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Diversification and Portfolio Risk
Asset Allocation With Two Risky Assets
6-1
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Combinations of risky assetsWhen we put stocks in a portfolio, p <
Why?
When Stock 1 has a return E[r1] it is likely that Stock 2 has a return E[r2] so that rp that contains stocks 1 and 2 remains close to
What statistics measure the tendency for r1 to be above expected when r2 is below expected?
Covariance and Correlation
(Wii)
E[rp]
<
>
n = # securities in the portfolio
Averaging principle
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Portfolio Variance and Standard Deviation
Q
1I
Q
1JJIJI
2p )]r,Cov(r W[Wσ
portfolio the in stocks of number total The Q
lyrespective J and I stock in invested portfolio total the of PercentageW,W JI
J Stock and I Stock of returns the of Covariance)r,Cov(r JI
)r,r(Cov)r,Cov(r & σ )r,(r Cov then J I If IJJI2
IJI
22
222121
21
21
2 2 W)r,r(CovWWWp
Variance of a Two Stock Portfolio:
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Covariance Calculation• Ex ante. Using scenario analysis with probabilities the
covariance can be calculated with the following formula:
• Ex post. Using a time series of returns, the covariance can be calculated with the following formula:
1
( , ) ( ) ( ) ( )S
S B S S B Bi
Cov r r p i r i r r i r
6-4
N
1T21 n
)r(r)r(r
1n
n)r,Cov(r
2T2,1T1,
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Covariance and correlation
• The problem with covariance
Covariance does not tell us the intensity of the comovement of the stock returns, only the direction.
We can standardize the covariance however and calculate the correlation coefficient which will tell us not only the direction but provides a scale to estimate the degree to which the stocks move together.
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Measuring the correlation coefficient
• Standardized covariance is called the _____________________
For Stock 1 and Stock 2
21
21(1,2) σσ
)r,Cov(rρ
correlation coefficient or
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Correlation Coefficients: Possible Values
If If = 1.0, the securities would be perfectly = 1.0, the securities would be perfectly positively correlated.positively correlated.
If If = - 1.0, the securities would be perfectly = - 1.0, the securities would be perfectly negatively correlated. negatively correlated.
The closer The closer is to -1, the better diversification.is to -1, the better diversification.
Range of values for correlation coefficients:
-1.0 < < 1.0
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and diversification in a 2 stock portfolio
•
•
•
•
•
Typically is greater than ____________________
(1,2) = (2,1) and the same is true for the COV
The covariance between any stock such as Stock 1 and itself is simply the variance of Stock 1,
(1,1) = +1.0 by definition
We have no measure for how three or more stocks move together.
zero and less than 1.0
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The effects of correlation & covariance on diversification
Asset A
Asset B
Portfolio AB
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6-10
The effects of correlation & covariance on diversification
Asset C
Asset D
Portfolio CD
