6. Dual Feasible Regions
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Decision Making Process
Portfolio selection
• Solve for efficient frontier
• Plot efficient frontier
expRet
stdDev
• Show to investor
• Investor selects most preferred point on frontier
as optimal portfolio
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In Matrix Form
In multi-criterion matrix form, the problem of portfolio
selection is
min xT S x
max m T x
s.t. x  S
where
x
S
m
S
n-vector of portfolio weights
n x n covariance matrix
n-vector of expected returns
set of all x-vectors that abide by the constraints
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The Two Feasible Regions
Solution of a portfolio selection problem involves points
from two feasible regions
min xT S x
max m T x
s.t. x  S
S  Rn
feasible region
in decision space
Z  Rk
feasible region in
criterion space
Endeavor is to compute all nondominated points in Z
and the x-vectors in S that go with them.
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Others Names
S  Rn
feasible region
in decision space
Z  Rk
feasible region in
criterion space
objective space
evaluation space
results space
attribute space
goal value space
predicted outcome space
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Simplest Case …
Simplest case of S and Z when n = 2.
min xT S x
max m T x
s.t. x  S
expRet
x2
z1
Z is just the curve
x2
Z
S
z2
x1
x1
stdDev
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The red are inverse images of the green. With
x1 + x2 = 1
the only constraint, there is nothing to hold back short
selling. The green are images of the red.
expRet
x2
z1
x2
Z
S
z2
x1
x1
stdDev
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Slightly More Complicated Case
Slightly more complicated case of S and Z when n = 3.
min xT S x
max m T x
s.t. x  S
expRet
x2
z1
x2
z3
S
Z
x1
x1
x3
z3
stdDev
x3
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Effect of xi ≥ 0 Constraints
The red are the inverse images of the green. With
x1 + x2 + x3 = 1
xi ≥ 0 (these preclude short selling)
all feasible x-vectors are on the plane as shown in the
nonnegative orthant of R3.
expRet
x2
z1
x2
z6
z2
x6
x5
x4
S
z5
z4
x1
x1
x3
Z
z3
stdDev
x3
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When Solver is Needed
In any portfolio problem
min xT S x
max m T x
s.t. x  S
we always have the constraint
x1  x2   xn1  xn  1
(mandatory)
But if there are other constraints such as
x2
 x6
 x8  .08
(sector constraint)
all xi  0
(no short selling)
Problem can only be solved by mathematical programming
(i.e., Solver).
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Dotted Representations …
When there is more than just the x1 + x2 +…+ xn = 1
constraint, mathematical programming can take time.
Efficient frontiers typically presented in the form of
dotted representations.
Requires a separate Solver run to get each dot. Tedious.
expRet
stdDev
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r-Constraint Approach
Start by obtaining top and bottom points on the efficient
frontier
max m T x
s.t. x  S
min xT S x
s.t. x  S
Solver, of course, needed
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1st intermediate optimization
min xT S x
s.t. x  S
m T x  .018
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2nd intermediate optimization
min xT S x
s.t. x  S
m T x  .016
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3rd intermediate optimization
min xT S x
s.t. x  S
m T x  .014
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4th intermediate optimization
min xT S x
s.t. x  S
m T x  .012
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5th intermediate optimization
min xT S x
s.t. x  S
m T x  .010
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6th intermediate optimization
min xT S x
s.t. x  S
m T x  .008
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Done
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In Reality
In reality, we would probably run for 50, 100, 200 points.
On problems hundreds of securities, can take considerable
computer resources.
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As We Add Constraints
What happens to efficient frontier when we add constraints.
expRet
stdDev
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Typically gets smaller and moves inward.
expRet
stdDev
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Multiple Objectives in Portfolio Selection
Suppose that in addition to
max { risk }
max { return }
There are additional objectives such as
max { dividends }
max { liquidity }
max { growth in sales }
max { social responsibility }
min { turnover }
etc.