### the same expected return from the portfolio with reduced risk

```Portfolio Theory
0
What and how do you select any
investment opportunity?
 Investors are concerned with

 Returns
 Risk

How to choose the optimal portfolio?
1
Mean-Variance Portfolio Theory
•We consider a risk-based approach and select an
investment portfolio
by minimizing risk for a given level of expected return
by maximizing expected return for a given level of risk.
•We estimate different measure of risk.
Variance is one of them and enjoys desirable
mathematical properties.
•The problem here is how can we select an investment
opportunity, which minimizes risk measured or
described by variance?
2
Mean-Variance Portfolio Theory
Harry Markowitz, a Nobel Laureate in Economics,
develops the celebrated mean-variance portfolio
selection model, which is also known as the modern
portfolio theory in some finance textbooks and
literature.
3
Mean-Variance Portfolio Theory
The key idea of the Markowitz mean-variance portfolio
selection model is Diversification, which is just the
same as an old saying “Don't put all of your eggs in a
basket”.
In the context of finance, diversification means that an
investor can reduce the risk from his/her portfolio by
holding a portfolio of assets, which are not perfectly
correlated.
Diversification also allows for the same expected return
from the portfolio with reduced risk.
4
What are investment returns?
Investment returns measure the
financial results of an investment.
Returns may be historical or
prospective (anticipated).
Returns can be expressed in:
Dollar terms.
Percentage terms.
5
What is the return on an investment that
costs \$1,000 and is sold after 1 year for
\$1,100?
Dollar return:
\$ Received - \$ Invested
\$1,100
\$1,000
= \$100.
Percentage return:
\$ Return/\$ Invested
\$100/\$1,000
= 0.10 = 10%.
6
What is investment risk?
Typically, investment returns are not
known with certainty.
Investment risk pertains to the
probability of earning a return less than
that expected return.
The greater the chance of a return far
below the expected return, the greater
the risk.
7
Probability distribution
Stock X
Stock Y
-20
0
15
50
Rate of
return (%)
 Which stock is riskier? Why?
8
Individual Securities

The characteristics of individual securities
that are of interest are the:
 Expected
Return
 Variance and Standard Deviation
 Covariance and Correlation
9
Calculate the expected rate of
return on each alternative.
r = expected rate of return.
n
r=
 ri Pi .
i =1
There are n situations.
ri means the returns when the situation is at i
and its probability is P .
10
What is the standard deviation
of returns for each alternative?
  Standard deviation
 

Variance

2
n
2P .


r

r
 i
i
i 1
11
Standard deviation measures the
stand-alone risk of an investment.
 The larger the standard deviation, the
higher the probability that returns will
be far below the expected return.
 Coefficient of variation is an
alternative measure of stand-alone
risk.

12
Expected Return, Variance, and Covariance
Rate of Return
Scenario Probability Stock fund Bond fund
Recession
33.3%
-7%
17%
Normal
33.3%
12%
7%
Boom
33.3%
28%
-3%
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.
