Robust Portfolio Selection in Multiattribute Capital Budgeting

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Robust Portfolio Selection in Multiattribute
Capital Budgeting

• Pekka Mild and Ahti Salo
• Systems Analysis Laboratory
• Helsinki University of Technology
• P.O. Box 1100, 02150 HUT, Finland
• http//www.sal.hut.fi

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Background

• Multiattribute capital budgeting
• Several projects evaluated w.r.t several
  attributes (e.g., 6-12 attributes)
• Project value as weighted sum of attribute
  specific scores
• Only some of the projects can be started
• E.g. RD project portfolios
• E.g., Kleinmuntz Kleinmuntz (2001), Stummer
  Heidenberg (2003)
• Incomplete information in MCDM
• Imprecise attribute weights in additive overall
  value
• Hard to acquire precise weights
• Group settings, multiple stakeholders with
  different preferences
• Sensitivity analysis, e.g. allow 5 fluctuation
  of each weight
• E.g., Arbel (1989) Salo Hämäläinen (1992,
  1995, 2001) Kim Han (2000)

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Multiattribute capital budgeting

• Large number (e.g. m 50) of multiattribute
  projects
• Portfolio denoted by binary vector
• Attributes, i 1,,n, scores denoted by
• Additive aggregate value, i.e. a weighted sum
• Constraints
• Budget constraint
• Other constraints, e.g., mutually exclusive
  projects, portfolio balance
• Let PF denote the set of feasible portfolios
• Solve p to maximize V(p,w)
• Binary programming with fixed scores and weights

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Incomplete weight information (1/2)

• Interval bounds on attribute weights
• Feasible weight region
• Non-negative
• Sum up to one
• Different weights lead to different optimal
  portfolios
• Objective function coefficients vary with weights
• Generate a set of good candidate portfolios

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Incomplete weight information (2/2)

• Potentially optimal portfolios
• Optimal for some weights
• Set of potentially optimal portfolios PPO
• Pairwise dominance
• pk at least as good as pl for all feasible
  weights, better for some weights
• Non-dominated portfolios
• Portfolios not dominated by any other portfolio
• Set of non-dominated portfolios PND
• PPO ? PND

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Conceptual ideas

• Incomplete information in multiattribute capital
  budgeting
• Optimality replaced by
• Potential optimality
• Non-dominated portfolios
• Decision recommendations through the application
  of decision rules
• E.g., maximax, maximin, minimax regret
• Robust portfolio selection
• Reasonable performance across the full range of
  permissible parameter values
• Accounts for the lack of complete attribute
  weight information
• What portfolios can be defended - knowing that
  we have only incomplete information about
  weights?

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Computational issues in portfolio optimization

• Dominance checks require pairwise comparisons
• Number of possible portfolios is high
• m projects lead to 2m possible combinations
• Typically high number of feasible portfolios as
  well
• Usually far fewer truly interesting portfolios
• Brute force enumeration of all possibilities not
  computationally attractive
• Need for a dedicated portfolio algorithm
• First determine potentially optimal portfolios
• Repeat the algorithm to determine non-dominated
  portfolios

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Determination of potentially optimal portfolios
(1/3)

• Algorithm computes potentially optimal portfolios
• Two-phase algorithm based on linear programming
  and linear algebra
• Extreme point optimality implications (e.g.,
  Arbel, 1989 Carrizosa et.al., 1995)
• Either weight is fixed or portfolio is fixed

Treats feasible weight region according to fixed
portfolios. Defines subsets and determines
extreme points.
Projects score matrix (fixed)
Attribute weight coefficients, w?S0
Computes optimal portfolio with fixed weight
vectors(extreme points). Fixed LP objective
function.
Portfolio indicator vector
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Determination of potentially optimal portfolios
(2/3)

• Splits feasible weight region into disjoint
  subsets
• Each subset is either divided in two or
  considered done
• New subsets by additional constraints
• Subsets defined explicitly by extreme points
• For each (sub)set Sk the basic steps are

1. Calculate optimal portfolio at each extreme
point of Sk 2. i) If each extreme point has the
same optimal portfolio, conclude that this
portfolio is optimal in the entire subset Sk
ii) If some of the extremes have different
optimal portfolios, divide the respective
subset in two with a hyperplane exhibiting equal
value for the two portfolios chosen to
define the division
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Determination of potentially optimal portfolios
(3/3)

• The portfolios are constructed in descending
  value
• Only feasible portfolios are constructed
• No all inconclusive computations
• Constructed portfolios are potentially optimal
• No cross-checks and later rejections
• Extreme points of the subsets are generated by
  utilizing the extremes of the parent set

