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Transcript of Finding the Efficient Set
8/13/2019 Finding the Efficient Set
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Finding the Efficient Set(Chapter 5)
Feasible Portfolios
Minimum Variance Set & the Efficient Set
Minimum Variance Set Without Short-Selling Key Properties of the Minimum Variance Set
Relationships Between Return, Beta, Standard
Deviation, and the Correlation Coefficient
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FEASIBLE PORTFOLIOS When dealing with 3 or more securities, a complete
mass of feasible portfolios may be generated by varying
the weights of the securities:
0
5
10
15
20
25
0 10 20 30 40
Standard Deviation of Returns (%)
Expected Rate of Return (%)
Stock 1
Stock 2
Portfolio of Stocks 1 & 2
Stock 3Portfolio of Stocks 2 & 3
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Minimum Variance Set and the
Efficient Set Minimum Variance Set: Identifies those portfolios that
have the lowest level of risk for a given expected rate ofreturn.
Efficient Set: Identifies those portfolios that have the
highest expected rate of return for a given level of risk.-
0
5
10
15
20
25
0 20 40
Expected Rate of Return (%)
Standard Deviation of Returns
Efficient Set (top half of theMinimum Variance Set)
Minimum Variance SetMVP
Note: MVP is the global minimum
variance portfolio (one with the
lowest level of risk)
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Finding the Efficient Set
In practice, a computer is used to perform the
numerous mathematical calculations required. To
illustrate the process employed by the computer,
discussion that follows focuses on:
1. Weights in a three-stock portfolio, where:
– Weight of Stock A = xA
– Weight of Stock B = xB
– Weight of Stock C =1 - xA - xB
•
and the sum of the weights equals 1.0 2. Iso-Expected Return Lines
3. Iso-Variance Ellipses
4. The Critical Line
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Weights in a Three-Stock Portfolio(Data Below Pertains to the Graph That Follows)
Point on Graph
____________
a
b
c
d
ef
g
h
I j
k
l
m
n
xA
______
0
1.0
0
.5
.50
.25
0
01.5
-.5
-.5
-.5
1.8
xB
______
1.0
0
0
.5
0.5
.25
1.5
-.50
0
-.5
1.8
-.3
xC
______
0
0
1.0
0
.5
.5
.5
-.5
1.5-.5
1.5
2.0
-.3
-.5
Invest in only one stock
(Corners of the triangle)
Invest in only two stocks
(Perimeter of the triangle)
Invest in all three stocks
(Inside the triangle)
Short-selling occurswhen you are outside
the triangle
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Iso-Expected Return Lines
In the graph below, the iso-expected return line
is a line on which all portfolios have the same
expected return.
Given xA = weight of stock (A), and xB = weightof stock (B), the iso-expected return line is:
xB = a0 + a1xA
Once a0 + a
1 have been determined, we can
solve for a value of xB and an implied value of
xC, for any given value of xA
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Iso-Expected Return Line
(A graphical representation)
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
-2 -1 0 1 2 3
Weight of Stock B
Weight of
Stock A
Iso-Expected
Return Line
xB = a0 + a1xA
a0 = the intercept
a1 = the slope
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Computing the Intercept and Slope of an
Iso-Expected Return Line
AB
AB
ACB
AC
CB
CpB
p
CBA
10
A
CB
AC
CB
CpB
CBCACBBAA
CBABBAAp
x2.00.40x
][x1510
515
1510
1513x
][x)E(r)E(r)E(r)E(r
)E(r)E(r)E(r)E(rx
13%)E(rforLineReturnExpectedIso-
15%,)E(r10%,)E(r5%,)E(r:Example
a a
][x)E(r)E(r
)E(r)E(r
)E(r)E(r
)E(r)E(rx
:llyalgebraicagRearrangin
)E(rx)E(rx)E(r)E(rx)E(rx=
)E(r)xx(1)E(rx)E(rx)E(r
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Iso-Expected Return Line for a
Portfolio Return of 13%
xA
_____
-.5
0
.5
1.0
xB = .40 - 2.00xA
_____________
1.4
.4
-.6
-1.6
xC = 1 - xA - xB
____________
.1
.6
1.1
1.6
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-2 -1 0 1 2 3
xB
xA
Iso-Expected Return Line
For E(r p) = 13%
A S i f I E d R i
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A Series of Iso-Expected Return Lines By varying the value of portfolio expected return, E(rp),
and repeating the process above many times, we could
generate a series of iso-expected return lines.
Note: When E(rp) is changed, the intercept (a0)
changes, but the slope (a1
) remains unchanged.
