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



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 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)



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

____________

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



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



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


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



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


For E(r p) = 13%

A S i f I E d R i



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



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



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



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



p2

A2A

p2

A2A

A

A

CCBCACCBBBAB

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

:equationquadratictheGiven

2

B

B2B




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




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

Fi di th Mi i V i P tf li (MVP)



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



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


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 Passes

Through 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


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 Pass

Through the Triangle

Mi i V i S t Wh Sh t S lli i



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


With Short-Selling

Without Short Selling



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



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 theirexpected (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

B

A



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)



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.



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



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



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



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

finding the efficient set

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