DISCUSSION
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FIN5823FINANCIALMODELINGAPPLICATIONS1.pptxFIN 5823 FINANCIAL MODELING ______________________________________________ Financial Modeling Applications
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FINANCIAL MODELING APPLICATIONS
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For targeting purposes, the problem statement is: READ SLIDE
The strategy propose to solve this problem is:
First, READ SLIDE
Then, READ SLIDE
Finally, after the first two steps are completed, READ SLIDE
The advantage of this approach that it does not require any further division of the temperature intervals
And that heat pumping temperatures are treated as variables instead of fixed values
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Scope of Discussion
We will discuss the definition and management of financial risk in
in any design process or decision making paradigm, like…
Extensions that are emerging are the treatment of other risks
in a multiobjective (?) framework, including for example
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
Environmental Risks
Accident Risks (other than those than can be expressed
as financial risk)
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Introduction – Understanding Risk
Consider two investment plans, designs, or operational decisions
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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).
The decision maker has two simultaneous objectives:
There is a need to develop new models that allow not only assessing but managing
financial risk.
Maximize Expected Profit.
Minimize Risk Exposure
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What does Risk Management mean?
REDUCE THESE FREQUENCIES
OR…
INCREASE THESE FREQUENCIES
One wants to modify the profit distribution in order to satisfy the preferences of the decision maker
OR BOTH!!!!
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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.
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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:
Characteristics of Two-Stage
Stochastic Optimization Models
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
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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.
Characteristics of Two-Stage
Stochastic Optimization Models
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Two-Stage Stochastic Formulation
LINEAR MODEL SP
s.t.
First-Stage Constraints
Second-Stage Constraints
Recourse
Function
First-Stage
Cost
First stage variables
Second Stage Variables
Technology matrix
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.
Let us leave it linear because as is it is complex enough.!!!
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Robust Optimization Using Variance (Mulvey et al., 1995)
Previous Approaches to Risk Management
0.0
0.1
0.2
0.3
0.4
0.5
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0.7
0.8
0.9
-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
Profit PDF
Expected
Profit
Variance is a measure
for the dispersion
of the distribution
Desirable Penalty
Maximize E[Profit] - ·V[Profit]
Underlying Assumption: Risk is monotonic with variability
Undesirable Penalty
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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*).
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Robust Optimization using Upper Partial Mean (Ahmed and Sahinidis, 1998)
Previous Approaches to Risk Management
D(x)
0.0
0.1
0.2
0.3
0.4
-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
Profit PDF
UPM
= E[
D(x)
]
E[
x
]
UPM
= 0.50
UPM
= 0.44
Maximize E[Profit] - ·UPM
Underlying Assumption: Risk is monotonic with lower variability
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Robust Optimization using the UPM
Advantages
Linear measure
Robust Optimization using the UPM
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.
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Robust Optimization using the UPM
| = 3 | Profits | s | ||
| 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 |
Objective Function: Maximize E[Profit] - ·UPM
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.
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Effect of Non-Optimal Second-Stage Decisions
Robust Optimization using the UPM
Both technologies are able to produce two products with different production cost and at different yield per unit of installed capacity
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OTHER APPROACHES
Cheng, Subrahmanian and Westerberg (2002, unpublished)
This paper proposes a new design paradigm of which risk is just one component.
We will revisit this issue later in the talk.
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.
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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.
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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.
Previous Approaches to Risk Management
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Financial Risk = Probability that a plan or design does not meet a certain profit target
Probabilistic Definition of Risk
zs is a new binary variable
Formal Definition of Financial Risk
Scenarios are independent events
For each scenario the profit is either
greater/equal or smaller than the target
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Financial Risk Interpretation
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Cumulative Risk Curve
Our intention is to modify the shape and location of this
curve according to the attitude towards risk of the decision maker
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Risk Preferences and Risk Curves
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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.
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Minimizing Risk: a Multi-Objective Problem
s.t.
.
.
.
Multiple Objectives:
At each profit we want minimize the associated risk
We also want to maximize the expected profit
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Restricted Risk MODEL
Risk Management
Constraints
s.t.
Forces Risk to be lower
than a specified level
Parametric Representations of the
Multi-Objective Model – Restricted Risk
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Parametric Representations of the
Multi-Objective Model – Penalty for Risk
s.t.
Penalty Term
Risk Penalty MODEL
Risk Management
Constraints
Define several profit
Targets and penalty
weights to solve the
model using a multi-
parametric approach
STRATEGY
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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.
