Chapter 5
The Mathematics of Diversification
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1
Introduction
The reason for portfolio theory
mathematics:
• To show why diversification is a good idea
• To show why diversification makes sense
logically
2
Introduction (cont’d)
Harry Markowitz’s efficient portfolios:
• Those portfolios providing the maximum return
for their level of risk
• Those portfolios providing the minimum risk
for a certain level of return
3
Introduction
A portfolio’s performance is the result of
the performance of its components
• The return realized on a portfolio is a linear
combination of the returns on the individual
investments
• The variance of the portfolio is not a linear
combination of component variances
4
Return
The expected return of a portfolio is a
weighted average of the expected returns of
the components:
n
E ( R p )    xi E ( Ri ) 
i 1
where xi  proportion of portfolio
invested in security i and
n
x
i 1
i
1
5
Variance
Introduction
Two-security case
Minimum variance portfolio
Correlation and risk reduction
The n-security case
6
Introduction
Understanding portfolio variance is the
essence of understanding the mathematics
of diversification
• The variance of a linear combination of random
variables is not a weighted average of the
component variances
7
Introduction (cont’d)
For an n-security portfolio, the portfolio
variance is:
n
n
   xi x j ij i j
2
p
i 1 j 1
where xi  proportion of total investment in Security i
ij  correlation coefficient between
Security i and Security j
8
Two-Security Case
For a two-security portfolio containing
Stock A and Stock B, the variance is:
  x   x   2 xA xB  AB A B
2
p
2
A
2
A
2
B
2
B
9
Two Security Case (cont’d)
Example
Assume the following statistics for Stock A and Stock B:
Stock A
Stock B
Expected return
Variance
Standard deviation
.015
.050
.224
.020
.060
.245
Weight
Correlation coefficient
40%
60%
.50
10
Two Security Case (cont’d)
Example (cont’d)
Solution: The expected return of this two-security
portfolio is:
n
E ( R p )    xi E ( Ri ) 
i 1
  x A E ( RA )    xB E ( RB ) 
  0.4(0.015)    0.6(0.020) 
 0.018  1.80%
11
Two Security Case (cont’d)
Example (cont’d)
Solution (cont’d): The variance of this two-security
portfolio is:
 2p  xA2 A2  xB2 B2  2 xA xB  AB A B
 (.4) (.05)  (.6) (.06)  2(.4)(.6)(.5)(.224)(.245)
2
2
 .0080  .0216  .0132
 .0428
12
Minimum Variance Portfolio
The minimum variance portfolio is the
particular combination of securities that will
result in the least possible variance
Solving for the minimum variance portfolio
requires basic calculus
13
Minimum Variance
Portfolio (cont’d)
For a two-security minimum variance
portfolio, the proportions invested in stocks
A and B are:
   A B  AB
xA  2
2
 A   B  2 A B  AB
2
B
xB  1  x A
14
Minimum Variance
Portfolio (cont’d)
Example (cont’d)
Solution: The weights of the minimum variance portfolios
in the previous case are:
 B2   A B  AB
.06  (.224)(.245)(.5)
xA  2

