FINANCIAL RISK
CHE 5480
Miguel Bagajewicz
University of Oklahoma
School of Chemical Engineering and Materials Science
1
Scope of Discussion
We will discuss the definition and management of financial risk in
in any design process or decision making paradigm, like…
• Investment Planning
• Scheduling and more in general, operations planning
• Supply Chain modeling, scheduling and control
• Short term scheduling (including cash flow management)
• Design of process systems
• Product Design
Extensions that are emerging are the treatment of other risks
in a multiobjective (?) framework, including for example
• Environmental Risks
• Accident Risks (other than those than can be expressed
as financial risk)
2
Introduction – Understanding Risk
Consider two investment plans, designs, or operational decisions
Probability
Profit Histogram
0.45
0.40
Investment Plan I - E[Profit] = 338
Investment Plan II - E[Profit] = 335
0.35
0.30
0.25
0.20
0.15
Probability of Loss
for Plan I = 12%
0.10
0.05
0.00
-300 -200
-100
0
100
200
300
400
500
600
700
800
Profit (M$)
3
Conclusions
Risk can only be assessed after a plan has been selected but it cannot be
managed during the optimization stage (even when stochastic optimization
including uncertainty has been performed).
There is a need to develop new models that allow not only assessing but managing
financial risk.
The decision maker has two simultaneous objectives:
•
•
Maximize Expected Profit.
Minimize Risk Exposure
4
What does Risk Management mean?
One wants to modify the profit distribution in order to satisfy
the preferences of the decision maker
Probability
0.20
OR…
0.18
REDUCE THESE
FREQUENCIES
INCREASE
THESE
FREQUENCIES
0.16
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
-300 -200 -100
0
100
200
300
Profit x
400
500
600
700
800
OR
BOTH!!!! 5
Characteristics of Two-Stage
Stochastic Optimization Models
Philosophy
• Maximize the Expected Value of the objective over all possible realizations of
uncertain parameters.
• Typically, the objective is Expected Profit , usually Net Present Value.
• Sometimes the minimization of Cost is an alternative objective.
Uncertainty
• Typically, the uncertain parameters are: market demands, availabilities,
prices, process yields, rate of interest, inflation, etc.
• In Two-Stage Programming, uncertainty is modeled through a finite number
of independent Scenarios.
• Scenarios are typically formed by random samples taken from the probability
distributions of the uncertain parameters.
6
Characteristics of Two-Stage
Stochastic Optimization Models
First-Stage Decisions
• Taken before the uncertainty is revealed. They usually correspond to structural
decisions (not operational).
• Also called “Here and Now” decisions.
• Represented by “Design” Variables.
• Examples:
−To build a plant or not. How much capacity should be added, etc.
−To place an order now.
−To sign contracts or buy options.
−To pick a reactor volume, to pick a certain number of trays and size
the condenser and the reboiler of a column, etc
7
Characteristics of Two-Stage
Stochastic Optimization Models
Second-Stage Decisions
• Taken in order to adapt the plan or design to the uncertain parameters
realization.
• Also called “Recourse” decisions.
• Represented by “Control” Variables.