][x)E(r)E(r
)E(r)E(r
)E(r)E(r
)E(r)E(rx A
CB
AC
CB
CpB
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-2 -1 0 1 2 3
xB
xA
17 15 13 11
Series of Iso-Expected Return
Lines in Percent
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Iso-Variance Ellipse
(A Set of Portfolios With Equal Variances)
First, note that the formula for portfolio
variance can be rearranged algebraically in
order to create the following quadratic
equation:
)r,Cov(r)xx(1x2+
)r,Cov(r)xx(1x2+
)r,Cov(rxx2+
)(rσ)xx(1)(rσx)(rσx)(rσ
CBBAB
CABAA
BABA
C22
BAB22
BA22
Ap2
:followsasfoundbecanc,andb,a,
:where
0cbxax B2B
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Iso-Variance Ellipse (Continued)
Next, the equation can be simplified further by
substituting the values for individual security
variances and covariances into the formula.
)(rσ)(rσ)](rσ-
)r,[Cov(rx2)]r,Cov(r2)(rσ)(r[σxc
)](rσ)r,2[Cov(r+
)]r,Cov(r)r,Cov(r)(rσ)r,[Cov(rx2b
)r,Cov(r2)(rσ)(rσa
p2
C2
C2
CAACAC
2
A
22
A
C2
CB
CBCAC2
BAA
CBC
2
B
2
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Iso-Variance Ellipse (An Example)
Given the covariance matrix for Stocks A, B, and C:
Therefore, in terms of axB2 + bxB + c = 0
Now, for a given 2(rp), we can create an iso-varianceellipse.
)(rσ-.28+x.22-x.19=
)(rσ-.28+.28)-(.17x2+2(.17)]-.28+[.25x=c
.38-x.34=
.28)-2(.09+.09)-.17-.28+(.15x2=b
.31=2(.09)-.28+.21a
.28 .09 .17
.09 .21 .15
.17 .15 .25
)r,Cov(r )r,Cov(r )r,Cov(r
)r,Cov(r )r,Cov(r )r,Cov(r
)r,Cov(r )r,Cov(r )r,Cov(r
p2
A2A
p2
A2A
A
A
CCBCACCBBBAB
CABAAA
0)](rσ.28x.22x[.19x.38]x[.34x.31 p2
A2ABA
2B
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Generating the Iso-Variance Ellipse for a
Portfolio Variance of .21
1. Select a value for xA
2. Solve for the two values of xB
Review of Algebra:
3. Repeat steps 1 and 2 many times for
numerous values of xA
a2
ac4bbx
0cbxax
:equationquadratictheGiven
2
B
B2B
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Generating the Iso-Variance Ellipse for a
Portfolio Variance of .21 (Continued)
Example: xA = .5
0)](rσ.28x.22x[.19x.38]x[.34x.31 p2
A2ABA
2B
.0382(.31)
75)4(.31)(.00.21)(.21)(x
.642(.31)
75)4(.31)(.00.21)(.21)(x
0.0075x.21x.31
0.21].28.22(.5)[.19(.5)x.38][.34(.5)x.31
2
B
2
B
B2B
2B
2B
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Generating the Iso-Variance Ellipse for a
Portfolio Variance of .21 (Continued)
A weight of .5 is simply one possible value for the
weight of Stock (A). For numerous values of xA you
could solve for the values of xB and plot the points in xB
xA space:
0
0.2
0.4
0.6
0.8
1
1.2
-1 -0.5 0 0.5 1 1.5
xB
xA
Iso-Variance Ellipse for2(rp) = .21.21
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Series of Iso-Variance Ellipses By varying the value of portfolio variance and
repeating the process many times, we could generate a
series of iso-variance ellipses. These ellipses will
converge on the MVP (the single portfolio with the
lowest level of variance).
0
0.5
1
1.5
-1 -0.5 0 0.5 1
xB
xA
.21.19 .17
MVP
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The Critical Line
Shows the portfolio weights for the portfolios in the minimum
variance set. Points of tangency between the iso-expected return
lines and the iso-variance ellipses. (Mathematically, these points of
tangency occur when the 1st derivative of the iso-variance formula
is equal to the 1st derivative of the iso-expected return line.)
