Calculating two-asset portfolio expected returns and standard deviations

7 PORTFOLIO THEORY

Glen Arnold: Corporate Financial Management, Second edition© Pearson Education Limited 2002

OHT 7.1

• Calculating two-asset portfolio expected returns and standard deviations

• Estimating measures of the extent of interaction – covariance and correlation coefficients

• Being able to describe dominance, identify efficient portfolios and then apply utility theory to obtain optimum portfolios

• Recognise the properties of the multi-asset portfolio set and demonstrate the theory behind the capital market line

LEARNING OBJECTIVES

7 PORTFOLIO THEORY


OHT 7.2

Holding period returns

One year:

Where: s = semi-annual rateR = annual rate

For three year holding period

7 PORTFOLIO THEORY


OHT 7.3

EXAMPLE:Initial share price = £1.00Share price three years later = £1.20Dividends: year 1 = 6p, year 2 = 7p, year 3 = 8p

7 PORTFOLIO THEORY


OHT 7.4

Ace plc

EXPECTED RETURNS AND STANDARD DEVIATION FOR SHARES

Event Estimated Estimated Return Probabilityselling price, P1 dividend, D1 Ri

Economic boom 114p 6p +20% 0.2Normal growth 100p 5p +5% 0.6Recession 86p 4p –10% 0.2

1.0

A share costs 100p to purchase now and the estimates of returns for thenext year are as follows:

7 PORTFOLIO THEORY


OHT 7.5

THE EXPECTED RETURN

where

R = expected return

Ri = return if event i occurs

pi

= probability of event i occurring

n = number of events

Expected return, Ace plc

Event Probability of Returnevent pi Ri Ri pi

Boom 0.2 +20 4Growth 0.6 +5 3Recession 0.2 –10 –2

Expected returns 5 or 5%

R = Rip

i

n

i = 1

7 PORTFOLIO THEORY


OHT 7.6

R i )Ri

Standard deviation, Ace plc

STANDARD DEVIATION

= n

i =1( R i – R )

2 pi

Probability Return Expected return Deviation

0.2 20% 5% 15 450.6 5% 5% 0 00.2 –10% 5% –15 45

Variance 2 90

Standard deviation 9.49%

Ri– RiRipi

Deviationsquared probability

– (Ripi

2

7 PORTFOLIO THEORY


OHT 7.7

Ri

Returns for a share in Bravo plc

Expected return on Bravo(–15 0.2) + (5 0.6) + (25 0.2) = 5 per cent

Event Return Probability

Boom –15% 0.2Growth +5% 0.6Recession +25% 0.2

1.0

Standard deviation, Bravo plc

0.2 –15% 5% –20 800.6 +5% 5% 0 00.2 +25% 5% +20 801.0 Variance 2

160

Standard deviation 12.65%

piRi

Probability Return Expected return DeviationRi

– RiRipi


– (Ri Ri ) pi2

7 PORTFOLIO THEORY


OHT 7.8

COMBINATIONS OF INVESTMENTS Hypothetical pattern of return for Ace plc

Time(years)

1 2 3 4 5 6 7 8

–10–

Ret

urn

%

+

5

20

–15

Hypothetical pattern of returns for Bravo plc

1 2 3 4 5 6 7 8 Time(years)

25

5

0

–

Ret

urn

%

+

7 PORTFOLIO THEORY


OHT 7.9

Returns over one year from placing £571 in Ace and £429 in Bravo

Event Returns Returns Overall returns PercentageAce Bravo on £1,000 returns

£ £

Boom 571(1.2) = 685 429 – 429(0.15) = 365 1,050 5%

Growth 571(1.05) = 600 429(1.05) = 450 1,050 5%

Recession 571 – 571(0.1) = 514 429 (1.25) = 536 1,050 5%

Hypothetical pattern of returns for Ace, Bravo and the two-asset portfolio

–15

5

25

0

–10

20

Bravo

Portfolio

Ace

1 2 3 4 5 6 7 8 Time(years)–

Ret

urn

%

+

7 PORTFOLIO THEORY


OHT 7.10

Returns over a one-year period from placing £500 in Ace and £500 in Clara

PERFECT NEGATIVE CORRELATION

PERFECT POSITIVE CORRELATION Annual returns on Ace and Clara

Event Returns Returns Overall Percentagei Ace Clara return on return

£ £ £1,000

Boom 600 750 1,350 35%Growth 525 575 1,100 10%Recession 450 400 850 –15%

Event Probability Returns Returnsi pi on Ace on Clara

% %

Boom 0.2 +20 +50Growth 0.6 +5 +15Recession 0.2 –10 –20

7 PORTFOLIO THEORY


OHT 7.11

Hypothetical patterns of returns for Ace and Clara

1 2 3 4 5 6 7 8 Time(years)