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= W1 + W2 W1 = Proportion of funds in Security 1W2 = Proportion of funds in Security 2
= Expected return on Security 1= Expected return on Security 2
Two-Security Portfolio: Return
r1
E( )rp
r2
r1 r2
portfolio the in securities # n ;rW)rE(n
1i
iip
Wii=1
n
= 1Wii=1
n
WiWii=1i=1
n
= 1
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p2
= W121
2 + W222
2 + 2W1W2 Cov(r1r2)p2
= W121
2 + W222
2 + 2W1W2 Cov(r1r2)
12 = Variance of Security 112 = Variance of Security 1
22 = Variance of Security 222 = Variance of Security 2
Cov(r1r2) = Covariance of returns for Security 1 and Security 2Cov(r1r2) = Covariance of returns for Security 1 and Security 2
Two-Security Portfolio: Risk
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Example 1: Calculating portfolio risk using a time series of returns
– next 4 slides
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Returns ABC XYZ 1 0.2515 -0.2255 2 0.4322 0.3144 3 -0.2845 -0.0645 4 -0.1433 -0.5114 5 0.5534 0.3378 6 0.6843 0.3295 7 -0.1514 0.7019 8 0.2533 0.2763 9 -0.4432 -0.4879
10 -0.2245 0.5263 AAR 0.09278 0.11969
Squared deviations
from average ABC XYZ
0.025192 0.119156 0.115206 0.037912 0.14234 0.033926 0.055734 0.398275 0.212171 0.047572 0.349896 0.04402 0.059624 0.338968 0.025767 0.024527 0.287275 0.369166 0.100667 0.165332
Sum 1.37387 1.578853 Average 0.137387 0.157885
2ABC =
ABC =
2XYZ =
XYZ =
1.37387 / (10-1) = 0.15265
39.07%
1.57885 / (10-1) = 0.17543
41.88%
Calculating Variance and CovarianceEx post
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6-15
Returns ABC XYZ 1 0.2515 -0.2255 2 0.4322 0.3144 3 -0.2845 -0.0645 4 -0.1433 -0.5114 5 0.5534 0.3378 6 0.6843 0.3295 7 -0.1514 0.7019 8 0.2533 0.2763 9 -0.4432 -0.4879
10 -0.2245 0.5263 AAR 0.09278 0.11969
COV(ABC,XYZ) =
ABC,XYZ =
ABC,XYZ =
Deviation from average
ABC XYZ 0.15872 -0.34519 0.33942 0.19471
-0.37728 -0.18419 -0.23608 -0.63109 0.46062 0.21811 0.59152 0.20981
-0.24418 0.58221 0.16052 0.15661
-0.53598 -0.60759 -0.31728 0.40661
Product of
deviations -0.05479 0.066088 0.069491 0.148988 0.100466 0.124107 -0.14216 0.025139 0.325656 -0.12901
Sum 0.533973 Average 0.053397
0.533973 / (10-1) = 0.059330
COV / (ABCXYZ) =
0.3626ABC = 39.07%
XYZ = 41.88%
0.059330 / (0.3907 x 0.4188)
N
1T21 n
)r(r)r(r
1n
n)r,Cov(r
2T2,1T1,
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E(rp) = W1r1 + W2r2
W1 =
W2 =
=
=
Two-Security Portfolio Return
E(rp) = 0.6(9.28%) + 0.4(11.97%) = 10.36%
Wi = % of total money invested in security i
0.6
0.4
9.28%
11.97%
r1
r2
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p2 =
p2
=
p2
=
p =
p <
Two-Security Portfolio Risk
Q
1I
Q
1JJI
2p J)]Cov(I, W[Wσ
W121
2 + 2W1W2 Cov(r1r2) + W222
2
0.36(0.15265) +
0.1115019 = variance of the portfolio
33.39%
Let W1 = 60% and W2 = 40% Stock 1 = ABC; Stock 2 = XYZ
40.20%
ABC = 39.07%
XYZ = 41.88%
2(.6)(.4)(0.05933) + 0.16(0.17543)
33.39% < [0.60(0.3907) + 0.40(0.4188)] =
W11 + W22
6-17
2ABC = 0.15265
2XYZ = 0.17543
COV(ABC,XYZ) = 0.05933
ABC,XYZ = 0.3626
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Example 2: Calculating portfolio risk using scenario analysis with probabilities
– next 5 slides
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Scenario Probability Stock Fund Return Bond Fund ReturnRecession 0.3 - 11% 16%Normal 0.4 13% 6%Boom 0.3 27% - 4%