• Example: the operating level; the production slate of a plant.
• Sometimes first stage decisions can be treated as second stage decisions.
In such case the problem is called a multiple stage problem.
Shortcomings
• The
model is unable to perform risk management decisions.
8
Two-Stage Stochastic Formulation
Let us leave it linear
because as is it is
complex enough.!!!
LINEAR MODEL SP
Max  ps qsT ys  cT x
s
Technology matrix
Recourse
Function
s.t.
Ax b
First-Stage Constraints
Ts x Wy s  hs
Second-Stage Constraints
x 0
First stage variables
First-Stage
Cost
x X
Second Stage Variables
ys  0
Recourse matrix (Fixed Recourse)
Sometimes not fixed (Interest rates in Portfolio Optimization)
Complete recourse: the
recourse cost (or profit) for
every possible uncertainty
realization remains finite,
independently of the first-stage
decisions (x).
Relatively complete recourse:
the recourse cost (or profit) is
feasible for the set of feasible
first-stage decisions. This
condition means that for every
feasible first-stage decision,
there is a way of adapting the
plan to the realization of
uncertain parameters.
We also have found that one
can sacrifice efficiency for
certain scenarios to improve
risk management. We do not
know how to call this yet.
9
Previous Approaches to Risk Management
Robust Optimization Using Variance (Mulvey et al., 1995)
Maximize E[Profit] - ·V[Profit]
Profit PDF
0.9
Expected
Profit
0.8
0.7
0.6
0.5
0.4
0.3
Desirable Penalty
Undesirable Penalty
Variance is a measure
for the dispersion
of the distribution
0.2
0.1
0.0
-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5
1.0 1.5 2.0 2.5 3.0 3.5 4.0
4.5 5.0 5.5
Profit
Underlying Assumption: Risk is monotonic with variability
10
Robust Optimization Using Variance
Drawbacks
• Variance is a symmetric risk measure: profits both above and below the target
level are penalized equally. We only want to penalize profits below the target.
• Introduces non-linearities in the model, which results in serious computational
difficulties, specially in large-scale problems.
• The model may render solutions that are stochastically dominated by others.
This is known in the literature as not showing Pareto-Optimality. In other words
there is a better solution (ys,x*) than the one obtained (ys*,x*).
11
Previous Approaches to Risk Management
Robust Optimization using Upper Partial Mean (Ahmed and Sahinidis, 1998)
Maximize E[Profit] - ·UPM
E[x ]
Profit PDF
0.4
UPM = 0.50
UPM = 0.44
0.3
0.2
UPM = E[D(x) ]
D(x)
0.1
0.0
-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5
1.0
1.5 2.0 2.5 3.0
3.5 4.0
4.5 5.0 5.5
Profit x
Underlying Assumption: Risk is monotonic with lower variability
12
Robust Optimization using the UPM
Robust Optimization using the UPM
Advantages
• Linear measure
Disadvantages
• The UPM may misleadingly favor non-optimal second-stage decisions.
• Consequently, financial risk is not managed properly and solutions with higher risk
than the one obtained using the traditional two-stage formulation may be obtained.
• The model losses its scenario-decomposable structure and stochastic decomposition
methods can no longer be used to solve it.
13
Robust Optimization using the UPM
Objective Function: Maximize E[Profit] - ·UPM


