7. Calculating Expected Returns and
Standard Deviations for Portfolios
7.1 What is a “Portfolio?”
• A portfolio is a list of securities indicating how much is (or
will be) invested in each one;
• The question is: which portfolio should be selected?
7.2 The Return on a Portfolio
• You have funds W0 to invest in a portfolio at the beginning
of the period;
• By definition we have W1 = W0(1 + rp), where rp is the
return on your portfolio;
• But at time 0, you do not know what W1 will be for any
given portfolio because you do not know what rp will be for
that portfolio;
7-2
• At time 1 you know what is W1;
• Hence, rp is a random variable ex ante but is known ex post.
7.3 How Should You Evaluate Alternative Portfolios?
• What is the “good thing” you want from your portfolio?
Return. You want “return to be high;”
• What is the “bad thing” you want to avoid in your your
portfolio? Risk. You want “risk to be low;”
• Hence, how should you evaluate alternative portfolios? By
examining their expected returns and their standard
deviations.
• Observe, therefore, that you have two conflicting objectives:
(1) maximize the expected return on your portfolio; and
(2) minimize the standard deviation of your portfolio.
7-3
• Harry Markowitz and his paper “Portfolio Selection”
[Journal of Finance (1952)] are generally viewed as the
origin of modern portfolio theory. He was awarded the 1990
Nobel Prize with William Sharpe and Merton Miller for
their pioneering work in the theory of financial economics.
7.4 Calculating the Expected Return of a Portfolio
• The expected return on a portfolio is the weighted average
of the expected returns on the securities included in that
portfolio;
• What is the weight of a security in a portfolio? It is the
percentage of wealth invested in that security;
7-4
• The formula to compute the expected return on a
portfolio of N securities is
N
r p  X i r i
,
i1
where
X i is the weight of security i; and
ri
is the expected return on security i.
7-5
Example
On January 25, 1999, you bought the following stocks:
Table 7.1
Quantity
Price
Stock
200
171
Microsoft
500
185 5/8
IBM
1,000
62 1/2 Coca-Cola
You expect to earn the following (annual) returns on the
stocks you bought: 20% on Microsoft, 12% on IBM, and
15% on Coca-Cola. What is the expected return on your
portfolio?
7-6
First, we need to compute the weight of each security.
The total wealth invested is equal to
W0 =($171*200) + ($185.625*500)+($62.50*1,000)
=$34,200+$92,812.50+$62,500
=$189,512.50.
Hence,
X1=18.05%[=$34,200/$189,512.50];
X2=48.97%[= $92,812.50 /$189,512.50];
X3=32.98%[= $62,500 /$189,512.50].
[Note that X3= 1-X1-X2]
7-7
Using the formula
3
r p  X i r i
,
i1
the expected return on your portfolio is 14.43%
[=0.1805*20%+0.4897*12%+0.3298*15%].
7-8
7.5 The Covariance Between Two Rates of Return
• The covariance is a (statistical) measure of how two random
variables (in this case, the returns of two securities) “move
together;”
• A positive covariance between the returns of two securities
indicates that the returns of the two securities tend to move
in the same direction, that is, better-than-expected returns for
one security are likely to occur when better-than-expected
returns occur for the other security;
• A negative covariance between the returns of two securities
indicates that the returns of the two securities tend to move
in opposite directions, that is, better-than-expected returns
for one security are likely to occur when worse-thanexpected returns occur for the other security;
7-9
• A relatively small or zero covariance between the returns of
two securities indicates that there is little or no relationship
between the returns of the two securities;
• We denote the covariance between the return of security i
and the return of security j by  ij (the Greek letter sigma);
• Note that ij ji ;
• You may use the Excel function “COVAR” to compute the
covariance between the returns of two securities.
7-10
7.6 The Correlation Coefficient
• The correlation coefficient is a statistical measure closely
related with the covariance;
• The interpretation of the correlation coefficient is that of a
“normalized covariance;”
• We denote the correlation coefficient between the return
of security i and the return of security j by ij (the Greek
letter pho);
• The relation between covariance and correlation is given
by the following equation:  ij ij  i  j ;
• The correlation coefficient between the return of security i
and the return of security j lies between -1 and 1;
.
7-11
• If the correlation coefficient between the returns of two
securities is positive, then the returns of the two securities
tend to move in the same direction, that is, better-thanexpected returns for one security are likely to occur when
better-than-expected returns occur for the other security;
• If it is negative, then the returns of the two securities tend to
move in opposite directions, that is, better-than-expected
returns for one security are likely to occur when worse-thanexpected returns occur for the other security;
• If it is close to 0, then there is little or no relationship
between the returns of the two securities;
• You may use the Excel function “CORREL” to compute the
correlation coefficient between the returns of two securities.
7-12
50%
30%
10%
-40%
-20%
0%
20%
40%
60%
-10%
-30%
Figure 7.1
Weekly returns on Amazon.com (y axis) versus Ebay (x axis)
in the period October 98-January 99
7-13
25-Jan
18-Jan
11-Jan
4-Jan
28-Dec
21-Dec
14-Dec
7-Dec
30-Nov
23-Nov
16-Nov
9-Nov
2-Nov
26-Oct
Ebay
19-Oct
Weakly Return
70%
60%
50%
40%
30%
20%
10%
0%
-10%
-20%
-30%
Amazon.com
Figure 7.2
Weekly returns on two internet stocks in the period October
98-January 99
7-14
12%
8%
4%
-30%
-10%
10%
0%
30%
50%
70%
-4%
Figure 7.3
Weekly returns on GM (y axis) and Ebay (x axis) in the
period October 98-January 99
7-15
GM
Figure 7.4
Weekly returns on Ebay and GM in the period October
98-January 99
7-16
25-Jan
18-Jan
11-Jan
4-Jan
28-Dec
21-Dec
14-Dec
7-Dec
30-Nov
23-Nov
16-Nov
9-Nov
2-Nov
26-Oct
Ebay
19-Oct
Weakly Return
70%
60%
50%
40%
30%
20%
10%
0%
-10%
-20%
-30%
• Using the Excel formula “CORREL” the correlation
coefficient between the weekly returns of Amazon.com and
Ebay in the period of October 98-January 99 turned out to
be 0.48;
• The correlation coefficient between the weekly returns of
GM and Ebay in that period was 0.16.
7.7 Calculating the Standard Deviation of a Portfolio
• The formula to compute the standard deviation of a
portfolio of N securities is

