finding the efficient set (chapter 5) feasible portfolios minimum variance set & the efficient...
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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
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
Minimum Variance Set and theEfficient Set
Minimum Variance Set: Identifies those portfolios that have the lowest level of risk for a given expected rate of return.
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 minimumvariance portfolio (one with the lowest level of risk)
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
Weights in a Three-Stock Portfolio(Data Below Pertains to the Graph That Follows)
Point on Graph____________
abcdefghIjkl
mn
xA
______0
1.00.5.50
.2500
1.5-.5-.5-.51.8
xB
______1.000.50.5
.251.5-.500
-.51.8-.3
xC
______00
1.00.5.5.5-.51.5-.51.52.0-.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 occurs when you are outside the triangle
Weights in a Three-Stock Portfolio (Continued)
(Graph of Preceding Data)
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
-2 -1 0 1 2 3
m
h
a
f
kc
g
e
d
b j
nil
Weight of Stock B
Weight of Stock A
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 = weight of stock (B), the iso-expected return line is:
xB = a0 + a1xA
Once a0 + a1 have been determined, we can solve for a value of xB and an implied value of xC, for any given value of xA
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
Computing the Intercept and Slope of an
Iso-Expected Return Line
AB
AB
ACB
AC
CB
CpB
p
CBA
10
ACB
AC
CB
CpB
CBCACBBAA
CBABBAAp
x2.00.40 x
][x1510
515
1510
1513x
][x)E(r)E(r
)E(r)E(r
)E(r)E(r
)E(r)E(rx
13%)E(rfor Line Return ExpectedIso-
15%,)E(r 10%,)E(r 5%,)E(r :Example
a a
][x)E(r)E(r
)E(r)E(r
)E(r)E(r
)E(r)E(rx
:llyalgebraica gRearrangin
)E(rx)E(rx)E(r)E(rx)E(rx =
)E(r)xx(1)E(rx)E(rx)E(r
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 LineFor E(rp) = 13%
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 ReturnLines in Percent
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
:follows as found be can c, and b, a,
:where
0cbxax B2B
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
CAACAC2
A22
A
C2
CB
CBCAC2
BAA
CBC2
B2
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-variance ellipse.
)(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 + .21 a
.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
CCBCAC
CBBBAB
CABAAA
0)](rσ.28x.22x[.19x.38]x[.34x.31 p2
A2ABA
2B
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
:equation quadratic the Given
2
B
B2B
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
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
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
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
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
)(rσAB
B
p2
0.22x.38x.34x
)(rσAB
A
p2
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
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
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 PassesThrough the Triangle
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)
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 PassThrough the Triangle
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
The Minimum Variance Set:(Property I)
If we combine two or more portfolios on the minimum variance set, we get another portfolio on the minimum variance set.
Example: Suppose you have $1,000 to invest. You sell portfolio (N) short $1,000 and invest the total $2,000 in portfolio (M). What are the security weights for your new portfolio (Z)?
Portfolio N: xA = -1.0, xB = 1.0, xC = 1.0
Portfolio M: xA = 1.0, xB = 0, xC = 0
Portfolio 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
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
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 their expected (or average) returns if and only if the betas are computed using a minimum variance market index portfolio.
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
BA
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)
C
A
BM
E(rZ)C
A
BM
E(rZ)
Notes on Property II
The intercept of a line drawn tangent to the bullet 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 is equal to 1.0 (see the following graph).
Given E(rZ) and the fact that Z = 0, and E(rM) and the fact that M = 1.0, the linear relationship between return and beta can be determined.
Notes on Property IICONTINUED
-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
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 that portfolios 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
jMj,
M2
MjMj,
M2
Mjj
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 = .7j,M = .5
M, A, B
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
CM**
A
B
M
E(r)
(r)
E(r)
CM**
A
B