D s  Max 0 ;  pk Profitk  Profits 
 kS

UPM   ps D s
sS
=3
Ds
Profits
Case I
Case II
Case I
Case II
S1
150
100
0
0
S2
125
100
0
0
S3
75
75
25
6.25
S4
50
50
50
31.25
E[Profit]
100.00
81.25
UPM
18.75
9.38
Objective
43.75
53.13
Downside scenarios are the same, but the UPM is affected by
the change in expected profit due to a different upside distribution.
As a result a wrong choice is made.
14
Robust Optimization using the UPM
Effect of Non-Optimal Second-Stage Decisions
E[Profit ]
P1
A
UPM
-200
18
-220
16
-240
14
Ro bustness So lutio n
-260
12
-280
P2
B
Ro bustness So lutio n with
Optimal Seco nd-Stage Decisio ns
-300
Ro bustness So lutio n with
Optimal Seco nd-Stage Decisio ns
8
-320
6
-340
4
-360
2
-380
Both technologies are able to produce
two products with different
production cost and at different yield
per unit of installed capacity
Ro bustness So lutio n
10
0
0
20
40
60
80
100
120

140
160
180
200
220
0
20
40
60
80
100

120
140
160
180
200
220
E[Profit ]
-200
-220
-240
-260
-280
Ro bustness So lution
-300
Ro bustness So lution with
Optimal Seco nd-Stage Decisions
-320
-340
-360
-380
0
1
2
3
4
5
6
7
8
9
UPM
10
11
12
13
14
15
16
17
15
OTHER APPROACHES
Cheng, Subrahmanian and Westerberg (2002, unpublished)
− Multiobjective Approach: Considers Downside Risk, ENPV and Process
Life Cycle as alternative Objectives.
− Multiperiod Decision process modeled as a Markov decision process
with recourse.
− The problem is sometimes amenable to be reformulated as a sequence
of single-period sub-problems, each being a two-stage stochastic program
with recourse. These can often be solved backwards in time to obtain
Pareto Optimal solutions.
This paper proposes a new design paradigm of which risk is just one component.
We will revisit this issue later in the talk.
16
OTHER APPROACHES
Risk Premium (Applequist, Pekny and Reklaitis, 2000)
− Observation: Rate of return varies linearly with variability. The
of such dependance is called Risk Premium.
− They suggest to benchmark new investments against the historical
− risk premium by using a two objective (risk premium and profit)
− problem.
−The technique relies on using variance as a measure of variability.
17
Previous Approaches to Risk Management
Conclusions
• The minimization of Variance penalizes both sides of the mean.
• The Robust Optimization Approach using Variance or UPM is not suitable
for risk management.
• The Risk Premium Approach (Applequist et al.) has the same problems
as the penalization of variance.
THUS,
• Risk should be properly defined and directly incorporated in the models to
manage it.
• The multiobjective Markov decision process (Applequist et al, 2000)
is very closely related to ours and can be considered complementary. In
fact (Westerberg dixit) it can be extended to match ours in the definition
of risk and its multilevel parametrization.
18
Probabilistic Definition of Risk
Financial Risk = Probability that a plan or
design does not meet a certain profit target
Scenarios are independent events
Risk ( x,)  P(Profit  )
Risk ( x,)   ps P(Profits  )
s
For each scenario the profit is either
greater/equal or smaller than the target
 1 If Profits  
P( Profits   )  
 0 else
zs is a new binary variable
P(Profits  )  zs
Formal Definition of Financial Risk
Risk ( x,)   ps zs
s
19
Financial Risk Interpretation
Probability
0.20

0.18
0.16
0.14
Cumulative Probability = Risk (x , )
0.12
0.10
0.08
0.06
0.04
0.02
0.00
Profit x
20
Cumulative Risk Curve
Risk
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
250
500
750
1000 1250 1500 1750 2000 2250 2500 2750 3000 3250
Profit (M$)
Our intention is to modify the shape and location of this
curve according to the attitude towards risk of the decision maker