–20

–15

–10

5

50

20

+

Ace

Clara

Portfolio

–

7 PORTFOLIO THEORY


OHT 7.12

Expected returns for shares in X and shares in Y

INDEPENDENT INVESTMENTS

Standard deviations for X or Y as single investments

Expected return for shares in X

Return Probability– 25 0.5 = –12.5

35 0.5 = 17.55.0%

–25% 0.5 5% –30 45035% 0.5 5% 30 450

Variance 900Standard deviation 30%

–25 0.5 = –12.535 0.5 = 17.5

5.0%

Return Probability

Expected return for shares in Y

Deviationssquared probabilityReturn Probability Expected return Deviations

RipiRi Ri – R (Ri – R)2 pi

7 PORTFOLIO THEORY


OHT 7.13

Exhibit 7.17 A mixed portfolio: 50 per cent of the fund invested in X and 50 per cent in Y, expected return

Standard deviation, mixed portfolio

–25 0.25 5 –30 2255 0.50 5 0 0

35 0.25 5 30 225

45021.21%

Possible Joint Joint Return outcome returns probability probabilitycombinations

Both firmsdo badly –25 0.5 0.5 = 0.25 –25 0.25 = –6.25

X does badlyY does well 5 0.5 0.5 = 0.25 5 0.25 = 1.25

X does wellY does badly 5 0.5 0.5 = 0.25 5 0.25 = 1.25

Both firmsdo well 35 0.5 0.5 = 0.25 35 0.25 = 8.75

1.00 Expected return 5.00%

Deviationssquared probabilityReturn Probability Expected return Deviations

RpiRi Ri – R (Ri – R)2 pi

Variance Standard deviation

7 PORTFOLIO THEORY


OHT 7.14

Correlation scale

So long as the returns of constituent assets of a portfolio are not perfectly positively correlated, diversification can reduce risk. The degree of risk reduction depends on:

• the extent of statistical interdependence between the returns of the different investments: the more negative the better; and• the number of securities over which to spread the risk: the greater the number, the lower the risk.

A CORRELATION SCALE

IndependentPerfectnegative

correlation

Perfectpositive

correlation

0–1 +1

7 PORTFOLIO THEORY


OHT 7.15

Company A: Standard deviation

THE EFFECTS OF DIVERSIFICATION WHEN SECURITY RETURNS ARE NOT PERFECTLY CORRELATED Returns on shares A and B for alternative economic states

Company A: Expected return

Event i Probability Return on A Return on BState of the economy pi RA RB

Boom 0.3 20% 3%Growth 0.4 10% 35%Recession 0.3 0% –5%

Probability Returnpi RA

0.3 20 60.4 10 40.3 0 0

10%


ExpectedProbability Return return Deviation

0.3 20 10 10 300.4 10 10 0 00.3 0 10 –10 30

60

7.75%

RApi

RApi RA (RA – RA) (RA – RA)2 pi

Variance

Standard deviation

7 PORTFOLIO THEORY


OHT 7.16

Summary table: Expected returns and standard deviations for Companies A and B

Company B: Expected return

Company B: Standard deviation

Probability Returnreturn

Deviation

pi RB (RB – RB)

0.3 3 13.4 10.4 32.450.4 35 13.4 21.6 186.620.3 –5 13.4 –18.4 101.57

320.64 = 17.91%

Probability Return

0.3 3 0.90.4 35 14.0

0.3 –5 –1.5

13.4%


Expected

RB(RB – RB)2 pi

Variance

pi RBRB pi

Company ACompany B

Expected return Standard deviation

10%13.4%

7.75%17.91%

Standard deviation

7 PORTFOLIO THEORY


OHT 7.17

Exhibit: 7.26 Return and standard deviation for shares in firms A and B

A general rule in portfolio theory:

Portfolio returns are a weighted average of the expected returns on the individual investment…

BUT…Portfolio standard deviation is less than the weighted average risk of the individual investments, except for perfectly positively correlated investments.