 59.07%
2
 A   B  2 A B  AB .05  .06  2(.224)(.245)(.5)
xB  1  xA  1  .5907  40.93%
15
Minimum Variance
Portfolio (cont’d)
Example (cont’d)
1.2
Weight A
1
0.8
0.6
0.4
0.2
0
0
0.01
0.02
0.03
0.04
Portfolio Variance
0.05
0.06
16
Correlation and
Risk Reduction
Portfolio risk decreases as the correlation
coefficient in the returns of two securities
decreases
Risk reduction is greatest when the
securities are perfectly negatively correlated
If the securities are perfectly positively
correlated, there is no risk reduction
17
The n-Security Case
For an n-security portfolio, the variance is:
n
n
   xi x j ij i j
2
p
i 1 j 1
where xi  proportion of total investment in Security i
ij  correlation coefficient between
Security i and Security j
18
The n-Security Case (cont’d)
A covariance matrix is a tabular
presentation of the pairwise combinations of
all portfolio components
• The required number of covariances to compute
a portfolio variance is (n2 – n)/2
• Any portfolio construction technique using the
full covariance matrix is called a Markowitz
model
19
Example of Variance-Covariance
Matrix Computation in Excel
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
B
C
D
E
F
G
H
I
J
CALCULATING THE VARIANCE-COVARIANCE MATRIX FROM EXCESS RETURNS
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
Mean
AMR
-0.3505
0.7083
0.7329
-0.2034
0.1663
-0.2659
0.0124
-0.0264
1.0642
0.1942
0.2032
BS
-0.1154
0.2472
0.3665
-0.4271
-0.0452
0.0158
0.4751
-0.2042
-0.1493
0.3680
0.0531
GE
-0.4246
0.3719
0.2550
-0.0490
-0.0573
0.0898
0.3350
-0.0275
0.6968
0.3110
0.1501
HR
-0.2107
0.2227
0.5815
-0.0938
0.2751
0.0793
-0.1894
-0.7427
-0.2615
1.8682
0.1529
MO
-0.0758
0.0213
0.1276
0.0712
0.1372
0.0215
0.2002
0.0913
0.2243
0.2066
0.1025
UK
0.2331
0.3569
0.0781
-0.2721
-0.1346
0.2254
0.3657
0.0479
0.0456
0.2640
0.1210 <-- =AVERAGE(G4:G13)
20
A
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
AMR
BS
GE
HR
MO
UK
B
C
Excess return matrix
AMR
BS
-0.5537
-0.1686
0.5051
0.1940
0.5297
0.3134
-0.4066
-0.4802
-0.0369
-0.0984
-0.4691
-0.0374
-0.1908
0.4220
-0.2296
-0.2574
0.8610
-0.2024
-0.0090
0.3149
D
E
F
G
H
GE
-0.5747
0.2218
0.1049
-0.1991
-0.2074
-0.0603
0.1849
-0.1777
0.5467
0.1609
HR
-0.3635
0.0698
0.4286
-0.2466
0.1222
-0.0736
-0.3423
-0.8956
-0.4144
1.7154
MO
-0.1784
-0.0812
0.0250
-0.0313
0.0347
-0.0810
0.0977
-0.0112
0.1217
0.1041
UK
0.1121
0.2359
-0.0429
-0.3931
-0.2555
0.1044
0.2447
-0.0731
-0.0754 <-- =G12-$G$14
0.1430 <-- =G13-$G$14
Transpose of excess return matrix
1974
1975
1976
1977
-0.5537
0.5051
0.5297
-0.4066
-0.1686
0.1940
0.3134
-0.4802
-0.5747
0.2218
0.1049
-0.1991
-0.3635
0.0698
0.4286
-0.2466
-0.1784
-0.0812
0.0250
-0.0313
0.1121
0.2359
-0.0429
-0.3931
1978
-0.0369
-0.0984
-0.2074
0.1222
0.0347
-0.2555
1979
-0.4691
-0.0374
-0.0603
-0.0736
-0.0810
0.1044
1980
-0.1908
0.4220
0.1849
-0.3423
0.0977
0.2447
I
1981
-0.2296
-0.2574
-0.1777
-0.8956
-0.0112
-0.0731
J
K
1982
0.8610
-0.2024
0.5467
-0.4144
0.1217
-0.0754
1983
-0.0090
0.3149
0.1609
1.7154
0.1041
0.1430
Cells B31:K36 contain the array formula =TRANSPOSE(B18:G27). To
enter this formula:
1. Mark the area B31:K36
2. Type =TRANSPOSE(B18:G27)
3. Instead of [Enter], finish with [Ctrl]-[Shift]-[Enter]
The formula will appear as {=TRANSPOSE(B18:G27)}
21
A
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
B
C
D
E
F
G
Product of transpose[excess return] times [excess return] / 10
AMR
BS
GE
HR
MO
UK
AMR
0.2060
0.0375
0.1077
0.0493
0.0208
0.0059
BS
0.0375
0.0790
0.0355
0.1028
0.0089
0.0406
GE
0.1077
0.0355
0.0867
0.0443
0.0194
0.0148
HR
0.0493
0.1028
0.0443
0.4435
0.0193
0.0274
MO
0.0208
0.0089
0.0194
0.0193
0.0083 -0.0015
UK
0.0059
0.0406
0.0148
0.0274 -0.0015 0.0392
H
Cells B47:G52 contain the array formula =MMULT(B31:K36,B18:G27)/10 . To
enter this formula:
1. Mark the whole area
2. Type =MMULT(B31:K36,B18:G27)/10
3. Instead of [Enter], finish with [Ctrl]-[Shift]-[Enter]
The formula will appear as {=MMULT(B31:K36,B18:G27)/10}
22
Portfolio Mathematics (Matrix Form)
Define w as the (vertical) vector of weights on the
different assets.
Define  the (vertical) vector of expected returns
Let V be their variance-covariance matrix
The variance of the portfolio is thus:
  w 'Vw
2
p
Portfolio optimization consists of minimizing this
variance subject to the constraint of achieving a
given expected return.
23
Portfolio Variance in the 2-asset case
We have:
 wA 
w 
 wB 
and
  A2  AB 
V 
2 