21
Risk Preferences and Risk Curves
Risk
1.0
0 .9
Risk-Averse
Investor's Choice
E[Profit ] = 0.4
0 .8
0 .7
Risk-T aker
Investor's Choice
E[Profit ] = 1.0
0 .6
0 .5
0 .4
0 .3
0 .2
0 .1
0 .0
- 2 .0
- 1.5
- 1.0
- 0 .5
0 .0
0 .5
1.0
1.5
2 .0
2 .5
3 .0
3 .5
4 .0
Profit
22
Risk Curve Properties
A plan or design with Maximum E[Profit] (i.e. optimal in Model SP) sets a
theoretical limit for financial risk: it is impossible to find a feasible plan/design
having a risk curve entirely beneath this curve.
Risk
1.0
0.9
0.8
Possible
curve
0.7
0.6
0.5
0.4
Impossible
curve
Maximum
E[Profit ]
0.3
0.2
0.1
0.0
Profit
23
Minimizing Risk: a Multi-Objective Problem
Risk
1
1.0
0.9
2
3
4
x fixed
0.8
Min Risk (x ,4)
Max EProfit   ps qs ys  cT x
s
Min Risk 1  
..
.
0.7
0.6
Min Risk (x ,3 )
0.5
Min Risk i  
pz
s s1
s
pz
s si
s
0.4
Max E[Profit (x )]
0.3
0.2
s.t.
Ax b
0.1
Min Risk (x ,2 )
0.0
Min Risk (x ,1 )
Target Profit 
Ts x Wy s  hs
x 0
x X
ys  0
Multiple Objectives:
• At each profit we want minimize the associated risk
• We also want to maximize the expected profit
qsT ys  cT x   U s z s
qsT ys  cT x   U s (1 z s )
z s (0,1)
24
Parametric Representations of the
Multi-Objective Model – Restricted Risk
Restricted Risk MODEL
Max  ps qs ys  cT x
s.t.
s
Ax b
Ts x Wy s  hs
x 0
x X
ys  0
 ps zsi  εi
s
Forces Risk to be lower
than a specified level
qsT ys  cT x  i U s z si
qsT ys  cT x  i U s (1 z si )
Risk Management
Constraints
z s (0,1)
25
Parametric Representations of the
Multi-Objective Model – Penalty for Risk
Risk Penalty MODEL
Max  ps qsT ys  cT x  i  ps z si
s
i
s
Penalty Term
s.t.
Ax b
Ts x Wy s  hs
x 0
STRATEGY
Define several profit
Targets and penalty
weights to solve the
model using a multiparametric approach
x X
ys  0
qsT ys  cT x  i U s z si
qsT ys  cT x  i U s (1 z si )
z s (0,1)
Risk Management
Constraints
26
Risk Management using the New Models
Advantages
• Risk can be effectively managed according to the decision maker’s criteria.
• The models can adapt to risk-averse or risk-taker decision makers, and their
risk preferences are easily matched using the risk curves.
• A full spectrum of solutions is obtained. These solutions always have
optimal second-stage decisions.
• Model Risk Penalty conserves all the properties of the standard two-stage
stochastic formulation.
Disadvantages
• The use of binary variables is required, which increases the computational
time to get a solution. This is a major limitation for large-scale problems.
27
Risk Management using the New Models
Computational Issues
• The most efficient methods to solve stochastic optimization problems reported
in the literature exploit the decomposable structure of the model.
• This property means that each scenario defines an independent second-stage
problem that can be solved separately from the other scenarios once the firststage variables are fixed.
• The Risk Penalty Model is decomposable whereas Model Restricted Risk is not.
Thus, the first one is model is preferable.
• Even using decomposition methods, the presence of binary variables in both
models constitutes a major computational limitation to solve large-scale problems.
• It would be more convenient to measure risk indirectly such that binary variables
in the second stage are avoided.
28
Downside Risk
Downside Risk (Eppen et al, 1989) =
Expected Value of the Positive
Profit Deviation from the target
DRisk (x,) E(x,)
Positive Profit Deviation from
Target 
   Profit (x ) If Profit (x ) 
(x, ) 
Otherwise
0
The Positive Profit Deviation is
also defined for each scenario
   Profits
s  
0
Formal definition of Downside Risk
DRisk (x,)   ps  s
If Profits  
Otherwise
s
29
Downside Risk Interpretation
Profit PDF f (x )

