Risk - FINANCIAL RISK CHE 5480 Miguel Bagajewicz University...

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Unformatted text preview: 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 0.45 0.40 0.35 0.30 0.25 0.20 0.1 5 0.1 0 0.05 0.00 -300 -200 -1 00 0 1 00 200 300 400 500 600 700 800 Profit Histogram Investment Plan I - E[Profit] = 338 Investment Plan II - E[Profit] = 335 Probability of Loss for Plan I = 12% 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 0.1 8 OR... INCREASE THESE FREQUENCIES REDUCE THESE FREQUENCIES 0.1 6 0.1 4 0.1 2 0.1 0 0.08 0.06 0.04 0.02 0.00 -300 -200 -1 00 0 1 00 200 300 400 500 600 700 800 Profit OR BOTH!!!! 5 Characteristics of TwoStage 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 TwoStage 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 TwoStage 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 TwoStage Stochastic Formulation Let us leave it linear because as is it is complex enough.!!! LINEAR MODEL SP T Max ps qs ys - cT x s Technology matrix s.t. Recourse Function First-Stage Cost Complete recourse: the recourse cost (or profit) for every possible uncertainty realization remains finite, independently of the first-stage decisions (x). Ax = b First-Stage Constraints Second-Stage Constraints Ts x +Wys = hs x 0 First stage variables x X Second Stage Variables 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. ys 0 Recourse matrix (Fixed Recourse) Sometimes not fixed (Interest rates in Portfolio Optimization) 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 0.8 0.7 0.6 0.5 0.4 0.3 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 Expected Profit Desirable Penalty Undesirable Penalty Variance is a measure for the dispersion of the distribution 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 s * s* there is a better solution (y ,x ) than the one obtained (y ,x*). 11 Previous Approaches to Risk Management Robust Optimization using Upper Partial Mean (Ahmed and Sahinidis, 1998) Maximize E[Profit] - UPM Profit PDF 0.4 E[ ] UPM = 0.50 UPM = 0.44 0.3 0.2 UPM = E[ ( )] ( ) 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 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 UPM = ps s sS s = Max 0 ; pk Profit k - Profit s kS s s =3 S1 S2 S3 S4 E[Profit] UPM Objective Profit Case I 150 125 75 50 100.00 18.75 43.75 Case II 100 100 75 50 81.25 9.38 53.13 Case I 0 0 25 50 Case II 0 0 6.25 31.25 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 ] -200 UPM 1 8 1 6 1 4 Ro bustness So lutio n P1 A -220 -240 -260 -280 1 2 1 0 Ro bustness So lutio n P2 B -300 -320 -340 -360 -380 Ro bustness So lutio n with Optimal Seco nd-Stage Decisio ns 8 6 4 2 0 Ro bustness So lutio n with Optimal Seco nd-Stage Decisio ns Both technologies are able to produce two products with different production cost and at different yield per unit of installed capacity 0 20 40 60 80 1 00 1 20 1 40 1 60 1 80 200 220 0 20 40 60 80 100 120 1 40 160 1 80 200 220 E[Profit ] -200 -220 -240 -260 -280 -300 -320 -340 -360 -380 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 R o bustness So lutio n with Optim al Seco nd-Stage D ecisio ns R o bustness So lutio n UPM 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( Profit s ) s For each scenario the profit is either greater/equal or smaller than the target 1 If Profit s < P( Profit s < ) = 0 else s z is a new binary variable P( Profit s > ) = z s Formal Definition of Financial Risk Risk ( x,) = ps z s s 19 Financial Risk Interpretation Probability Profit PDF f ( ) 0.20 0.1 4 0.1 8 0.1 2 0.1 6 0.1 0 0.1 4 0.1 2 0.08 0.1 0 0.06 0.08 0.06 0.04 0.04 0.02 0.02 0.00 x fixed Cumulative Probability = Risk (x , ) Area = Risk (x , ) Profit