FINANCIAL RISK. CHE 5480 Miguel Bagajewicz. University of Oklahoma School of Chemical Engineering and Materials Science

Transcription

1 FINANCIAL RISK CHE 5480 Miguel Bagajewicz Univerity of Oklahoma School of Chemical Engineering and Material Science 1

2 Scope of Dicuion We will dicu the definition and management of financial rik in in any deign proce or deciion making paradigm, like Invetment Planning Scheduling and more in general, operation planning Supply Chain modeling, cheduling and control Short term cheduling (including cah flow management) Deign of proce ytem Product Deign Extenion that are emerging are the treatment of other rik in a multiobjective (?) framework, including for example Environmental Rik Accident Rik (other than thoe than can be expreed a financial rik) 2

3 Introduction Undertanding Rik Conider two invetment plan, deign, or operational deciion Probability Profit Hitogram Invetment Plan I - E[Profit] = 338 Invetment Plan II - E[Profit] = Probability of Lo for Plan I = 12% Profit (M$) 3

4 Concluion Rik can only be aeed after a plan ha been elected but it cannot be managed during the optimization tage (even when tochatic optimization including uncertainty ha been performed). There i a need to develop new model that allow not only aeing but managing financial rik. The deciion maker ha two imultaneou objective: Maximize Expected Profit. Minimize Rik Expoure 4

5 What doe Rik Management mean? One want to modify the profit ditribution in order to atify the preference of the deciion maker REDUCE THESE FREQUENCIES Probability OR INCREASE THESE FREQUENCIES Profit ξ OR BOTH!!!! 5

6 Characteritic of Two-Stage Stochatic Optimization Model Philoophy Maximize the Expected Value of the objective over all poible realization of uncertain parameter. Typically, the objective i Expected Profit, uually Net Preent Value. Sometime the minimization of Cot i an alternative objective. Uncertainty Typically, the uncertain parameter are: market demand, availabilitie, price, proce yield, rate of interet, inflation, etc. In Two-Stage Programming, uncertainty i modeled through a finite number of independent Scenario. Scenario are typically formed by random ample taken from the probability ditribution of the uncertain parameter. 6

7 Characteritic of Two-Stage Stochatic Optimization Model Firt-Stage Deciion Taken before the uncertainty i revealed. They uually correpond d to tructural deciion (not operational). Alo called Here and Now deciion. Repreented by Deign Variable. Example: To build a plant or not. How much capacity hould be added, etc. To place an order now. To ign contract or buy option. To pick a reactor volume, to pick a certain number of tray and ize the condener and the reboiler of a column, etc 7

8 Characteritic of Two-Stage Stochatic Optimization Model Second-Stage Stage Deciion Taken in order to adapt the plan or deign to the uncertain parameter realization. Alo called Recoure deciion. Repreented by Control Variable. Example: the operating level; the production late of a plant. Sometime firt tage deciion can be treated a econd tage deciion. d In uch cae the problem i called a multiple tage problem. Shortcoming The model i unable to perform rik management deciion. 8

9 Two-Stage Stochatic Formulation Let u leave it linear becaue a i it i complex enough.!!! Technology matrix Firt tage variable Max.t. Ax=bb x 0 LINEAR MODEL SP p q T x + Wy = h y 00 y T + Wy h y x X Recoure Function c T x Recoure matrix (Fixed Recoure) Firt-Stage Cot Firt-Stage Contraint Second-Stage Stage Contraint Second Stage Variable Sometime not fixed (Interet rate in Portfolio Optimization) Complete recoure: the recoure cot (or profit) for every poible uncertainty realization remain finite, independently of the firt-tage deciion (x). Relatively complete recoure: the recoure cot (or profit) i feaible for the et of feaible firt-tage deciion. Thi condition mean that for every feaible firt-tage deciion, there i a way of adapting the plan to the realization of uncertain parameter. We alo have found that one can acrifice efficiency for certain cenario to improve rik management. We do not know how to call thi yet. 9