0.14
x fixed
0.12
0.10
0.08
0.06
0.04
0.02
DRisk (x ,) = E[(x ,)]


DRisk ( x, )  ò(  x) f (x) dx
ò
0.00
Profit x
30
Downside Risk & Probabilistic Risk
31
Two-Stage Model using Downside Risk
MODEL DRisk
Max   ps qsT ys  cT x    ps  s
s
 s
Penalty Term
s.t.
Ax b
Ts x Wy s  hs
x 0
Advantages
• Same as models using Risk
• Does not require the use of
binary variables
• Potential benefits from the
use of decomposition methods
x X
Strategy
ys  0
Solve the model using different
 s    (qsT ys  cT x)
s  0
Downside
Risk Constraints
profit targets to get a full spectrum
of solutions. Use the risk curves to
select the solution that better suits
the decision maker’s preference
32
Two-Stage Model using Downside Risk
Warning: The same risk may imply different Downside Risks.
Risk
1.0
0.9
0.8
DRisk (Design I , 0.5) = 0.2
Risk (Design I , 0.5) = 0.5
0.7
0.6
0.5
DRisk (Design II , 0.5) = 0.2
Risk (Design II , 0.5) = 0.309
0.4
0.3
0.2
0.1
0.0
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Profit
Immediate Consequence:
Minimizing downside risk does not guarantee minimizing risk.
33
Commercial Software
Riskoptimizer (Palisades) and CrystalBall (Decisioneering)
• Use excell models
• Allow uncertainty in a form of distribution
• Perform Montecarlo Simulations or use genetic algorithms
to optimize (Maximize ENPV, Minimize Variance, etc.)
Financial Software. Large variety
•Some use the concept of downside risk
• In most of these software, Risk is mentioned but not manipulated directly.
34
Process Planning Under Uncertainty
GIVEN:
Process Network
Forecasted Data
DETERMINE:
Set of Processes
Set of Chemicals
B
A
Demands & Availabilities
Costs & Prices
Capital Budget
Network Expansions
2
C
3
D
1
Timing
Sizing
Location
Production Levels
OBJECTIVES:
Maximize Expected Net Present Value
Minimize Financial Risk
35
Process Planning Under Uncertainty
Design Variables: to be decided before the uncertainty reveals
x
{Yit , Eit , Qit }
Y: Decision of building process i in period t
E: Capacity expansion of process i in period t
Q: Total capacity of process i in period t
Control Variables: selected after the uncertain parameters become known
ys 
{ Sjlts , Pjlts , Wits}
S: Sales of product j in market l at time t and scenario s
P: Purchase of raw mat. j in market l at time t and scenario s
W: Operating level of of process i in period t and scenario s
36
Example
Uncertain Parameters: Demands, Availabilities, Sales Price, Purchase Price
Total of 400 Scenarios
Project Staged in 3 Time Periods of 2, 2.5, 3.5 years
Chemical 5
Chemical 1
Process 1
Process 2
Chemical 6
Process 5
Chemical 8
Chemical 2
Chemical 7
Process 3
Chemical 4
Chemical 3
Process 4
37
Example – Solution with Max ENPV
Period 321
2 years
2.5
3.5
years
14.95 kton/yr
Chemical
5 55
Chemical
Chemical
5.27
4.71 kton/yr
kton/yr
29.49 kton/yr
Chemical
Chemical 11
44.44
kton/yr
5.27
4.71 kton/yr
kton/yr
Process 2
Process111
Process
Process
Chemical 6
29.49 kton/yr
80.77 kton/yr
10.23
kton/yr
10.23
80.77 kton/yr
Chemical 2
29.49 kton/yr
Chemical
7
Process
Process 33
Chemical
Chemical333
Chemical
19.60
41.75
kton/yr
43.77kton/yr
kton/yr
21.88kton/yr
kton/yr
20.87
19.60
Chemical7kton/yr
7
Chemical
Process 5
22.73
22.73 kton/yr
kton/yr
22.73
22.73 kton/yr
ton/yr
Chemical 8
21.88
20.87 kton/yr
kton/yr
Process 4
22.73 kton/yr
Chemical
Chemical44
21.88
20.87kton/yr
kton/yr
38
Example – Solution with Min DRisk(=900)
2.39 kton/yr
Period 321
2 years
2.5
3.5
years
Chemical 5
5.15 kton/yr
Chemical 1
Process 1
Process 2
Chemical 5
7.54 kton/yr
10.85 kton/yr
4.99 kton/yr
Process 1
Chemical 1
10.85 kton/yr
5.59 kton/yr
5.15 kton/yr
Chemical
5
4.99 kton/yr
10.85 kton/yr
Chemical 2
Chemical 1
Chemical 6
5.59 kton/yr
5.15 kton/yr
Process 1
10.85 kton/yr
21.77 kton/yr
20.85
kton/yr7
Chemical
Chemical 3
Chemical 3
41.70
43.54kton/yr
kton/yr
Process 3
Process 3
22.37
kton/yr
22.37
kton/yr
Chemical 3
Process
Process 44
19.30 kton/yr
22.37
22.37 kton/yr
kton/yr
Chemical 7
Process 3
Chemical 7
Process 5
Process
19.305 kton/yr
22.77kton/yr
ton/yr
22.43
Chemical 8
Chemical 8
21.77 kton/yr
20.85 kton/yr
22.37 kton/yr
Chemical44
Chemical
20.85
kton/yr
21.77 kton/yr
Same final structure, different production capacities.
39
Example – Solution with Max ENPV
Risk
1.0
PP solution
0.9
0.8
0.7
0.6
0.5
0.4
E[NPV ] = 1140 M$
0.3
0.2
0.1
0.0
250
500
750
1000
1250
1500
1750
2000
2250
2500
2750
3000
3250
NPV (M$)
40
Example – Risk Management Solutions
NPV
Risk
Risk PDF f (x)
1.0
0.0026
0.0024
0.9
0.0022
0.8
0.0020
 = 900
 = 900
ENPV = 908
 = 1100
ENPV = 1074
 increases
0.7
0.0018
PP
ENPV =1140

