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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
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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
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ObjectivesObjectives
• To understand the process of To understand the process of personal portfolio selection in theory personal portfolio selection in theory and practiceand practice
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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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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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……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
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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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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
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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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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
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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
Stock_2_YearBond_2_Year
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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
Stock_5_YearBond_5_Year
Mode = 122
Mode = 135
Median= 126Mean = 128
Median= 165Mean = 182
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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
Stock_40_YearBond_40_Year
Mode =503
Mode =1,102
Median=650
Mean =739
Median=5,460Mean =12,151
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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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Combining the Riskless Combining the Riskless Asset and a Single Risky Asset and a Single Risky AssetAsset
– 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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Combining the Riskless Combining the Riskless Asset and a Single Risky Asset and a Single Risky AssetAsset
– 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
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Combining the Riskless Combining the Riskless Asset and a Single Risky Asset and a Single Risky AssetAsset
– 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
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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
Long risky and short risk-free
Long both risky and risk-free
100% Risky
100% Risk-less
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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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To obtain a 20% ReturnTo obtain a 20% Return
• 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
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To obtain a 20% ReturnTo obtain a 20% Return
• 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%
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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
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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
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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
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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
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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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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
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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
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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
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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
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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
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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
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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
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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