The Mathematics of Diversification
Chapter Six
The Mathematics of Diversification
ANSWERS TO QUESTIONS
1. Selling stock short brings cash in rather than requiring a cash outflow. In the absence
of margin requirements (which is arguably true for large institutional investors), this
means there is no initial investment, and any gain on no investment is an infinite
return.
2. Student response
3. The two-security portfolio is preferable, as it has higher expected return per unit of
risk.
4. Covariance is the expected value of the product of two numbers. Each of the two
numbers is a value minus its mean. Some values lie below the mean, some above.
Consequently, each number can be positive or negative, and the product can therefore
be positive or negative. Depending on the nature of the dispersions around the means,
the expected value of the product can be positive or negative.
~
~
~
5. E[( a~  a )( b  b )]  cov( a~, b ) and E[( b  b )( a~  a )]  cov( b , a ) . By the commutative
law for multiplication, ab = ba. This means the order of the two products inside the
~
~
expected value operator can be reversed and cov( a~, b )  cov( b , a~ ).
6.  a   am a m
Where
 am = correlation between security a and the market
 a = standard deviation of security a
 m = standard deviation of the market
7. 0.25 x (4).5 x (6).5= 1.225
8. The size of the error term approaches zero as the number of portfolio components
increases.
9. Standard deviations can only be positive, so a negative correlation means the
covariance is also negative.
38
The Mathematics of Diversification
10. In a prediction model, R squared can only increase if additional explanatory variables
are added. You cannot lose predictive ability by including additional data.
ANSWERS TO PROBLEMS
1.
(n 2  n) 1700 2  1700

 1,444,150
2
2
Note: The first printing of this book had errors in Table 6-5. The covariance values
should be as shown in the table below.
A
.280
.215
.136
.249
A
B
C
D
xa 
2.
B
.215
.360
.170
.185
C
.136
.170
.203
.114
D
.149
.185
.114
.226
 B2   A B  AB
 A2   B2  2 A B  AB
.36  (.28) .5 (.36) .5  AB

.28  .36  2(.28) .5 (.36) .5  AB
 AB 
xA 
=
4
~ ~
cov( A, B )
 A B

.215
 0.677
(.28) .5 (.36) .5
.36  (.28) .5 (.36) .5 (.677)
.28  .36  2(.28) .5 (.36) .5 (.677)
0.1450
= 69.2%
0.2101
1
4
3.  p   x1  i  (1.05 + 1.20 + 0.90 + 0.95) = 1.025
i 1
4.  2p   p2 m 2   ep2 . The error term approaches zero, so  p2  (1.025) 2 (.25)  0.263.
39
The Mathematics of Diversification
5. xA = -.30 xB = .50
xC = .80
N
~
~
E ( R p )   x i E ( Ri )
i 1
= (-.30)(.14) + (.50)(.16) + (.80)(.12) = .1340 = 13.4%
N
N
 p2   x i x j  ij i j
i 1 j 1
 x A2  A2  x B2  B2  xC2  C2  x A xC  AC a C  x A x B  AB A B  x B xC  BC B C
= (-.3)2 (.28) + (.5)2 (.36) + (.8)2 (.203) + (-.3)(.8)  AC (.28).5(.203).5 +
(-.3)(.5)  AB (.28).5(.36).5 +(.5)(.8)  BC (.36).5(.203).5
= .0252 + .09 + .1299 - .0572  AC - .0476  AB + .1081  BC
 AB 
 AC 
 BC 
~ ~
cov( A, B )
 A B

.215
 0.677
(.28) .5 (.36) .5

.136
 0.570
(.28) .5 (.203) .5

.170
 0.629
(.36) .5 (.203) .5
~ ~
cov( A, C )
 A C
~ ~
cov( B , C )
 B C
 2p  .0252  .09  .1299  .0572(.570)  .0476(.677)  .1081(.629)  0.2483
~ ~
6. cov(C, D)   C  D m2
= (0.90)(0.95)(.32) = 0.274
7.  2  25%  .25
 =(.25).5 = .5 = 50%
8.  BC 
9.
40
~ ~
cov( B , C )
 B C

.170
= .629
(.36) .5 (.203) .5
cov(1,2)   1  2 m2
The Mathematics of Diversification
 m2 
cov(1,2)
1  2

