Introduction to Multiple Criteria Portfolio Selection Part 1: Conventional Portfolio Selection

1
Introduction to Multiple Criteria Portfolio
SelectionPart 1 Conventional Portfolio
Selection

• Ralph E. Steuer (University of Georgia)
• Yue York Qi (University of Georgia)
• Markus Hirschberger (University of
  Eichstätt-Ingolstadt)

2
Overview

• Conventional portfolio selection.
• Solve mean-variance formulation for all efficient
  points.
• Plot efficient frontier.
• Show to investor.
• Investor selects most preferred point on frontier
  as optimal portfolio.
• What is called the efficient set will be called
  the nondominated set.
• Developments in multiple criteria portfolio
  selection.

Because conventional portfolio selection is
discussed to form a basis for multiple criteria
portfolio selection, it will be motivated a bit
differently.
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Conventional Portfolio Selection

• Assume
• A sum to be invested
• Beginning of holding period, end of holding
  period
• n securities in which it is possible to invest
• x (x1,, xn) is an investment proportion vector
  (also called a portfolio)
• As for the number of securities n
• many large institutions have approved lists
  where n is anywhere from several hundred to a
  thousand
• when attempting to form a portfolio to mimic a
  large broad based index (like SP500, EAFE,
  Wilshire 5000), n can be up to several thousand.

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• Securities have attributes over the holding
  period
• ri is the percent return of i-th security
• Other attributes could be
• di is dividends of i-th security
• si is social responsibility of i-th security
• qi is liquidity of i-th security
• gi is growth in sales of i-th security
• ai is amount invested in RD of i-th security
• etc.
• Conventional portfolio selection hardly pays any
  attention to attributes other than ri .
• Note that all of the above-mentioned attributes
  are stochastic (which we will assume are random
  variables that come from distributions)

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• Trying to make as much money as possible and
  picking from the n securities, investors problem
  is to solve
• where
• rp is percent return on a portfolio over the
  holding period.
• S, the feasible region in decision space, is
  often as simple as
• What makes portfolio selection interesting is
  that the quantity rp to be maximized in (Orig) is
  a random variable (as rp is a function of the
  individual-security ri random variables).
• Hence, (Orig) is a stochastic programming problem
  -- and how to solve a stochastic programming
  problem is not well-defined.

(Orig)
or
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• Strategy is to convert the stochastic program
• to an equivalent deterministic problem.
• But to do so, decisions must be made.
• Those made in conventional portfolio selection
  include
• that the means mi , variances sii and
  covariances sij of the ri are known (this is a
  strong assumption but is standardly done in the
  theory of conventional portfolio selection) at
  the beginning of the holding period
• that maximizing expected utility leads to a
  optimal solution
• Also, conventional portfolio selection assumes
  that
• it suffices for expected utility to be given by
  EU(rp) Erp l Varrp
• which means that optimal portfolio involves can
  be identified by a trade-off between the mean and
  variance of random variable rp
• each investors l ? 0 may be different and is not
  known beforehand
• l can be viewed as a risk tolerance parameter

(Orig)
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• Assuming U is Taylor series expandable, theory
  guarantees the validity of EU(rp) Erp l
  Varrp whenever
• U is increasing and quadratic, or
• r (r1,, rn) follows the multinormal
  distribution
• First condition is proved quite easily. Proof of
  second condition is more difficult.
• To maximize expected utility EU(rp) Erp l
  Varrp , the problem is to solve
• (1) is to be solved for all l ? 0 because the
  true value of l to use only becomes revealed
  after all x-solutions have been processed.

(1)
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• Processing begins by forming the (Stdrp, Erp)
  combination for each of the x-solutions generated
  by (1).
• In portfolio optimization, theory and computation
  are generally conducted in terms of variance
  while solutions are displayed to investors in
  terms of standard deviation.
• In multiple criteria optimization terms, a
  (Stdrp, Erp) combination is a criterion
  vector
• The set of all (Stdrp, Erp) criterion vectors
  resulting from (1) form precisely the set of all
  such criterion vectors that are nondominated in
  (Stdrp, Erp) space.
• A combination is nondominated iff there exists no
  x ? S whose combination is the same or better in
  all components, and strictly better in at least
  one.