B 
 AB
Hence:
2


 AB   wA 
2
A
 p  w 'Vw   wA wB  

2 
w
 AB  B   B 
 p2  wA2 A2  wB2 B2  2wA wB AB
 p2  wA2 A2  wB2 B2  2wAwB  AB A B
24
Covariance Between Two Portfolios
(Matrix Form)
Define w1 as the (vertical) vector of weights on
the different assets in portfolio P1.
Define w2 as the (vertical) vector of weights on
the different assets in portfolio P2.
Define  the (vertical) vector of expected returns
Let V be their variance-covariance matrix
The covariance between the two portfolios is:
 P , P  w1 'Vw2  w2 'Vw1
1
2
(by symmetry)
25
The Optimization Problem
Minimize w 'Vw
w
Subject to:

w 1



 ' w  E ( Rp )
1'
where E(Rp) is the desired (target) expected return on the
portfolio and is a vector of ones and the vector  is
1
defined as:
 1   E ( R1 ) 

     

 n   E ( Rn ) 
26
Min
w
Lagrangian Method
1
L  w 'Vw   E ( R )  w '     1  w '1 


2
p
 
1


Or: Min L  w 'Vw   E ( Rp )  w '  , 1  w '   

  
2
w
1
 1
1
Thus: Min L  w 'Vw   E ( Rp ),1  w '  ,

2
w
 1
where the notation  ,
 1 1
  1
indicates the matrix  2 





1
 n 
  
  
27
Taking Derivatives
 1
 1
 L
 
 
1
 Vw   ,    0  w  V  ,  

 
 
 w

 L 

 
 0       E ( R ),1  w '  ,  0, 0
 
 p 
 0   L 

  

 1
(1)
(2)
Plugging (1) into (2) yields:
  ' 1
 E ( Rp ),1   ,     V  ,
 
1'
 1  0, 0
28
And so we have:
   ' 1
 ,     E ( Rp ),1    V  ,
  
In other words:
1'
 1



1
 1 '  1
1
  
 E ( Rp )
1

     , V  ,   1 
 


Plugging the last expression back into (1) finally yields:
 1  1 '
w  V 1   ,
( n1)
( n n )


   ,  V 1   ,
( nn )

( n2)
 (2n )
( n2)
( n2)
 1
1

  E ( Rp )
 

1

 

(21)
(22)
( n1)
29
The last equation solves the mean-variance
portfolio problem. The equation gives us the
optimal weights achieving the lowest
portfolio variance given a desired expected
portfolio return.
Finally, plugging the optimal portfolio
2

weights back into the variance p  w 'Vw
gives us the efficient portfolio frontier:
 1 'V  ,1
   E ( Rp ),1   ,

2
p
1
  E ( Rp ) 
  1 
1
30
Global Minimum Variance Portfolio
In a similar fashion, we can solve for the global
minimum variance portfolio:
*
1'
V 

