statistical calculus in finding standard deviation

7. Calculating Expected Returns and Standard Deviations for Portfolios

7-2

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;

7-3

• 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.

7-4

• 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;

7-5

r X rp ii

N

i ,

1

• The formula to compute the expected return on a portfolio of N securities is

where

r i

Xi

is the expected return on security i.

is the weight of security i; and

7-6

ExampleOn January 25, 1999, you bought the following stocks:

Quantity Price Stock200 171 Microsoft500 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?

Table 7.1

7-7

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]

7-8

r rp ii

i X ,

1

3

Using the formula

the expected return on your portfolio is 14.43%

[=0.1805*20%+0.4897*12%+0.3298*15%].

7-9

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-than-expected returns occur for the other security;

7-10

• 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 (the Greek letter sigma);

• Note that ;• You may use the Excel function “COVAR” to compute the

covariance between the returns of two securities.

ij

ij ji

7-11

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 (the Greek letter pho);

• The relation between covariance and correlation is given by the following equation: ;

• The correlation coefficient between the return of security i and the return of security j lies between -1 and 1;

.

ij

ij ij i j

7-12

• 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-than-expected 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-than-expected 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.

7-13

Figure 7.1 Weekly returns on Amazon.com (y axis) versus Ebay (x axis) in the period October 98-January 99

-30%

-10%

10%

30%

50%

-40% -20% 0% 20% 40% 60%

7-14

Figure 7.2Weekly returns on two internet stocks in the period October 98-January 99

-30%-20%-10%

0%10%20%30%40%50%60%70%

19-O

ct

26-O

ct

2-N

ov

9-N

ov

16-N

ov

23-N

ov

30-N

ov

7-D

ec

14-D

ec

21-D

ec

28-D

ec

4-Ja

n

11-J

an

18-J

an

25-J

an

Wea

kly

Ret

urn

Ebay

Amazon.com

7-15

Figure 7.3Weekly returns on GM (y axis) and Ebay (x axis) in the period October 98-January 99

-4%

0%

4%

8%

12%

-30% -10% 10% 30% 50% 70%

7-16

-30%-20%-10%

0%10%20%30%40%50%60%70%

19-O

ct

26-O

ct

2-N

ov

9-N

ov

16-N

ov

23-N

ov

30-N

ov

7-D

ec

14-D

ec

21-D

ec

28-D

ec

4-Ja

n

11-J

an

18-J

an

25-J

an

Wea

kly

Ret

urn

Ebay

GM

Figure 7.4Weekly returns on Ebay and GM in the period October 98-January 99

7-17

• 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 ij

N

ji

N

ijX

X

1/2

.

11

7-18

• The standard deviation is the square root of the variance.

7.8 Important Special Cases• Note that ;• Hence, ;• 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,Hence,

ii1 ii ii i i i 2

11 11 1 1 12

22 22 2 2 22 and .

7-19

p ij

ji

ijX

X

1/2

1

2

1

2

X X X X X X X X1 1 11 1 2 12 2 1 21 2 2 22 1/2

X X X X12

12

22

22

1 2 122 1/2

X X X X12

12

12

22

1 1 121 2 1 ( ) ( )1/2

.

7-20

ExampleYou 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 X X X 12

12

12

22

1 1 121 2 1( ) ( )1/2

.

7-21

p ( . ) ( . ) ( . ) ( . ) ( . )( . )( . )0 4 018 0 6 0 31 2 0 4 0 6 0 019532 2 2 2 1/2

Note that the covariance between the returns of GE and GM is0.01953

Hence,

[ . * . * . ]. GEGM GEGM GE GM 0 35 018 0 31

22 17%..