1 chapter 12: portfolio selection and diversification copyright © prentice hall inc. 2000. author:...

1

Chapter 12: Portfolio Chapter 12: Portfolio Selection and Selection and DiversificationDiversification

Copyright © Prentice Hall Inc. 2000. Author: Nick Bagley, bdellaSoft, Inc.

ObjectiveTo understand the theory of personal

portfolio selection in theory and in practice

2

Chapter 12 ContentsChapter 12 Contents

• 12.1 The process of personal 12.1 The process of personal portfolio selectionportfolio selection

• 12.2 The trade-off between 12.2 The trade-off between expected return and riskexpected return and risk

• 12.3 Efficient diversification with 12.3 Efficient diversification with many risky assetsmany risky assets

3

ObjectivesObjectives

• To understand the process of To understand the process of personal portfolio selection in theory personal portfolio selection in theory and practiceand practice

4

Security Prices

10

100

1000

10000

100000

0 5 10 15 20 25 30 35 40

Years

Val

ue

(Lo

g)

StockBond

Stock_MuBond_Mu

5

Security Prices

10

100

1000

10000

100000

0 5 10 15 20 25 30 35 40

Years

Val

ue

(Lo

g)

StockBond

Stock_MuBond_Mu

6

Security Prices

10

100

1000

10000

100000

0 5 10 15 20 25 30 35 40

Years

Val

ue

(Lo

g)

StockBond

Stock_MuBond_Mu

7

……and Lots More!and Lots More!Security Prices

10

100

1000

10000

100000

0 5 10 15 20 25 30 35 40

Years

Val

ue

(Lo

g)

StockBondStock_MuBond_Mu

Security Prices

10

100

1000

10000

100000

0 5 10 15 20 25 30 35 40

Years

Val

ue

(Lo

g)

StockBond

Stock_MuBond_Mu

Security Prices

10

100

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10000

100000

0 5 10 15 20 25 30 35 40

Years

Val

ue

(Lo

g)

StockBond

Stock_MuBond_Mu

Security Prices

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10000

100000

0 5 10 15 20 25 30 35 40

Years

Val

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(Lo

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StockBond

Stock_MuBond_Mu

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Security Prices

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0 5 10 15 20 25 30 35 40

Years

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(Lo

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StockBond

Stock_MuBond_Mu

9

Probability of Future Price

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0 50 100 150 200 250 300

Value

Pro

bab

ilit

y D

ensi

ty

Prob_Stock_2Prob_Bond_2Prob_Stock_5Prob_Bond_5Prob_Stock_10Prob_Bond_10Prob_Stock_40Prob_Bond_40

10

Probabilistic Stock Price Changes Over Time

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

0.018

0.020

0 200 400 600 800

Price

Pro

bab

ilit

y D

ensi

ty

Stock_Year_1Stock_Year_2Stock_Year_3Stock_Year_4Stock_Year_5Stock_Year_6Stock_Year_7Stock_Year_8Stock_Year_9Stock_Year_10

11

Probabilistic Bond Price Changes over Time

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.045

0 100 200 300 400

Price

Pro

bab

ilit

y D

ensi

ty

Bond_Year_1Bond_Year_2Bond_Year_3Bond_Year_4Bond_Year_5Bond_Year_6Bond_Year_7Bond_Year_8Bond_Year_9Bond_Year_10

12

1-Year Out

0.0000

0.0050

0.0100

0.0150

0.0200

0.0250

0.0300

0.0350

0.0400

0.0450

0 20 40 60 80 100 120 140 160 180 200

Price

Den

sity

Stock_1_YearBond_1_Year

Mode =104Mode =106

Median=104Mean =104Median=111Mean = 113

13

Two Years Out

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0 20 40 60 80 100 120 140 160 180 200

Price

Den

sity


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5-Years Out

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

0.018

0.020

0 100 200 300 400 500

Price

Den

sity


Mode = 122

Mode = 135

Median= 126Mean = 128

Median= 165Mean = 182

15

10-Years Out

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0 200 400 600 800 1,000

Value

Den

sity

Stock_10_Year

Bond_10_Year

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40 Years Out

0.000

0.000

0.000

0.001

0.001

0.001

0.001

0.001

0.002

0 5,000 10,000 15,000 20,000 25,000 30,000

Value

Den

sity


Mode =503

Mode =1,102

Median=650

Mean =739

Median=5,460Mean =12,151

17

Value of Central Tendency Statistics for the LogNormal

1_Year 2_Years 5_Years 10_Years 40_Years

Assume: Sig = 0.20, Mu = 0.12mode $106.18 $112.75 $134.99 $182.21 $1,102.32median $110.52 $122.14 $164.87 $271.83 $5,459.82mean $112.75 $127.12 $182.21 $332.01 $12,151.04

