Robust Portfolio Selection in Multiattribute Capital Budgeting

1. 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

2. 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)

3. 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 PFdenote the set of feasible portfolios • Solve pto maximize V(p,w) • Binary programming with fixed scores and weights

4. Coeffs. for binary vector p 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

5. V(pk,w) V(pk,w) p1 p2 p3 w1 p4 0 1 w2 1 0 Incomplete weight information (2/2) • Potentially optimal portfolios • Optimal for some weights: • Set of potentially optimal portfolios PPO • Pairwise dominance • pkat least as good as plfor 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

6. 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?”

7. Computational issues in portfolio optimization • Dominance checks require pairwise comparisons • Number of possible portfolios is high • m projects lead to 2mpossible 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

8. 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

9. 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

10. V(pk,w) V(pk,w) pinfeas p1 p2 w1 0 1 w2 1 0 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

11. x1 x2 x3 x4 x5 An example: potentially optimal portfolios (1/3) v1(xj) v2(xj) v3(xj) c(xj) = cT = Q

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

13. p1 p3 p2 p3 p1 An example: potentially optimal portfolios (3/3)

14. 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

15. w1 0 1 w2 1 0 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

16. 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

17. 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

18. 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

19. 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

20. 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 R&D Portfolio Analysis with Project Interdependencies and Time Profiles of Multiple Objectives, IEEE Trans. on Engineering Management, Vol. 50, pp. 175 - 183.