Profit 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 1 000 1 250 1 500 1 750 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 0 .8 0 .7 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 Risk-Averse Investor's Choice E[Profit ] = 0.4 Risk-T aker Investor's Choice E[Profit ] = 1.0 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 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 Possible curve Maximum E[Profit ] Impossible curve Profit 23 Minimizing Risk: a MultiObjective Problem Risk 1 .0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 1 x fixed 2 3 4 Max E[ Profit ] = ps qs y s - cT x s Min Risk (x ,4) Min Risk [ 1 ] = Min Risk (x ,3 ) Min Risk [ i ] = . . . pz s s s1 pz s s si Max E[Profit (x )] s.t. Ax= b Min Risk (x ,2 ) Min Risk (x ,1 ) Target Profit Ts x +Wys = hs x0 ys 0 x X Multiple Objectives: At each profit we want minimize the associated risk We also want to maximize the expected profit T qs ys - cT x -U s z s T qs ys - cT x +U s (1- z s ) z s (0,1) 24 Parametric Representations of the MultiObjective Model Restricted Risk Restricted Risk MODEL Max ps qs ys - cT x s.t. s Ax= b Ts x +Wys = hs x 0 ys 0 x X Forces Risk to be lower than a specified level ps z si i s T qs ys - cT x i -U s z si T qs ys - cT x i +U s (1- z si ) Risk Management Constraints z s (0,1) 25 Parametric Representations of the MultiObjective Model Penalty for Risk Risk Penalty MODEL T Max ps qs ys - cT x - i ps z si s i s STRATEGY Define several profit Targets and penalty weights to solve the model using a multiparametric approach s.t. Ax= b Ts x +Wys = hs Penalty Term x 0 ys 0 x X T qs ys - cT x i -U s z si T qs 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 Positive Profit Deviation from Target DRisk ( x, ) = E [ ( x, ) ] - Profit ( x ) If Profit ( x ) < ( x , ) = Otherwise 0 The Positive Profit Deviation is also defined for each scenario - Profit s s = 0 If Profit s < Otherwise Formal definition of Downside Risk DRisk ( x, ) = ps s s 29 Downside Risk Interpretation Profit PDF f ( ) 0.1 4 x fixed 0.1 2 0.1 0 0.08 0.06 0.04 DRisk (x ,) = E[(x ,)] DRisk ( x, ) = ( - ) f ( ) d - 0.02 0.00 Profit 30 Downside Risk & Probabilistic Risk Risk (x ,) 1 .0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 x fixed Area = DRisk (x ,) DRisk ( x, = Risk ( x, ) d ) - Profit 31 TwoStage Model using Downside Risk MODEL DRisk T Max ps qs ys - cT x - ps s s s Penalty Term s.t. Advantages Same as models using Risk Does not require the use of binary variables Potential benefits from the use of decomposition methods Ax= b Ts x +Wys = hs x 0 ys 0 x X Strategy ys - c x) T T s - (qs s 0 Downside Risk Constraints 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 32 TwoStage Model using Downside Risk Warning: The same risk may imply different Downside Risks. Risk 1 .0 0.9 0.8 0.7 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 DRisk (Design I , 0.5) = 0.2 Risk (Design I , 0.5) = 0.5 DRisk (Design II , 0.5) = 0.2 Risk (Design II , 0.5) = 0.309 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. directly 34 Process Planning Under Uncertainty GIVEN: Process Network Set of Processes Set of Chemicals B A 2 3 C 1 D Forecasted Data Demands & Availabilities Costs & Prices Capital Budget DETERMINE: Network Expansions Production Levels Timing Sizing Location 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 Chemical 2 Chemical 7 Process 3 Chemical 3 Process 4 Chemical 4 Process 5 Chemical 8 37 Example Solution with Max ENPV Period 1 3 2 3.5 2.5 years 2 years 14.95 kton/yr Chemical 5 5 Chemical 5 Chemical 5.27 kton/yr 4.71 kton/yr 29.49 kton/yr Chemical 1 Chemical 1 44.44kton/yr 5.27 kton/yr 4.71 kton/yr Process 1 Process 