1 chapter 12: portfolio selection and diversification copyright © prentice hall inc. 1999. author:...

74
1 Chapter 12: Portfolio Chapter 12: Portfolio Selection and Selection and Diversification Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of perso portfolio selection in theory and in practice

Upload: roderick-sweeney

Post on 01-Apr-2015

214 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

1

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

Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley

ObjectiveTo understand the theory of personal

portfolio selection in theory and in practice

Page 2: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

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

Page 3: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

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

Page 4: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

4

IntroductionIntroduction

• How should you invest your wealth How should you invest your wealth optimally?optimally?– Portfolio selectionPortfolio selection

• Your wealth portfolio containsYour wealth portfolio contains– Stock, bonds, shares of unincorporated Stock, bonds, shares of unincorporated

businesses, houses, pension benefits, businesses, houses, pension benefits, insurance policies, and all liabilitiesinsurance policies, and all liabilities

Page 5: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

5

Portfolio Selection Portfolio Selection StrategyStrategy

• There are general principles to guide There are general principles to guide you, but the implementation will depend you, but the implementation will depend such factors as your (and your spouse’s)such factors as your (and your spouse’s)– age, existing wealth, existing and target age, existing wealth, existing and target

level of education, health, future earnings level of education, health, future earnings potential, consumption preferences, risk potential, consumption preferences, risk preferences, life goals, your children’s preferences, life goals, your children’s educational needs, obligations to older educational needs, obligations to older family members, and a host of other factorsfamily members, and a host of other factors

Page 6: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

6

12.1 The Process of 12.1 The Process of Personal Portfolio Personal Portfolio SelectionSelection• Portfolio selectionPortfolio selection

– the study of how people should invest their the study of how people should invest their wealthwealth

– process of trading off risk & expected return process of trading off risk & expected return to find the best portfolio of assets & liabilitiesto find the best portfolio of assets & liabilities• Narrower dfn: consider only securitiesNarrower dfn: consider only securities

• Wider dfn: house purchase, insurance, debtWider dfn: house purchase, insurance, debt

• Broad dfn: human capital, educationBroad dfn: human capital, education

Page 7: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

7

12.1.1 The Life Cycle12.1.1 The Life Cycle

• The risk exposure you should accept The risk exposure you should accept depends upon your agedepends upon your age

• Consider two investments (rho=0.2)Consider two investments (rho=0.2)– Security 1 has a volatility of 20% and Security 1 has a volatility of 20% and

an expected return of 12%an expected return of 12%

– Security 2 has a volatility of 8% and an Security 2 has a volatility of 8% and an expected return of 5%expected return of 5%

Page 8: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

8

Price TrajectoriesPrice Trajectories

• The following graph show the the price The following graph show the the price of the two securities generated by a of the two securities generated by a bivariate normal distribution for returnsbivariate normal distribution for returns– The more risky security may be thought of The more risky security may be thought of

as a share of common stock or a stock as a share of common stock or a stock mutual fundmutual fund

– The less risky security may be thought of The less risky security may be thought of as a bond or a bond mutual fundas a bond or a bond mutual fund

Page 9: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

9

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

Page 10: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

10

Interpretation of the Interpretation of the GraphGraph

• The graph is plotted on a log scale in so The graph is plotted on a log scale in so that you can see the important featuresthat you can see the important features

• The The magentamagenta bond trajectory is clearly bond trajectory is clearly less risky than the less risky than the navy-bluenavy-blue stock stock trajectorytrajectory

• The expected prices of the The expected prices of the bondbond and the and the stockstock are straight lines on a log scale are straight lines on a log scale

Page 11: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

11

Interpretation of the Interpretation of the GraphGraph

• Recall the log scale: the volatility Recall the log scale: the volatility increases with the length of the increases with the length of the investmentinvestment

• You begin to form the conjecture You begin to form the conjecture that the chances of the stock price that the chances of the stock price being less than the bond price is being less than the bond price is higher in earlier yearshigher in earlier years

Page 12: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

12

Generating More Generating More TrajectoriesTrajectories

• This was just one of an infinite number This was just one of an infinite number of trajectories generated by the same 2 of trajectories generated by the same 2 means, 2 volatilities, and the correlationmeans, 2 volatilities, and the correlation– I have not cheated you, this was indeed the I have not cheated you, this was indeed the

first trajectory generated by the statisticsfirst trajectory generated by the statistics

– the following trajectories are not reordered the following trajectories are not reordered nor editednor edited

• Instructor: On slower computers there may be a delayInstructor: On slower computers there may be a delay