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• Plotting all (Stdrp, Erp) images of all of
  the x-solutions of (1) results in the efficient
  frontier -- but what we will call the
  nondominated frontier.
• After viewing the nondominated frontier, the
  investor selects his or her most preferred
  criterion vector on the curve.
• Taking an inverse image of the selected (Stdrp,
  Erp) criterion vector, we obtain an x ? S that
  specifies the investors optimal portfolio.
• People are sometimes interested in the minimum
  variance boundary of the set of all (Stdrp,
  Erp) criterion vectors.

Erp
nondominated portion of the set of all (Stdrp,
Erp) criterion vectors
Stdrp
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• Plotting all (Stdrp, Erp) images of all of
  the x-solutions of (1) results in the efficient
  frontier -- but what we will call the
  nondominated frontier.
• After viewing the nondominated frontier, the
  investor selects his or her most preferred
  criterion vector on the curve.
• Taking an inverse image of the selected (Stdrp,
  Erp) criterion vector, we obtain an x ? S that
  specifies the investors optimal portfolio.
• People are sometimes interested in the minimum
  variance boundary of the set of all (Stdrp,
  Erp) criterion vectors.

Erp
minimum variance boundary of the set of all
(Stdrp, Erp) criterion vectors
Stdrp
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• There are several ways to solve

• where
• S is the n ? n covariance matrix (of the sij)
• m the n-vector of the mi

• One is use use Markowitzs critical line
  algorithm, which is a form of parametric
  quadratic programming. Many people find this
  algorithm difficult to learn.
• Other than for Markowitzs team, finance and
  operations research have largely taken a pass on
  this algorithm.

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• Another is to view (1) as a multi-criteria
  optimization program

feasible region in criterion space
feasible region in decision space
where the endeavor is to compute all efficient
points in S in order to obtain the nondominated
portion of Z.
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• S and Z when
• and n 2 securities
• feasible region

Erp
Stdrp

• Extent to which Z bows toward the vertical axis
  depends upon how uncorrelated the returns of the
  two securities are.
• Z is a hyperbola (Stdrp, Erp) space, or a
  parabola in (Varrp, Erp) space.

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• S and Z when
• and n 2 securities
• feasible region

Erp
Stdrp

• Dont need mathematical programming whenever
• Everything can be computed by formulas.

15
Let
Then, for instance,
and
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• S and Z when
• and n 2 securities
• feasible region

Erp
Stdrp

• where z1 (Stdr1, Er1) plots the standard
  deviation and expected value of the return of the
  1-st security. Same for z2 but for the 2-nd
  security.

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• S and Z when
• and n 2 securities
• feasible region

Erp
Stdrp

• Need mathematical programming whenever S is more
  complicated than

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• S and Z when
• and n 3 securities
• feasible region

Erp
Stdrp
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• S and Z when
• and n 3 securities
• feasible region

Erp
Stdrp
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• S and Z when
• and n 3 securities
• feasible region

Erp
Stdrp

• Efficient set is piecewise linear, while
  nondominated set is piecewise hyperbolic.
• All portfolios on the same nondominated
  hyperbolic segment (or on the same efficient
  linear segment) involve the same securities, just
  in different proportions.

21
Introduction to Multiple Criteria Portfolio
SelectionPart 2 Traditional Dotted
Representations

• Ralph E. Steuer (University of Georgia)
• Yue York Qi (University of Georgia)
• Markus Hirschberger (University of
  Eichstätt-Ingolstadt)

22

• However, the most popular approach is to attempt
  a solution of

(1)
by systematically varying r across the interval
a, b in the e-constraint formulation
(2)
where a, b is a large enough interval to
generate the nondominated frontier.

