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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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FinanceFinance School of Management School of Management
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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FinanceFinance School of Management School of Management
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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FinanceFinance School of Management School of Management
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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FinanceFinance School of Management School of Management
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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FinanceFinance School of Management School of Management
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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FinanceFinance School of Management School of Management
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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FinanceFinance School of Management School of Management
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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FinanceFinance School of Management School of Management
Professional Asset Managers Professional Asset Managers
Investment advisors & “finished products” from a financial intermediary
Specialization, information and cost advantages
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FinanceFinance School of Management School of Management
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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FinanceFinance School of Management School of Management
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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FinanceFinance School of Management School of Management
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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FinanceFinance School of Management School of Management
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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FinanceFinance School of Management School of Management
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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FinanceFinance School of Management School of Management
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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FinanceFinance School of Management School of Management
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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FinanceFinance School of Management School of Management
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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FinanceFinance School of Management School of Management
Short SellingShort Selling
– Ik < 0 : short selling (borrowing) asset k
1,0 ki
ik wthenIIf
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FinanceFinance School of Management School of Management
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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FinanceFinance School of Management School of Management
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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FinanceFinance School of Management School of Management
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
22
FinanceFinance School of Management School of Management
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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FinanceFinance School of Management School of Management
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
– standard deviation 2 = 0.25
– correlation of returns = 0.90
– initial price 1 = $57.25
– initial price 2 = $72.625
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FinanceFinance School of Management School of Management
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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FinanceFinance School of Management School of Management
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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FinanceFinance School of Management School of Management
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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FinanceFinance School of Management School of Management
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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FinanceFinance School of Management School of Management
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
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FinanceFinance School of Management School of Management
where
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FinanceFinance School of Management School of Management
Portfolio of many risky assets
Standard Deviation (%)
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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FinanceFinance School of Management School of Management
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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FinanceFinance School of Management School of Management
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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FinanceFinance School of Management School of Management
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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FinanceFinance School of Management School of Management
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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FinanceFinance School of Management School of Management
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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FinanceFinance School of Management School of Management
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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FinanceFinance School of Management School of Management
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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FinanceFinance School of Management School of Management
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
39
FinanceFinance School of Management School of Management
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
– mean return 1 = 0.14
– mean return 2 = 0.08
– standard deviation 1 = 0.20
– standard deviation 2 = 0.15
– correlation of returns = 0
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FinanceFinance School of Management School of Management
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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FinanceFinance School of Management School of Management
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
42
FinanceFinance School of Management School of Management
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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FinanceFinance School of Management School of Management
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
FinanceFinance School of Management School of Management
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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FinanceFinance School of Management School of Management
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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FinanceFinance School of Management School of Management
Portfolio of many risky assets and the riskless asset
Standard Deviation (%)
Expected Return (%)
rfTangent Portfolio
Efficient frontier
Short sell
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FinanceFinance School of Management School of Management
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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FinanceFinance School of Management School of Management
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.