Page 13: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

13

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

Page 14: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

14

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

Page 15: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

15

……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

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

1000

10000

100000

0 5 10 15 20 25 30 35 40

Years

Val

ue

(Lo

g)

StockBond

Stock_MuBond_Mu

Page 16: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

16

From Conjecture to From Conjecture to HypothesisHypothesis

• You are probably ready to make the You are probably ready to make the hypothesis thathypothesis that– the probability of the high-risk, high-the probability of the high-risk, high-

return security will out-perform the return security will out-perform the low-risk, low-return increases with timelow-risk, low-return increases with time

Page 17: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

17

But:But:

• I promised to be perfectly frank and I promised to be perfectly frank and honest (pfah) with you about the honest (pfah) with you about the ordering of the simulated ordering of the simulated trajectoriestrajectories

• The next trajectory truly was the The next trajectory truly was the next trajectory in the sequence, next trajectory in the sequence, honest!honest!

Page 18: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

18

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

Page 19: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

19

Implication for InvestorsImplication for Investors

– If you are older, the average remaining If you are older, the average remaining life of the investment is relatively short, life of the investment is relatively short, and there is a larger probability that an and there is a larger probability that an investment in the risky security will investment in the risky security will result in a lossresult in a loss

– This is not serious if you have This is not serious if you have substantial assets, in which case you substantial assets, in which case you can afford to take the risk, and enjoy can afford to take the risk, and enjoy higher expected returnshigher expected returns

Page 20: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

20

Implication for InvestorsImplication for Investors

– If you are younger, the average If you are younger, the average remaining life of retirement investment remaining life of retirement investment is longer, and there is only a small is longer, and there is only a small probability that an investment in the probability that an investment in the risky security will be less than the risky security will be less than the “safer” one“safer” one

– Investing in the less risky security will Investing in the less risky security will almost always result in a significantly almost always result in a significantly smaller retirement incomesmaller retirement income

Page 21: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

21

Implication for InvestorsImplication for Investors

– Relatively early during a typical life Relatively early during a typical life cycle, there may be a need to liquidate cycle, there may be a need to liquidate some invested funds, perhaps for a some invested funds, perhaps for a house deposit, a child’s education, or an house deposit, a child’s education, or an uninsured medical emergencyuninsured medical emergency

– In the case where liquidating an In the case where liquidating an investment early may damage long-investment early may damage long-term goals, some precautionary funds term goals, some precautionary funds should be kept in lower-risk securitiesshould be kept in lower-risk securities

Page 22: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

22

12.1.2 Time Horizons12.1.2 Time Horizons

– Planning horizonPlanning horizon• The total length of time for which one plansThe total length of time for which one plans

– Decision horizonDecision horizon• The length of time between decisions to The length of time between decisions to

revise a portfoliorevise a portfolio

– Trading horizonTrading horizon• The shortest possible time interval over The shortest possible time interval over

which investors may revise their portfolioswhich investors may revise their portfolios

Page 23: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

23

12.1.3 Risk Tolerance12.1.3 Risk Tolerance

• Your tolerance for bearing risk is a Your tolerance for bearing risk is a major determinant of portfolio major determinant of portfolio choiceschoices– It is the mirror image of risk aversionIt is the mirror image of risk aversion

– Whatever its cause, we do not Whatever its cause, we do not distinguish between capacity to bear distinguish between capacity to bear risk and attitude towards riskrisk and attitude towards risk

Page 24: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

24

12.1.4 Role of Professional 12.1.4 Role of Professional Asset ManagersAsset Managers• Most people have neither the time Most people have neither the time

nor the skill necessary to optimize a nor the skill necessary to optimize a portfolio for risk and returnportfolio for risk and return– Professional fund managers provide this Professional fund managers provide this

service asservice as• individually designed solutions to the individually designed solutions to the

precise needs of a customer ($$$$)precise needs of a customer ($$$$)

• a set of financial products which may be a set of financial products which may be used together to satisfy most customer used together to satisfy most customer goals ($$)goals ($$)

Page 25: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

25

12. 2 Trade-Off between 12. 2 Trade-Off between Expected Return and RiskExpected Return and Risk

• Assume a world with a single risky asset Assume a world with a single risky asset and a single riskless assetand a single riskless asset

• The risky asset is, in the real world, a The risky asset is, in the real world, a portfolio of risky assetsportfolio of risky assets

• The risk-free asset is a default-free bond The risk-free asset is a default-free bond with the same maturity as the investor’s with the same maturity as the investor’s decision (or possibly the trading) horizondecision (or possibly the trading) horizon

