1

Chapter 5

The Mathematics of Diversification

2

Introduction The reason for portfolio theory

mathematics:• To show why diversification is a good idea

• To show why diversification makes sense logically

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

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

5

Return The expected return of a portfolio is a

weighted average of the expected returns of the components:

1

1

( ) ( )

where proportion of portfolio

invested in security and

1

n

p i ii

i

n

ii

E R x E R

x

i

x

6

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

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

8

Introduction (cont’d) For an n-security portfolio, the portfolio

variance is:

2

1 1

where proportion of total investment in Security

correlation coefficient between

Security and Security

n n

p i j ij i ji j

i

ij

x x

x i

i j

9

Two-Security Case For a two-security portfolio containing

Stock A and Stock B, the variance is:

2 2 2 2 2 2p A A B B A B AB A Bx x x x

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

11

Two Security Case (cont’d)Example (cont’d)

Solution: The expected return of this two-security portfolio is:

1

( ) ( )

( ) ( )

0.4(0.015) 0.6(0.020)

0.018 1.80%

n

p i ii

A A B B

E R x E R

x E R x E R

12

Two Security Case (cont’d)Example (cont’d)

Solution (cont’d): The variance of this two-security portfolio is:

2 2 2 2 2

2 2

2

(.4) (.05) (.6) (.06) 2(.4)(.6)(.5)(.224)(.245)

.0080 .0216 .0132

.0428

p A A B B A B AB A Bx x x x

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

14

Minimum Variance Portfolio (cont’d)

For a two-security minimum variance portfolio, the proportions invested in stocks A and B are:

2

2 2 2

1

B A B ABA

A B A B AB

B A

x

x x

15

Minimum Variance Portfolio (cont’d)

Example (cont’d)

Solution: The weights of the minimum variance portfolios in the previous case are:

2

2 2

.06 (.224)(.245)(.5)59.07%

2 .05 .06 2(.224)(.245)(.5)

1 1 .5907 40.93%

B A B ABA

A B A B AB

B A

x

x x

16

Minimum Variance Portfolio (cont’d)

Example (cont’d)

0

0.2

0.4

0.6

0.8

1

1.2

0 0.01 0.02 0.03 0.04 0.05 0.06

Wei

ght A

Portfolio Variance

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

18

The n-Security Case For an n-security portfolio, the variance is:

2

1 1

where proportion of total investment in Security

correlation coefficient between

Security and Security

n n

p i j ij i ji j

i

ij

x x

x i

i j

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

20

Example of Variance-Covariance Matrix Computation in Excel

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1011121314

A B C D E F G H I J

CALCULATING THE VARIANCE-COVARIANCE MATRIX FROM EXCESS RETURNS

AMR BS GE HR MO UK1974 -0.3505 -0.1154 -0.4246 -0.2107 -0.0758 0.23311975 0.7083 0.2472 0.3719 0.2227 0.0213 0.35691976 0.7329 0.3665 0.2550 0.5815 0.1276 0.07811977 -0.2034 -0.4271 -0.0490 -0.0938 0.0712 -0.27211978 0.1663 -0.0452 -0.0573 0.2751 0.1372 -0.13461979 -0.2659 0.0158 0.0898 0.0793 0.0215 0.22541980 0.0124 0.4751 0.3350 -0.1894 0.2002 0.36571981 -0.0264 -0.2042 -0.0275 -0.7427 0.0913 0.04791982 1.0642 -0.1493 0.6968 -0.2615 0.2243 0.04561983 0.1942 0.3680 0.3110 1.8682 0.2066 0.2640Mean 0.2032 0.0531 0.1501 0.1529 0.1025 0.1210 <-- =AVERAGE(G4:G13)

