investment analysis and portfolio management lecture 3 gareth myles

Investment Analysis and Portfolio Management

Lecture 3

Gareth Myles

FT 100 Index

£ and $

Risk

VarianceThe standard measure of risk is the

variance of returnor Its square root: the standard deviation

Sample variance The value obtained from past data

Population variance The value from the true model of the data

Sample Variance

General Motors Stock Price 1962-2008

Sample Variance

Year 93-94 94-95 95-96 96-97 97-98

Return % 36.0 -9.2 17.6 7.2 34.1

Year 98-99 99-00 00-01 01-02 02-03

Return % -1.2 25.3 -16.6 12.7 -40.9

Return on General Motors Stock 1993-2003

Sample Variance

-50

-40

-30

-20

-10

0

10

20

30

40

93-94

94-95

95-96

96-97

97-98

98-99

99-00

00-01

01-02

02-03

Graph of return

Sample Variance

With T observations sample variance is

The standard deviation is

Both these are biased estimators The unbiased estimators are

T

tt rr

T 1

22 1

T

tt rr

T 1

21 0

02

T

ttT rr

T 1

21 1

1

T

ttT rr

T 1

221 1

1

Sample Variance For the returns on the General Motors stock,

the mean return is 6.5 Using this value, the deviations from the

mean and their squares are given by

Year 93-94 94-95 95-96 96-97 97-98

29.5 -15.7 11.1 0.7 27.6

870.25 246.49 123.21 0.49 761.76

Year 98-99 99-00 00-01 01-02 02-03

-7.7 18.8 -23.1 6.2 -47.4

59.29 353.44 533.61 38.44 2246.76

rrt 2rrt

2rrt

rrt

Sample Variance

After summing and averaging, the variance is

The standard deviation is

This information can be used to compare different securities

A security has a mean return and a variance of the return

4.5232

88.224.523

Sample Covariance

The covariance measures the way the returns on two assets vary relative to each other Positive: the returns on the assets tend to rise and fall

together Negative: the returns tend to change in opposite

directions Covariance has important consequences for

portfolios

Asset Return in 2001 Return in 2002

A 10 2

B 2 10

Sample Covariance

Mean return on each stock = 6 Variances of the returns are Portfolio: 1/2 of asset A and 1/2 of asset B

Return in 2001:

Return in 2002:

Variance of return on portfolio is 0

1622 BA

6221

1021 pr

61021

221 pr

Sample Covariance

The covariance of the return is

It is always true that

i.

ii.

T

tBBtAAtAB rrrr

T 1

1

BAAB

2iii

Sample Covariance

Example. The table provides the returns on three assets over three years

Mean returns

Year 1 Year 2 Year 3

A 10 12 11

B 10 14 12

C 12 6 9

9,12,11 CBA rrr

Sample Covariance

Covariance between A and B is

Covariance between A and C is

333.1

12121111121411121210111031

AB

2

991111961112912111031

AC

Variance-Covariance Matrix

Covariance between B and C is

The matrix is symmetric

2

2

2

CBCAC

BAB

A

C

B

A

CBA

4

99121296121491212103

1

CB

Variance-Covariance Matrix

For the example the variance-covariance matrix is

642

66.2333.1

666.0

C

B

A

CBA

Population Return and Variance

Expectations: assign probabilities to outcomes Rolling a dice: any integer between 1 and 6 with

probability 1/6 Outcomes and probabilities are:

{1,1/6}, {2,1/6}, {3,1/6}, {4,1/6}, {5,1/6}, {6,1/6} Expected value: average outcome if experiment

repeated

5.3

661

561

461

361

261

161

][

XE


Formally: M possible outcomes Outcome j is a value xj with probability j

Expected value of the random variable X is

The sample mean is the best estimate of the expected value

M

jjj xXE

1][


After market analysis of Esso an analyst determines possible returns in 2010

The expected return on Esso stock using this data is

E[rEsso] = .2(2) + .3(6) + .3(9) + .2(12)

= 7.3

Return 2 6 9 12

Probability 0.2 0.3 0.3 0.2


The expectation can be applied to functions of X For the dice example applied to X2

