Robust Portfolio Modeling for Scenario-Based Project Appraisal

1
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, 02150 TKK, Finland
• http//www.sal.tkk.fi
• firstname.lastname_at_tkk.fi

2
Project portfolio selection under uncertainty

• 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
• 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 DMs risk attitude, too

3
RPM with s cenarios (1/2)

• Projects evaluated in each scenario
• Projects ,
  outcomes
• Scenario probabilities
• Projects expected value
• Portfolio is a subset of the available projects
• Outcome of portfolio p in ith scenario
• Expected portfolio value
• A feasible portfolio satisfies a system of linear
  constraints

4
RPM with scenarios (2/2)

• Problem for a risk neutral DM with known
  probabilities
• Example n5 scenarios, m10 projects

5
Incomplete information on probabilities (1/2)

• Incomplete information on probability estimates
• Set of feasible probabilities
• Convex polytope bounded by linear constraints
• Several probability distributions consistent with
  this information
• E.g. scenario 1 is the most likely out of three

6
Dominance concept for a risk neutral DM

• Portfolio p dominates p if the expected value of
  p is greater than that of p for all feasible
  probabilities
• Set of non-dominated portfolios
• Multi-objective zero-one linear programming
  problem
• MOZOLP algorithms Bitran (1977), Villareal and
  Karwan (1980), Deckro and Winkofsky (1983),
  Liesiö et al. (2005)

7
Identification of robust projects and portfolios

• 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
• 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

8
Consideration of risk

• 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
• 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

9
Additional dominance concepts (1/3)

• Stochastic dominance
• Probability of obtaining a portfolio value at
  most t
• First degree
• Second degree
• Stochastic dominance checks computationally
  straightforward
• Cumulative distributions are step-functions with
  steps
• Check only required at the extreme points of
  feasible probability set

10
Additional dominance concepts (2/3)

• Stochastically non-dominated portfolios
• A feasible portfolio is non-dominated iff it is
  not dominated by any other feasible portfolio

11
Additional dominance concepts (3/3)

• Properties
• If then has a greater
  outcome in each scenario
• Thus, for any set of feasible probabilities
• Therefore
• Computation of stochastically non-dominated
  portfolios
• Solve the MOZOLP problem to obtain
• Use pair-wise stochastic dominance checks to
  obtain or

12
Example (1/3)

• Underlying precise probabilities
• Approximated by incomplete probability
  information

13
Example (2/3)

• Maximin , Minimax regret

14
Example (3/3)

• 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
• 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

15
Conclusions

• 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
• 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
  
• Current research questions
• Consideration of interval-valued multi-attribute
  project outcomes in scenarios
• Explicit modeling of the DMs risk preferences

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. 204-223
• Bitran, G.R., (1977). Linear Multiple Objective
  Programs with Zero-One Variables, Mathematical
  Programming, Vol. 13, pp. 121-139.
• Decro, R.F., Winkofsky, E.P. (1983). Solving
  Zero-One Multiple Objective Programs through
  implicit enumeration, EJOR, Vol. 12, pp. 362-374

17
Several Time Periods

• 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
• Maximization of expected value in the last period
• Portfolios are compared through their performance
  in the last time period
• LP model includes both continuos and binary
  variables
• Multiple Objective Mixed Zero-One Programming
  (Mavrotas and Diakoulaki 1998)

18
Additional dominance concepts

• First degree stochastic dominance
• Sufficient and necessary condition
• Stochastically non-dominated portfolios

19
How to model risk attitude? (2/3)

• Computation of stochastically non-dominated
  portfolios
• For any set of feasible probabilities
• since
• portfolio p
  has a greater value than p in each scenario
• Algorithm
• Solve the MOZOLP problem to obtain
• Use pair-wise stochastic dominance checks to
  obtain

20
How to model risk attitude? (3/3)

• Similar treatment for second degree stochastic
  dominance
• Additional information on probabilities or DMs
  risk attitude narrows the set of good
  portfolios
• For any set of feasible probabilities