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


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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

jiij xvwpwpV

1 1

)(),(

mjpppp jm ,...,11,0),,...,( 1

)]([][ jiji xvqQ

m

jjjj jcbudgetcp

1

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

Fllk PpwpVwpVSw ),,(),(s.t.0

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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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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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)

1

,0

3

85.0

2

3/2

3

11

32

31

21

21

i

i

w

iw

ww

ww

ww

ww

10Tpc= Q = cT

v1(xj) v2(xj) v3(xj) c(xj)

x1

x2

x3

x4

x5

854

231

371

435

732

6

1

4

5

4


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An example: potentially optimal portfolios (2/3)An example: potentially optimal portfolios (2/3)

3w

2w

1w

)0,1,0(

)0,0,1(

)1,0,0(

0S

21 2ww 21 67.0 ww

31 85.0 ww 32 3ww

1w2w

3w

4w

)0,1,0,1,1(11 pw)0,1,1,0,1(22 pw

)0,1,1,1,0(33 pw)0,1,1,1,0(34 pw

3w

2w

1w

)0,0,1(

)1,0,0(

21 pp

2S

1S

6w

1w

4w

5w

0S

2S1S)0,1,0,1,1(11 pw)0,1,0,1,1(15 pw

)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

8w

3S

0S

2S1S

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

3w

2w

1w

)0,0,1(

)0,1,0(

)1,0,0(

3S

4S

5S

6S

3S

0S

2S1S

4S 6S5S p1 p3 p2 p3

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.

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