Assume: Sig = 0.08, Mu = 0.05mode $104.12 $108.42 $122.38 $149.78 $503.29median $104.79 $109.81 $126.36 $159.68 $650.13mean $105.13 $110.52 $128.40 $164.87 $738.91

mode The most probable pricemedian 50% of prices are equal or lower that thismean The expected or average price

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Mortality Table Male Female

Age MDePm MExLife FDePm FExLife60 16.08 17.51 9.47 21.2561 17.54 16.79 10.13 20.4465 25.42 14.04 14.59 17.3270 39.51 10.96 22.11 13.6775 64.19 8.31 38.24 10.3280 98.84 6.18 65.99 7.4885 152.95 4.46 116.1 5.1890 221.77 3.18 190.75 3.4595 329.96 1.87 317.32 1.91

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Deaths Per Thousand M & F

0

50

100

150

200

250

300

350

60 65 70 75 80 85 90 95

Age

Dea

ths

/ 10

00

MDePm

FDePm

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Life Expection

0

5

10

15

20

25

60 65 70 75 80 85 90 95

Age

Rem

ain

ing

Exp

ecte

d L

ife

MExLife

FExLife

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Combining the Riskless Combining the Riskless Asset and a Single Risky Asset and a Single Risky AssetAsset

– The expected return of the portfolio is The expected return of the portfolio is the weighted average of the the weighted average of the component returnscomponent returns

p = p = WW1*1*1 + 1 + WW2*2*2 2

p = p = WW1*1*1 + 1 + (1- W(1- W11))**2 2

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– The volatility of the portfolio is not The volatility of the portfolio is not quite as simple:quite as simple:

p = ((p = ((WW1* 1* 1)1)22 + + 22WW1* 1* 1* 1* WW2* 2*

2 + (2 + (WW2* 2* 2)2)22))1/21/2

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– We know something special about the We know something special about the portfolio, namely that security 2 is portfolio, namely that security 2 is

riskless, so riskless, so 2 = 0, and 2 = 0, and p becomes:p becomes:

p = ((p = ((WW1* 1* 1)1)22 + + 22WW1* 1* 1* 1* WW2* 2* 00

+ (+ (WW2* 2* 00))22))1/21/2

p = |p = |WW1| * 1| * 11

24


– In summaryIn summary

p = |p = |WW1| * 1| * 1, And:1, And:

p = p = WW1*1*1 + 1 + (1- W(1- W11))**rrff , So: , So:

If If WW1>0, 1>0, p = [(p = [(rrff - -1)/ 1)/ 1]*1]*p + p + rrff

Else Else p = [(p = [(1-1-rrff )/ )/ 1]*1]*p + p + rrff

25

A Portfolio of a Risky and a Riskless Security

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.00 0.10 0.20 0.30 0.40 0.50

Volatility

Ret

urn

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Capital Market Line

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

Volatility

Ret

urn

Long risky and short risk-free

Long both risky and risk-free

100% Risky

100% Risk-less

27

Mutual Fund Average % Mutual Fund Average % Total ReturnsTotal Returns

YTD 1-Yr 3-Yrs 5-Yrs 10-Yrs Life

14.81 30.40 15.87 14.15 16.53 16.96

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To obtain a 20% ReturnTo obtain a 20% Return

• You settle on a 20% return, and decide You settle on a 20% return, and decide not to pursue on the computational not to pursue on the computational issueissue

– Recall: Recall: p = p = WW1*1*1 + 1 + (1- W(1- W11))**rrff

– Your portfolio: Your portfolio: = 20%, = 20%, = 15%, rf = 5% = 15%, rf = 5%

– So: So: WW1 = (1 = (p - p - rrff)/()/(1 - 1 - rrff) )

= (0.20 - 0.05)/(0.15 - 0.05) = 150%= (0.20 - 0.05)/(0.15 - 0.05) = 150%

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• Assume that your manage a Assume that your manage a $50,000,000 portfolio$50,000,000 portfolio

• A W1 of 1.5 or 150% means you invest A W1 of 1.5 or 150% means you invest (go long) $75,000,000, and borrow (go long) $75,000,000, and borrow (short) $25,000,000 to finance the (short) $25,000,000 to finance the differencedifference

