chapter 12: choosing an investment portfolio

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FinanceFinance School of Management School of Management

Chapter 12: Choosing an Chapter 12: Choosing an Investment PortfolioInvestment Portfolio

Objective• To understand the theory of personal

portfolio selection in theory and in practice

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Chapter 12: ContentsChapter 12: Contents

The process of personal portfolio selection The trade-off between expected return and risk Efficient diversification with many risky assets

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The Concept of ‘Portfolio’The Concept of ‘Portfolio’

A person’s wealth portfolio includes– Assets: stocks, bonds, shares in unincorporated

business, houses or apartments, pensions benefits, insurance policies, etc.

– Liabilities: student loans, auto loans, home mortgages, etc.

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Portfolio Selection Portfolio Selection

A study of how people should invest their wealth optimally

A process of trading off risk and expected return to find the best portfolio of assets and liabilities

Narrow and broad definitions:– How much to invest in stocks, bonds, and other securities– Whether to buy or rent one’s house– What types and amounts of insurance to purchase– How to manage one’s liabilities– How much to invest in one’s human capital

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Portfolio Selection Portfolio Selection

Although there are some general rules for portfolio selection that apply to virtually everyone, there is no single portfolio or portfolio strategy that is best for everyone.

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The Life CycleThe Life Cycle

In portfolio selection, the best strategy depends on an individual’s personal circumstances (family status, occupation, income, wealth).

Illustrations– Young couple: buy a house and take out a mortagage loan /

older couple: sell house and invest in assets provding a steady stream of income.

– Investing in stock market: Chang (30, a security analyst) / Obi (30, an English teacher).

– Buying insurance policies: Miriam (a parent with dependent children) / Sanjiv (a single person with no dependents).

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Time HorizonTime Horizon

In formulating a plan for portfolio selection, you begin by determining your goals and time horizons.– Planning horizon: the total length of time for which one plans

– Decision horizon: the length of time between decisions to revise the portfolio

– Trading horizon: the minimum time interval over which investors can revise their portfolios / its determination and impacts

– Investment strategy & trading horizon: portfolio insurance or dynamic portfolio strategy.

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Risk ToleranceRisk Tolerance

A major determinant of portfolio choices It is influenced by such characteristics as

– age, family status, job status, wealth, and

– other attributes that affect a person’s ability to maintain his standard of living in the face of adverse movements in the market value of his investment portfolio

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Professional Asset Managers Professional Asset Managers

Investment advisors & “finished products” from a financial intermediary

Specialization, information and cost advantages

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The Trade-off between Expected The Trade-off between Expected Return and RiskReturn and Risk

The objective is to find the portfolio which offers investors the highest expected rate of return for the degree of risk they are willing to tolerate.

Two step process:– find the optimal combination of risky assets.

– mix this optimal risk-asset with the riskless asset.

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Riskless AssetRiskless Asset

A security that offers a perfectly predictable rate of return in terms of the unit of account selected for the analysis and the length of the investor’s decision horizon.– For example, if the U.S dollars is taken as the unit of

account and the decision horizon is half a year, the riskless rate is the interest rate on U.S Treasury bills maturing after half a year.

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Rates of Return on Risky AssetsRates of Return on Risky Assets

Required return depends on the risk of the investment.

– Greater the risk, greater the return

– Risk premium

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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

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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

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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

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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

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Measuring Portfolio ReturnMeasuring Portfolio Return

– Ii : the initial investment in asset i (if Ii <0, short selling)

– wi: the proportion of the portfolio investing in asset I

– ri : the rate of return on asset I

– rp: the rate of return on the portfolio

Portfolio of n risky assets

1

(1 )1

i i ni

p i ii

I rr w r

I

i

iw 1

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Short SellingShort Selling

– Ik < 0 : short selling (borrowing) asset k

1,0 ki

ik wthenIIf

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Mean and Variance of Portfolio ReturnMean and Variance of Portfolio Return

i

ii

n

iiipp rwrEwrEr

1)()(

i j

jiijjip ww 2

– : the expected value of ri

– : the standard deviation of ri

– : the correlation between ri and rj

iij

ir

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Variance with 2 Securities

2,1212122

22

21

21

2 2 wwwwp

Variance with 3 Securities

2 2 2 2 2 2 21 1 2 2 3 3 1 2 1 2 1,2

1 3 1 3 1,3 2 3 2 3 2,3

2

2 2

p w w w w w

w w w w

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Suppose you invest $6000 in Bristol-Myers at an expected return of 15%, and $4000 in Ford Motor at an expected return of 21%.

The standard deviation of the return on BM’s stock is 18.6%, while the standard deviation of the return on FM is 28%.

The correlation between the returns is 0.4.%4.1721.40.15.60. pr

An Example: A Portfolio of BM and FMAn Example: A Portfolio of BM and FM

0493.28.186.4.4.6.228.40.186.60. 22222 p

%4.220493. p

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Portfolios of BM and FMPortfolios of BM and FM

Ford Motor

Standard Deviation (%)

Bristol-Myers

Expected Return (%)

●

●

40% F M60% BM

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Portfolios of Two Correlated Portfolios of Two Correlated Common StockCommon Stock

Two common stock with these statistics:– mean return 1 = 0.15

– mean return 2 = 0.10

– standard deviation 1 = 0.20


– correlation of returns = 0.90

– initial price 1 = $57.25

– initial price 2 = $72.625

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Share Prices

0

50

100

150

200

250

300

350

0 1 2 3 4 5 6 7 8 9 10

Years

Val

ue

(ad

just

ed f

or

Sp

lits

)

ShareP_1

ShareP_2

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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

Portfolio of Two Securities

Security 1

Security 2

Efficient Portfolio

Sub-optimal Portfolio

Minimum Variance Portfolio

Is one “better”?