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An example potentially optimal portfolios (1/3)
v1(xj) v2(xj) v3(xj)
c(xj)
cT
Q
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An example potentially optimal portfolios (2/3)
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An example potentially optimal portfolios (3/3)
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From potentially optimal to non-dominated

• Potentially optimal portfolios not necessarily
  robust
• Optimal for some weights, lower bound omitted
• Missing a portfolio that is the second best for
  all weights
• Non-dominated portfolios are of interest
• The best portfolio is among the set of
  non-dominated
• No dominated portfolio can perform better
• Set of non-dominated portfolios still
  considerably focused
• Search for potentially optimal can be utilized
• Add constraints to exclude higher value
  portfolios (higher layers)
• Peeling off layers of portfolios, descending
  portfolio value
• Linearity with respect to the weights is essential

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Determination of non-dominated portfolios (1/2)
1. Calculate potentially optimal portfolios
on entire S0 2. Add constraints to exclude
portfolios generated thus far 3. Calculate
potentially optimal portfolios on entire S0
with additional constraints of step 2 4.
Check dominance for the candidate portfolios
of step 3. Accept portfolios that are not
dominated by any upper layer portfolio
V(pk,w)
V(pk,w)
pinfeas
p1
p2
p3
p4
p1 dominates p4
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Determination of non-dominated portfolios (2/2)

• The portfolios on the topmost layer are
  potentially optimal
• The portfolios accepted on lower layers are
  non-dominated
• Rules for early termination
• Only one new candidate portfolio on a new layer
• Each new candidate absolutely dominated by some
  upper layer portfolio
• Fewer computational rounds
• Dominance check required for each lower layer
  portfolio
• Pairwise check with all portfolios already
  generated on upper layers
• Number of pairwise comparisons still considerably
  lower compared to mechanical search through all
  pairs of possible portfolios

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Measures of portfolio performance

• Large number of non-dominated portfolios
• A set of good portfolios is of interest
• Performance measures required
• Convenient to calculate the measures only for the
  good portfolios
• Decision rules
• Maximax, Maximin, Central values, Minimax regret
• Measures based on weight regions
• Assuming a probability distribution on weights
• E.g., portfolio pk is optimal in 65 of the
  feasible weight region

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Portfolio-oriented project evaluation

• Core of a non-dominated portfolio
• Consists of projects included in all
  non-dominated portfolios
• Share of non-dominated portfolios in which a
  project is included
• Measures derived in the portfolio context - and
  not in isolation
• Implications for project choice
• Select core projects
• Discard projects that are not included in any
  non-dominated portfolio
• Reconsider remaining projects

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Uses of methodology

• Consensus-seeking in group decision making
• Consideration of multiple stakeholders interests
  (incomplete weights)
• Select a portfolio that best satisfies all views
• E.g. no-one has to give up more than 30 of their
  individual optimum
• Robust decision making in scenario analysis
• Attributes interpreted as scenarios
• Weights interpreted as probabilities
• Sequential project selection
• Core projects
• Additional constraints
• Sensitivity analysis
• Effect of small changes in the weights
• Displaying the emerging potential portfolios at
  once

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References

• Arbel, A., (1989). Approximate Articulation of
  Preference and Priority Derivation, EJOR, Vol.
  43, pp. 317-326.
• Carrizosa, E., Conde, E., Fernández, F. R.,
  Puerto, J., (1995). Multi-Criteria Analysis with
  Partial Information about the Weighting
  Coefficients, EJOR, Vol. 81, pp 291-301.
• Kim, S. H., Han, C. H., (2000). Establishing
  Dominance between Alternatives with Incomplete
  Information in a Hierarchically Structured Value
  Tree, EJOR, Vol. 122, pp. 79-90.
• Salo, A., Hämäläinen, R. P., (1992). Preference
  Assessment by Imprecise Ratio Statements,
  Operations Research, Vol. 40, pp. 1053-1060.
• Salo, A., Hämäläinen, R. P., (1995). Preference
  Programming Through Approximate Ratio
  Comparisons, EJOR, Vol. 82, pp. 458-475.
• Salo, A., Hämäläinen, R. P., (2001). Preference
  Ratios in Multiattribute Evaluation (PRIME) -
  Elicitation and Decision Procedures under
  Incomplete Information, IEEE Transactions on SMC,
  Vol. 31, pp. 533-545.
• Stummer, C., Heidenberg, K., (2003). Interactive
  RD Portfolio Analysis with Project
  Interdependencies and Time Profiles of Multiple
  Objectives, IEEE Trans. on Engineering
  Management, Vol. 50, pp. 175 - 183.