Step 1: Calculate the expected return for the each fund using our formula from Chap.5 for discrete random variables:
Spreadsheet #1
)s(r)s(p)r(Es
1i
Column B Column C Column E Stock Fund Bond Fund
Scenario Probability Rate of Return Col B x Col C
Rate of Return Col B x Col E
Recession 0.3 -11 -3.3 16 4.8 Normal 0.4 13 5.2 6 2.4 Boom 0.3 27 8.1 -4 -1.2 Expected or Mean Return: SUM: 10.0 SUM: 6.0
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Scenario Probability Stock Fund Return Bond Fund ReturnRecession 0.3 - 11% 16%Normal 0.4 13% 6%Boom 0.3 27% - 4%
Step 2: Calculate the risk (i.e., variance and standard deviation) for the each fund using formulas for discrete random variables:
Spreadsheet #2
)r(Var and )]r(E)s(r)[(s(p)r(Vars
1i
22
12345678910
A B C D E F G H I J Stock Fund Bond Fund Deviation Deviation
Rate from Column B Rate from Column Bof Expected Squared x of Expected Squared x
Scenario Prob. Return Return Deviation Column E Return Return Deviation Column IRecession 0.3 -11 -21 441 132.3 16 10 100 30Normal 0.4 13 3 9 3.6 6 0 0 0Boom 0.3 27 17 289 86.7 -4 -10 100 30
Variance = SUM 222.6 Variance: 60Standard deviation = SQRT(Variance) 14.92 Std. Dev.: 7.75
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Scenario Probability Stock Fund Return Bond Fund ReturnRecession 0.3 - 11% 16%Normal 0.4 13% 6%Boom 0.3 27% - 4%
Step 3: Calculate the covariance and correlation coefficient of the 2 funds’ returns. These are formulas for discrete random variables that we haven’t seen before:
Spreadsheet #3
s
1iBBSSBS )]r)i(r)][(r)i(r)[(i(p)r,r(CovCovariance
12345678
A B C D E F GDeviation from Mean Return Covariance
Scenario Probability Stock Fund Bond Fund Product of Dev Col B x Col ERecession 0.3 -21 10 -210 -63Normal 0.4 3 0 0 0Boom 0.3 17 -10 -170 -51
Covariance = SUM: -114Correlation coefficient = Covariance/(StdDev(stocks)*StdDev(bonds)) = -0.99
BS
BSSB
)r,r(CovtCoefficien nCorrelatio
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Step 4: Calculate the expected return of a PORTFOLIO that invests in the stock and bond funds:
rp = wB rB + wS rS
For example, let’s calculate the return for a portfolio that has 60% of its money invested in the stock fund and 40% of the portfolio invested in the bond fund:
rp = wB rB + wS rS
= (0.4)(6.0%) + (0.6)(10.0%)
= (2.4%) + (6.0%)
= 8.4%
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Step 5: Calculate the portfolio the risk (i.e., variance and standard deviation) of a PORTFOLIO that invests in the stock and bond funds:
For a portfolio that has 60% of its money invested in the stock fund and 40% of the portfolio invested in the bond fund:
s 2P = (0.4) 2(0.0775) 2 + (0.6) 2(0.1492) 2 + (2)(0.4)(0.6)(0.0775)(0.1492) (-0.99)
= 0.008014 + 0.000961 - 0.00549
= 0.00348, or 0.348%
Standard deviation ( s P ) = √ 0.00348 = 0.059, or 5.9%
Deviation Standard Portfolio and VariancePortfolio 22pp
SBSBSB2
S2
S2
B2
B2
p ww2ww
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= +1
= .3
E(r)
13%
8%
12% 20% St. Dev
TWO-SECURITY PORTFOLIOS WITH DIFFERENT CORRELATIONS
Stock A Stock B
WA = 0%
WB = 100%
WA = 100%
WB = 0%
= 0
= -150%A
50%B
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Summary: Portfolio Risk/Return Two Security Portfolio
• Amount of risk reduction depends critically on _________________________.
• Adding securities with correlations _____ will result in risk reduction.