 p  
 i1
N
N

j1

X i X j ij 

7-17
1/2
.
• The standard deviation is the square root of the variance.
7.8 Important Special Cases
• Note that ii 1;
2







• Hence, ii ii i i i ;
• What is the correlation coefficient between a riskfree
security and a stock? Zero!
• Therefore, the covariance between a riskfree security and a
stock is also zero!
• What is the formula for the standard deviation of a portfolio
with two securities (say, security 1 and security 2)? Note
that X2 =1-X1,  11  11  1  1   21 and  22  22  2  2   22 .
Hence,
7-18
 2
 p  
 i1
2

j1
 1/2
Xi X j ij 

  X 1 X 1  11  X 1 X 2  12  X 2 X 1  21  X 2 X 2  22 

 X21  21 X22  22 2 X1 X2 12 

1/2
 X   (1 X 1 )   2 X 1 (1 X 1 )  12
2
1
2
1
2
2
2
7-19

1/2
.
1/2
Example
You estimated that the standard deviations of General
Electric and General Motors stocks are 18% and 31%,
respectively. Also, you estimated that the correlation
coefficient between the return of the two securities is 0.35.
What is the standard deviation of a portfolio (including the
two securities) with GE having a weight of 40%?
The formula in the last slide for the standard deviation of a
portfolio with two securities is

 p  X   (1 X 1 )   2 X 1 (1 X 1 )  12
2
1
2
1
2
2
2
7-20

1/2
.
Note that the covariance between the returns of GE and GM is
0.01953 [   GEGM   GEGM  GE  GM 0.35*0.18*0.31].
Hence,

 p  ( 0 .4 ) ( 0 .1 8 )  ( 0 .6 ) ( 0 .3 1 )  2 ( 0 .4 ) ( 0 .6 ) ( 0 .0 1 9 5 3 )
2
2
2
2
 2 2 .1 7 % .
7-21

1/2