5 10 15 20

Standard deviation %

5

10

15

20

A

BP

Q

Exp

ecte

d re

turn

R %

7 PORTFOLIO THEORY


OHT 7.18

PORTFOLIO EXPECTED RETURNS AND STANDARD DEVIATION

• Proportion of funds in A = a = 0.90

where

Expected returns, two-asset portfolio

Rp = aRA + (1 – a)RB

Rp = 0.90 10 + 0.10 13.4 = 10.34%

p = a2 2A + (1 – a)2 2

B + 2a (1 – a) cov (RA, RB)

• Proportion of funds in B = 1 – a = 0.10

p = portfolio standard deviation

A = variance of investment A

B= variance of investment Bcov (RA, RB) = covariance of A and B

• 90 per cent of the portfolio funds are placed in A• 10 per cent are placed in B

7 PORTFOLIO THEORY


OHT 7.19

The covariance formula is:

COVARIANCE

Exhibit 7.27 Covariance

Event and Expected Deviation of A probability of Returns returns Deviations deviation of B probability

event pi RA R

B

RA

RB

(RA

– RA

)(RB – R

B

)pi

Boom 0.3 20 3 10 13.4 10 –10.4 10 –10.4 0.3 = –31.2Growth 0.4 10 35 10 13.4 0 21.6 0 21.6 0.4 = 0Recession 0.3 0 –5 10 13.4 –10 –18.4 –10 –18.4 0.3 = 55.2

Covariance of A and B, cov ( RA

, RB

) = +24

cov (RA, RB) = {(RA – RA)(RB – RB)pi}n

i = 1

RA – R

A

RB – R

B

7 PORTFOLIO THEORY


OHT 7.20

STANDARD DEVIATION

Exhibit 7.28 Summary table: expected return and standard deviation

Expected return (%) Standard deviation (%)

All invested in Company A 10 7.75

All invested in Company B 13.4 17.91

10.34 7.49Invested in a portfolio(90% in A, 10% in B)

p = a2 2A + (1 – a)2 2

B + 2a (1 – a) cov (RA, RB)

p = 0.902 60 + 0.102 320.64 + 2 0.90 0.10 24

p = 48.6 + 3.206 + 4.32

p = 7.49%


Exp

ecte

d re

turn

R %

Portfolio (A=90%, B=10%)20

15

10

5

2015105

Exhibit 7.29 Expected returns and standard deviation for A and B and a 90:10 portfolio of A and B

AB

7 PORTFOLIO THEORY


OHT 7.21

CORRELATION COEFFICIENT

RAB =

RAB = = +0.1729

If RAB = then cov (RARB) =

p = a22A + (1 – a)2 2

B + 2a (1 – a)

cov (RARB)RABAB

24

7.75 17.91

cov (RARB)

AB

AB

RABAB

7 PORTFOLIO THEORY


OHT 7.22

Exhibit 7.32 Zero correlation coefficient

Exhibit 7.30 Perfect positive correlation

Exhibit 7.31 Perfect negative correlation

Returns on G

Returns on F

Returns on G

Returns on F

Returns on G

Returns on F

7 PORTFOLIO THEORY


OHT 7.23

DOMINANCE AND THE EFFICIENT FRONTIER

Exhibit 7.33 Returns on shares in Augustus and Brown

Event(weather

for season)

Probabilityof event

Returns onBrown

Warm

pi

20% –10%Average

0.215% 22%

Wet0.6

10% 44%

Returns onAugustus

Probability Returns Returnspi on Augustus on Brown

RA

0.2 20 5 –10 180.00.6 15 0 22 2.40.2 10 5 44 115.2

10 297.6

3.162 17.25

(RA – RA)2 pi (RB – RB)2 pi

Variance,

Standard deviation,

Variance, B

Standard deviation, B

Expectedreturn

0.2

RA

15%

RB

20%

Exhibit 7.34 Standard deviation for Augustus and Brown

7 PORTFOLIO THEORY


OHT 7.24

24

2

Exhibit 7.35 Covariance

ExpectedDeviation of A

returns Deviationsdeviation of B probability

Probability Returnspi

0.2 20 –10 15 20 5 –30 5 –30 0.2 = –30

0.6 15 22 15 20 0 2 0 0.6 = 0

0.2 10 44 15 20 –5 24 –5 0.2 = –24

RAB =

–54RAB = = –0.99

cov (RA, RB)

AB

3.162 17.25

RA RB RA RB RA – RA RB – RB (RA – RA)( RB – RB)pi

Covariance (RA RB) –54

7 PORTFOLIO THEORY


OHT 7.25E

xhib

it 7

.36

Ris

k-r

etu

rn c

orre

lati

ons:

tw

o-as

set

por

tfol

ios

for

Au

gust

us

and

Bro

wn

Por

tfol

ioSt

anda

rd d

evia

tion

Exp

ecte

dre

turn

(%

)A

ugus

tus

wei

ghti

ng(%

)

Bro

wn

wei

ghti

ng

(%)