PP
0.0016
0.6
500
0.0014
0.5
0.0012
600
0.4
0.0010
900
700
800
 = 1100
0.3
0.0008
0.0006
0.2
PP
1100
1200
1300
0.0004
0.1
0.0002
0.0
0.0000
250 0 500
1000
1400
1500
750500 1000
1250
1000 1500
1750
1500 2000
2250
2000 2500
2750
2500 3000
3250
3000
NPV
NPV ((M$)
x , M$ )
41
Process Planning with Inventory
PROBLEM DESCRIPTION:
D
B
2
A
1
D
3
MODEL:
The mass balance is modified such that now a certain level
of inventory for raw materials and products is allowed
A storage cost is included in the objective
OBJECTIVES:
Maximize Expected Net Present Value
Minimize Financial Risk
42
Example with Inventory – SP Solution
Period 321
2 years
2.5
3.5
years
33.90
43.14
kton/yr
kton/yr
39.04
kton/yr
2.88
kton/yr
Chemical
0.42 1
kton/yr
Chemical 1
6.80
kton
5.75
kton
Chemical 1
1.94
kton/yr
2.09
1.18kton/yr
kton/yr
16.28
13.61
11.80
kton/yr
kton/yr
kton/yr
Chemical 5
Chemical 55
Chemical
Process
11
Process
Process 1
51.95 kton/yr
76.81 kton/yr
51.95 kton/yr
12.48
30.44
kton/yr
kton/yr
27.24
kton/yr
7.32
10.28
kton
kton 5.14
kton/yr
12.48
31.47
kton/yr
kton/yr
27.24
kton/yr
Process
22
Process
Process 2
22.36 kton/yr
76.81 kton/yr
1.03
76.81 kton/yr
kton/yr
1.05
kton/yr
3.86
0.60
kton
kton/yr 2.11
Chemical
2
kton
Chemical 2
11.67
kton/yr
26.34
31.47
kton/yr
kton/yr
Chemical 6
Chemical 6
1.10
Chemical 6kton/yr
0.81
kton/yr
0.90
kton/yr
3.86
kton
1.62
kton
Chemical 2
0.04 kton/yr
44.13
kton/yr
35.74
kton/yr
3
ChemicalChemical
3
4.77
kton/yr
11.91
kton
3.40
kton/yr
31.09
kton/yr
Process 3
Process 3
36.45 kton/yr
36.45 kton/yr
Process 4
11.64
kton
Chemical 7
Chemical 7
22.12
kton/yr
3.29
4.65
kton/yr
kton/yr
Chemical 8
Process 5
25.41
kton/yr
26.77 kton/yr
25.41
kton/yr
Chemical 4
26.77 kton/yr
43
Example with Inventory
Solution with Min DRisk (=900)
Period 321
2 years
2.5
3.5
years
0.26 kton/yr
7.48
kton/yr
5.73
kton/yr
Chemical 1
0.02
kton/yr
2.39
kton/yr
6.63
5.61
kton/yr
kton/yr
Process 1
0.90
kton
0.64
Chemical 5
kton
5.80
Chemical0.10
5
5.39
Chemical
5
kton/yr
kton/yr
kton/yr
Process 2
0.51
Process 1
5.39
11.23 kton/yr
kton/yr
Process
1
kton/yr
1.07
Chemical 1
kton
Chemical
1
11.23 kton/yr
0.30
1.01 kton/yr
11.23 kton/yr
Chemical 2
kton
41.68
kton/yr
43.72
kton/yr
Chemical 3
Chemical 3
7.27
kton
1.29
4.05
kton/yr
kton
1.16
kton/yr
5.39
kton/yr
0.32
kton/yr
Chemical 6
11.23 kton/yr
20.54
7.38
kton/yr
kton
23.00
3.37
1.60
kton/yr
kton
Chemical 7
kton/yr
18.46
3.69
Chemical
0.96 7
kton/yr
kton/yr
20.58
Chemical 7
kton/yr
kton/yr
25.79 3
Process 5
Process
Chemical 8
22.04
kton/yr
Chemical 8
kton/yr
Process 3 22.18
Process 5
Process
22.153kton/yr
23.38 kton/yr
1.17
kton/yr
Chemical 3
kton/yr
22.15 kton/yr
22.15 kton/yr
Chemical 423.38 kton/yr
22.85
1.64
3.64 Process 4
kton/yr
0.15
4.11
kton/yr
kton/yr
kton/yr
kton
Process
23.384kton/yr
Chemical0.20
4 kton/yr 0.51
23.38 kton/yr
kton
44
Example with Inventory - Solutions
Risk
Risk
1.0
1
.0
0.9
DRisk
0.8
0.8
 solution
= 900
PP
ENPV = 980
E[NPV ] = 1140 M$
0.7
PPI solution
E[NPV ] = 1237 M$
WithPPI
Inventory
Without
PPI
Inventory
0.6
0.6
ENPV = 1237
ENPV = 1140
0.5
0.4
0.3
DRisk
0.2
 = 1400
0.1
ENPV = 1184
0.0
0
250 500 750 1000 1250 1500 1750 2000 2250 2500 2750 3000 3250 3500 3750 4000
NPV
NPV
(M$)
(M$)
45
Downside Expected Profit

