The document discusses portfolio theory and diversification from a mathematical perspective. It introduces portfolio variance and how diversifying investments reduces risk. The variance of a portfolio is not a linear combination of the component variances due to correlation between investments. Harry Markowitz's efficient portfolios provide the maximum return for a given level of risk or minimum risk for a given level of return through diversification.

1. Chapter 5
The Mathematics of Diversification
1

2. Introduction
The reason for portfolio theory
mathematics:
• To show why diversification is a good idea
• To show why diversification makes sense
logically
2

3. 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

4. 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

5. 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

6. Variance
Introduction
Two-security case
Minimum variance portfolio
Correlation and risk reduction
The n-security case
6

7. 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

8. Introduction (cont’d)
For an n-security portfolio, the portfolio
variance is:
n n
σ = ∑∑ xi x j ρijσ iσ j
2
p
i =1 j =1
where xi = proportion of total investment in Security i
ρij = correlation coefficient between
Security i and Security j
8

9. Two-Security Case
For a two-security portfolio containing
Stock A and Stock B, the variance is:
σ = x σ + x σ + 2 x A xB ρ ABσ Aσ B
2
p
2
A
2
A
2
B
2
B
9

10. Two Security Case (cont’d)
Example
Assume the following statistics for Stock A and Stock B:
Stock A Stock B
Expected return .015 .020
Variance .050 .060
Standard deviation .224 .245
Weight 40% 60%
Correlation coefficient .50
10

11. 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

12. Two Security Case (cont’d)
Example (cont’d)
Solution (cont’d): The variance of this two-security
portfolio is:
σ 2 = x Aσ A + xBσ B + 2 x A xB ρ ABσ Aσ B
p
2 2 2 2
= (.4) (.05) + (.6) (.06) + 2(.4)(.6)(.5)(.224)(.245)
2 2
= .0080 + .0216 + .0132
= .0428
12

13. 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

14. Minimum Variance
Portfolio (cont’d)
For a two-security minimum variance
portfolio, the proportions invested in stocks
A and B are:
σ − σ Aσ B ρ AB
2
xA = 2 B
σ A + σ B − 2σ Aσ B ρ AB
2
xB = 1 − x A
14

15. Minimum Variance
Portfolio (cont’d)
Example (cont’d)
Solution: The weights of the minimum variance portfolios
in the previous case are:
σ B − σ Aσ B ρ AB
2
.06 − (.224)(.245)(.5)
xA = 2 = = 59.07%
σ A + σ B − 2σ Aσ B ρ AB .05 + .06 − 2(.224)(.245)(.5)
2
xB = 1 − x A = 1 − .5907 = 40.93%
15

16. Minimum Variance
Portfolio (cont’d)
Example (cont’d)
1.2
1
0.8
0.6
At hg e W
0.4
0.2
i
0
0 0.01 0.02 0.03 0.04 0.05 0.06
Portfolio Variance 16

17. 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

18. The n-Security Case
For an n-security portfolio, the variance is:
n n
σ = ∑∑ xi x j ρijσ iσ j
2
p
i =1 j =1
where xi = proportion of total investment in Security i
ρij = correlation coefficient between
Security i and Security j
18

19. 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

20. Example of Variance-Covariance
Matrix Computation in Excel
A B C D E F G H I J
1 CALCULATING THE VARIANCE-COVARIANCE MATRIX FROM EXCESS RETURNS
2
3 AMR BS GE HR MO UK
4 1974 -0.3505 -0.1154 -0.4246 -0.2107 -0.0758 0.2331
5 1975 0.7083 0.2472 0.3719 0.2227 0.0213 0.3569
6 1976 0.7329 0.3665 0.2550 0.5815 0.1276 0.0781
7 1977 -0.2034 -0.4271 -0.0490 -0.0938 0.0712 -0.2721
8 1978 0.1663 -0.0452 -0.0573 0.2751 0.1372 -0.1346
9 1979 -0.2659 0.0158 0.0898 0.0793 0.0215 0.2254
10 1980 0.0124 0.4751 0.3350 -0.1894 0.2002 0.3657
11 1981 -0.0264 -0.2042 -0.0275 -0.7427 0.0913 0.0479
12 1982 1.0642 -0.1493 0.6968 -0.2615 0.2243 0.0456
13 1983 0.1942 0.3680 0.3110 1.8682 0.2066 0.2640
14 Mean 0.2032 0.0531 0.1501 0.1529 0.1025 0.1210 <-- =AVERAGE(G4:G13)
20