And to X3

167.15

3661

2561

1661

961

461

161

][ 2

XE

5.73

21661

12561

6461

2761

861

161

][ 3

XE


The expected value of the square of the deviation from the mean is

This is the population variance

9167.2

5.366

15.35

6

15.34

6

1

5.336

15.32

6

15.31

6

1]][[

222

2222

XEXE

Modelling Returns

States of the world Provide a summary of the information about

future return on an asset A way of modelling the randomness in asset

returns Not intended as a practical description

Modelling Returns

Let there be M states of the world Return on an asset in state j is rj

Probability of state j occurring is j

Expected return on asset i is

M

jjj

MM

r

rrrE

1

11 ...][

Modelling Returns

Example: The temperature next year may be hot, warm or cold

The return on stock in a food production company in each state

If each states occurs with probability 1/3, the expected return on the stock is

State Hot Warm Cold

Return 10 12 18

333.131831

1231

1031

][ rE

Portfolio Expected Return

N assets M states of the world Return on asset i in state j is rij

Probability of state j occurring is j

Xi proportion of the portfolio in asset i Return on the portfolio in state j

N

iijiPj rXr

1

Portfolio Expected Return The expected return on the portfolio

Using returns on individual assets

Collecting terms this is

So

PMMPP rrrE ...11

N

iiMiM

N

iiiP rXrXrE

1111 ...

N

iiMMiiP rrXrE

111 ...

N

iiiP rXr

1


Example: Portfolio of asset A (20%), asset B (80%)

Returns in the 5 possible states and probabilities are:

State 1 2 3 4 5

Probability 0.1 0.2 0.4 0.1 0.2

Return on A 2 6 9 1 2

Return on B 5 1 0 4 3


For the two assets the expected returns are

For the portfolio the expected return is

7.132.041.004.012.051.0

5.522.011.094.062.021.0

B

A

r

r

46.27.18.05.52.0 Pr

Population Variance and Covariance

Population variance

The sample variance is an estimate of this Population covariance

The sample covariance is an estimate of this

22iii rErE

jjiiij rErrErE


M states of the world, return in state j is rij

Probability of state j is j

Population variance is

Population standard deviation is

M

jiijji rr

1

22

M

jiijji rr

1

2


Example: The table details returns in five possible states and the probabilities

The population variance is

State 1 2 3 4 5

Return 5 2 -1 6 3

Probability 0.1 0.2 0.4 0.1 0.2

9.7

332.361.314.322.351. 222222

Portfolio Variance

Two assets A and B Proportions XA and XB

Return on the portfolio rP

Mean return Portfolio variance

Pr

22PPP rErE

Population covariance between A and B is

For M states with probabilities j

M

jBBjAAjjAB rrrr

1

BBAAAB rErrErE

Portfolio Variance

Portfolio Variance

The portfolio return is

So

Collecting terms

BBAAP rXrXr BBAAP rXrXr

22BBAABBAAP rXrXrXrXE

22BBBAAAP rrXrrXE

Squaring

Separate the expectations

Hence

BBAABABBBAAAP rrrrXXrrXrrXE 222222

BBAABA

BBBAAAP

rrrrEXX

rrEXrrEX

2

22222

ABBABBAAP XXXX 222222

Portfolio Variance

Portfolio Variance

Example: Portfolio consisting of 1/3 asset A 2/3 asset B

The variances/covariance are

The portfolio variance is333.3,333.8,333.2 22 ABBA

148.9

333.332

31

2333.832

333.231 22

2

P

Correlation Coefficient

The correlation coefficient is defined by

Value satisfies

perfect positive correlation

BA

ABAB

11 AB

1AB

rA

rB


perfect negative correlation

Variance of the return of a portfolio

BAABBABBAAP XXXX 222222

rB

rA

1AB


Example: Portfolio consisting of 1/4 asset A 3/4 asset B

The variances/correlation are

The portfolio variance is

5.0,9,16 22 ABBA

8125.3

)3)(4)(5.0(4

3

4

129

4

316

4

122

2

P

General Formula

N assets, proportions Xi

Portfolio variance is

But so

N

i

N

ikkikkiiiP XXX

1 ,1

222

N

i

N

kikkiP XX

1 1

2

2iii

Effect of Diversification

Diversification: a means of reducing risk Consider holding N assets Proportions Xi = 1/N Variance of portfolio

N

i

N

ikkikiP NN1 ,1

22

22 11


N terms in the first summation, N[ N-1] in the second

Gives

Define

Then

N

i

N

ikk

ikN

iiP NNN

NNN 1 11

22

11111

N

iia N1

22 1

N

i

N

ikk

ikab NN1 1 11

abaP N

N

N

11 22


Let N tend to infinity (extreme diversification) Then

Hence

In a well-diversified portfolio only the covariance between assets counts for portfolio variance

abP 2

01 2

aN ababN

N

1

investment analysis and portfolio management lecture 3 gareth myles

Documents

mean return

expected return

x3population return

return isit

future return

expected value population

possible returns

value xj