Multi-Objective Portfolio Optimization

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Multi-Objective Portfolio Optimization
Jeremy Eckhause AMSC 698S Professor S.
Gabriel 6 December 2004
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Portfolio Optimization Overview
Objective I want to maximize a quantity that
measures historical portfolio performance subject
to constraints such as 1. Not too much
concentration in any one sector, industry, or
individual stock. 2. Reasonable capital
usage 3. Low correlation with market
indices, Etc. The quantities that change will be
the weights of the portfolio constituents.
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Portfolio Optimization Overview

• Constraints
• Not too much concentration in any one sector,
  industry, or individual stock
• Reasonable capital usage
• Low correlation with market indices

(Linear Constraints)
(Linear Constraints)
(Change to covariance to get convex constraints.
This constraint will be added, but not in time
for the results of this presentation. These
constraints are not difficult to add, despite
being nonlinear, as they are convex.)
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Portfolio Optimization Overview
Multi-Objective Optimization I want to
maximize a quantity that measures historical
portfolio performance subject to
constraints. Quantity is the Sharpe
Ratio. Sharpe Ratio mean / standard deviation
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Portfolio Optimization Overview
Multi-Objective Optimization I want to
maximize a quantity that measures historical
portfolio performance subject to
constraints. Quantity is the Sharpe
Ratio. Sharpe Ratio mean / standard deviation
Maximize return while minimizing risk. Using
Markowitzs method, we can generate a Pareto
curve!
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Portfolio Optimization Definitions
Return The monthly historic returns over the
past five years for each possible
investment. In reality, the optimization uses
historic data to show the best portfolio over the
past five years. The hope is that these trends
will continue in the future.
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Portfolio Optimization Notation
Data
return of investment i in month j capital
usage of investment i (inverse of standard
deviation over five year returns) maximum capital
usage allowed if investment i is in in sector
k threshold level for sector k
directional bias within some range no. of
months vector of 1 (long) and 1 (short) for each
investment
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Portfolio Optimization Formulation
min
Make second objective function a constraint!
max
s.t.
(1)
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Portfolio Optimization Approach
min
s.t.
(2)
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Portfolio Optimization Approach
We know that from the constraint method
that If is a binding constraint,
then the optimal solution to (2) is along the
efficient Pareto frontier for (1). What are
those values for where the constraint
is binding?
Objective function changes for 3.23 lt l lt 6.702
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Portfolio Optimization Model

• Optimization model built in LINGO
• Change value for lower bound return
• Quickly calculates the global optimum for even
  large- scale problems
• Output to Excel to generate graphs of Pareto
  optimal values

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Portfolio Optimization Model
LINGO Code
!Objective function MIN SHARPEVAR !Min
variance SHARPEVAR (_at_SUM(PD(J) ((Y(J) -
MEANY)2))/N) !Constraints _at_FOR(PD(I)
_at_SUM(ST(J) M5(I,J)X(J) ) Y(I)) !Assigns
y_i to the sum of each row MEANY (_at_SUM(PD(I)
Y(I))/N) !Mean of y is the average
return MEANY gt LB !Return above
some lower bound _at_SUM(ST(I) CU(I)XX(I)) lt
MAXCU !Below some maximum capital usage
CU(1)X(1) CU(4)X(4) lt THR(1) !Threshold
limits for each sector CU(2)X(2)
lt THR(2) CU(3)X(3) lt THR(3)
_at_SUM(ST(I) CU(I)X(I)) lt DBMAX _at_SUM(ST(I)
CU(I)X(I)) gt DBMIN ! Within some directional
bias range
Etc.
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Portfolio Optimization Data
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Portfolio Optimization Results
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Portfolio Optimization Conclusions
Multi-objective optimization tool for maximizing
return while minimizing variance works
well. Relatively easy to minimize variance and
find the knee of the Pareto frontier when
changing the minimum return. Allows the decision
maker on Wall Street to pick his or her risk
tolerance vs. expected payoff.
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Portfolio Optimization Future Work
Add additional constraints, such as considering
some minimum on recent return. Some may require
using integer variables, although this is
expected to be a small number. Perform analysis
on actual data (in progress). will be in the
100 variable range. Matrix should be about
100x100. Additional constraints may add another
100-500 variables. All easily handled by a LINGO
or similar solver. Enhance model to generate
Pareto frontier efficiently, automatically
providing each strategy (i.e., ) during the
output phase.