• If risk is reduced by more than expected return, what happens to the return per unit of risk (the Sharpe ratio)?
correlations or covariances
< 1
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2p = W1
2122
p = W121
2 + W22
+ W22
+ W323
2+ W323
2
+ 2W1W2+ 2W1W2 Cov(r1r2) Cov(r1r2)
Cov(r1r3) Cov(r1r3)+ 2W1W3+ 2W1W3
Cov(r2r3) Cov(r2r3)+ 2W2W3+ 2W2W3
Three-Security Portfolio n or Q = 3
Q
1I
Q
1JJIJI
2p )]r,Cov(r W[Wσ
6-26
For an n security portfolio there would be _ variances and _____ covariance terms.
The ___________ are the dominant effect on
nn(n-1)
covariances
2p
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Possible Risky Investments
Using data from example 2, we calculate the return and risk (standard deviation) of portfolios that invests in different weights of stock and bond funds:
12345
67891011121314151617181920212223
A B C D EInput data
E(rS) E(rB) S
B
SB
10 6 14.92 7.75 -0.99 Portfolio Weights Expected Return
wS wB = 1 - wS E(rP) = Col A x A3 + Col B x B3 Std Deviation*
0.0 1.0 6.00 7.750.1 0.9 6.40 5.500.2 0.8 6.80 3.270.3 0.7 7.20 1.180.4 0.6 7.60 1.510.5 0.5 8.00 3.660.6 0.4 8.40 5.900.7 0.3 8.80 8.150.8 0.2 9.20 10.400.9 0.1 9.60 12.661.0 0.0 10.00 14.92
* The formula for portfolio standard deviation is: SQRT[ (Col A*$C$3)^2 + (Col B*$D$3)^2 + 2*$E$3*Col A*$C$3*Col B*$D$3 ]
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Possible Risky Investments (continued) Graph the return and risk (standard deviation) of portfolios that invests in
different weights of stock and bond funds:
Investment Opportunity Set for Stocks and Bonds Funds Calculated in Class
0.00
2.00
4.00
6.00
8.00
10.00
12.00
0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00
Standard Deviation (%)
Exp
ecte
d R
etu
rn (
%)
100% in Bond Fund
100% in Stock Fund
The minimum variance portfolio
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Possible Risky Investments (continued) Question: Would you ever want to invest in a portfolio that had a higher % of $ invested
in the bond fund than that of the “minimum variance portfolio?Answer: No. You would expect a lower return for risk than you expect in other
combinations!
Investment Opportunity Set for Stocks and Bonds Funds Calculated in Class
0.00
2.00
4.00
6.00
8.00
10.00
12.00
0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00
Standard Deviation (%)
Exp
ecte
d R
etu
rn (
%)
100% in Bond Fund
100% in Stock Fund
The minimum variance portfolio
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Minimum Variance Combinations -1< < +1
11 22
- Cov(r1r2) - Cov(r1r2)
W1W1==
++ - 2Cov(r1r2) - 2Cov(r1r2)
22
W2W2 = (1 - W1)= (1 - W1)
2
2
22 2
2
Choosing weights to minimize the portfolio variance
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11
Minimum Variance Combinations -1< < +1