Page 26: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

26

Trade-Off between Trade-Off between Expected Return and RiskExpected Return and Risk

• The assumption of a risky and The assumption of a risky and riskless security simplifies the riskless security simplifies the analysisanalysis

Page 27: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

27

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

• Assume that you invest w Assume that you invest w proportion of your wealth in a risky proportion of your wealth in a risky security and (1-w) proportion of security and (1-w) proportion of your wealth in a riskless security your wealth in a riskless security

• Let rLet rss and r and rf f be the returns on the be the returns on the risky security and the riskless risky security and the riskless security, respectively. security, respectively.

Page 28: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

28

Combining the Riskless Combining the Riskless Asset and a Single Risky Asset and a Single Risky AssetAsset• Your statistics background tells you how Your statistics background tells you how

to determine the expected return and to determine the expected return and volatility of any two-security portfoliovolatility of any two-security portfolio

• Form a new random variable, the return Form a new random variable, the return of the portfolio,of the portfolio, r rPP, from the two security , from the two security

return variables, return variables, rrss and and rrff

rrPP = w*r = w*rss + (1-w) r + (1-w) rff

Page 29: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

29

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 expected the weighted average of the expected returns on the securities:returns on the securities:

E(rE(rPP)= w*E(r)= w*E(rss) + (1-w)*r) + (1-w)*rff

or,or,

E(rE(rPP)= r)= rff + w*(E(r + w*(E(rss) - r) - rff))

Page 30: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

30

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

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

Variance = Variance = PP = (w * = (w * ss))22 + +

2w*(1-w)2w*(1-w)ss**ff+ ((1-w)* + ((1-w)* ff))22

Std. dev. = Std. dev. = PP = ((w * = ((w * ss))22 + 2w*(1- + 2w*(1-w)w)ss**ff+ ((1-w)* + ((1-w)* ff))22))1/21/2

Page 31: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

31

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

– We know something special about We know something special about the portfolio, namely that riskless the portfolio, namely that riskless security has zero standard deviation security has zero standard deviation or or ff = 0, and = 0, and PP becomes: becomes:

PP = ((w * = ((w * ss))22 + 2w*(1-w) + 2w*(1-w)ss*0 + *0 + ((1-w)* 0)((1-w)* 0)22))1/21/2

PP = |w| * = |w| * ss and w = and w = PPs s

Page 32: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

32

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

Case 1: w > 0Case 1: w > 0

Substituting w = Substituting w = PPs s from the from the

previous slide into the last previous slide into the last equation on slide 29 we get:equation on slide 29 we get:

E(rE(rPP)= r)= rff + [(E(r + [(E(rss) - r) - rff)/)/ss] ] P P

Page 33: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

33

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

Case 2: w < 0Case 2: w < 0

Substituting w = - Substituting w = - PPs s from slide from slide

31 into the last equation on slide 31 into the last equation on slide 29 we get:29 we get:

E(rE(rPP)= r)= rff - [(E(r - [(E(rss) - r) - rff)/)/ss] ] P P

Page 34: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

34

IllustrationIllustration

• Consider the set of all portfolios that Consider the set of all portfolios that may be formed by investing (long may be formed by investing (long and or short) in and or short) in – a risky security with a volatility of 20% a risky security with a volatility of 20%

and an expected return of 15%and an expected return of 15%

– a riskless security with a volatility of a riskless security with a volatility of 0% and a known return of 5%0% and a known return of 5%

Page 35: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

35

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

Page 36: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

36

Sub-Optimal InvestmentsSub-Optimal Investments

• Investments on the higher part of the Investments on the higher part of the line (i.e., case 1 on slide 32) are always line (i.e., case 1 on slide 32) are always preferred to investments on the lower preferred to investments on the lower part of the line (i.e., case 2 on slide 33) part of the line (i.e., case 2 on slide 33) so for our current purposes we may so for our current purposes we may ignore the lower line. That is, we will ignore the lower line. That is, we will not sell the risky asset short and invest not sell the risky asset short and invest the proceeds in the riskless securitythe proceeds in the riskless security

Page 37: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

37

ObservationsObservations– An investor with a low risk tolerance may An investor with a low risk tolerance may

invest in a portfolio containing a small % invest in a portfolio containing a small % of risky securities, and a correspondingly of risky securities, and a correspondingly higher % of riskless securitieshigher % of riskless securities

– An investor with a high tolerance for risk An investor with a high tolerance for risk may sell risk-free securities he does not may sell risk-free securities he does not own (also called short selling), and invest own (also called short selling), and invest the proceeding in the risky investmentthe proceeding in the risky investment

– They both use the same two securitiesThey both use the same two securities

Page 38: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

38

Achieving a Target Achieving a Target Expected Return (1)Expected Return (1)

• Suppose you want an expected return of Suppose you want an expected return of 20% on your portfolio. What should be 20% on your portfolio. What should be the allocation between the risky security the allocation between the risky security and the riskless security?and the riskless security?

