1 helsinki university of technology systems analysis laboratory robust portfolio selection in...
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Helsinki University of Technology Systems Analysis Laboratory
Robust Portfolio Selection in Multiattribute Robust Portfolio Selection in Multiattribute
Capital BudgetingCapital Budgeting
Pekka Mild and Ahti SaloSystems Analysis Laboratory
Helsinki University of TechnologyP.O. Box 1100, 02150 HUT, Finland
http://www.sal.hut.fi
Helsinki University of Technology Systems Analysis Laboratory
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Background 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. R&D 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 budgetingMultiattribute 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
m
j
n
i
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)(),(
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)]([][ jiji xvqQ
m
jjjj jcbudgetcp
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project ofcost where,
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Incomplete weight information (1/2)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
1,,|0 iiiijijijij wuwlwuwwlwS
m
j
n
i
jiijp xvwp
1 11,0 )(max
Coeffs. for binary vector p
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Incomplete weight information (2/2)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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0)],(),([min0
wpVwpV lkSw
w2 1 0
V(pk,w) V(pk,w)
w1 0 1
p1
p2
p4
p3
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Conceptual ideas 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 optimizationComputational 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)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
Computes optimal portfolio with fixed weight vectors(extreme points). Fixed LP objective function.
Treats feasible weight region according to fixed portfolios.Defines subsets anddetermines extreme points.
TpQwwpV ),(
Portfolio indicator vector
Attribute weightcoefficients, wS0
Projects’ scorematrix (fixed)
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Determination of potentially optimal portfolios (2/3)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)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
V(pk,w) V(pk,w)
w10 1w21 0
pinfeas
p1
p2
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An example: potentially optimal portfolios (1/3)An example: potentially optimal portfolios (1/3)
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An example: potentially optimal portfolios (2/3)An example: potentially optimal portfolios (2/3)
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)0,1,1,1,0(33 pw)0,1,1,1,0(34 pw
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21 pp
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6w
1w
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)0,1,1,1,0(36 pw)0,1,1,1,0(34 pw
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An example: potentially optimal portfolios (3/3)An example: potentially optimal portfolios (3/3)3w
2w
1w
)0,0,1(
)0,1,0(
)1,0,0(
31 pp
4S
3S 2S1w
5w
7w
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4S p1
)0,1,0,1,1(11 pw)0,1,0,1,1(15 pw
)0,1,0,1,1(17 pw)0,1,0,1,1(18 pw
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3S
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32 pp
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From potentially optimal to non-dominatedFrom 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)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)
w10 1w21 0
pinfeas
p1
p2
p4
p3
p1 dominates p4
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Determination of non-dominated portfolios (2/2)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
),(max),(min00
wpVwpV lSw
kSw
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Measures of portfolio performanceMeasures 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 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 methodologyUses 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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ReferencesReferences
» 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 R&D Portfolio Analysis with Project Interdependencies and Time Profiles of Multiple Objectives, IEEE Trans. on Engineering Management, Vol. 50, pp. 175 - 183.