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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

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