• Borrowing at the risk-free rate is mootBorrowing at the risk-free rate is moot

30


• How risky is this strategy?How risky is this strategy?

p = |p = |WW1| * 1| * 1 = 1.5 * 0.20 = 0.301 = 1.5 * 0.20 = 0.30

• The portfolio has a volatility of 30%The portfolio has a volatility of 30%

31

Portfolio of Two Risky Portfolio of Two Risky AssetsAssets

• Recall from statistics, that two Recall from statistics, that two random variables, such as two random variables, such as two security returns, may be combined security returns, may be combined to form a new random variableto form a new random variable

• A reasonable assumption for returns A reasonable assumption for returns on different securities is the linear on different securities is the linear model:model: 1 with ; 212211 wwrwrwrp

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Equations for Two SharesEquations for Two Shares

• The sum of the weights w1 and w2 The sum of the weights w1 and w2 being 1 is not necessary for the being 1 is not necessary for the validity of the following equations, validity of the following equations, for portfolios it happens to be truefor portfolios it happens to be true

• The expected return on the portfolio The expected return on the portfolio is the sum of its weighted is the sum of its weighted expectationsexpectations 2211 wwp

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Equations for Two SharesEquations for Two Shares

• Ideally, we would like to have a Ideally, we would like to have a similar result for risksimilar result for risk

– Later we discover a measure of risk Later we discover a measure of risk with this property, but for standard with this property, but for standard deviation:deviation:

(wrong) 2211 wwp

22

222,12121

21

21

2 2 wwwwp

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MnemonicMnemonic

• There is a mnemonic that will help There is a mnemonic that will help you remember the volatility you remember the volatility equations for two or more securitiesequations for two or more securities

• To obtain the formula, move To obtain the formula, move through each cell in the table, through each cell in the table, multiplying it by the row heading by multiplying it by the row heading by the column heading, and summingthe column heading, and summing

35

Variance with 2 SecuritiesVariance with 2 SecuritiesW1*Sig1 W2*Sig2

W1*Sig1 1 Rho(1,2)

W2*Sig2 Rho(2,1) 1

2,1212122

22

21

21

2 2 wwwwp

36

Variance with 3 SecuritiesVariance with 3 SecuritiesW1*Sig1 W2*Sig2 W3*Sig3

W1*Sig1 1 Rho(1,2) Rho(1,3)

W2*Sig2 Rho(2,1) 1 Rho(2,3)

W3*Sig3 Rho(3,1) Rho(3,2) 1

3,232323,13131

2,1212123

23

22

22

21

21

2

22

2

wwww

wwwwwp

37

Correlated Common StockCorrelated Common Stock

• The next slide shows statistics of The next slide shows statistics of two common stock with these two common stock with these statistics:statistics:

– mean return 1 = 0.15mean return 1 = 0.15– mean return 2 = 0.10mean return 2 = 0.10– standard deviation 1 = 0.20standard deviation 1 = 0.20– standard deviation 2 = 0.25standard deviation 2 = 0.25– correlation of returns = 0.90correlation of returns = 0.90– initial price 1 = $57.25initial price 1 = $57.25– Initial price 2 = $72.625Initial price 2 = $72.625

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2-Shares: Is One "Better?"

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 0.05 0.1 0.15 0.2 0.25 0.3

Standard Deviation

Exp

ecte

d R

etu

rn

39

Share Prices

0

50

100

150

200

250

300

350

0 1 2 3 4 5 6 7 8 9 10

Years

Val

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(ad

just

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or

Sp

lits

)

ShareP_1

ShareP_2

40

Portfolio of Two Securities

0.00

0.05

0.10

0.15

0.20

0.25

0.15 0.17 0.19 0.21 0.23 0.25 0.27 0.29

Standard Deviation

Exp

ecte

d R

etu

rn

Share 1

Share 2

Efficient

Sub-optima

l

MinimumVariance

41

Fragments of the Output Fragments of the Output TableTable

Data For two securitiesThis data has been constructed to produce the mean-varience paradox

mu_1 15.00%mu_2 10.00%sig_1 20.00%sig_2 25.00%rho 90.00%

w_1 w_2 Port_Sig Port_Mu-2.50 3.50 0.4776 -0.0250-2.40 3.40 0.4674 -0.0200-2.30 3.30 0.4573 -0.0150-2.20 3.20 0.4472 -0.0100-2.10 3.10 0.4372 -0.0050-2.00 3.00 0.4272 0.0000-1.90 2.90 0.4173 0.0050-1.80 2.80 0.4074 0.0100-1.70 2.70 0.3976 0.0150