22E(r2) = .14E(r2) = .14 = .20= .20Stk 2Stk 2 1212 = .2= .2E(r1) = .10E(r1) = .10 = .15= .15Stk 1Stk 1
11 22
- Cov(r1r2)- Cov(r1r2)
W1W1==
++ - 2Cov(r1r2)- 2Cov(r1r2)
22
W2W2 = (1 - W1)= (1 - W1)
22
22 2211 22
- Cov(r1r2)- Cov(r1r2)
W1W1==
++ - 2Cov(r1r2)- 2Cov(r1r2)
22
W2W2 = (1 - W1)= (1 - W1)
22
22 22WW11
==(.2)(.2)22 -- (.2)(.15)(.2)(.2)(.15)(.2)
(.15)(.15)22 + (.2)+ (.2)22 -- 2(.2)(.15)(.2)2(.2)(.15)(.2)
WW11 = .6733= .6733
WW22 = (1 = (1 -- .6733) = .3267.6733) = .3267
WW11==
(.2)(.2)22 -- (.2)(.15)(.2)(.2)(.15)(.2)
(.15)(.15)22 + (.2)+ (.2)22 -- 2(.2)(.15)(.2)2(.2)(.15)(.2)
WW11 = .6733= .6733
WW22 = (1 = (1 -- .6733) = .3267.6733) = .3267
WW11==
(.2)(.2)22 -- (.2)(.15)(.2)(.2)(.15)(.2)
(.15)(.15)22 + (.2)+ (.2)22 -- 2(.2)(.15)(.2)2(.2)(.15)(.2)
WW11 = .6733= .6733
WW22 = (1 = (1 -- .6733) = .3267.6733) = .3267
WW11==
(.2)(.2)22 -- (.2)(.15)(.2)(.2)(.15)(.2)
(.15)(.15)22 + (.2)+ (.2)22 -- 2(.2)(.15)(.2)2(.2)(.15)(.2)
WW11 = .6733= .6733
WW22 = (1 = (1 -- .6733) = .3267.6733) = .3267Cov(r1r2) = 1122
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E[rp] =
Minimum Variance: Return and Risk with = .2
22E(r2) = .14E(r2) = .14 = .20= .20Stk 2Stk 2 1212 = .2= .2E(r1) = .10E(r1) = .10 = .15= .15Stk 1Stk 1
22E(r2) = .14E(r2) = .14 = .20= .20Stk 2Stk 2 1212 = .2= .2
E(r1) = .10E(r1) = .10 = .15= .15Stk 1Stk 1 E(r1) = .10E(r1) = .10 = .15= .15Stk 1Stk 1
1/22222p (0.2) (0.15) (0.2) (0.3267) (0.6733) 2 )(0.2 )(0.3267 )(0.15 )(0.6733σ
p2
=p2
=
%.. /p 081301710 21
WW11==
(.2)(.2)22 -- (.2)(.15)(.2)(.2)(.15)(.2)
(.15)(.15)22 + (.2)+ (.2)22 -- 2(.2)(.15)(.2)2(.2)(.15)(.2)
WW11 = .6733= .6733
WW22 = (1 = (1 -- .6733) = .3267.6733) = .3267
WW11==
(.2)(.2)22 -- (.2)(.15)(.2)(.2)(.15)(.2)
(.15)(.15)22 + (.2)+ (.2)22 -- 2(.2)(.15)(.2)2(.2)(.15)(.2)
WW11 = .6733= .6733
WW22 = (1 = (1 -- .6733) = .3267.6733) = .3267
1
.6733(.10) + .3267(.14) = .1131 or 11.31%
W121
2 + W222
2 + 2W1W2 1,212
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WW11==
(.2)(.2)22 -- (.2)(.15)((.2)(.15)(--.3).3)
(.15)(.15)22 + (.2)+ (.2)22 -- 2(.2)(.15)(2(.2)(.15)(--.3).3)
WW11 = .6087= .6087
WW22 = (1 = (1 -- .6087) = .3913.6087) = .3913
WW11==
(.2)(.2)22 -- (.2)(.15)((.2)(.15)(--.3).3)
(.15)(.15)22 + (.2)+ (.2)22 -- 2(.2)(.15)(2(.2)(.15)(--.3).3)
WW11 = .6087= .6087
WW22 = (1 = (1 -- .6087) = .3913.6087) = .3913
Minimum Variance Combination with = -.3
11 22
- Cov(r1r2)- Cov(r1r2)
W1W1==
++ - 2Cov(r1r2)- 2Cov(r1r2)
22
W2W2 = (1 - W1)= (1 - W1)
22
22 2211 22
- Cov(r1r2)- Cov(r1r2)
W1W1==
++ - 2Cov(r1r2)- 2Cov(r1r2)
22
W2W2 = (1 - W1)= (1 - W1)
22
22 22
22E(r2) = .14E(r2) = .14 = .20= .20Stk 2Stk 2 1212 = .2= .2E(r1) = .10E(r1) = .10 = .15= .15Stk 1Stk 1
22E(r2) = .14E(r2) = .14 = .20= .20Stk 2Stk 2 1212 = .2= .2
E(r1) = .10E(r1) = .10 = .15= .15Stk 1Stk 1 E(r1) = .10E(r1) = .10 = .15= .15Stk 1Stk 1 -.31