1.55
 1.127
(1.10)(1.25)
10. See the computer printouts below.
ENTER UP TO 100 RETURNS FOR UP TO FIVE
SECURITIES
IN LOTUS COLUMNS B, C, D, E, AND
F
HIT ESC to enter data or ALT S for menu.
Return #
EXAMPLE
1
2
3
4
5
6
7
8
Sec 1
10.00%
0.0270
0.0120
-0.0220
0.0130
-0.0110
-0.0330
0.0290
0.0550
Sec 2
Sec 3
Sec 4
-8.00% 20.00% 10.00%
-0.0230 0.0560 0.0020
0.0000 0.0130 0.0040
-0.0100 -0.0150 0.0020
0.0340 0.0150 0.0100
-0.0230 0.0120 -0.0290
-0.0610 -0.0350 -0.0220
0.0260 0.0020 0.0000
0.0450 0.0470 0.0200
Sec 5
0.00%
0.0330
0.0170
-0.0450
0.0080
-0.0190
-0.0240
-0.0010
0.0560
Security
Statisti
cs
Sec 1
0.88%
0.02736
Sec 2
-0.15%
0.03301
Sec 3
Sec 4
Sec 5
1.19% -0.16%
0.31%
0.02786 0.01512 0.03054
mean
std dev
varianc
e
7.49E-04 1.09E-03 7.76E-042.28E-049.33E-04
COVARIANCE MATRIX
Sec
Sec
Sec
Sec
Sec
1
2
3
4
5
Sec 1
7.49E-04
7.05E-04
6.28E-04
3.06E-04
7.53E-04
Sec 2
Sec 3
Sec 4
Sec 5
1.09E-03
4.43E-04 7.76E-04
3.95E-04 2.25E-042.28E-04
5.49E-04 7.26E-042.95E-049.33E-04
41
The Mathematics of Diversification
CORRELATION MATRIX
Sec
Sec
Sec
Sec
Sec
1
2
3
4
5
Sec 1
1.000
0.781
0.824
0.739
0.901
Sec 2
Sec 3
1.000
0.481
0.792
0.545
1.000
0.534
0.853
Sec 4
1.000
0.640
Sec 5
1.000
~
E ( R p )  .30(-.15%) + .70(1.19%) = 0.788%
 2p  x22 22  x32 32  x2 x3 23 2 3
= (.3)2 (.0011) + (.7)2 (.0008) + (.3)(.7)(.481)(.03301)( .02786)
= 0.0006
11. Because we have the entire set of data, we can simply compute each periodic
portfolio return and then determine the mean and variance of this series.
Security
1
0.0270
0.0120
-0.0220
0.0130
-0.0110
-0.0330
0.0290
0.0550
Security
2
-0.0230
0.0000
-0.0100
0.0340
-0.0230
-0.0610
0.0260
0.0450
Security
3
0.0560
0.0130
-0.0150
0.0150
0.0120
-0.0350
0.0020
0.0470
Security
4
0.0020
0.0040
0.0020
0.0100
-0.0290
-0.0220
0.0000
0.0200
Security
5
0.0330
0.0170
-0.0450
0.0080
-0.0190
-0.0240
-0.0010
0.0560
Portfolio
mean
Portfolio
variance
~
E ( R p )  0.0041
Average
0.0190
0.0092
-0.0180
0.0160
-0.0140
-0.0350
0.0112
0.0446
0.0041
0.0005
 p2 = 0.0005
12. The minimum variance combinations are as follows:
Pair
1,2
1,3
42
Proportion in the First
100%
98.72%
Proportion in the Second
0%
1.28%
The Mathematics of Diversification
1,4
1,5
2,3
2,4
2,5
3,4
3,5
4,5
100%
100%
34%
100%
41.5%
0.61%
80.5%
100%
0%
0%
66%
0%
58.5%
99.39%
19.5%
0%
13. See the spreadsheet extracts below from the COV and MINVAR3 templates. The
minimum variance portfolio involves going short security 5 and buying securities 3
and 4.
ENTER UP TO 100 RETURNS FOR UP TO
FIVE SECURITIES
IN LOTUS COLUMNS B, C, D, E,
AND F
Run the \S macro for
the main menu.
4
5
Return
3
#
EXAMPL 10.00% -8.00% 20.00% 10.00%
E
1 0.0560 0.0020 0.0330
2 0.0130 0.0040 0.0170
3 -0.0150 0.0020 -0.0450
4 0.0150 0.0100 0.0080
5 0.0120 -0.0290 -0.0190
6 -0.0350 -0.0220 -0.0240
7 0.0020 0.0000 -0.0010
8 0.0470 0.0200 0.0560
Security
Statistics
3
4
5
mean
1.19% -0.16%
0.31%
std dev 0.02786 0.01512 0.03054
varianc 7.76E- 2.28E- 9.33Ee
04
04
04
43
The Mathematics of Diversification
CORRELATION MATRIX
3
4
5
3
1.000
0.534
0.853
4
5
1.000
0.640
1.000
DETERMINING THE MINIMUM VARIANCE
THREE-SECURITY PORTFOLIO
Enter the INPUT in the boxes provided below
:Variance of Stock A:
Variance of Stock B:
Variance of Stock C:
0.0008
0.0002
0.0009
Correlation between Stocks A and B:
Correlation between Stocks B and C:
Correlation between Stocks A and C:
(Standard deviation =
(Standard deviation =
(Standard deviation =
0.5340
0.8530
0.6400
The Minimum Variance Combination is
:
14.65% <-Stock A
135.65% <-Stock B
COV(A,B)=
COV(B,C)=
COV(A,C)=
0.0002
0.0004
0.0005
-50.30% <-Stock C
With these proportions, Portfolio variance is :->
Portfolio standard deviation is :->
44
0.0279
0.0151
0.0305
0.0001
0.0120
The Mathematics of Diversification
45