Definition: DENPV ( x, p ) 
x
f ( x, x ) dx   Risk ( x, )  DRisk ( x, )

CEP (M$)
Risk
1250
1.0
1125
0.9
PP solution
PP
 = 900
1000
 = 1100
0.8
E[NPV ] = 1140 M$
875
0.7
750
0.6
625
0.5
 = 900
500
0.4
0.3
375
0.2
250
0.1
125
E[NPV ] = 908 M$
0
0.0
250
500
750
1000
1250
1500
1750
2000
NPV (M$)
2250
2500
2750
3000
3250
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Risk
Up to 50% of risk (confidence?) the lower ENPV solution has higher profit
expectations.
46
Value at Risk
Definition: VaR is given by the difference between the mean value of the
profit and the profit value corresponding to the p-quantile.
VaR( x, p )
E[ Profit ( x)]
VaR( x, p )  E[ Profit ( x)]  Risk 1 ( x, )
VaR=zp for symmetric distributions (Portfolio optimization)
47
COMPUTATIONAL APPROACHES
Sampling Average Approximation Method:
−Solve M times the problem using only N scenarios.
−If multiple solutions are obtained, use the first stage variables to solve the
problem with a large number of scenarios N’>>N to determine the optimum.
Generalized Benders Decomposition Algorithm
− First Stage variables are complicating variables.
− This leaves a primal over second stage variables, which is decomposable.
48
Example

Generate a model to:

Evaluate a large network of natural gas supplierto-market transportation alternatives

Identify the most profitable alternative(s)

Manage financial risk
2005
2030
49
Network of Alternatives
Suppliers
Australia
Indonesia
Iran
Kazakhstan
Malaysia
Qatar
Russia
Transportation
Methods
Pipeline
Markets
China
LNG
India
CNG
Japan
GTL
S. Korea
Ammonia
Thailand
Methanol
United States
50
Network of Alternatives
51
Results
Risk Management (Downside Risk):
1.0
1.0
1s
4.666
0.9
Malaysia
GTL
0.9
0.8
0.8
0.7
0.7
DR-200s
0.6
0.6
Mala-GTL
4.640
Ships: 4 & 2
ENPV:4.570
DR@ 4: 0.157
DR@ 3.5: 0.058
0.50.5
0.40.4
Indo-GTL
200s
Ships:
5&3
4.678
ENPV:4.633
DR@ 4: 0.190
DR@ 3.5: 0.086
0.30.3
0.20.2
Thailand
China
0.10.1
0.00.0
0
0
1
1
2
2
3
3
44
55
66
77
8
8
99
10
10
52
Value at Risk (VaR):
VaR is the expected loss for a certain
confidence level usually set at 5%
VaR =ENPV – NPV @ p-quantile
Opportunity Value (OV) or Upper Potential (UP):
OV = NPV @ (1-p)-quantile – ENPV
53
Results
Value at Risk (VaR) and Opportunity Value (OV):
1.0
OV @ 95%: 1.42
OV @ 95%: 1.75
0.9
0.8
0.7
0.6
Mala-GTL
Ships: 4 & 2
ENPV:4.570
DR@ 4: 0.157
DR@ 3.5: 0.058
0.5
0.4
Indo-GTL
Ships: 5 & 3
ENPV:4.633
DR@ 4: 0.190
DR@ 3.5: 0.086
0.3
0.2
0.1
VaR @ 5%: 1.82
VaR @ 5%: 1.49
0.0
0
1
2
3
Reduction in VaR: 18.1%
4
5
6
7
8
9
10
Reduction in OV: 18.9%
54
Results
Risk /Upside Potential Loss Ratio
1.0
OV @ 95%: 1.42
OV @ 95%: 1.75
0.9
0.8
0.7
O-Area: 0.116
0.6
Mala-GTL
Ships: 4 & 2
ENPV:4.570
DR@ 4: 0.157
DR@ 3.5: 0.058
0.5
0.4
Indo-GTL
Ships: 5 & 3
ENPV:4.633
DR@ 4: 0.190
DR@ 3.5: 0.086
0.3
R-Area: 0.053
0.2
0.1
VaR @ 5%: 1.82
VaR @ 5%: 1.49
0.0
0
1
2
3
4
5
6
Risk /Upside Potential Loss Ratio: 2.2
7
8
9
10
55
Risk /Upside Potential Loss Ratio

