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, 02150 TKK, Finland
http://www.sal.tkk.fi
firstname.lastname@tkk.fi
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Helsinki University of Technology
Systems Analysis Laboratory
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 DM’s risk attitude, too
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Helsinki University of Technology
Systems Analysis Laboratory
RPM with scenarios (1/2)

Projects evaluated in each scenario
x j  X , j  1,..., m , outcomes vij  [v]ij
– Scenario probabilities s  s ,...., s T
1
n
– Projects
– Project’s expected value EV ( x ) 
j
n
s v
i 1

j
i i
Portfolio is a subset of the available projects p  X  p  P  2 X
– Outcome of portfolio p in ith scenario Vi ( p ) 
x jp
n
– Expected portfolio value
j
v
i
EV ( p )   si  vij 
i 1
x jp
 EV ( x
j
)
x jp
– A feasible portfolio satisfies a system of linear constraints
PF  { p  P | Az( p)  B}
1, x j  p
z j ( p)  
j
0
,
x
p

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Helsinki University of Technology
Systems Analysis Laboratory
RPM with scenarios (2/2)

Problem for a risk neutral DM with known probabilities
max EV ( p)  max
pPF

z ( p ){0 ,1}
m
z( p)
T

vs | Az ( p)  B
Example: n=5 scenarios, m=10 projects
Scenario
probabilities
Projects
0.35
43
71
81
120
61
145
73
125
75
95
0.30
50
108
93
109
112
79
48
112
92
126
0.20
32
93
148
154
106
68
107
127
98
93
0.10
53
43
134
92
69
122
90
112
118
66
0.05
33
70
69
70
79
69
30
74
120
104
Budget
Costs
1
31
88
84
33
88
48
56
62
66
278
Optim al z(p)
1
1
0
0
1
1
0
1
0
1
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Helsinki University of Technology
Systems Analysis Laboratory
Incomplete information on probabilities (1/2)

Incomplete information on probability estimates
– Set of feasible probabilities
n
S  S  {s |  si  1, si  0}
0
i 1
– Convex polytope bounded by linear constraints
– Several probability distributions consistent with this information

E.g. scenario 1 is the most likely out of three:
S  {s  S 0 | s1  s2 , s1  s3}
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Helsinki University of Technology
Systems Analysis Laboratory
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:
p  S p'  EV ( p)  EV ( p' )  s  S

Set of non-dominated portfolios
PN (S )  { p  PF | p' PF s.t. p'  S p}

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
Identification of robust projects and portfolios

Core Index of projects
– Share of non-dominated portfolios that include the project
CI ( x j , S ) | { p  PN ( S ) | x j  p} | / | PN ( S ) |
– 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
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Helsinki University of Technology
Systems Analysis Laboratory
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
not exceed
for any feasible probabilities:

Vmust
V   M  V ( p )  V  (1   ) M  i  1,..., n
i
i
i
 sii    s  ext(S )
  {0,1} i  1,..., n
i
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Helsinki University of Technology
Systems Analysis Laboratory
Additional dominance concepts (1/3)

Stochastic dominance
– Probability of obtaining a portfolio value at most t:
– First degree:
Fps (t ) 
s
Vi ( p ) t
i
s
s
p  FSD
p
'

F
(
t
)

F
S
p
p ' (t )  t  R, s  S
t
– Second degree:
p  SSD
p' 
S
s
s
[
F
(
y
)

F
p ' ( y ) ]dy  0  t  R , s  S
 p


Stochastic dominance checks computationally straightforward
– Cumulative distributions are step-functions with
n
steps
– Check only required at the extreme points of feasible probability set
S
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Helsinki University of Technology
Systems Analysis Laboratory
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
PFSD ( S )  { p  PF | p' PF s.t. p' 
FSD
S
p}
PSSD ( S )  { p  PF | p' PF s.t. p'  SSD
p}
S
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Helsinki University of Technology
Systems Analysis Laboratory
Additional dominance concepts (3/3)

Properties
– If
p  S 0 p'
then
p
has a greater outcome in each scenario
S  S0
p'  p  SSD
p'  p  S p '
S
– Thus, for any set of feasible probabilities
p  S 0 p'  p  FSD
S
– Therefore

PN (S 0 )  PFSD (S )  PSSD (S )  PN (S )
Computation of stochastically non-dominated portfolios
– Solve the MOZOLP problem to obtain
PN ( S 0 )
– Use pair-wise stochastic dominance checks to obtain
PFSD (S )
or
PSSD (S )
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Helsinki University of Technology
Systems Analysis Laboratory
Example (1/3)


Underlying precise probabilities sˆ  [0.35, 0.3, 0.2, 0.1,0.05]T
Approximated by incomplete probability information
S rank  {s  S 0 | s1  s2  s3  s4  s5 }
Scenarion
probabilities
Projects
0.35
43
71
81
120
61
145
73
125
75
95
0.30
50
108
93
109
112
79
48
112
92
126
0.20
32
93
148
154
106
68
107
127
98
93
0.10
53
43
134
92
69
122
90
112
118
66
0.05
33
70
69
70
79
69
30
74
120
104
Budget
Costs
1
31
88
84
33
88
48
56
62
66
278
Optim al z(p)
1
1
0
0
1
1
0
1
0
1
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Helsinki University of Technology
Systems Analysis Laboratory
Example (2/3)
| PN (S 0 ) | 12
| PFSD ( S rank ) | 11
| PSSD ( S rank ) | 8
| PEVD ( S rank ) | 4
| PN ({sˆ}) | 1

Maximin
, Minimax regret
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Helsinki University of Technology
Systems Analysis Laboratory
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
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Helsinki University of Technology
Systems Analysis Laboratory
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 DM’s risk preferences
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Helsinki University of Technology
Systems Analysis Laboratory
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
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Helsinki University of Technology
Systems Analysis Laboratory
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)
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Helsinki University of Technology
Systems Analysis Laboratory
Additional dominance concepts

First degree stochastic dominance
p FSDS p' 

Vi ( p ) y
si 
s
i
Vi ( p ') y
 s  S, y  R
– Sufficient and necessary condition:
 s  ext(S ), y {V1 ( p),..., Vn ( p), V1 ( p' ),..., Vn ( p' )}

Stochastically non-dominated portfolios
PFSD ( S )  { p  PF | p' PF s.t. p' FSDS p}
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Helsinki University of Technology
Systems Analysis Laboratory
How to model risk attitude? (2/3)

Computation of stochastically non-dominated portfolios
– For any set of feasible probabilities S
PFSD (S )  PEVD (S 0 )
since
p EVDS 0 p'  p FSDS p'
–

p EVDS 0 p ' 
portfolio p has a greater value than p’ in each scenario
Algorithm
– Solve the MOZOLP problem to obtain
PEVD (S 0 )
– Use pair-wise stochastic dominance checks to obtain
PFSD (S )
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Helsinki University of Technology
Systems Analysis Laboratory
How to model risk attitude? (3/3)

Similar treatment for second degree stochastic dominance
p SSDS p'  
t



Vi ( p )  y
si 
s
i
Vi ( p ')  y
dy  0  s  S , t  R
Additional information on probabilities or DM’s risk attitude
narrows the set of ‘good’ portfolios
~
– For any set of feasible probabilities S 0  S  S
PEVD ( S 0 )  PFSD ( S )  PSSD ( S )  PEVD ( S )
|
|
|
~
~
~
PFSD ( S )  PSSD ( S )  PEVD ( S )
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