1.30 -0.30 0.1953 0.16501.40 -0.40 0.1949 0.17001.50 -0.50 0.1953 0.17501.60 -0.60 0.1962 0.18001.70 -0.70 0.1978 0.18501.80 -0.80 0.2000 0.19001.90 -0.90 0.2028 0.19502.00 -1.00 0.2062 0.20002.10 -1.10 0.2101 0.20502.20 -1.20 0.2145 0.21002.30 -1.30 0.2194 0.21502.40 -1.40 0.2247 0.22002.50 -1.50 0.2305 0.2250

-0.30 1.30 0.2723 0.0850-0.20 1.20 0.2646 0.0900-0.10 1.10 0.2571 0.09500.00 1.00 0.2500 0.10000.10 0.90 0.2432 0.10500.20 0.80 0.2366 0.11000.30 0.70 0.2305 0.1150

42

Sample of the Excel Sample of the Excel FormulaeFormulae

w_1 w_2 Port_Sig Port_Mu-2.5 =1-A14 =SQRT(w_1^2*sig_1^2 + 2*w_1*w_2*sig_1*sig_2*rho + w_2^2*sig_2^2) =w_1*mu_1 + w_2*mu_2=A14+0.1 =1-A15 =SQRT(w_1^2*sig_1^2 + 2*w_1*w_2*sig_1*sig_2*rho + w_2^2*sig_2^2) =w_1*mu_1 + w_2*mu_2=A15+0.1 =1-A16 =SQRT(w_1^2*sig_1^2 + 2*w_1*w_2*sig_1*sig_2*rho + w_2^2*sig_2^2) =w_1*mu_1 + w_2*mu_2=A16+0.1 =1-A17 =SQRT(w_1^2*sig_1^2 + 2*w_1*w_2*sig_1*sig_2*rho + w_2^2*sig_2^2) =w_1*mu_1 + w_2*mu_2

=SQRT(w_1^2*sig_1^2 + 2*w_1*w_2*sig_1*sig_2*rho + w_2^2*sig_2^2)

=w_1*mu_1 + w_2*mu_2

43

Formulae for Minimum Formulae for Minimum Variance PortfolioVariance Portfolio

*1

22212,1

21

212,121*

2

22212,1

21

212,122*

1

1

2

2

w

w

w

44

Formulae for Tangent Formulae for Tangent PortfolioPortfolio

32tan

2

32tan

1

22

2tan1

1tan2

221212,121

212

212,12221tan

1

1

2

25.0*10.025.0*20.0*90.0*05.010.020.0*05.0

25.0*20.0*90.0*05.025.0*10.0

1

w

w

w

ww

rrrr

rrw

ffff

ff

45

Example: What’s the Best Example: What’s the Best Return given a 10% SD?Return given a 10% SD?

1261.005.010.02409.0

05.02333.0

2409.0

90.0*25.0*2.0*3

5

3

8225.0

3

520.0

3

8

2

2333.0

10.03

515.0

3

8

tan

tan

tan

22

22

2tan

2,121tan2

tan1

22

2tan2

21

2tan1

2tan

tan

tan

2tan21

tan1tan

ff rr

wwww

ww

46

Achieving the Target Achieving the Target Expected Return (2): Expected Return (2): WeightsWeights

• Assume that the investment Assume that the investment criterion is to generate a 30% returncriterion is to generate a 30% return

• This is the weight of the risky This is the weight of the risky portfolio on the CMLportfolio on the CML

3636.105.02333.0

05.030.0

1

1

11

ftangent

fcriterion

ftangentcriterion

r

rw

wrw

47

Achieving the Target Achieving the Target Expected Return Expected Return (2):Volatility(2):Volatility

• Now determine the volatility Now determine the volatility associated with this portfolioassociated with this portfolio

• This is the volatility of the portfolio This is the volatility of the portfolio we seekwe seek

3285.02409.0*3636.11 tangentw

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Achieving the Target Achieving the Target Expected Return (2): Expected Return (2): Portfolio WeightsPortfolio Weights

COMPUTATION WEIGHT

RISKLESS -0.3636 -0.3636

ASSET 1 1.3636*2.6667 3.6363

ASSET 2 1.3636*(-1.6667) -2.2727

TOTAL 1.0000

1 chapter 12: portfolio selection and diversification copyright © prentice hall inc. 2000. author:...

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