21. A B C D E F G H I J K
16 Excess return matrix
17 AMR BS GE HR MO UK
18 1974 -0.5537 -0.1686 -0.5747 -0.3635 -0.1784 0.1121
19 1975 0.5051 0.1940 0.2218 0.0698 -0.0812 0.2359
20 1976 0.5297 0.3134 0.1049 0.4286 0.0250 -0.0429
21 1977 -0.4066 -0.4802 -0.1991 -0.2466 -0.0313 -0.3931
22 1978 -0.0369 -0.0984 -0.2074 0.1222 0.0347 -0.2555
23 1979 -0.4691 -0.0374 -0.0603 -0.0736 -0.0810 0.1044
24 1980 -0.1908 0.4220 0.1849 -0.3423 0.0977 0.2447
25 1981 -0.2296 -0.2574 -0.1777 -0.8956 -0.0112 -0.0731
26 1982 0.8610 -0.2024 0.5467 -0.4144 0.1217 -0.0754 <-- =G12-$G$14
27 1983 -0.0090 0.3149 0.1609 1.7154 0.1041 0.1430 <-- =G13-$G$14
28
29 Transpose of excess return matrix
30 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983
31 AMR -0.5537 0.5051 0.5297 -0.4066 -0.0369 -0.4691 -0.1908 -0.2296 0.8610 -0.0090
32 BS -0.1686 0.1940 0.3134 -0.4802 -0.0984 -0.0374 0.4220 -0.2574 -0.2024 0.3149
33 GE -0.5747 0.2218 0.1049 -0.1991 -0.2074 -0.0603 0.1849 -0.1777 0.5467 0.1609
34 HR -0.3635 0.0698 0.4286 -0.2466 0.1222 -0.0736 -0.3423 -0.8956 -0.4144 1.7154
35 MO -0.1784 -0.0812 0.0250 -0.0313 0.0347 -0.0810 0.0977 -0.0112 0.1217 0.1041
36 UK 0.1121 0.2359 -0.0429 -0.3931 -0.2555 0.1044 0.2447 -0.0731 -0.0754 0.1430
37 Cells B31:K36 contain the array formula =TRANSPOSE(B18:G27). To
38 enter this formula:
39 1. Mark the area B31:K36
40 2. Type =TRANSPOSE(B18:G27)
41 3. Instead of [Enter], finish with [Ctrl]-[Shift]-[Enter]
42 The formula will appear as {=TRANSPOSE(B18:G27)}
43
21

22. A B C D E F G H
45 Product of transpose[excess return] times [excess return] / 10
46 AMR BS GE HR MO UK
47 AMR 0.2060 0.0375 0.1077 0.0493 0.0208 0.0059
48 BS 0.0375 0.0790 0.0355 0.1028 0.0089 0.0406
49 GE 0.1077 0.0355 0.0867 0.0443 0.0194 0.0148
50 HR 0.0493 0.1028 0.0443 0.4435 0.0193 0.0274
51 MO 0.0208 0.0089 0.0194 0.0193 0.0083 -0.0015
52 UK 0.0059 0.0406 0.0148 0.0274 -0.0015 0.0392
53
Cells B47:G52 contain the array formula =MMULT(B31:K36,B18:G27)/10 . To
54
enter this formula:
55
1. Mark the whole area
56
2. Type =MMULT(B31:K36,B18:G27)/10
57
3. Instead of [Enter], finish with [Ctrl]-[Shift]-[Enter]
58
The formula will appear as {=MMULT(B31:K36,B18:G27)/10}
59
22