A10

00

15=

3.1

6

J90

1015

.5=

1.1

6

K85

1515

.75

= 0

.39

L80

2016

.0=

1.0

1

M50

5017

.5=

7.0

6

N25

7518

.75

=12

.2

B0

100

20=

17.2

5

0.92

10

+ 0

.12

297

.6 +

2

0.9

0

.1

–54

0.85

2 1

0 +

0.1

52 2

97.6

+ 2

0

.85

0

.15

–

54

0.82

10

+ 0

.22

297

.6 +

2

0.8

0

.2

–54

0.52

10

+ 0

.52

297

.6 +

2

0.5

0

.5

–54

0.25

2 1

0 +

0.7

52 2

97.6

+ 2

0

.25

0

.75

–

54

7 PORTFOLIO THEORY


OHT 7.26

Exhibit 7.37 Risk-return profile for alternative portfolios of Augustus and Brown

Efficiency frontier

p Standard deviation

1 2 543 8 107 9 11 126 13 15 1614 17 18

15

16

17

18

20

19

21

M

N

B

L

JK

A

Ret

urn

Rp %

7 PORTFOLIO THEORY


OHT 7.27

FINDING THE MINIMUM STANDARD DEVIATION FOR COMBINATIONS OF

TWO SECURITIES

• If a fund is to be split between two securities, A and B, and a is the fraction to be allocated to A, then the value for a which results in the lowest standard deviation is given by:

7 PORTFOLIO THEORY


OHT 7.28

INDIFFERENCE CURVES

Exhibit 7.38 Indifference curve for Mr Chisholm


16 20

Y

W

Z

Indifference curve I 105

X

Ret

urn

%

10

14

7 PORTFOLIO THEORY


OHT 7.29

Exhibit 7.39 A map of indifference curves


Ret

urn

%

16 20

10

14

South-west

W

S

TZ

I 105

I 107

I 110

I 121

I 129

North-west

7 PORTFOLIO THEORY


OHT 7.30

Exhibit 7.40 Intersecting indifference curves


Ret

urn

%

I101

I101

I105

I105

M

7 PORTFOLIO THEORY


OHT 7.31

Exhibit 7.41 Varying degrees of risk aversion as represented by indifference curves




(a) Moderate risk aversion (b) Low risk aversion (c) High risk aversion

Ret

urn

%

Ret

urn

%

Ret

urn

%

7 PORTFOLIO THEORY


OHT 7.32

Exhibit 7.42 Optimal combination of Augustus and Brown

CHOOSING THE OPTIMAL PORTFOLIO

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

15

16

17

18

19

20

21

AJ

L

K

M

N

B

Efficiency frontier

p Standard deviation %

I1

I2

I3

Ret

urn

%

7 PORTFOLIO THEORY


OHT 7.33

Exhibit 7.44 The boundaries of diversification

THE BOUNDARIES OF DIVERSIFICATION

1 2 3 4 5 6 7 8 9 10

15

16

17

18

19

20

21

C

p Standard deviation %

Return %

22

F

GE

H

D

= +1

= –1

= 0

RCD

RCD

RCD

Ret

urn

%

7 PORTFOLIO THEORY


OHT 7.34

Exhibit 7.47 A three-asset portfolio

EXTENSION TO A LARGE NUMBER OF SECURITIES

Standard deviation

B

A

C

23

4 1

Ret

urn

7 PORTFOLIO THEORY


OHT 7.35

Exhibit 7.48 The opportunity set for multi-security portfolios and portfolio selection for a highly risk-averse person and for a slightly risk-averse person

Standard deviation

Ret

urn

Inefficient region

Inefficient region

Efficiency frontierIL3

IL2

IL1VIH1

IH2

IH3

U

7 PORTFOLIO THEORY


OHT 7.36

THE CAPITAL MARKET LINE

Exhibit 7.53 Combining risk-free and risky investments

Standard deviation

Ret

urn

rf

FC

B

M

A

7 PORTFOLIO THEORY


OHT 7.37

Exhibit 7.54 Indifference curves applied to combinations of the market portfolio and the risk-free asset

Standard deviation

Ret

urn

rf

M

XY

7 PORTFOLIO THEORY


OHT 7.38

Exhibit 7.55 The capital market line

Standard deviation

Ret

urn M

rf

GH

S

T

N

Capital market line

7 PORTFOLIO THEORY


OHT 7.39

Problems with portfolio theory:

• relies on past data to predict future risk and return • involves complicated calculations • indifference curve generation is difficult • few investment managers use computer programs because of the nonsense results they frequently produce

Calculating two-asset portfolio expected returns and standard deviations

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