Cov(r1r2) = 1122
WW11==
(.2)(.2)22 -- ((--.3)(.15)(.2).3)(.15)(.2)
(.15)(.15)22 + (.2)+ (.2)22 -- 2(2(--.3)(.15)(.2).3)(.15)(.2)WW11
==(.2)(.2)22 -- ((--.3)(.15)(.2).3)(.15)(.2)
(.15)(.15)22 + (.2)+ (.2)22 -- 2(2(--.3)(.15)(.2).3)(.15)(.2)
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WW11==
(.2)(.2)22 -- (.2)(.15)((.2)(.15)(--.3).3)
(.15)(.15)22 + (.2)+ (.2)22 -- 2(.2)(.15)(2(.2)(.15)(--.3).3)
WW11 = .6087= .6087
WW22 = (1 = (1 -- .6087) = .3913.6087) = .3913
WW11==
(.2)(.2)22 -- (.2)(.15)((.2)(.15)(--.3).3)
(.15)(.15)22 + (.2)+ (.2)22 -- 2(.2)(.15)(2(.2)(.15)(--.3).3)
WW11 = .6087= .6087
WW22 = (1 = (1 -- .6087) = .3913.6087) = .3913
Minimum Variance Combination with = -.3
22E(r2) = .14E(r2) = .14 = .20= .20Stk 2Stk 2 1212 = .2= .2E(r1) = .10E(r1) = .10 = .15= .15Stk 1Stk 1
22E(r2) = .14E(r2) = .14 = .20= .20Stk 2Stk 2 1212 = .2= .2
E(r1) = .10E(r1) = .10 = .15= .15Stk 1Stk 1 E(r1) = .10E(r1) = .10 = .15= .15Stk 1Stk 1 -.3
E[rp] =
1/22222p (0.2) (0.15) (-0.3) (0.3913) (0.6087) 2 )(0.2 )(0.3913 )(0.15 )(0.6087σ
p2
= p2
=
%.. /p 091001020 21
0.6087(.10) + 0.3913(.14) = .1157 = 11.57%
W121
2 + W222
2 + 2W1W2 1,212
1
Notice lower portfolio standard deviation but higher expected return with smaller
12 = .2
E(rp) = 11.31%
p = 13.08%
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Extending Concepts to All Securities
Consider all possible combinations of securities, with all possible different weightings and keep track of combinations that provide more return for less risk or the least risk for a given level of return and graph the result.
The set of portfolios that provide the optimal trade-offs are described as the efficient frontier.
The efficient frontier portfolios are dominant or the best diversified possible combinations.
All investors should want a portfolio on the efficient frontier. … Until we add the
riskless asset6-35
•
•
•
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E(r)E(r)
The minimum-variance frontier of The minimum-variance frontier of risky assetsrisky assets Efficient Frontier is the best diversified set of
investments with the highest returns
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GlobalGlobalminimumminimumvariancevarianceportfolioportfolio
EfficientEfficientfrontierfrontier
IndividualIndividualassetsassets
MinimumMinimumvariancevariancefrontierfrontier
St. Dev.
Found by forming portfolios of securities with the lowest covariances at a given E(r) level.
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E(r)E(r)The EF and asset allocation
EfficientEfficientfrontierfrontier
St. Dev.
20% Stocks80% Bonds
100% Stocks
EF including international & alternative investments
Ex-Post 2000-2002
80% Stocks20% Bonds
60% Stocks40% Bonds40%
Stocks60% Bonds
100% Stocks
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