23. Portfolio Mathematics (Matrix Form)
Define w as the (vertical) vector of weights on the
different assets.
Define µ the (vertical) vector of expected returns
Let V be their variance-covariance matrix
The variance of the portfolio is thus:
σ = w 'Vw
2
p
Portfolio optimization consists of minimizing this
variance subject to the constraint of achieving a
given expected return.
23

24. Portfolio Variance in the 2-asset case
We have:
 wA   σ A σ AB 
2
w=  and V = 2 
 wB  σ AB σ B 
Hence:
 σ A σ AB   wA 
2
σ p = w 'Vw = [ wA wB ] 
2
2 
σ AB σ B   wB 
σ p = wAσ A + wBσ B + 2wA wBσ AB
2 2 2 2 2
σ p = wAσ A + wBσ B + 2wA wB ρ ABσ Aσ B
2 2 2 2 2
24

25. 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.
µ
Define the (vertical) vector of expected returns
Let V be their variance-covariance matrix
The covariance between the two portfolios is:
σ P1 , P2 = w1 'Vw2 = w2 'Vw1 (by symmetry)
25

26. The Optimization Problem
Minimize w 'Vw
w
Subject to: 


1'
w =1
µ ' w = E ( Rp )

where E(Rp) is the desired (target) expected return on the
1
portfolio and is a vector of ones and the vector µ is
 µ1   E ( R1 ) 
defined as:
µ= M = M 
   
 µn   E ( Rn ) 
   
26

27. Lagrangian Method
1
Min L = w 'Vw +  E ( R ) − w ' µ  λ + 1 − w '1 γ
   
w 2
p
 
1  E ( R ) − w ' µ , 1 − w '  λ 
Or: Min L = w 'Vw +  1
2  p  γ 
 
w
w
1
Thus: Min L = w 'Vw + ( E ( R p ),1) − w ' µ ,
2 
 ( 1)  λ 
 γ 
 
 µ1 1
µ 1
( 1)
where the notation µ , indicates the matrix  2
M

M
 
 µn 1
27

28. Taking Derivatives
 ∂L

 ∂w
( 1) λ 
= Vw − µ ,   = 0 ⇒ w = V µ ,  
γ 
−1 λ 
γ 
( 1) (1)
  ∂L 
  
  0   ∂L  ( 1)
  0  =  ∂λ  ⇒ E ( R ),1 − w ' µ , = 0, 0
( p ) ( ) (2)
   
  ∂γ 
Plugging (1) into (2) yields:
 µ ' −1
1'  
 
( 1) = ( 0, 0)
( E ( Rp ),1) − [ λ , γ ]   V µ ,
28

29. And so we have:
−1
  µ ' −1 
1'

 
 
( 1)
[ λ , γ ] = ( E ( Rp ),1)    V µ , 



In other words:
λ  
( 1) ' ( 1)   E ( Rp )
−1
γ  =  µ , V µ ,   1 
−1
     
Plugging the last expression back into (1) finally yields:
−1
 
{ { ( 1) ( 1) '
−1 
w = V × µ , × µ , ×{ × µ ,  ×
V −1
( 1)   E (Rp )
 1 3 ( n×n ) {  4 3 1 
( n×1) ( n× n ) { 2 1 24
1 24  (2×n ) 2444 
4 ( n×3 1444
2) ( n×2)
3 (2×1)
( n×2)
14444444 (2×2) 4 244444444 3
( n×1) 29

30. 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
weights back into the variance σ p = w 'Vw
2
gives us the efficient portfolio frontier:
( 1) 'V ( µ,1)   E ( Rp ) 
−1
σ = ( E ( R p ),1)  µ ,
2 −1
p

   1 ÷
  
30

31. Global Minimum Variance Portfolio
In a similar fashion, we can solve for the global
minimum variance portfolio:
1'V µ 1
(1'V 1)
−1 −1
V −1
µ* = σ =
2 −1
with w* =
1'V 1
−1
*
1'V 1
−1
The global minimum variance portfolio is the
efficient frontier portfolio that displays the
absolute minimum variance.
31