1'V 1
1
1
 
2
*
1'V 1
1
1
with w*
1

1'V 1
V 1
1
The global minimum variance portfolio is the
efficient frontier portfolio that displays the
absolute minimum variance.
31
Another Way to Derive the MeanVariance Efficient Portfolio Frontier
Make use of the following property: if two
portfolios lie on the efficient frontier, any
linear combination of these portfolios will
also lie on the frontier. Therefore, just find
two mean-variance efficient portfolios, and
compute/plot the mean and standard
deviation of various linear combinations of
these portfolios.
32
A
B
C
D
E
F
G
H
I
J
K
1
EXAMPLE OF A FOUR-ASSET PORTFOLIO PROBLEM
2
3
Variance-covariance
Mean returns
4
0.10
0.01
0.03
0.05
6%
5
0.01
0.30
0.06
-0.04
8%
6
0.03
0.06
0.40
0.02
10%
7
0.05
-0.04
0.02
0.50
15%
8 Assume you have found two portfolios on the mean-variance efficient frontier, having the following weights:
9
Portfolio 1
0.2
0.3
0.4
0.1
10
Portfolio 2
0.2
0.1
0.1
0.6
11 Thus
12
Portfolio 1
Portfolio 2
13
Mean
9.10%
Mean
12.00% <-- =MMULT(C10:F10,$G$4:$G$7)
14
Variance
12.16%
Variance 20.34% <-- =MMULT(C10:F10,MMULT(B4:E7,D21:D24))
15
16
Covariance 0.0714 <-- =MMULT(C9:F9,MMULT(B4:E7,D21:D24))
17
Correlation 0.4540 <-- =C16/SQRT(C14*F14)
18
19
Transposes
20
Portfolio 1
Portfolio 2
21
0.2
0.2
22
0.3
0.1
23
0.4
0.1
24
0.1
0.6
33
A
C
D
E
F
G
H
I
J
Calculating returns of combinations of Portfolio 1 and Portfolio 2
Proportion of Portfolio 1
Mean return
Variance of return
Stand. dev. of return
0.3
11.13% <-- =B27*C13+(1-B27)*F13
14.06% <-- =B27^2*C14+(1-B27)^2*F14+2*B27*(1-B27)*C16
37.50% <-- =SQRT(B29)
Table of returns (uses this example and Data|Table)
Proportion Stand. dev.
37.50%
0
45.10%
0.1
42.29%
0.2
39.74%
0.3
37.50%
0.4
35.63%
0.5
34.20%
0.6
33.26%
0.7
32.84%
0.8
32.99%
0.9
33.67%
1
34.87%
1.1
36.53%
1.2
38.60%
Mean
11.13% <--the content of these cells is given below:
12.00%
<-- =B30
11.71%
<-- =B28
11.42%
11.13%
Four-Asset Portfolio Returns
10.84%
10.55%
13.0%
10.26%
12.0%
9.97%
9.68%
11.0%
9.39%
10.0%
9.10%
9.0%
8.81%
8.52%
8.0%
Mean return
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
B
30.0%
35.0%
40.0%
45.0%
50.0%
Standard deviation
34
K
Some Excel Tips
To give a name to an array (i.e., to name a
matrix or a vector):
• Highlight the array (the numbers defining the
matrix)
• Click on ‘Insert’, then ‘Name’, and finally
‘Define’ and type in the desired name.
35
Excel Tips (Cont’d)
To compute the inverse of a matrix
previously named (as an example) “V”:
• Type the following formula: ‘=minverse(V)’
and click ENTER.
• Re-select the cell where you just entered the
formula, and highlight a larger area/array of the
size that you predict the inverse matrix will
take.
• Press F2, then CTRL + SHIFT + ENTER
36
Excel Tips (end)
To multiply two matrices named “V” and
“W”:
• Type the following formula: ‘=mmult(V,W)’
and click ENTER.
• Re-select the cell where you just entered the
formula, and highlight a larger area/array of the
size that you predict the product matrix will
take.
• Press F2, then CTRL + SHIFT + ENTER
37
Single-Index Model
Computational Advantages
The single-index model compares all
securities to a single benchmark
• An alternative to comparing a security to each
of the others
• By observing how two independent securities
behave relative to a third value, we learn
something about how the securities are likely to
behave relative to each other
38
Computational
Advantages (cont’d)
A single index drastically reduces the
number of computations needed to
determine portfolio variance
• A security’s beta is an example:
i 
COV ( Ri , Rm )
 m2
where Rm  return on the market index
 m2  variance of the market returns
Ri  return on Security i
39
Portfolio Statistics With the
Single-Index Model
Beta of a portfolio:
n
 p   xi  i
i 1
Variance of a portfolio:
 2p   p2 m2   ep2
  p2 m2
40
Proof
Ri  R f  i ( Rm  R f )  ei
n
n
n
i 1
i 1
i 1
R p   xi Ri R f   xi  i ( Rm  R f )   xi ei
p
ep
n
n
n
i 1
i 1
i 1
R p  R f   xi  i Rm   xi  i R f   xi ei
p
p
ep
2


n
n


 p2    xi  i   m2   xi2 ie2   p2 m2   ep2   p2 m2
i 1
 i 1

  p 
41
Portfolio Statistics With the
Single-Index Model (cont’d)
Variance of a portfolio component:
    
2
i
2
i
2
m
2
ei
Covariance of two portfolio components:
 AB   A  B m2
42
Proof
Ri  R f  i Rm  i R f  ei
 i2  i2 m2   ei2
 A, B  Cov( RA , RB )  Cov( R f   A Rm   A R f  eA , R f   B Rm   B R f  eB )
 A, B  Cov(  A Rm  eA ,  B Rm  eB )
 A, B  Cov(  A Rm ,  B Rm )  Cov(eA ,  B Rm )  Cov(  A Rm , eB )  Cov(eA , eB )
 A, B   A  B Cov( Rm , Rm )   A  B m2
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Multi-Index Model
A multi-index model considers independent
variables other than the performance of an
overall market index
• Of particular interest are industry effects
– Factors associated with a particular line of business
– E.g., the performance of grocery stores vs. steel
companies in a recession
44
Multi-Index Model (cont’d)
The general form of a multi-index model:
Ri  ai   im I m   i1 I1   i 2 I 2  ...   in I n
where ai  constant
I m  return on the market index
I j  return on an industry index
 ij  Security i's beta for industry index j
 im  Security i's market beta
Ri  return on Security i
45