• Assume Assume E(rE(rss) = 15%, ) = 15%, ss = 20%, and r= 20%, and rff

= 5%= 5%

• Compute w and 1-wCompute w and 1-w

Page 39: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

39

Achieving a Target Achieving a Target Expected Return (1)Expected Return (1)

• Rearranging the last equation on Rearranging the last equation on slide 29 we get: slide 29 we get: w = (E(rw = (E(rPP) - r) - rff)/(E(r)/(E(rss) - r) - rff))

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

• 1-w = 100% - 150% = -50%1-w = 100% - 150% = -50%

Page 40: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

40

Achieving a Target Achieving a Target Expected Return (1)Expected Return (1)• Assume that you manage a Assume that you manage a

$50,000,000 portfolio$50,000,000 portfolio

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

• Borrowing at the risk-free rate is Borrowing at the risk-free rate is requiredrequired

Page 41: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

41

Achieving a Target Achieving a Target Expected Return (1)Expected Return (1)

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

PP = |w| * = |w| * s = 1.5 * 0.20 = 0.30s = 1.5 * 0.20 = 0.30

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

Page 42: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

42

Important ObservationImportant Observation

• It doesn’t require much skill to It doesn’t require much skill to leverage a portfolio; stockbrokers will leverage a portfolio; stockbrokers will let most investors trade “on margin”let most investors trade “on margin”

• When evaluating an investment’s When evaluating an investment’s performance, you performance, you mustmust examine examine bothboth the risk the risk andand the expected the expected returnreturn

Page 43: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

43

Returning to the ExampleReturning to the Example

• You can leverage the funds You can leverage the funds expected returns up or downexpected returns up or down

• If you want an expected returns of If you want an expected returns of 10%, or, 20%, 30%, 40%, 50%, 60%10%, or, 20%, 30%, 40%, 50%, 60%… you can have it (under the … you can have it (under the condition you can continue to condition you can continue to borrow at the risk-free rate)borrow at the risk-free rate)

Page 44: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

44

Portfolio EfficiencyPortfolio Efficiency

• An efficient portfolio is defined as An efficient portfolio is defined as the portfolio that offers the investor the portfolio that offers the investor the highest possible expected rate the highest possible expected rate of return at a specific riskof return at a specific risk

• We now investigate more than one We now investigate more than one risky asset in a portfoliorisky asset in a portfolio

Page 45: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

45

12.3 Efficient 12.3 Efficient Diversification with Many Diversification with Many AssetsAssets• We have considered We have considered

– Investments with a single risky, and a Investments with a single risky, and a single riskless, securitysingle riskless, security

– Investments where each security shares Investments where each security shares the same underlying return statisticsthe same underlying return statistics

• We will now investigate investments We will now investigate investments with more than one (heterogeneous) with more than one (heterogeneous) stockstock

Page 46: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

46

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

Page 47: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

47

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

Page 48: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

48

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

Page 49: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

49

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

Page 50: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

50

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

Page 51: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

51

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

Page 52: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

52

Note:Note:

• The correlation of a with b is equal to The correlation of a with b is equal to the correlation of b with athe correlation of b with a

• For every element in the upper triangle, For every element in the upper triangle, there is an element in the lower trianglethere is an element in the lower triangle

– so compute each upper triangle element so compute each upper triangle element once, and just double itonce, and just double it

• This generalizes in the expected mannerThis generalizes in the expected manner

Page 53: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

53

Correlated Common Correlated Common StocksStocks

• 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

Page 54: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

54

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

Page 55: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

55

CorrelationCorrelation

• The two shares are highly correlatedThe two shares are highly correlated– They track each other closely, but even They track each other closely, but even

adjusting for the different average adjusting for the different average returns, they have some individual returns, they have some individual behaviorbehavior