32. Another Way to Derive the Mean-
Variance 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

33. 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

34. A B C D E F G H I J K
26 Calculating returns of combinations of Portfolio 1 and Portfolio 2
27 Proportion of Portfolio 1 0.3
28 Mean return 11.13% <-- =B27*C13+(1-B27)*F13
29 Variance of return 14.06% <-- =B27^2*C14+(1-B27)^2*F14+2*B27*(1-B27)*C16
30 Stand. dev. of return 37.50% <-- =SQRT(B29)
31
32
33 Table of returns (uses this example and Data|Table)
34
35 Proportion Stand. dev. Mean
36 37.50% 11.13% <--the content of these cells is given below:
37 0 45.10% 12.00% <-- =B30
38 0.1 42.29% 11.71% <-- =B28
39 0.2 39.74% 11.42%
40 0.3 37.50% 11.13%
41 0.4 35.63% 10.84% Four-Asset Portfolio Returns
42 0.5 34.20% 10.55% 13.0%
43 0.6 33.26% 10.26%
44 0.7 32.84% 9.97% 12.0%
Mean return
45 0.8 32.99% 9.68% 11.0%
46 0.9 33.67% 9.39%
10.0%
47 1 34.87% 9.10%
48 1.1 36.53% 8.81% 9.0%
49 1.2 38.60% 8.52% 8.0%
50 30.0% 35.0% 40.0% 45.0% 50.0%
51 Standard deviation
52
34

35. 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

36. 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

37. 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

38. 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

39. 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:
% %
COV ( Ri , Rm )
βi =
σm2
%
where R = return on the market index
m
σ m = variance of the market returns
2
%
Ri = return on Security i
39

40. Portfolio Statistics With the
Single-Index Model
Beta of a portfolio:
n
β p = ∑ xi β i
i =1
Variance of a portfolio:
σ 2 = β pσ m + σ ep
p
2 2 2
≈ β pσ m
2 2
40

41. Proof
Ri = R f + βi ( Rm − R f ) + ei
n n n
R p = ∑ xi Ri =R f + ∑ xi β i ( Rm − R f ) + ∑ xi ei
i =1 i =1 i =1
123
4 4 1 32
βp ep
n n n
R p = R f + ∑ xi β i Rm − ∑ xi β i R f + ∑ xi ei
i =1 i =1 i =1
123
4 4 123
4 4 1 32
βp βp ep
2
 
 n  2 n 2 2
σ p =  ∑ xi β i  σ m + ∑ xi σ ie = β pσ m + σ ep ≈ β pσ m
2 2 2 2 2 2
 123 
i =1
4 4 i =1
 βp 
  41

42. Portfolio Statistics With the
Single-Index Model (cont’d)
Variance of a portfolio component:
σ = β σ +σ
i
2
i
2 2
m
2
ei
Covariance of two portfolio components:
σ AB = β A β Bσ m
2
42

43. Proof
Ri = R f + β i Rm − β i R f + ei
σ i2 = β i2σ m + σ ei
2 2
σ A, B = Cov( RA , RB ) = Cov( R f + β A Rm − β A R f + eA , R f + β B Rm − β B R f + eB )
σ A, B = Cov( β A Rm + eA , β B Rm + eB )
σ A, B = Cov( β A Rm , β B Rm ) + Cov(eA , β B Rm ) + Cov( β A Rm , eB ) + Cov(eA , eB )
σ A, B = β A β B Cov( Rm , Rm ) = β A β Bσ m
2
43

44. 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

45. Multi-Index Model (cont’d)
The general form of a multi-index model:
% % % % %
Ri = ai + β im I m + β i1 I1 + β i 2 I 2 + ... + β in I n
where ai = constant
%
I m = return on the market index
%
I = return on an industry index
j
β ij = Security i's beta for industry index j
β im = Security i's market beta
%
Ri = return on Security i
45