1 Helsinki University of Technology Systems Analysis Laboratory Robust Portfolio Selection in Multiattribute Capital Budgeting Pekka Mild and Ahti Salo Systems Analysis Laboratory Helsinki University of Technology P.O. Box 1100, HUT, Finland

Helsinki University of Technology Systems Analysis Laboratory 2 Background n 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) n 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)

Helsinki University of Technology Systems Analysis Laboratory 3 Multiattribute capital budgeting n 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 n Constraints –Budget constraint –Other constraints, e.g., mutually exclusive projects, portfolio balance –Let P F denote the set of feasible portfolios n Solve p to maximize V(p,w) –Binary programming with fixed scores and weights

Helsinki University of Technology Systems Analysis Laboratory 4 Incomplete weight information (1/2) n Interval bounds on attribute weights –Feasible weight region »Non-negative »Sum up to one n Different weights lead to different optimal portfolios –Objective function coefficients vary with weights –Generate a set of “good” candidate portfolios Coeffs. for binary vector p

Helsinki University of Technology Systems Analysis Laboratory 5 Incomplete weight information (2/2) n Potentially optimal portfolios –Optimal for some weights: –Set of potentially optimal portfolios P PO n Pairwise dominance –p k at least as good as p l for all feasible weights, better for some weights – n Non-dominated portfolios –Portfolios not dominated by any other portfolio –Set of non-dominated portfolios P ND –P PO  P ND w2w2 10 V(p k,w) w1w1 01 p1p1 p2p2 p4p4 p3p3

Helsinki University of Technology Systems Analysis Laboratory 6 Conceptual ideas n 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 n 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?”

Helsinki University of Technology Systems Analysis Laboratory 7 Computational issues in portfolio optimization n Dominance checks require pairwise comparisons n Number of possible portfolios is high –m projects lead to 2 m 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 n Need for a dedicated portfolio algorithm –First determine potentially optimal portfolios –Repeat the algorithm to determine non-dominated portfolios

Helsinki University of Technology Systems Analysis Laboratory 8 Determination of potentially optimal portfolios (1/3) n 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 and determines extreme points. Portfolio indicator vector Attribute weight coefficients, w  S 0 Projects’ score matrix (fixed)

Helsinki University of Technology Systems Analysis Laboratory 9 Determination of potentially optimal portfolios (2/3) n 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 n For each (sub)set S k the basic steps are 1. Calculate optimal portfolio at each extreme point of S k 2. i) If each extreme point has the same optimal portfolio, conclude that this portfolio is optimal in the entire subset S k 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

Helsinki University of Technology Systems Analysis Laboratory 10 Determination of potentially optimal portfolios (3/3) n The portfolios are constructed in descending value –Only feasible portfolios are constructed n No all inconclusive computations –Constructed portfolios are potentially optimal –No cross-checks and later rejections n Extreme points of the subsets are generated by utilizing the extremes of the parent set V(p k,w) w1w1 01 w2w2 10 p infeas p1p1 p2p2

Helsinki University of Technology Systems Analysis Laboratory 11 An example: potentially optimal portfolios (1/3) = Q = c T v 1 (x j ) v 2 (x j ) v 3 (x j ) c(x j ) x1x2x3x4x5x1x2x3x4x5

Helsinki University of Technology Systems Analysis Laboratory 12 An example: potentially optimal portfolios (2/3)

Helsinki University of Technology Systems Analysis Laboratory 13 An example: potentially optimal portfolios (3/3) p 1 p 3 p 2 p 3

Helsinki University of Technology Systems Analysis Laboratory 14 From potentially optimal to non-dominated n Potentially optimal portfolios not necessarily robust –Optimal for some weights, lower bound omitted –Missing a portfolio that is the second best for all weights n 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 n 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

Helsinki University of Technology Systems Analysis Laboratory 15 Determination of non-dominated portfolios (1/2) 1. Calculate potentially optimal portfolios on entire S 0 2. Add constraints to exclude portfolios generated thus far 3. Calculate potentially optimal portfolios on entire S 0 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(p k,w) w1w1 01 w2w2 10 p infeas p1p1 p2p2 p4p4 p3p3 p 1 dominates p 4

Helsinki University of Technology Systems Analysis Laboratory 16 Determination of non-dominated portfolios (2/2) n The portfolios on the topmost layer are potentially optimal n The portfolios accepted on lower layers are non-dominated n Rules for early termination –Only one new candidate portfolio on a new layer –Each new candidate absolutely dominated by some upper layer portfolio » n 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

Helsinki University of Technology Systems Analysis Laboratory 17 Measures of portfolio performance n 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 n Decision rules –Maximax, Maximin, Central values, Minimax regret n Measures based on weight regions –Assuming a probability distribution on weights –E.g., portfolio p k is optimal in 65% of the feasible weight region

Helsinki University of Technology Systems Analysis Laboratory 18 Portfolio-oriented project evaluation n 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 n Implications for project choice –Select core projects –Discard projects that are not included in any non-dominated portfolio –Reconsider remaining projects

Helsinki University of Technology Systems Analysis Laboratory 19 Uses of methodology n 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 n Robust decision making in scenario analysis –Attributes interpreted as scenarios –Weights interpreted as probabilities n Sequential project selection –Core projects –Additional constraints n Sensitivity analysis –Effect of small changes in the weights –Displaying the emerging potential portfolios at once

Helsinki University of Technology Systems Analysis Laboratory 20 References »Arbel, A., (1989). Approximate Articulation of Preference and Priority Derivation, EJOR, Vol. 43, pp »Carrizosa, E., Conde, E., Fernández, F. R., Puerto, J., (1995). Multi-Criteria Analysis with Partial Information about the Weighting Coefficients, EJOR, Vol. 81, pp »Kim, S. H., Han, C. H., (2000). Establishing Dominance between Alternatives with Incomplete Information in a Hierarchically Structured Value Tree, EJOR, Vol. 122, pp »Salo, A., Hämäläinen, R. P., (1992). Preference Assessment by Imprecise Ratio Statements, Operations Research, Vol. 40, pp »Salo, A., Hämäläinen, R. P., (1995). Preference Programming Through Approximate Ratio Comparisons, EJOR, Vol. 82, pp »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 »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