Page 56: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

56

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

Page 57: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

57

ObservationObservation

• Shorting the high-risk, low-return stock, Shorting the high-risk, low-return stock, and re-investing in the low-risk, high-and re-investing in the low-risk, high-return stock, creates efficient portfoliosreturn stock, creates efficient portfolios– Shorting high-risk by 80% of the net wealth Shorting high-risk by 80% of the net wealth

crates a portfolio with a volatility of 20% crates a portfolio with a volatility of 20% and a return of 19% (c.f. 15% on security 1)and a return of 19% (c.f. 15% on security 1)

– Shorting by 180% gives a volatility of 25%, Shorting by 180% gives a volatility of 25%, and a return of 24% (c.f. 10% on security 2)and a return of 24% (c.f. 10% on security 2)

Page 58: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

58

ObservationObservation

• In order to generate a portfolio that In order to generate a portfolio that generates the same risk, but with a generates the same risk, but with a higher returnhigher return– Compute the weights of the minimum Compute the weights of the minimum

portfolio, Wportfolio, W1 (min-var)1 (min-var), W, W2 (min-var)2 (min-var)

• (Formulae given later)(Formulae given later)

– Use the relationship Use the relationship •WWi (sub-opt)i (sub-opt) +W +Wi (opt)i (opt) = 2 * W = 2 * Wi (min-var)i (min-var)

Page 59: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

59

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

Page 60: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

60

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

Page 61: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

61

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

Page 62: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

62

Selecting the Preferred Selecting the Preferred PortfolioPortfolio

• The procedure is as followsThe procedure is as follows– Find the portfolio weights of the tangent Find the portfolio weights of the tangent

portfolio of the line (CML) through (0, rf)portfolio of the line (CML) through (0, rf)

– Determine the standard deviation and Determine the standard deviation and expectation of this pointexpectation of this point

– Construct the equation of the CMLConstruct the equation of the CML

– Apply investment criterionApply investment criterion

Page 63: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

63

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

Page 64: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

64

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

Page 65: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

65

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

Page 66: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

66

Investment StrategiesInvestment Strategies

• We have examined two strategies in We have examined two strategies in detail whendetail when

– the volatility is specifiedthe volatility is specified

– the return is specifiedthe return is specified

• Additionally, one of the graphs indicated Additionally, one of the graphs indicated an approach to take when presented an approach to take when presented with investor’s risk/return preferenceswith investor’s risk/return preferences

Page 67: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

67

Portfolio of Many Risky Portfolio of Many Risky AssetsAssets

• In order to solve problems with more In order to solve problems with more than two securities requires tools such than two securities requires tools such as quadratic programmingas quadratic programming

• The “Solve” function in Excel may be The “Solve” function in Excel may be used to obtain solutions, but it is used to obtain solutions, but it is generally better to use a software generally better to use a software package such as the one that came with package such as the one that came with this bookthis book

Page 68: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

68

Chapter AssumptionsChapter Assumptions

• The theory underlying this chapter The theory underlying this chapter is essentially just probability theory, is essentially just probability theory, but there are financial assumptionsbut there are financial assumptions

Page 69: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

69

– We do not have to assume that the We do not have to assume that the generating process of returns is generating process of returns is normal, but we do assume that the normal, but we do assume that the process has a mean and a variance. process has a mean and a variance. This is may not be the case in real lifeThis is may not be the case in real life

Page 70: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

70

– We assumed that the process was We assumed that the process was generated without inter-temporal generated without inter-temporal correlations. Some investors believe correlations. Some investors believe that there is valuable information in old that there is valuable information in old price data that has not been price data that has not been incorporated into the current price--this incorporated into the current price--this runs counter to many rigorous runs counter to many rigorous empirical studies.empirical studies.

Page 71: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

71

– There are no “hidden variables” that There are no “hidden variables” that explain some of the noiseexplain some of the noise

Page 72: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

72

– Investors make decisions based on Investors make decisions based on mean-variances alone mean-variances alone • statistics such as skewness & kurtosis statistics such as skewness & kurtosis

have been ignored have been ignored

Page 73: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

73

• We have made the assumption the We have made the assumption the we can lend at the risk-free rate, we can lend at the risk-free rate, and that we can “short” common and that we can “short” common stock aggressively stock aggressively

Page 74: 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley Objective To understand the theory of personal

74

SummarySummary

• There is no single investment strategy There is no single investment strategy that is suitable for all investors; nor for that is suitable for all investors; nor for a single investor for his whole lifea single investor for his whole life

• Time makes risky investments more Time makes risky investments more attractive than safer investmentsattractive than safer investments

• In practice, diversification has In practice, diversification has somewhat limited power to reduce risksomewhat limited power to reduce risk