1. 6. Dual Feasible Regions

2. Decision Making Process
Portfolio selection
Solve for efficient frontier
Plot efficient frontier
Show to investor
Investor selects most preferred point on frontier as optimal portfolio
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3. In Matrix Form
In multi-criterion matrix form, the problem of portfolio selection is
where
x n-vector of portfolio weights
S n x n covariance matrix
m n-vector of expected returns
S set of all x-vectors that abide by the constraints

4. The Two Feasible Regions
Solution of a portfolio selection problem involves points from two feasible regions
feasible region in decision space
feasible region in criterion space
Endeavor is to compute all nondominated points in Z and the x-vectors in S that go with them.

5. Others Names feasible region in decision space
feasible region in criterion space
objective space
evaluation space
results space
attribute space
goal value space
predicted outcome space

6. Simplest case of S and Z when n = 2.
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Z is just the curve
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7. 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.
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8. Slightly More Complicated Case
Slightly more complicated case of S and Z when n = 3.
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9. Effect of xi ≥ 0 Constraints
The red are the inverse images of the green. With
x1 + x2 + x3 = 1
xi ≥ (these preclude short selling)
all feasible x-vectors are on the plane as shown in the nonnegative orthant of R3.
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10. When Solver is Needed In any portfolio problem
we always have the constraint
But if there are other constraints such as
Problem can only be solved by mathematical programming (i.e., Solver).

11. 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.
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12. r-Constraint Approach
Start by obtaining top and bottom points on the efficient frontier
Solver, of course, needed

13. 1st intermediate optimization

14. 2nd intermediate optimization

15. 3rd intermediate optimization

16. 4th intermediate optimization

17. 5th intermediate optimization

18. 6th intermediate optimization

19. Done

20. In Reality In reality, we would probably run for 50, 100, 200 points.
On problems hundreds of securities, can take considerable computer resources.

21. As We Add Constraints
What happens to efficient frontier when we add constraints.
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22. Typically gets smaller and moves inward.
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23. Multiple Objectives in Portfolio Selection
Suppose that in addition to
There are additional objectives such as
max { risk }
max { return }
max { dividends }
max { liquidity }
max { growth in sales }
max { social responsibility }
min { turnover }
etc.