1 Process 1 10.23 kton/yr 10.23 kton/yr 80.77 Process 2 80.77 kton/yr Chemical 2 29.49 7 Chemical kton/yr 21.88 kton/yr 20.87 kton/yr 19.60 Chemicalkton/yr Chemical 77 Chemical 6 29.49 kton/yr Process 3 Process 3 Chemical 33 Chemical Chemical 3 19.60 kton/yr 41.75 kton/yr 43.77 kton/yr 22.73 kton/yr 22.73 kton/yr Process 5 22.73 kton/yr 22.73 ton/yr Chemical 8 21.88 kton/yr 20.87 kton/yr Process 4 22.73 kton/yr Chemical 44 Chemical 21.88 kton/yr 20.87 kton/yr 38 Example Solution with Min DRisk( =900) Period 1 3 2 3.5 2.5 years 2 years Chemical 1 7.54 kton/yr Chemical 1 4.99 kton/yr Process 1 10.85 kton/yr Process 1 Chemical 1 10.85 kton/yr 5.59 kton/yr 2.39 kton/yr Chemical 5 5.15 kton/yr Chemical 5 Process 2 Chemical 6 5.15 kton/yr 5.59 kton/yr Chemical 5 4.99 kton/yr 10.85 kton/yr Chemical 2 5.15 kton/yr Process 1 10.85 kton/yr 21.77 kton/yr 20.85 kton/yr Chemical 7 Chemical 7 Process 3 22.37 kton/yr Chemical 4 Chemical 4 20.85kton/yr 21.77 kton/yr Chemical 7 Process 5 Process 5 kton/yr 19.30 22.77 ton/yr 22.43 kton/yr Chemical 8 Chemical 8 21.77 kton/yr 20.85 kton/yr Chemical 3 Chemical 3 41.70 kton/yr 43.54 kton/yr Process 3 Process 3 22.37 kton/yr 22.37 kton/yr Chemical 3 Process 4 Process 4 19.30 kton/yr 22.37 kton/yr 22.37 kton/yr Same final structure, different production capacities. 39 Example Solution with Max ENPV 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 1 000 1250 1 500 1 750 2000 2250 2500 2750 3000 3250 PP solution E[NPV ] = 1140 M$ NPV (M$) 40 Example Risk Management Solutions NPV PDF f () Risk Risk 1.0 0.0026 0.0024 0.9 0.0022 0.8 0.0020 0.7 8 0.001 0.001 0.6 6 0.001 4 0.5 0.001 2 0.4 0.001 0 0.3 0.0008 0.0006 0.2 0.0004 0.1 0.0002 0.0 0.0000 250 0 500 = 900 ENPV = 908 = 900 = 1100 ENPV = 1074 PP ENPV =1140 increases PP 500 600 700 800 900 1 000 PP = 1100 1 00 1 1 200 1 300 1 400 1 500 750 500 1000 1250 500 1 000 1 11 750 2000 500 2250 2500 2000 2750 2500 3000 3250 3000 NPV ((M$) ) NPV , M$ 41 Process Planning with Inventory PROBLEM DESCRIPTION: B D 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 Maximize Expected Net Present Value Minimize Financial Risk 42 OBJECTIVES: Example with Inventory SP Solution Period 1 3 2 3.5 2.5 years 2 years 33.90 43.14 kton/yr kton/yr 39.04 kton/yr 2.88 kton/yr Chemical 1 0.42 kton/yr Chemical 1 6.80 kton 5.75 kton Chemical 1 1.94 kton/yr 2.09 kton/yr 1.18 kton/yr 16.28 13.61 11.80 kton/yr kton/yr kton/yr Chemical 5 Chemical 5 Chemical 5 7.32 10.28 kton 5.14 kton kton/yr 11.67 kton/yr 26.34 31.47 kton/yr kton/yr Chemical 6 Chemical 6 Process 1 1 Process Process 1 51.95 kton/yr 76.81 kton/yr 51.95 kton/yr30.44 12.48 kton/yr kton/yr 27.24 kton/yr Process 2 2 Process Process 2 22.36 kton/yr 76.81 kton/yr 1.03 kton/yr 76.81 kton/yr 1.05 kton/yr 3.86 0.60 kton kton/yr 2.11 Chemical 2 2 kton Chemical Chemical 2 11.64 kton 3.29 4.65 kton/yr kton/yr 12.48 31.47 kton/yr kton/yr 27.24 kton/yr 1.10 Chemical 6 kton/yr 3.86 kton 1.62 kton 0.81 kton/yr 0.90 kton/yr 0.04 kton/yr 44.13 kton/yr 35.74 kton/yr ChemicalChemical 3 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 26.77 kton/yr Chemical 7 Chemical 7 22.12 kton/yr Chemical 8 Process 5 26.77 kton/yr 25.41 kton/yr 25.41 kton/yr Chemical 4 43 Example with Inventory Solution with Min DRisk ( =900) Period 1 3 2 3.5 2.5 years 2 years 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 Chemical 5 kton/yr 5.39 kton/yr kton/yr Process 2 11.23 kton/yr 0.26 kton/yr 5.39 kton/yr 0.32 kton/yr Chemical 6 0.51 Process 1 5.39 11.23 kton/yr kton/yr Process 1 kton/yr 1.07 Chemical 1 kton Chemical kton/yr 11.23 1 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 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 7 0.96 