10 Previou Approache to Rik Management Robut Optimization Uing Variance (Mulvey( et al., 1995) Maximize E[Profit] - ρ V[Profit] Profit PDF 0.9 Expected Profit Deirable Penalty Undeirable Penalty Variance i a meaure for the diperion of the ditribution Profit Underlying Aumption: Rik i monotonic with variability 10

11 Robut Optimization Uing Variance Drawback Variance i a ymmetric rik meaure: profit both above and below the target level are penalized equally. We only want to penalize profit below b the target. Introduce non-linearitie in the model, which reult in eriou computational difficultie, pecially in large-cale problem. The model may render olution that are tochatically dominated by other. Thi i known in the literature a not howing Pareto-Optimality. In other word there i a better olution (y(,x * ) than the one obtained (y *,x*). 11

12 Previou Approache to Rik Management Robut Optimization uing Upper Partial Mean (Ahmed and Sahinidi, 1998) Profit PDF 0.4 Maximize E[Profit] - ρ UPM E[ξ] 0.3 UPM = 0.50 UPM = UPM = E[ (ξ) ] (ξ) Profit ξ Underlying Aumption: Rik i monotonic with lower variability 12

13 Robut Optimization uing the UPM Robut Optimization uing the UPM Advantage Linear meaure Diadvantage The UPM may mileadingly favor non-optimal optimal econd-tage tage deciion. Conequently, financial rik i not managed properly and olution with higher rik than the one obtained uing the traditional two-tage formulation may be obtained. The model loe it cenario-decompoable tructure and tochatic decompoition method can no longer be ued to olve it. 13

14 Robut Optimization uing the UPM Objective Function: Maximize E[Profit] - ρ UPM UPM = S p = Max 0 ; k S p k Profit k Profit ρ = 3 Cae I Profit Cae II Cae I Cae II S S S S E[Profit] UPM Objective Downide cenario are the ame, but the UPM i affected by the change in expected profit due to a different upide ditribution. tion. A a reult a wrong choice i made. 14

15 Robut Optimization uing the UPM Effect of Non-Optimal Second-Stage Stage Deciion E[Profit ] -200 UPM 18 P1 A Robutne Solution 12 Robutne Solution P2 B Robutne Solution with Optimal Second-Stage Deciion Robutne Solution with Optimal Second-Stage Deciion Both technologie are able to produce two product with different production cot and at different yield per unit of intalled capacity ρ E[Profit ] ρ Robutne Solution Robutne Solution with Optimal Second-Stage Deciion UPM 15

16 OTHER APPROACHES Cheng, Subrahmanian and Weterberg (2002, unpublihed) Multiobjective Approach: Conider Downide Rik, ENPV and Proce Life Cycle a alternative Objective. Multiperiod Deciion proce modeled a a Markov deciion proce with recoure. The problem i ometime amenable to be reformulated a a equence of ingle-period ub-problem, each being a two-tage tochatic program with recoure. Thee can often be olved backward in time to obtain Pareto Optimal olution. Thi paper propoe a new deign paradigm of which rik i jut one component. We will reviit thi iue later in the talk. 16

17 OTHER APPROACHES Rik Premium (Applequit( Applequit, Pekny and Reklaiti, 2000) Obervation: Rate of return varie linearly with variability. The T of uch dependance i called Rik Premium. They ugget to benchmark new invetment againt the hitorical rik premium by uing a two objective (rik premium and profit) problem. The technique relie on uing variance a a meaure of variability. 17

18 Previou Approache to Rik Management Concluion The minimization of Variance penalize both ide of the mean. The Robut Optimization Approach uing Variance or UPM i not uitable for rik management. The Rik Premium Approach (Applequit( et al.) ha the ame problem a the penalization of variance. THUS, Rik hould be properly defined and directly incorporated in the model to manage it. The multiobjective Markov deciion proce (Applequit( et al, 2000) i very cloely related to our and can be conidered complementary. In fact (Weterberg dixit) it can be extended to match our in the definition of rik and it multilevel parametrization. 18

