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1 Helsinki University of Technology Systems Analysis Laboratory Robust Portfolio Modeling for Scenario-Based Project Appraisal Juuso Liesiö, Pekka Mild and Ahti Salo Systems Analysis Laboratory Helsinki University of Technology P.O. Box 1100, TKK, Finland

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Helsinki University of Technology Systems Analysis Laboratory 2 Project portfolio selection under uncertainty n Robust Portfolio Modeling in multi-attribute evaluation –A subset of projects to be selected subject to resource constraints –Projects evaluated with regard to several attributes –Allows for incomplete information about attribute weights and projects’ scores –Offers robust decision recommendations at project and portfolio level »Core Index values, decision rules n Use of RPM for project selection under uncertainty –Uncertainties captured through scenarios –Projects’ (single-attribute) outcomes known in each scenario –Incomplete information about scenario probabilities –Provides robust decision recommendations –Accounts for the DM’s risk attitude, too

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Helsinki University of Technology Systems Analysis Laboratory 3 RPM with scenarios (1/2) n Projects evaluated in each scenario –Projects, outcomes –Scenario probabilities –Project’s expected value n Portfolio is a subset of the available projects –Outcome of portfolio p in i th scenario –Expected portfolio value –A feasible portfolio satisfies a system of linear constraints

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Helsinki University of Technology Systems Analysis Laboratory 4 n Problem for a risk neutral DM with known probabilities n Example: n=5 scenarios, m=10 projects RPM with scenarios (2/2)

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Helsinki University of Technology Systems Analysis Laboratory 5 Incomplete information on probabilities (1/2) n Incomplete information on probability estimates –Set of feasible probabilities –Convex polytope bounded by linear constraints –Several probability distributions consistent with this information n E.g. scenario 1 is the most likely out of three:

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Helsinki University of Technology Systems Analysis Laboratory 6 Dominance concept for a risk neutral DM n Portfolio p dominates p’ if the expected value of p is greater than that of p’ for all feasible probabilities: n Set of non-dominated portfolios n Multi-objective zero-one linear programming problem –MOZOLP algorithms: Bitran (1977), Villareal and Karwan (1980), Deckro and Winkofsky (1983), Liesiö et al. (2005)

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Helsinki University of Technology Systems Analysis Laboratory 7 Identification of robust projects and portfolios n Core Index of projects –Share of non-dominated portfolios that include the project –CI(x)=1 x is recommended –CI(x)=0 x is not recommended n Examples of decision rules for portfolios –Maximin: ND portfolio with the maximal minimum expected value –Minimax-regret: ND portfolio for which the maximum expected value difference to other feasible portfolios is minimized

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Helsinki University of Technology Systems Analysis Laboratory 8 Consideration of risk n Accounting for risk aversion –The DM may be interested in portfolios that are dominated in the EV sense –We thus propose a less restrictive approach based on »extention of stochastic dominance concepts to incomplete probability information »introduction of constraints to rule out portfolios which do not satisfy risk requirements n Introduction of risk constraints –E.g., Value-at-Risk (VaR) : The probability of a portfolio value less than must not exceed for any feasible probabilities:

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Helsinki University of Technology Systems Analysis Laboratory 9 Additional dominance concepts (1/3) n Stochastic dominance –Probability of obtaining a portfolio value at most t: –First degree: –Second degree: n Stochastic dominance checks computationally straightforward –Cumulative distributions are step-functions with steps –Check only required at the extreme points of feasible probability set

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Helsinki University of Technology Systems Analysis Laboratory 10 Additional dominance concepts (2/3) n Stochastically non-dominated portfolios –A feasible portfolio is non-dominated iff it is not dominated by any other feasible portfolio

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Helsinki University of Technology Systems Analysis Laboratory 11 Additional dominance concepts (3/3) n Properties –If then has a greater outcome in each scenario –Thus, for any set of feasible probabilities –Therefore n Computation of stochastically non-dominated portfolios –Solve the MOZOLP problem to obtain –Use pair-wise stochastic dominance checks to obtain or

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Helsinki University of Technology Systems Analysis Laboratory 12 Example (1/3) n Underlying precise probabilities n Approximated by incomplete probability information

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Helsinki University of Technology Systems Analysis Laboratory 13 Example (2/3) n Maximin, Minimax regret

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Helsinki University of Technology Systems Analysis Laboratory 14 Example (3/3) n Core Index values for projects –Risk neutrality may be too strong of an assumption –For risk averse DM recommendation can be based on SSD »Projects that can be surely recommended: 1, 5 and 8 »Strong support for project 2 and lack of support for project 3 n Decision rules for portfolios –Maximin: projects 1, 2, 4, 5, 8, 10 –Minimax-regret: »FSD: 1, 5, 7, 8, 9, 10 »SSD: 1, 2, 5, 6, 8, 10 »Expected value: 1,2, 4, 5, 8, 10

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Helsinki University of Technology Systems Analysis Laboratory 15 Conclusions n RPM for scenario-based project selection –Admits incomplete probability information –Computes all (stochastically) non-dominated portfolios –Indicates projects that are robust choices in view of incomplete information n Decision support –The DM is presented with several portfolios that perform well –Core Indexes support the comparison of projects –Decision rules assist in comparison of portfolios n Current research questions –Consideration of interval-valued multi-attribute project outcomes in scenarios –Explicit modeling of the DM’s risk preferences

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Helsinki University of Technology Systems Analysis Laboratory 16 References »Liesiö, J., Mild, P., Salo, A., (2005). Preference Programming for Robust Portfolio Modelling and Project Selection, EJOR, (Conditionally Accepted). »Villareal, B., Karwan, M.H., (1981) Multicriteria Integer Programming: A Hybrid Dynamic Programming Recursive Algorithm, Mathematical Programming, Vol. 21, pp »Bitran, G.R., (1977). Linear Multiple Objective Programs with Zero-One Variables, Mathematical Programming, Vol. 13, pp »Decro, R.F., Winkofsky, E.P. (1983). Solving Zero-One Multiple Objective Programs through implicit enumeration, EJOR, Vol. 12, pp

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Helsinki University of Technology Systems Analysis Laboratory 17 Several Time Periods n Model remains linear (cf. CPP) –Each project corresponds to several time-period specific decision variables –Future options depend on decisions in preceding periods »Linear constraints –Resource flow variables transfer leftover resources from one period to another n Maximization of expected value in the last period –Portfolios are compared through their performance in the last time period n LP model includes both continuos and binary variables –Multiple Objective Mixed Zero-One Programming (Mavrotas and Diakoulaki 1998)

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Helsinki University of Technology Systems Analysis Laboratory 18 Additional dominance concepts n First degree stochastic dominance –Sufficient and necessary condition: n Stochastically non-dominated portfolios

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Helsinki University of Technology Systems Analysis Laboratory 19 How to model risk attitude? (2/3) n Computation of stochastically non-dominated portfolios –For any set of feasible probabilities since – portfolio p has a greater value than p’ in each scenario n Algorithm –Solve the MOZOLP problem to obtain –Use pair-wise stochastic dominance checks to obtain

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Helsinki University of Technology Systems Analysis Laboratory 20 How to model risk attitude? (3/3) n Similar treatment for second degree stochastic dominance n Additional information on probabilities or DM’s risk attitude narrows the set of ‘good’ portfolios –For any set of feasible probabilities

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