Risk /Upside Potential Loss Ratio

O_Area

R_Area
where:
Risk
1.0
0.9
Risk(x2,NPV)





 
0
0.7
dNPV


O_Area
0.8


 dNPV
  0
if
otherwise


Risk(x1,NPV)
0.6
 
0
0.5
 
0.4
R_Area
0.3
if
  0
otherwise


0.2
  Risk NGC ( NPV )  Risk NGC  DR ( NPV )
0.1
0.0
0
1
ENPV2
2
3
ENPV1
4
5
6
7
8
9
NPV
10
56
Upper and Lower Bound Risk
Curve
Upper Bound Risk Curve (Envelope):
The curve constructed by plotting the set of net present
values (NPV) for the best design under each scenario.
Risk
1
a) Possible
0.8
b) Possible
c) Impossible
0.6
E) Envelope
0.4
0.2
d) Impossible
0
0.00
0.0
2.00
2.0
4.00
4.0
6.00
6.0
8.00
8.0
10.00
10.0
57
Upper and Lower Bound Risk
Curve
Lower Bound Risk Curve:
The curve constructed by plotting the set of net present
values (NPV) for the worst (of the set of best designs)
under each scenario.
s\d
d1
d2
d3
:
:
dn
s1
s2
s3
NPVd1,s1 NPVd2,s1 NPVd3,s1
NPVd1,s2 NPVd2,s2 NPVd3,s2
NPVd1,s3 NPVd2,s3 NPVd3,s3
:
:
:
:
:
:
…
sn
… NPVdn,s1
… NPVdn,s2
… NPVdn,s3
:
:
NPVd1,sn NPVd2,sn NPVd3,sn … NPVdn,sn
Min NPVs1Min NPVs2Min NPVs3Min … NPVsnMin
58
Results
Upper and Lower Bound Risk
Curve
1.0
0.8
Mala-GTL
Ships: 4 & 3
ENPV:4.540
0.6
Lower Envelope
ENPV:3.654
0.4
Upper Envelope
ENPV:4.921
0.2
Indo-GTL
Ships: 6 & 3
ENPV:4.63
0.0
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
59
Conclusions
A probabilistic definition of Financial Risk has been introduced in the
framework of two-stage stochastic programming. Theoretical properties of
related to this definition were explored.
New formulations capable of managing financial risk have been introduced.
The multi-objective nature of the models allows the decision maker to choose
solutions according to his risk policy. The cumulative risk curve is used as a
tool for this purpose.
The models using the risk definition explicitly require second-stage binary
variables. This is a major limitation from a computational standpoint.
To overcome the mentioned computational difficulties, the concept of Downside
Risk was examined, finding that there is a close relationship between this
measure and the probabilistic definition of risk.
Using downside risk leads to a model that is decomposable in scenarios and that
allows the use of efficient solution algorithms. For this reason, it is suggested
that this model be used to manage financial risk.
An example illustrated the performance of the models, showing how the risk
curves can be changed in relation to the solution with maximum expected profit.
60