19 Probabilitic Definition of Rik Financial Rik = Probability that a plan or deign doe not meet a certain profit target ( x, Ω = P ( Profit <Ω ) Rik ) Scenario are independent event Rik ( x, Ω ) = p P ( Ω ) Scenario are independent event Profit For each cenario the profit i either greater/equal or maller than the target P( ( Profit <Ω ) = 1 If Profit 0 ele < Ω z i a new binary variable ( Profit >Ω ) z P = Formal Definition of Financial Rik = = Rik ( x, Ω ) p z 19

20 Financial Rik Interpretation Profit Probability PDF f (ξ ) Ω Ω x fixed Cumulative Probability = Rik (x,ω) Area = Rik (x,ω) Profit ξξ 20

21 Cumulative Rik Curve Rik Profit (M$) Our intention i to modify the hape and location of thi curve according to the attitude toward rik of the deciion maker 21

22 Rik Preference and Rik Curve Rik Rik-Avere Invetor' Choice E[Profit ] = 0.4 Rik-Taker Invetor' Choice E[Profit ] = Profit 22

23 Rik Curve Propertie A plan or deign with Maximum E[Profit] ] (i.e. optimal in Model SP) et a theoretical limit for financial rik: it i impoible to find a feaible plan/deign having a rik curve entirely beneath thi curve. Rik Poible curve Maximum E[Profit ] Impoible curve Profit 23

24 Minimizing Rik: a Multi-Objective Problem Rik Ω 1 Ω 2 Ω 3 Ω 4 x fixed Min Rik (x,ω 4 ) Min Rik (x,ω 3 ) Max E[Profit (x )] Min Rik (x,ω 2 ) Min Rik (x,ω 1 ) Target Profit Ω Max E.t. Ax=b T [ Profit] = p q y c x x 0 x X [ Ω1] = Min Rik p z 1 Min Rik. [ Ωi ] = T x+ Wy = h p z i Multiple Objective: At each profit we want minimize the aociated rik We alo want to maximize the expected profit y q q T T 0 y y c c T T z (0,1) x Ω U z x Ω+ U (1 z ) 24

25 Parametric Repreentation of the Multi-Objective Model Retricted Rik Retricted Rik MODEL Max.t. Ax=b p q T x+ Wy = h y x 0 x X y 0 c T x p z ε T q q z T y y i c c T T (0,1) i x Ω x Ω Force Rik to be lower than a pecified level i i U + U z i (1 z i ) Rik Management Contraint 25

26 Parametric Repreentation of the Multi-Objective Model Penalty for Rik Rik Penalty MODEL Max.t. Ax=b p q T y T x+ Wy = h c T x ρ i i p z i Penalty Term STRATEGY Define everal profit Target and penalty weight to olve the model uing a multi- parametric approach x 0 x X y 0 q q z T T y y c c T T (0,1) x Ω x Ω i i U + U z i (1 z i ) Rik Management Contraint 26

27 Rik Management uing the New Model Advantage Rik can be effectively managed according to the deciion maker criteria. The model can adapt to rik-avere or rik-taker deciion maker, and their rik preference are eaily matched uing the rik curve. A full pectrum of olution i obtained. Thee olution alway ay have optimal econd-tage tage deciion. Model Rik Penalty conerve all the propertie of the tandard d two-tage tochatic formulation. Diadvantage The ue of binary variable i required, which increae the computational c time to get a olution. Thi i a major limitation for large-cale problem. 27

28 Rik Management uing the New Model Computational Iue The mot efficient method to olve tochatic optimization problem reported in the literature exploit the decompoable tructure of the model. Thi property mean that each cenario define an independent econd cond-tage problem that can be olved eparately from the other cenario once o the firt- tage variable are fixed. The Rik Penalty Model i decompoable wherea Model Retricted Rik i not. Thu, the firt one i model i preferable. Even uing decompoition method, the preence of binary variable in both model contitute a major computational limitation to olve large rge-cale problem. It would be more convenient to meaure rik indirectly uch that binary variable in the econd tage are avoided. 28

29 Downide Rik Downide Rik (Eppen et al, 1989) = Expected Value of the Poitive Profit Deviation from the target DRik( ( x, Ω ) = E [ δ ( x, Ω ) ] Poitive Profit Deviation from Target Ω δ ( x, Ω ) = Ω Profit 0 ( x ) If Profit ( x ) Otherwie < Ω The Poitive Profit Deviation i alo defined for each cenario δ = Ω Profit 0 If Profit Otherwie < Ω Formal definition of Downide Rik DRik, ( x Ω ) = δ p 29