0
0.5
1
1.5
-1 -0.5 0 0.5 1
xB
xA
.21.19 .17
MVP
16.9 15.6 13.6 9.4 7.4 6.1
Critical Line
Fi di th Mi i V i P tf li (MVP)
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Finding the Minimum Variance Portfolio (MVP)
Previously, we generated the following quadratic equation:
Rearranging, we can state:
1. Take the 1st derivative with respect to xB, and set it equal to 0:
2. Take the 1st derivative with respect to xA, and set it equal to 0:
3. Simultaneously solving the above two derivatives for xA & xB:
xA = .06 xB = .58 xC = .36
0)](rσ.28x.22x[.19x.38]x[.34x.31 p2
A2ABA
2B
.28x.22x.19x.38xx.34x.31)(rσ A2ABBA
2Bp
2
0.38x.34x.62x
)(rσ
ABB
p2
0.22x.38x.34x
)(rσ
ABA
p2
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Relationship Between the Critical Line and
the Minimum Variance Set
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
-2 -1 0 1 2
*
xB
MVP
xA
Critical LineC
D
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Relationship Between the Critical Line and
the Minimum Variance Set (Continued)
0
0.05
0.1
0.15
0.2
0.25
0 0.1 0.2 0.3 0.4 0.5 0.6
*MVP
C
D
Expected Return
Standard Deviation of Returns
Minimum Variance Set
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Minimum Variance Set When Short-Selling is Not
Allowed (Critical Line Passes Through the
Triangle)
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
-2 -1 0 1 2
*
xB
MVP
xA
Critical Line Passes
Through the Triangle
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Minimum Variance Set When Short-Selling is Not
Allowed (Critical Line Passes Through the Triangle)
CONTINUED
0
0.05
0.1
0.15
0.2
0.25
0 0.1 0.2 0.3 0.4 0.5 0.6
*MVP
Expected Return
Standard Deviation of Returns
With Short-Selling
Without Short-Selling
Stock (C)
Stock (A)
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Minimum Variance Set When Short-Selling is
Not Allowed (Critical Line Does Not Pass
Through the Triangle)
-1
-0.5
0
0.5
1
1.5
2
-2 -1 0 1 2
xB
xA
Critical Line Does Not Pass
Through the Triangle
Mi i V i S t Wh Sh t S lli i
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Minimum Variance Set When Short-Selling is
Not Allowed (Critical Line Does Not Pass
Through the Triangle)
CONTINUED
0
0.05
0.1
0.15
0.2
0.25
0 0.1 0.2 0.3 0.4 0.5 0.6
Expected Return
Standard Deviation of Returns
With Short-Selling
Without Short Selling
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The Minimum Variance Set:
(Property I)
If we combine two or more portfolios on the minimumvariance set, we get another portfolio on the minimumvariance set.
Example : Suppose you have $1,000 to invest. You sellportfolio (N) short $1,000 and invest the total $2,000 in
portfolio (M). What are the security weights for yournew portfolio (Z)?
Portfol io N : xA = -1.0, xB = 1.0, xC = 1.0
Portfol io M : xA = 1.0, xB = 0, xC = 0
Portfol io Z : xA = -1(-1.0) + 2(1.0) = 3.0
xB = -1(1.0) + 2(0) = -1.0
xC = -1(1.0) + 2(0) = -1.0
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The Minimum Variance Set: (Property I)
CONTINUED
-1.5
-1
-0.5
0
0.5
1
1.5
-2 -1 0 1 2 3 4
XB
XA
N
M
Z
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The Minimum Variance Set:
(Property II)
Given a population of securities, there will be
a simple linear relationship between the beta
factors of different securities and theirexpected (or average) returns if and only if the
betas are computed using a minimum variance
market index portfolio.
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The Minimum Variance Set: (Property II)
CONTINUED
0
0.05
0.1
0.15
0.2
0.25
0.3
0 0.16 0.32 0.48
0
0.05
0.1
0.15
0.2
0.25
0.3
0 1 2
E(r) E(r)
(r)
E(rZ) E(rZ)
M C
B
A
CM
B
A
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The Minimum Variance Set: (Property II)
CONTINUED
0
0.05
0.1
0.15
0.2
0.25
0 0.16 0.32 0.48
0
0.05
0.1
0.15
0.2
0.25
-1 0 1 2
E(r) E(r)
(r)
CA
B
M
E(rZ)CA
B
M
E(rZ)
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Notes on Property II
The intercept of a line drawn tangent to thebullet at the position of the market index
portfolio indicates the return on a zero beta
security or portfolio, E(rZ).
By definition, the beta of the market portfolio isequal to 1.0 (see the following graph).
Given E(rZ) and the fact that Z = 0, and E(rM)
and the fact thatM
= 1.0, the linear
relationship between return and beta can be
determined.
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Notes on Property II
CONTINUED
-0.2
-0.1
0
0.1
0.2
0.3
-0.1 0 0.1 0.2 0.3
rM
rM
= M = 1.00
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Return, Beta, Standard Deviation, and the
Correlation Coefficient
In the following graph, portfolios M, A, and B,
all have the same return and the same beta.
Portfolios M, A, and B, have different standard
deviations, however. The reason for this is thatportfolios A and B are less than perfectly
positively correlated with the market portfolio
(M).
)σ(r
)σ(rρ
)(rσ
)σ(r)σ(rρ
)(rσ
)r,Cov(rβ
M
jM j,
M2
M jM j,
M2
M j j
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Return, Beta, Standard Deviation, and the
Correlation Coefficient (Continued)
0
0.05
0.1
0.15
0.2
0.25
0.3
-0.16 0 0.16 0.32 0.48
0
0.05
0.1
0.15
0.2
0.25
0.3
0 1 2
E(r)
E(rZ) E(rZ)
(r)
E(r)
M
A B
j,M = 1.0
j,M = .7 j,M = .5
M, A, B
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Return Versus Beta When the Market
Portfolio (M**) is Inefficient
0
0.05
0.1
0.15
0.2
0.25
0.3
0 0.16 0.32 0.48
0
0.05
0.1
0.15
0.2
0.25
0.3
0 1 2
C
M**
A
B
M
E(r)
(r)
E(r)
CM**
A
B