21

16171819202122232425262728293031323334353637383940414243

A B C D E F G H I J KExcess return matrix AMR BS GE HR MO UK

1974 -0.5537 -0.1686 -0.5747 -0.3635 -0.1784 0.11211975 0.5051 0.1940 0.2218 0.0698 -0.0812 0.23591976 0.5297 0.3134 0.1049 0.4286 0.0250 -0.04291977 -0.4066 -0.4802 -0.1991 -0.2466 -0.0313 -0.39311978 -0.0369 -0.0984 -0.2074 0.1222 0.0347 -0.25551979 -0.4691 -0.0374 -0.0603 -0.0736 -0.0810 0.10441980 -0.1908 0.4220 0.1849 -0.3423 0.0977 0.24471981 -0.2296 -0.2574 -0.1777 -0.8956 -0.0112 -0.07311982 0.8610 -0.2024 0.5467 -0.4144 0.1217 -0.0754 <-- =G12-$G$141983 -0.0090 0.3149 0.1609 1.7154 0.1041 0.1430 <-- =G13-$G$14

Transpose of excess return matrix1974 1975 1976 1977 1978 1979 1980 1981 1982 1983

AMR -0.5537 0.5051 0.5297 -0.4066 -0.0369 -0.4691 -0.1908 -0.2296 0.8610 -0.0090 BS -0.1686 0.1940 0.3134 -0.4802 -0.0984 -0.0374 0.4220 -0.2574 -0.2024 0.3149 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 0.1430

Cells B31:K36 contain the array formula =TRANSPOSE(B18:G27). To enter this formula: 1. Mark the area B31:K362. Type =TRANSPOSE(B18:G27)3. Instead of [Enter], finish with [Ctrl]-[Shift]-[Enter]The formula will appear as {=TRANSPOSE(B18:G27)}

22

454647484950515253545556575859

A B C D E F G HProduct 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

Cells B47:G52 contain the array formula =MMULT(B31:K36,B18:G27)/10 . To enter this formula:1. Mark the whole area2. 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}

23

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:

Portfolio optimization consists of minimizing this variance subject to the constraint of achieving a given expected return.

2 'p w Vw

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Portfolio Variance in the 2-asset caseWe have:

Hence:

2 2 2 2 2 2p A A B B A B ABw w w w

2

2and A A AB

B AB B

ww V

w

2

2

2' AA AB

p A BBAB B

ww Vw w w

w

2 2 2 2 2 2p A A B B A B AB A Bw w w w

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.

Define the (vertical) vector of expected returns Let V be their variance-covariance matrix The covariance between the two portfolios is:

1 2, 1 2 2 1' ' (by symmetry)P P w Vw w Vw

26

The Optimization Problem Minimize

Subject to:

where E(Rp) is the desired (target) expected return on the portfolio and is a vector of ones and the vector is

defined as:

'w Vww

1

' ( )

1'p

w

w E R

1 1 1( )

( )n n

E R

E R

27

Lagrangian MethodMin

Or: Min

Thus: Min

1' ( ) ' , 1 '

21pL w Vw E R w w

1' ( ) ' 1 '

21pL w Vw E R w w w

w

1' ( ),1 ' ,

21pL w Vw E R w

w

1

2

1

1where the notation , indicates the matrix

1

1

n

28

Taking Derivatives

1

1

, 0 , (1)

0( ),1 ' , 0,0 (2)

0

Plugging (1) into (2) yields:

'( ),1 , , 0,0

1 1

1

11'

p

p

LVw w V

w

L

E R wL

E R V

29

1

1

11

1

( 1) ( )( 2)

( 2)

And so we have:

', ( ),1 ,

In other words:

( ), ,

1

Plugging the last expression back into (1) finally yields:

,

11'

1 ' 1

1

p

p

n n nn

n

E R V

E RV

w V

1

1

( )( 2)(2 )

(2 1)

(2 2)

( 1)

( ), ,

11 ' 1 p

n nnn

n

E RV

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

gives us the efficient portfolio frontier:

2 'p w Vw

12 1 ( )

( ),1 , ,1

1 ' 1 pp p

E RE R V

31

Global Minimum Variance Portfolio In a similar fashion, we can solve for the global

minimum variance portfolio:

The global minimum variance portfolio is the efficient frontier portfolio that displays the absolute minimum variance.

1 11

2 1* * *1 1

1' 11' 11' 1 1' 1

V VV with w

V V

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.