13
Expected Return, Variance, and Covariance
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Stock fund
Rate of
Squared
Return Deviation
-7%
3.24%
12%
0.01%
28%
2.89%
11.00%
0.0205
14.3%
Bond Fund
Rate of
Squared
Return Deviation
17%
1.00%
7%
0.00%
-3%
1.00%
7.00%
0.0067
8.2%
E ( rS )  1  (7%)  1  (12%)  1  ( 28%)
3
3
3
E ( rS )  11%
14
Expected Return, Variance, and Covariance
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Stock fund
Rate of
Squared
Return Deviation
-7%
3.24%
12%
0.01%
28%
2.89%
11.00%
0.0205
14.3%
Bond Fund
Rate of
Squared
Return Deviation
17%
1.00%
7%
0.00%
-3%
1.00%
7.00%
0.0067
8.2%
E ( rB )  1  (17%)  1  (7%)  1  ( 3%)
3
3
3
E ( rB )  7%
15
Expected Return, Variance, and Covariance
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Stock fund
Rate of
Squared
Return Deviation
-7%
3.24%
12%
0.01%
28%
2.89%
11.00%
0.0205
14.3%
Bond Fund
Rate of
Squared
Return Deviation
17%
1.00%
7%
0.00%
-3%
1.00%
7.00%
0.0067
8.2%
(7%  11%)  3.24%
2
16
Expected Return, Variance, and Covariance
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Stock fund
Rate of
Squared
Return Deviation
-7%
3.24%
12%
0.01%
28%
2.89%
11.00%
0.0205
14.3%
Bond Fund
Rate of
Squared
Return Deviation
17%
1.00%
7%
0.00%
-3%
1.00%
7.00%
0.0067
8.2%
(12%  11%)  .01%
2
17
Expected Return, Variance, and Covariance
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Stock fund
Rate of
Squared
Return Deviation
-7%
3.24%
12%
0.01%
28%
2.89%
11.00%
0.0205
14.3%
Bond Fund
Rate of
Squared
Return Deviation
17%
1.00%
7%
0.00%
-3%
1.00%
7.00%
0.0067
8.2%
(28%  11%)  2.89%
2
18
Expected Return, Variance, and Covariance
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Stock fund
Rate of
Squared
Return Deviation
-7%
3.24%
12%
0.01%
28%
2.89%
11.00%
0.0205
14.3%
Bond Fund
Rate of
Squared
Return Deviation
17%
1.00%
7%
0.00%
-3%
1.00%
7.00%
0.0067
8.2%
1
2.05%  (3.24%  0.01%  2.89%)
3
19
Expected Return, Variance, and Covariance
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Stock fund
Rate of
Squared
Return Deviation
-7%
3.24%
12%
0.01%
28%
2.89%
11.00%
0.0205
14.3%
Bond Fund
Rate of
Squared
Return Deviation
17%
1.00%
7%
0.00%
-3%
1.00%
7.00%
0.0067
8.2%
14.3%  0.0205
20
The Return and Risk for Portfolios
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Stock fund
Rate of
Squared
Return Deviation
-7%
3.24%
12%
0.01%
28%
2.89%
11.00%
0.0205
14.3%
Bond Fund
Rate of
Squared
Return Deviation
17%
1.00%
7%
0.00%
-3%
1.00%
7.00%
0.0067
8.2%
Note that stocks have a higher expected return than bonds
and higher risk.
21
The Return and Risk for Portfolios
Rate of Return
Stock fund Bond fund Portfolio
-7%
17%
5.0%
12%
7%
9.5%
28%
-3%
12.5%
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
11.00%
0.0205
14.31%
7.00%
0.0067
8.16%
squared deviation
0.160%
0.003%
0.123%
9.0%
0.0010
3.08%
Let us turn now to the risk-return tradeoff of a portfolio that
is 50% invested in bonds and 50% invested in stocks.
The rate of return on the portfolio is a weighted average of
the returns on the stocks and bonds in the portfolio:
rP  w B rB  w S rS
5%  50%  (  7%)  50 %  (17%)
22
The Return and Risk for Portfolios
Rate of Return
Stock fund Bond fund Portfolio
-7%
17%
5.0%
12%
7%
9.5%
28%
-3%
12.5%
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
11.00%
0.0205
14.31%
7.00%
0.0067
8.16%
squared deviation
0.160%
0.003%
0.123%
9.0%
0.0010
3.08%
The rate of return on the portfolio is a weighted average of
the returns on the stocks and bonds in the portfolio:
rP  w B rB  w S rS
9 .5%  50%  (12%)  50%  (7 %)
23
The Return and Risk for Portfolios
Rate of Return
Stock fund Bond fund Portfolio
-7%
17%
5.0%
12%
7%
9.5%
28%
-3%
12.5%
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
11.00%
0.0205
14.31%
7.00%
0.0067
8.16%
squared deviation
0.160%
0.003%
0.123%
9.0%
0.0010
3.08%
The rate of return on the portfolio is a weighted average of
the returns on the stocks and bonds in the portfolio:
rP  w B rB  w S rS
12.5%  50%  ( 28 %)  50 %  (  3 %)
24
The Return and Risk for Portfolios
Rate of Return
Stock fund Bond fund Portfolio
-7%
17%
5.0%
12%
7%
9.5%
28%
-3%
12.5%
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
11.00%
0.0205
14.31%
7.00%
0.0067
8.16%
squared deviation
0.160%
0.003%
0.123%
9.0%
0.0010
3.08%
The expected rate of return on the portfolio is a weighted
average of the expected returns on the securities in the
portfolio.
