about the way the world acts
–Empirical testing shows that the CAPM works
reasonably well
71

72. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-72
Arbitrage Pricing Theory
APT background
The APT model
Comparison of the CAPM and the APT
72

73. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-73
APT Background
•Arbitrage pricing theory (APT) states that a number of
distinct factors determine the market return
–Roll and Ross state that a security’s long-run return is a
function of changes in:
•Inflation
•Industrial production
•Risk premiums
•The slope of the term structure of interest rates
73

74. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-74
APT Background (cont’d)
Not all analysts are concerned with the same set of economic
information
–A single market measure such as beta does not capture all
the information relevant to the price of a stock
74

75. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-75
The APT Model
General representation of the APT model:
75
1 1 2 2 3 3 4 4
( )
where actual return on Security
( ) expected return on Security
sensitivity of Security to factor
unanticipated change in factor
A A A A A A
A
A
iA
i
R E R b F b F b F b F
R A
E R A
b A i
F i
    







76. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-76Comparison of the
CAPM and the APT (cont’d)
The CAPM and APT complement each other rather than
compete
–Both models predict that positive returns will result from
factor sensitivities that move with the market and vice
versa
76

77. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-77
10.5 The Efficient Set for Many Securities
Consider a world with many risky assets; we can still identify
the opportunity set of risk-return combinations of various
portfolios.
r
e
t
u
r
n

P
Individual Assets

78. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-78
10.5 The Efficient Set for Many Securities
Given the opportunity set we can identify the minimum
variance portfolio.
r
e
t
u
r
n

P
minimum
variance
portfolio
Individual Assets

79. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-79
10.5 The Efficient Set for Many Securities
The section of the opportunity set above the minimum variance
portfolio is the efficient frontier.
r
e
t
u
r
n

P
minimum
variance
portfolio
efficient frontier
Individual Assets

80. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-80
Optimal Risky Portfolio with a Risk-Free Asset
In addition to stocks and bonds, consider a world that also has
risk-free securities like T-bills
100%
bonds
100%
stocks
r
f
r
e
t
u
r
n


81. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-81
10.7 Riskless Borrowing and Lending
Now investors can allocate their money across the T-bills and a
balanced mutual fund
100%
bonds
100%
stocks
r
f
r
e
t
u
r
n

Balanced
fund
C
M
L

82. McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-82
10.8 Market Equilibrium
With the capital allocation line identified, all investors
choose a point along the line—some combination of the
risk-free asset and the market portfolio M. In a world with
homogeneous expectations, M is the same for all investors.
r
e
t
u
r
n

P
efficient frontier
r
f
M
C
M
L