Risk Management using the New Models
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.
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Computational Issues
Risk Management using the New Models
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 first-
stage 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.
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Downside Risk (Eppen et al, 1989) =
Expected Value of the Positive
Profit Deviation from the target
Downside Risk
Positive Profit Deviation from
Target
Formal definition of Downside Risk
The Positive Profit Deviation is
also defined for each scenario
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Downside Risk Interpretation
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Downside Risk & Probabilistic Risk
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Two-Stage Model using Downside Risk
s.t.
Penalty Term
MODEL DRisk
Downside
Risk Constraints
Advantages
Same as models using Risk
Does not require the use of
binary variables
Potential benefits from the
use of decomposition methods
Strategy
Solve the model using different
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
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Two-Stage Model using Downside Risk
Warning: The same risk may imply different Downside Risks.
Immediate Consequence:
Minimizing downside risk does not guarantee minimizing risk.
Click to edit Master text styles
Second level
Third level
Fourth level
Fifth level
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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.
Commercial Software
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Process Planning Under Uncertainty
OBJECTIVES:
Maximize Expected Net Present Value
Minimize Financial Risk
Production Levels
DETERMINE:
Network Expansions
Timing
Sizing
Location
GIVEN:
Process Network
Set of Processes
Set of Chemicals
A
1
C
2
D
3
B
Forecasted Data
Demands & Availabilities
Costs & Prices
Capital Budget
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Process Planning Under Uncertainty
Design Variables: to be decided before the uncertainty reveals
{
}
x
=
E
it
Y
it
,
,
Q
it
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
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
{
}
ys
=
P
jlts
S
jlts
,
,
W
its
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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
Process 1
Chemical 1
Process 2
Chemical 5
Chemical 2
Chemical 6
Process 3
Chemical 3
Process 5
Chemical 7
Chemical 8
Process 4
Chemical 4
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Period 1
2 years
Period 2
2.5 years
Period 3
3.5 years
Process 1
Chemical 1
Chemical 5
Process 3
Chemical 3
Chemical 7
10.23 kton/yr
22.73 kton/yr
5.27 kton/yr
5.27 kton/yr
19.60 kton/yr
19.60 kton/yr
Process 1
Chemical 1
Chemical 5
Process 3
Chemical 3
Process 5
Chemical 7
Chemical 8
Process 4
Chemical 4
10.23 kton/yr
22.73 kton/yr
22.73 kton/yr
22.73 kton/yr
4.71 kton/yr
4.71 kton/yr
41.75 kton/yr
20.87 kton/yr
20.87 kton/yr
20.87 kton/yr
Chemical 1
Process 2
Chemical 5
Chemical 2
Chemical 6
Process 3
Chemical 3
Process 5
Chemical 7
Chemical 8
Process 4
Chemical 4
22.73 kton/yr
22.73 kton/yr
22.73 ton/yr
80.77 kton/yr
80.77 kton/yr
44.44 kton/yr
14.95 kton/yr
29.49 kton/yr
29.49 kton/yr
43.77 kton/yr
29.49 kton/yr
21.88 kton/yr
21.88 kton/yr
21.88 kton/yr
Process 1
Example – Solution with Max ENPV
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Period 1
2 years
Period 2
2.5 years
Period 3
3.5 years
Process 1
Chemical 1
Chemical 5
Process 3
Chemical 3
Chemical 7
10.85 kton/yr
22.37 kton/yr
5.59 kton/yr
5.59 kton/yr
19.30 kton/yr
19.30 kton/yr
Process 1
Chemical 1
Chemical 5
Process 3
Chemical 3
Process 5
Chemical 7
Chemical 8
Process 4
Chemical 4
10.85 kton/yr
22.37 kton/yr
22.37 kton/yr
22.43 kton/yr
4.99 kton/yr
4.99 kton/yr
41.70 kton/yr
20.85 kton/yr
20.85 kton/yr
20.85 kton/yr
Process 1
Chemical 1
Process 2
Chemical 5
Chemical 2
Chemical 6
Process 3
Chemical 3
Process 5
Chemical 7
Chemical 8
Process 4
Chemical 4
22.37 kton/yr
22.37 kton/yr
22.77 ton/yr
10.85 kton/yr
10.85 kton/yr
7.54 kton/yr
2.39 kton/yr
5.15 kton/yr
5.15 kton/yr
43.54 kton/yr
5.15 kton/yr
21.77 kton/yr
21.77 kton/yr
21.77 kton/yr
Same final structure, different production capacities.