• That is, solve (2) for about 10 to 20
  different values of r from a, b.
• With regard a, b, b is easy to obtain but a
  can be more difficult.
• For convenience, we will call (2) the
  r-constraint formulation.

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• Solving (2) repetitively results in a dotted
  representation of the nondominated frontier
• By enlarging a, b, minimum variance boundary
  can be obtained

Erp
Stdrp
Erp
bottom part requires r-constraint to be
Stdrp
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• Why have people opted for the r-constraint
  formulation?
• Because
• process of obtaining nondominated frontier is,
  perhaps, more transparent.
• by using (2), software from Solver, Matlab,
  Cplex, can be employed.
• Double duty can be obtained from (2). By
  converting the r-constraint in (2) to and
  assuming S x ? Rn 1Tx 1, everything about
  the nondominated frontier and minimum variance
  boundary can be computed in closed form.
• parametric quadratic programming, which involves
  advanced mathematical programming, can be avoided.

(2)
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r-Constraint Approach
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.16
27
.14
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.12
.12
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.10
.10
Can stop because Erp of the point computed is
less than r-value used.
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.08
.08
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connecting the dots
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Tangency Solution Approach
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.10
34
.10
35
.10
36
.07
37
.07
38
.07
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-.05
40
-.05
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-.05
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Cant get nose, though, because that would
require a c ?
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• But there are disadvantages of the two procedures
• each repetition winds up being solved from
  scratch
• while each dot produced is on the nondominated
  frontier, the full
• nondominated frontier is not computed
• in tangency solution approach, difficult to
  control for dots equally spaced by expected
  return
• Furthermore, the r-constraint approach of (2)
• does not reveal the interesting structure of the
  nondominated frontier
• is not easily generalizable to multiple criteria
  portfolio selection

44
Introduction to Multiple Criteria Portfolio
SelectionPart 3 Structure of Nondominated Sets

• Ralph E. Steuer (University of Georgia)
• Yue York Qi (University of Georgia)
• Markus Hirschberger (University of
  Eichstätt-Ingolstadt)

45

• Actual nondominated frontiers are piecewise
  hyperbolic. The dots on the nondominated
  frontier below are turning points, where one
  hyperbolic segment ends an another begins.
• When S is more complicated than x ? Rn 1Tx
  1, nondominated frontiers tend not to be as
  perfectly smooth as one gets the impression from
  textbooks.

Erp
from a problem with n 12
Stdrp
46

• Lets step back and look at the process we have
  been following.

to only make money
overall focus
random variable to be maximized
expected utility assumption
equivalent deterministic objectives

• How is it that Varrp and Erp are
  deterministic? Recall that all mi , sii and
  sij are assumed known at the beginning of the
  holding period
• Note that U has only one argument that being rp

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• One might wonder why not to incorporate multiple
  objectives into portfolio selection using
  constraints?
• For example, liquidity ?
• However, RHSs like are essentially judgment or
  guesses.
• Since adding extra constraints only reduces the
  size of the feasible region, new nondominated
  frontier can only be lower than the old one.
• Since nondominated set is a surface, what is our
  chance of guessing a line across the nondominated
  surface that hits an optimal criterion vector of
  an investor?

Erp
Stdrp
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• One might wonder why not to incorporate multiple
  objectives into portfolio selection using
  constraints?
• For example, liquidity ?
• However, RHSs like are essentially judgment or
  guesses.
• Since adding extra constraints only reduces the
  size of the feasible region, new nondominated
  frontier can only be lower than the old one.
• Since nondominated set is a surface, what is our
  chance of guessing a line across the nondominated
  surface that hits an optimal criterion vector of
  an investor?