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Formula for Minimum Variance Portfolio

*1

22212,1

21

212,121*

2

22212,1

21

212,122*

1

1

2

2

w

w

w

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Portfolio Selection with Portfolio Selection with nn Risky Assets Risky Assets

2

wmin w wp i j ij i ji j

w w

1

( ) ( ) w rn

p i ii

E r w E r

s.t.

1

1 1n

ii

w w

Harry Markowitz (1952): Portfolio Selection, Journal of Finance

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Solution：

w, ,

1min w w ( w r) (1 w 1)

2L

w r 1 0w

w r 0

1 w 1 0 w

L

L

L

29


where

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Portfolio of many risky assets


Expected Return (%)

efficient frontier

Efficient frontier: the set of portfolios offering the highest expected return for any given standard deviation.

minimum-variance portfolio

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Combining the Riskless Asset and a Combining the Riskless Asset and a Single Risky Asset: An illustrationSingle Risky Asset: An illustration

Let’s suppose that you have $100,000 to invest.

You are choosing between a riskless asset with a interest of 6% per year and a risky asset with an expected rate of return of 14% per year and a standard deviation of 20%.

How much of your $100,000 should you invest in the risky asset?

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Mean and Standard DeviationMean and Standard Deviation

PortfolioProportion

Invested in theRisky Asset

ProportionInvested in theRiskless Asset

ExpectedRate ofReturn

StandardDeviation

F 0 100% 0.06 0.00G 25% 75% 0.08 0.05H 50% 50% 0.10 0.10J 75% 25% 0.12 0.15S 100% 0 0.14 0.20

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The Risk-Return Trade-off LineThe Risk-Return Trade-off Line

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

S

JH

F

G R inefficient

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

We know something special about the portfolio, namely that security 2 is riskless, so σ2 = 0, and σp becomes

1121

1212

2211 020 wwwwwp

11 )( wrrrr ffp

pffp rrrrwIf ])([0 111

pffp rrrrelse ])([ 11

where

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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

100% 100% Risk-lessRisk-less

100% 100% RiskyRisky

Long risky and

short risk-free

Long both risky

and risk-free

CML

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Risk PremiumRisk Premium

The slope measure the extra expected return the market offers for each extra risk a investor is willing to bear

pf

fp

rrrr

1

1

11 )( frr

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Achieving a Target Expected ReturnAchieving a Target Expected Return

To find the portfolio corresponding to an expected rate of return of 0.11 per year, we substitute 0.11 for E(rp) and solve for w1.

Thus, the portfolio mix is 62.5% risky asset and 37.5% riskless asset.

375.0

08.006.011.0

1

1

w

w

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Portfolios of the Riskless Security Portfolios of the Riskless Security and Two Risky Securitiesand Two Risky Securities

The riskless security and two risky securities with the following statistics:

– riskless rate of return rf = 0.06





– correlation of returns = 0

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The Optimal Combination of the The Optimal Combination of the Three SecuritiesThree Securities

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

S

R

T

E

◆Tangent Portfolio

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Formula for Tangent Portfolio

%77.30,%23.69

15.20.002.008.020.02.15.08.

15.20.002.15.08.

1

tan2

tan1

22

2tan1

1tan2

212,121212

221

212,12221tan

1

ww

w

ww

rrrrrrrr

rrrrw

ffff

ff

12154.0)(E Tr 14595.0T

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Efficient Trade-off LineEfficient Trade-off Line

New efficient trade-off line:

Compare the old trade-off line connecting points F and S.

Clearly the investor is better off.

ppT

fTfp

rrrr

42165.06.

ppr 4.06.

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Achieving a Target Expected ReturnAchieving a Target Expected Return

The investment criterion is to generate a 10% expected rate of return.

Thus, the portfolio mix is 35% riskless asset and 65% tangent portfolio, namely 45% risky security 1 and 20% risky security 2.

09487.14595.65.

65.

)06.12154(.06.010.0

p

T

T

w

w

44


Selecting the Preferred PortfolioSelecting the Preferred Portfolio

It is important to note that in finding the optimal combination of risky assets, we do not need to know anything about investor preferences.

There is always a particular optimal portfolio of risky assets that all risk-averse investors who share the same forecasts of rates of return will combine with the riskless asset to reach their most-preferred portfolio.

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The Rationale for Portfolio SelectionThe Rationale for Portfolio Selection

Return

Risk

Low Risk

High Return

High Risk

High Return

Low Risk

Low Return

High Risk

Low Return

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Portfolio of many risky assets and the riskless asset


Expected Return (%)

rfTangent Portfolio

Efficient frontier

Short sell

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Efficient FrontierEfficient Frontier The jelly fish shape contains all possible combinations of risk and

return: The feasible set. The red line constitutes the efficient frontier of portfolios of risky

assets: Highest return for given risk. The tangent portfolio T is the optimal portfolio of risky assets

that all risk-averse investors will combine with the riskless asset.

Standard Deviation

Expected Return

Two-Fund Separation Two-Fund Separation Theorem (Tobin, 1958)Theorem (Tobin, 1958)

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Theory & PracticeTheory & Practice

The static mean-variance model & elementary theory of mutual fund financial intermediation.

Dynamic versions integrating intertemporal optimization of the life-cycle consumption-saving decisions with the allocation of those savings among alternative investments & a richer theory for the role of securities and financial intermediation.

Optimal combination of assets & optimal hedging portfolio more tailored to the needs of different clienteles.

chapter 12: choosing an investment portfolio

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