Example – Solution with Min DRisk(=900)
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Example – Solution with Max ENPV
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Example – Risk Management Solutions
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Process Planning with Inventory
OBJECTIVES:
Maximize Expected Net Present Value
Minimize Financial Risk
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
PROBLEM DESCRIPTION:
A
1
2
D
3
B
D
MODEL:
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Period 1
2 years
Period 2
2.5 years
Period 3
3.5 years
Chemical 5
Chemical 2
Chemical 6
51.95 kton/yr
22.36 kton/yr
Process 1
Process 2
5.14 kton/yr
12.48 kton/yr
1.05 kton/yr
16.28 kton/yr
Chemical 1
33.90
kton/yr
2.88 kton/yr
11.67 kton/yr
0.81 kton/yr
12.48 kton/yr
4.77 kton/yr
Chemical 6
Process 3
36.45 kton/yr
51.95 kton/yr
76.81 kton/yr
Process 1
1.62
kton
Chemical 5
10.28
kton
11.80 kton/yr
Chemical 2
2.11
kton
27.24 kton/yr
0.60 kton/yr
Chemical 7
4.65 kton/yr
31.09 kton/yr
Chemical 1
Chemical 3
39.04
kton/yr
35.74
kton/yr
5.75
kton
0.42 kton/yr
26.34 kton/yr
0.90 kton/yr
27.24 kton/yr
Process 2
1.18 kton/yr
Chemical 1
Chemical 6
Process 3
Chemical 3
Process 5
Chemical 8
Process 4
Chemical 4
26.77 kton/yr
36.45 kton/yr
26.77 kton/yr
76.81 kton/yr
76.81 kton/yr
43.14
kton/yr
25.41
kton/yr
Process 1
Process 2
3.86
kton
Chemical 7
11.64
kton
25.41 kton/yr
Chemical 5
7.32
kton
13.61 kton/yr
Chemical 2
3.86
kton
30.44 kton/yr
3.29 kton/yr
0.04 kton/yr
6.80
kton
1.94 kton/yr
11.91
kton
3.40 kton/yr
44.13 kton/yr
2.09 kton/yr
31.47 kton/yr
22.12 kton/yr
1.10 kton/yr
31.47 kton/yr
1.03 kton/yr
Example with Inventory – SP Solution
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Example with Inventory
Solution with Min DRisk (=900)
3.64 kton/yr
Process 3
22.15 kton/yr
11.23 kton/yr
Process 1
Chemical 5
5.80 kton/yr
Chemical 7
3.69 kton/yr
18.46 kton/yr
Chemical 1
Chemical 3
6.63
kton/yr
25.79
kton/yr
0.51 kton/yr
0.32 kton/yr
Chemical 1
Process 3
Chemical 3
Process 5
Chemical 8
Process 4
Chemical 4
23.38 kton/yr
22.15 kton/yr
23.38 kton/yr
11.23 kton/yr
5.73
kton/yr
1.64
kton/yr
Process 1
Chemical 7
7.38
kton
22.18 kton/yr
Chemical 5
0.64
kton
5.61 kton/yr
1.60 kton/yr
1.01
kton
0.02 kton/yr
7.27
kton
1.29 kton/yr
41.68 kton/yr
20.58 kton/yr
0.20 kton/yr
0.10 kton/yr
20.54 kton/yr
Chemical 1
Chemical 6
Process 3
Chemical 3
Process 5
Process 4
23.38 kton/yr
22.15 kton/yr
23.38 kton/yr
11.23 kton/yr
11.23 kton/yr
7.48
kton/yr
Process 1
Process 2
Chemical 7
3.37
kton
22.85 kton/yr
Chemical 5
0.90
kton
2.39 kton/yr
Chemical 2
5.39 kton/yr
0.96 kton/yr
1.07
kton
0.30 kton/yr
4.05
kton
1.16 kton/yr
43.72 kton/yr
0.26 kton/yr
5.39 kton/yr
22.04 kton/yr
5.39 kton/yr
Chemical 4
0.51
kton
Chemical 8
1.17
kton/yr
4.11
kton
23.00 kton/yr
0.15 kton/yr
Period 1
2 years
Period 2
2.5 years
Period 3
3.5 years
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Example with Inventory - Solutions
With Inventory
Without
Inventory
DRisk
DRisk
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Downside Expected Profit
Definition:
Up to 50% of risk (confidence?) the lower ENPV solution has higher profit expectations.