E ( rP )  w B E ( rB )  w S E ( rS )
9%  50%  (11%)  50 %  (7 %)
25
The Return and Risk for Portfolios
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Rate of Return
Stock fund Bond fund Portfolio
-7%
17%
5.0%
12%
7%
9.5%
28%
-3%
12.5%
11.00%
0.0205
14.31%
7.00%
0.0067
8.16%
squared deviation
0.160%
0.003%
0.123%
9.0%
0.0010
3.08%
The variance of the rate of return on the two risky assets
portfolio is
σ P2  (w B σ B ) 2  (w S σ S ) 2  2(w B σ B )(w S σ S )ρ BS
where BS is the correlation coefficient between the returns
on the stock and bond funds.
26
The Return and Risk for Portfolios
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Rate of Return
Stock fund Bond fund Portfolio
-7%
17%
5.0%
12%
7%
9.5%
28%
-3%
12.5%
11.00%
0.0205
14.31%
7.00%
0.0067
8.16%
squared deviation
0.160%
0.003%
0.123%
9.0%
0.0010
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 stocks or bonds held in
isolation.
27
% in stocks
Risk
Return
0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
50.00%
55%
60%
65%
70%
75%
80%
85%
90%
95%
100%
8.2%
7.0%
5.9%
4.8%
3.7%
2.6%
1.4%
0.4%
0.9%
2.0%
3.08%
4.2%
5.3%
6.4%
7.6%
8.7%
9.8%
10.9%
12.1%
13.2%
14.3%
7.0%
7.2%
7.4%
7.6%
7.8%
8.0%
8.2%
8.4%
8.6%
8.8%
9.00%
9.2%
9.4%
9.6%
9.8%
10.0%
10.2%
10.4%
10.6%
10.8%
11.0%
Portfolio Return
The Efficient Set for Two Assets
Portfolo Risk and Return Combinations
12.0%
11.0%
10.0%
9.0%
8.0%
7.0%
6.0%
5.0%
0.0%
100%
stocks
100%
bonds
5.0%
10.0%
15.0%
20.0%
Portfolio Risk (standard deviation)
We can consider other
portfolio weights besides
50% in stocks and 50% in
bonds …
28
% in stocks
Risk
Return
0%
0%
5%
5%
10%
10%
15%
15%
20%
20%
25%
25%
30%
30%
35%
35%
40%
40%
45%
45%
50%
50%
55%
55%
60%
60%
65%
65%
70%
70%
75%
75%
80%
80%
85%
85%
90%
90%
95%
95%
100%
100%
8.2%
8.2%
7.0%
7.0%
5.9%
5.9%
4.8%
4.8%
3.7%
3.7%
2.6%
2.6%
1.4%
1.4%
0.4%
0.4%
0.9%
0.9%
2.0%
2.0%
3.1%
3.1%
4.2%
4.2%
5.3%
5.3%
6.4%
6.4%
7.6%
7.6%
8.7%
8.7%
9.8%
9.8%
10.9%
10.9%
12.1%
12.1%
13.2%
13.2%
14.3%
14.3%
7.0%
7.0%
7.2%
7.2%
7.4%
7.4%
7.6%
7.6%
7.8%
7.8%
8.0%
8.0%
8.2%
8.2%
8.4%
8.4%
8.6%
8.6%
8.8%
8.8%
9.0%
9.0%
9.2%
9.2%
9.4%
9.4%
9.6%
9.6%
9.8%
9.8%
10.0%
10.0%
10.2%
10.2%
10.4%
10.4%
10.6%
10.6%
10.8%
10.8%
11.0%
11.0%
Portfolio Return
The Efficient Set for Two Assets
Portfolo Risk and Return Combinations
12.0%
11.0%
100%
stocks
10.0%
9.0%
8.0%
7.0%
6.0%
100%
bonds
5.0%
0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0%
Portfolio Risk (standard deviation)
We can consider other
portfolio weights besides
50% in stocks and 50% in
bonds …
29
% in stocks
Risk
Return
0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
50%
55%
60%
65%
70%
75%
80%
85%
90%