33

123456789101112131415161718192021222324

A B C D E F G H I J K

EXAMPLE OF A FOUR-ASSET PORTFOLIO PROBLEM

Variance-covariance Mean returns0.10 0.01 0.03 0.05 6%0.01 0.30 0.06 -0.04 8%0.03 0.06 0.40 0.02 10%0.05 -0.04 0.02 0.50 15%

Assume you have found two portfolios on the mean-variance efficient frontier, having the following weights:Portfolio 1 0.2 0.3 0.4 0.1Portfolio 2 0.2 0.1 0.1 0.6

ThusPortfolio 1 Portfolio 2Mean 9.10% Mean 12.00% <-- =MMULT(C10:F10,$G$4:$G$7)Variance 12.16% Variance 20.34% <-- =MMULT(C10:F10,MMULT(B4:E7,D21:D24))

Covariance 0.0714 <-- =MMULT(C9:F9,MMULT(B4:E7,D21:D24))Correlation 0.4540 <-- =C16/SQRT(C14*F14)

TransposesPortfolio 1 Portfolio 2

0.2 0.20.3 0.10.4 0.10.1 0.6

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262728293031323334353637383940414243444546474849505152

A B C D E F G H I J K

Calculating returns of combinations of Portfolio 1 and Portfolio 2Proportion of Portfolio 1 0.3Mean return 11.13% <-- =B27*C13+(1-B27)*F13Variance of return 14.06% <-- =B27^2*C14+(1-B27)^2*F14+2*B27*(1-B27)*C16Stand. dev. of return 37.50% <-- =SQRT(B29)

Table of returns (uses this example and Data|Table)

Proportion Stand. dev. Mean37.50% 11.13% <--the content of these cells is given below:

0 45.10% 12.00% <-- =B300.1 42.29% 11.71% <-- =B280.2 39.74% 11.42%0.3 37.50% 11.13%0.4 35.63% 10.84%0.5 34.20% 10.55%0.6 33.26% 10.26%0.7 32.84% 9.97%0.8 32.99% 9.68%0.9 33.67% 9.39%

1 34.87% 9.10%1.1 36.53% 8.81%1.2 38.60% 8.52%

Four-Asset Portfolio Returns

8.0%

9.0%

10.0%

11.0%

12.0%

13.0%

30.0% 35.0% 40.0% 45.0% 50.0%

Standard deviation

Mea

n r

etu

rn

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.

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

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

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

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:

2

2

( , )

where return on the market index

variance of the market returns

return on Security

i mi

m

m

m

i

COV R R

R

R i

40

Portfolio Statistics With the Single-Index Model

Beta of a portfolio:

Variance of a portfolio:1

n

p i ii

x

2 2 2 2

2 2

p p m ep

p m

41

Proof

1 1 1

1 1 1

2

2 2 2 2 2 2 2 2 2

1 1

( )

( )

p p

p p p

p

i f i m f i

n n n

p i i f i i m f i ii i i

e

n n n

p f i i m i i f i ii i i

e

n n

p i i m i ie p m ep p mi i

R R R R e

R x R R x R R x e

R R x R x R x e

x x

42

Portfolio Statistics With the Single-Index Model (cont’d)

Variance of a portfolio component:

Covariance of two portfolio components:

2 2 2 2i i m ei

2AB A B m

43

Proof

2 2 2 2

,

,

,

2,

( , ) ( , )

( , )

( , ) ( , ) ( , ) ( , )

( , )

i f i m i f i

i i m ei

A B A B f A m A f A f B m B f B

A B A m A B m B

A B A m B m A B m A m B A B

A B A B m m A B m

R R R R e

Cov R R Cov R R R e R R R e

Cov R e R e

Cov R R Cov e R Cov R e Cov e e

Cov R R

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

45

Multi-Index Model (cont’d) The general form of a multi-index model:

1 1 2 2 ...

where constant

return on the market index

return on an industry index

Security 's beta for industry index

Security 's market beta

retur

i i im m i i in n

i

m

j

ij

im

i

R a I I I I

a

I

I

i j

i

R

n on Security i