Erp
Stdrp
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Multiple Objective Portfolio Selection

• Say that in addition to
• investor has additional stochastic
  objectives such as

max rp portfolio return
max dp dividends max qp liquidity max
gp growth in sales max sp social
responsibility max tp portfolio return over
some benchmark
50

• With multiple stochastic original objectives, the
  process becomes

build an optimal well rounded portfolio
overall focus
random variables to be maximized
expected utility assumption
equivalent deterministic objectives

• Distinguishing characteristic here is that U has
  multiple arguments.

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• Conventional (1-quadratic 1-linear)

Erp
piece-wise hyperbolic
Stdrp
efficient set is a path of connected line
segments in S

• Multiple criteria (1-quadratic 2-linear)

Erp
platelet-wise hyperboloidic
Edp
Stdrp
efficient set is a set of connected polyhedral
subsets in S
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• To compute all hyperboloidic platelets of a
  1-quadratic q-linear program, q 2, we have been
  writing a computer code. Intended for the
  academic public domain (not ready yet), it is
  called CIOPS.
• Some results for the 1-quadratic 1-linear case
  (when the covariance matrices are 100 dense and
  diagonal) are

• Times in seconds on Dell 2.13GHz Centrino laptop
  (sample size 10 for all cells)
• Until early 90s, because of CPU time limitations,
  about the only portfolio problems larger than n
  500 that could be solved were those that
  possessed covariance matrices sparse in nonzero
  elements.

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Introduction to Multiple Criteria Portfolio
SelectionPart 4 Emerging Research

• Ralph E. Steuer (University of Georgia)
• Yue York Qi (University of Georgia)
• Markus Hirschberger (University of
  Eichstätt-Ingolstadt)

54

• One might ask where did all of the covariance
  matrices come from for the tests.
• From a random covariance matrix generator.
• Distributions of variances and covariances in a
  typical real portfolio-selection covariance
  matrix.

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?
Assume investor is interested in optimizing both
portfolio return and dividends
?
Under appropriate assumptions, expected utility
is given by
?
Because of two unknown parameters, equivalent
deterministic problem is
?
Erp

• actual nondominated surface of
• 111 platelets
• 130 platelet corner points
• for an n 50 problem

Stdrp
E(dp)
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• With multiple stochastic original objectives, the
  process becomes

build an optimal well rounded portfolio
overall focus
random variables to be maximized
expected utility assumption
equivalent deterministic objectives

• Distinguishing characteristic here is that U has
  multiple arguments.

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• For 1-quadratic 2-linear problems, we have
• Are able to compute all platelet corner points
  (which are the images of the polyhedral subsets
  of S corresponding to the platelets.
• Number of unique platelet corner points is nearly
  the same as number of nondominated platelets.

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• Consider the 2-quadratic 1-linear
• Strategy for generating discrete representations
  of the nondominated set
• When S1 and S2 are positive definite, any convex
  combination
• is positive definite.
• Thus, nondominated set of (b) is a subset of
  nondominated set of (a)
• Solve (b) for a hundred different convex
  combinations of S1 and S2

(a)
(b)
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• Consider the 2-quadratic 2-linear
• Strategy for generating discrete representations
  of the nondominated set
• Let be a convex combination of S1 and S2
  
• Then, nondominated set of (d) is a subset of
  nondominated set of (c)

(c)
(d)
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Future Research

• We think strategy can be applied to 2-quadratic
  2-linear, and beyond.

• We believe hundreds of thousands of points can be
  searched using

versions of the following interactive procedures
from multiple criteria optimization

• projected line search (Korhonen and Karaivanova)
• dispersed probing (Tchebycheff Method)
• classification (Miettinen)
• interactive decision maps (Lotov)

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• Other original objectives that are deterministic,
  but harder to implement than stochastic
  objectives. Note their portfolio-as-a-whole
  nature.

min pa deviations from given asset allocation
percentages min pn number of securities in
portfolio min pt turnover min pw
maximum investment proportion weight min ps
amount of short selling min ph number
securities sold short
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The End