A B C D E F G

Data

1

σS σB

E(rS) E(rB)

2

10 6 25 12

3

Portfolio Standard Deviation for Given Correlation (ρ)

Weight in Portfolio

4

σP=[(A*C3)^2+((1-A)*D3)^2+2*A*C3*(1-A)*D3*ρ]^.5

Stocks Expected Return

5

wS E(rP)=A*A3+(1-A)*B3 -1 0 0.2 0.5 1

6

0 6 12 12 12 12 12

7

0.2 6.8 4.6 10.8 11.7 12.9 14.6

8

0.4 7.6 2.8 12.3 13.4 15.0 17.2

9

0.6 8.4 10.2 15.7 16.6 17.9 19.8

10

0.8 9.2 17.6 20.1 20.6 21.3 22.4

11

1 10 25 25 25 25 25

12

w S(min) = (σB^2-σBσSρ)/(σS^2+σB^2-2*σBσSρ)

Minimum Variance Portfolio:

13

w S(min) 0.3243 0.1873 0.1294 -0.0128 -0.9231

14

E(rP) = wS(min)*A3+(1-wS(min))*B3 7.30 6.75 6.52 5.95 2.31

15

σP 0.00 10.82 11.54 11.997 0.00

16

Notes: (1) The standard deviation is calculated from equation 6.3 with the weights of the

17

minimum-variance portfolio:

18

σP=((w S(min)*C3)^2+((1-wS(min))*D3)^2+2*wS(min)*C3*(1-wS(min))*D3*ρ]^.5

19

(2) As the correlation coefficient grows, the minimum variance portfolio requires a smaller

20

position in stocks (even a negative position for higher correlations), and the performance

21

of this portfolio becomes less attractive.

22

(3) With correlation of .5, minimum variance is achieved with a short position in stocks.

23

The standard deviation is slightly lower than that of bonds, but with a slightly lower mean as well.

24

(4) With perfectly positive correlation you can get the standard deviation to zero by taking

25

a large, short position in stocks. The mean return is then as low as 2.31%.

26

Bodie’Kane’Marcus: II. Portfolio Theory 6. Efficient Diversification © The McGraw’Hill

Essentials of Investments, Companies, 2003

Fifth Edition

182 Part TWO Portfolio Theory

way to zero.2 With our data, this will happen when wB 67.57%. While exposing us to zero

risk, investing 32.43% in stocks (rather than placing all funds in bonds) will still increase the

portfolio expected return from 6% to 7.30%. Of course, we can hardly expect results this at-

tractive in reality.

>

2. Suppose that for some reason you are required to invest 50% of your portfolio in

Concept

bonds and 50% in stocks.

CHECK a. If the standard deviation of your portfolio is 15%, what must be the correlation

coefficient between stock and bond returns?

b. What is the expected rate of return on your portfolio?

c. Now suppose that the correlation between stock and bond returns is 0.22 but

that you are free to choose whatever portfolio proportions you desire. Are you

likely to be better or worse off than you were in part (a)?

Let™s return to the data for ABC and XYZ in Example 6.1. Using the spreadsheet estimates of

the means and standard deviations obtained from the AVERAGE and STDEV functions, and

6.3 EXAMPLE the estimate of the correlation coefficient we obtained in that example, we can compute the

risk-return trade-off for various portfolios formed from ABC and XYZ.

Using Historical

Columns E and F in the lower half of the spreadsheet on the following page are calculated

Data to Estimate

from Equations 6.2 and 6.3 respectively, and show the risk-return opportunities. These calcu-

the Investment

lations use the estimates of the stocks™ means in cells B16 and C16, the standard deviations in

Opportunity Set

cells B17 and C17, and the correlation coefficient in cell F10.

Examination of column E shows that the portfolio mean starts at XYZ™s mean of 11.97%

and moves toward ABC™s mean as we increase the weight of ABC and correspondingly reduce

that of XYZ. Examination of the standard deviation in column F shows that diversification

reduces the standard deviation until the proportion in ABC increases above 30%; thereafter,

standard deviation increases. Hence, the minimum-variance portfolio uses weights of approxi-

mately 30% in ABC and 70% in XYZ.

The exact proportion in ABC in the minimum-variance portfolio can be computed from the

formula shown in Spreadsheet 6.6. Note, however, that achieving a minimum-variance port-

folio is not a compelling goal. Investors may well be willing to take on more risk in order to

increase expected return. The investment opportunity set offered by stocks ABC and XYZ may

be found by graphing the expected return“standard deviation pairs in columns E and F.

>

3. The following tables present returns on various pairs of stocks in several periods.

Concept

In part A, we show you a scatter diagram of the returns on the first pair of stocks.

CHECK Draw (or prepare in Excel) similar scatter diagrams for cases B through E. Match

up the diagrams (A“E) to the following list of correlation coefficients by choosing

the correlation that best describes the relationship between the returns on the two

stocks: 1, 0, 0.2, 0.5, 1.0.

(continued)

2

The proportion in bonds that will drive the standard deviation to zero when 1 is:

S

wB

B S

Compare this formula to the formula in footnote 1 for the variance-minimizing proportions when 0.

Bodie’Kane’Marcus: II. Portfolio Theory 6. Efficient Diversification © The McGraw’Hill

Essentials of Investments, Companies, 2003

Fifth Edition

183

6 Efficient Diversification

Spreadsheet for Example 6.3

< Concept

A. % Return

CHECK

Scatter diagram A

Stock 1 Stock 2

6

5 1

1 1 5

4 3

4

Stock 2

2 3

3

3 5

2

1

0

0 1 2 3 4 5 6

Stock 1

(continued)

Bodie’Kane’Marcus: II. Portfolio Theory 6. Efficient Diversification © The McGraw’Hill

Essentials of Investments, Companies, 2003

Fifth Edition

184 Part TWO Portfolio Theory

>

(concluded)

Concept

CHECK B. % Return C. % Return D. % Return E. % Return

Stock 1 Stock 2 Stock 1 Stock 2 Stock 1 Stock 2 Stock 1 Stock 2

1 1 1 5 5 5 5 4

2 2 2 4 1 3 1 3

3 3 3 3 4 3 4 1

4 4 4 2 2 0 2 0

5 5 5 1 3 5 3 5

6.3 THE OPTIMAL RISKY PORTFOLIO

WITH A RISK-FREE ASSET

Now we can expand the asset allocation problem to include a risk-free asset. Let us continue

to use the input data from the bottom of Spreadsheet 6.5, but now assume a realistic correla-

tion coefficient between stocks and bonds of 0.20. Suppose then that we are still confined to

the risky bond and stock funds, but now can also invest in risk-free T-bills yielding 5%. Fig-

ure 6.5 shows the opportunity set generated from the bond and stock funds. This is the same

opportunity set as graphed in Figure 6.4 with BS 0.20.

Two possible capital allocation lines (CALs) are drawn from the risk-free rate (rf 5%) to

two feasible portfolios. The first possible CAL is drawn through the variance-minimizing port-

folio (A), which invests 87.06% in bonds and 12.94% in stocks. Portfolio A™s expected return

is 6.52% and its standard deviation is 11.54%. With a T-bill rate (rf) of 5%, the reward-to-

variability ratio of portfolio A (which is also the slope of the CAL that combines T-bills with

portfolio A) is

E(rA) rf 6.52 5

SA 0.13

11.54

A

F I G U R E 6.5 12

The opportunity set

11

using bonds and

stocks and two capital

10

allocation lines

Expected return (%)

Stocks

9

8

CALB

CALA

7

B

A

6

Bonds

5

4

0 5 10 15 20 25 30 35