95%
100%
8.2%
7.0%
5.9%
4.8%
3.7%
2.6%
1.4%
0.4%
0.9%
2.0%
3.1%
4.2%
5.3%
6.4%
7.6%
8.7%
9.8%
10.9%
12.1%
13.2%
14.3%
7.0%
7.2%
7.4%
7.6%
7.8%
8.0%
8.2%
8.4%
8.6%
8.8%
9.0%
9.2%
9.4%
9.6%
9.8%
10.0%
10.2%
10.4%
10.6%
10.8%
11.0%
Portfolio Return
The Efficient Set for Two Assets
Portfolo Risk and Return Combinations
12.0%
11.0%
10.0%
100%
stocks
9.0%
8.0%
7.0%
6.0%
100%
bonds
5.0%
0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0%
Portfolio Risk (standard deviation)
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.
30
Two-Security Portfolios with Various
return
Correlations
100%
stocks
 = -1.0
100%
bonds
 = 1.0
 = 0.2

31
Portfolio 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

32
Mean-Variance Example With Excel

A case of Three Stock
33
What would happen to the
risk of portfolio as more randomly
selected stocks were added?
 p would decrease because the added
stocks would not be perfectly correlated,
but rp would remain relatively constant.
34
Prob.
Large
2
1
0
15
Return
1  35% ; 2  20%.
35
Portfolio Risk as a Function of the Number of Stocks in the
Portfolio
In a large portfolio the variance terms are effectively

diversified away, but the covariance terms are not.
Diversifiable Risk;
Nonsystematic Risk;
Firm Specific Risk;
Unique Risk
Portfolio risk
Nondiversifiable risk;
Systematic Risk;
Market Risk
n
Thus diversification can eliminate some, but not all of the
risk of individual securities.
36
Total Risk




Total risk = systematic risk + unsystematic risk
The standard deviation of returns is a measure
of total risk.
For well-diversified portfolios, unsystematic risk
is very small.
Consequently, the total risk for a diversified
portfolio is essentially equivalent to the
systematic risk.
Stand-alone
risk
=
Market
risk
+
Diversifiable
risk
Market risk is that part of a security’s
stand-alone risk that cannot be
eliminated by diversification.
Firm-specific, or diversifiable, risk is
that part of a security’s stand-alone risk
that can be eliminated by
diversification.
38
Can an investor holding one stock earn
a return commensurate with its risk?
No. Rational investors will minimize risk
by holding portfolios.
 They bear only market risk, so prices and
returns reflect this lower risk.
 The one-stock investor bears higher
(stand-alone) risk, so the return is less
than that required by the risk.

39
Conclusions
As more stocks are added, each new
stock has a smaller risk-reducing
impact on the portfolio.
In general p falls very slowly after
about 40 stocks are included.
By forming well-diversified portfolios,
investors can eliminate part of the
riskiness of owning a single stock.
40
```