kton/yr 7 kton/yr 20.58 Chemical kton/yr kton/yr 25.79 Process 5 Process 3 Chemical 8 22.04 kton/yr Chemical 8 kton/yr Process 3 22.18 Process 5 Process 3kton/yr 22.15 23.38 kton/yr 1.17 kton/yr Chemical 3 kton/yr 22.15 kton/yr 22.15 kton/yr 23.38 kton/yr Chemical 4 22.85 1.64 3.64 Process 4 kton/yr 0.15 4.11 kton/yr kton/yr kton/yr kton Process 4kton/yr 23.38 0.20 Chemical 4 kton/yr 0.51 23.38 kton/yr kton 44 Example with Inventory Solutions Risk Risk 1.0 1.0 0.9 0.8 0.8 0.7 0.6 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0 250 500 750 1000 1 250 1 500 1750 2000 2250 2500 2750 3000 3250 3500 3750 4000 ENPV = 980 E[NPV ] = 1140 M$ Without PPI Inventory DRisk = 900 PP solution PPI solution WithPPIE[NPV ] = 1237 M$ Inventory ENPV = 1237 ENPV = 1140 DRisk = 1400 ENPV = 1184 NPV (M$) NPV (M$) 45 Downside Expected Profit Definition: DENPV ( x, p ) = - f ( x, ) d = Risk ( x, ) - DRisk ( x, ) Risk 1.0 CEP (M$) 1250 1125 0.9 = 900 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 250 500 750 1000 1250 1500 1750 PP PP solution E[NPV ] = 1140 M$ = 1100 1000 875 750 625 500 375 250 125 0 = 900 E[NPV ] = 908 M$ 2000 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 NPV (M$) 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. Profit PDF 0.14 x fixed 0.12 0.10 VaR( x, p ) 0.08 0.06 0.04 0.02 Area = Risk (x , ) Profit 0.00 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 supplier tomarket 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 LNG CNG GTL Ammonia Methanol Markets China India Japan S. Korea Thailand United States 50 Network of Alternatives 51 Results Risk Management (Downside Risk): 1.0 1.0 0.9 0.8 0.7 0.6 0.5 Mala-GTL 4.640 Ships: 4 & 2 ENPV:4.570 DR@ 4: 0.157 DR@ 3.5: 0.058 DR-200s Malaysia GTL 0.9 0.8 0.7 0.6 0.5 1s 4.666 0.4 0.4 0.3 0.3 0.2 0.2 Indo-GTL 200s Ships: 5 & 3 4.678 ENPV:4.633 DR@ 4: 0.190 DR@ 3.5: 0.086 Thailand China 0.1 0.1 0.0 0.0 0 0 1 1 2 2 3 3 44 55 66 7 7 8 8 9 9 10 10 52 Value at Risk (VaR): VaR is the expected loss for a certain confidence level usually set at 5% VaR =ENPV NPV @ pquantile Opportunity Value (OV) or Upper Potential (UP): OV = NPV @ (1p)quantile ENPV 53 Results Value at Risk (VaR) and Opportunity Value (OV): 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0 1 2 3 4 5 6 7 8 9 10 54 OV @ 95%: 1.42 OV @ 95%: 1.75 Mala-GTL Ships: 4 & 2 ENPV:4.570 DR@ 4: 0.157 DR@ 3.5: 0.058 Indo-GTL Ships: 5 & 3 ENPV:4.633 DR@ 4: 0.190 DR@ 3.5: 0.086 VaR @ 5%: 1.49 VaR @ 5%: 1.82 Reduction in VaR: 18.1% Reduction in OV: 18.9% Results Risk /Upside Potential Loss Ratio 1.0 0.9 0.8 0.7 OV @ 95%: 1.42 OV @ 95%: 1.75 O-Area: 0.116 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0 1 2 3 Mala-GTL Ships: 4 & 2 ENPV:4.570 DR@ 4: 0.157 DR@ 3.5: 0.058 R-Area: 0.053 VaR @ 5%: 1.82 Indo-GTL Ships: 5 & 3 ENPV:4.633 DR@ 4: 0.190 DR@ 3.5: 0.086 VaR @ 5%: 1.49 4 5 6 7 8 9 10 55 Risk /Upside Potential Loss Ratio: 2.2 Risk /Upside Potential Loss Ratio + Risk /Upside Potential Loss Ratio = O_Area Risk R_Area = - + - dNPV + where: Risk(x2,NPV) O_Area Risk(x1,NPV) - dNPV if 0 otherwise if < 0 otherwise 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0 1 ENPV2 2 3 ENPV1 4 5 6 7 8 9 NPV 10 R_Area = 0 + - - = 0 = Risk NGC ( NPV ) - Risk NGC - DR ( NPV ) 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 0.4 E) Envelope 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 : : Min Min Min Min ... NPVsn Min NPVs1 NPVs2 NPVs3 NPVd1,sn NPVd2,sn NPVd3,sn ... NPVdn,sn 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 Upper Envelope ENPV:4.921 Indo-GTL Ships: 6 & 3 ENPV:4.63 0.4 0.2 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 ...
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