30 Downide Rik Interpretation Profit PDF f (ξ ) Ω x fixed DRik (x,ω) = E[δ(x,Ω)] δ Ω DRik ( x, Ω) = ( Ω ξ) Profit ξ f ( ξ) dξ 30

31 Downide Rik & Probabilitic Rik Rik (x,ξ) x fixed Ω Profit ξ Area = DRik (x,ω) Ω DRik ( x, Ω) = Rik ( x, ξ) dξ 31

32 Two-Stage Model uing Downide Rik MODEL DRik Max.t. Ax=bb µ p q T y c T x pδ Penalty Term Advantage Same a model uing Rik Doe not require the ue of binary variable T x + Wy = h x 0 x X y δ δ + Wy h 0 T Ω ( q y c x) 0 T Downide Rik Contraint Potential benefit from the ue of decompoition method Strategy Solve the model uing different profit target to get a full pectrum of olution. Ue the rik curve to elect the olution that better uit the deciion maker preference 32

33 Two-Stage Model uing Downide Rik Warning: The ame rik may imply different Downide Rik. Rik DRik (Deign I, 0.5) = 0.2 Rik (Deign I, 0.5) = DRik (Deign II, 0.5) = 0.2 Rik (Deign II, 0.5) = Immediate Conequence: Profit Minimizing downide rik doe not guarantee minimizing rik. 33

34 Commercial Software Rikoptimizer (Paliade) and CrytalBall (Deciioneering) Ue excell model Allow uncertainty in a form of ditribution Perform Montecarlo Simulation or ue genetic algorithm to optimize (Maximize ENPV, Minimize Variance, etc.) Financial Software. Large variety Some ue the concept of downide rik In mot of thee oftware, Rik i mentioned but not manipulated directly. 34

35 Proce Planning Under Uncertainty GIVEN: Proce Network Set of Procee Set of Chemical A 1 B 2 C Forecated Data Demand & Availabilitie Cot & Price Capital Budget 3 D DETERMINE: Network Expanion Production Level Timing Sizing Location OBJECTIVES: Maximize Expected Net Preent Value Minimize Financial Rik 35

36 Proce Planning Under Uncertainty Deign Variable: to be decided before the uncertainty reveal x = { } Y it,, Q it Y: Deciion of building proce E it Y: Deciion of building proce i in period t E: Capacity expanion of proce i in period t Q: Total capacity of proce i in period t Control Variable: elected after the uncertain parameter become known y = { S jlt, } P jlt, W it S: Sale of product j in market l at time t and cenario P: Purchae of raw mat. j in market l at time t and cenario W: Operating level of of proce i in period t and cenario 36

37 Uncertain Parameter: Total of 400 Scenario Example Demand, Availabilitie, Sale Price, Purchae Price Project Staged in 3 Time Period of 2, 2.5, 3.5 year Chemical 5 Chemical 1 Proce 1 Proce 2 Chemical 6 Chemical 2 Chemical 7 Proce 3 Proce 5 Chemical 8 Chemical 3 Chemical 4 Proce 4 37

38 Example Solution with Max ENPV Period year Chemical Chemical 1 Proce 1 Proce Chemical 2 Chemical Chemical 77 Proce 3 Proce Chemical Proce 4 Chemical Chemical Chemical

39 Example Solution with Min DRik(Ω=900) Period year 2.39 Chemical Chemical 1 Proce 1 Proce 2 Chemical Chemical Chemical 1 Proce 1 Chemical Chemical 1 Proce Chemical 7 Chemical 7 Proce 3 Chemical 7 Proce 5 Proce 3 Proce Proce 3 Chemical Chemical Chemical Proce Chemical Chemical Chemical 8 Chemical Same final tructure, different production capacitie. 39

40 Example Solution with Max ENPV Rik PP olution E[NPV ] = 1140 M$ NPV (M$) 40

41 Example Rik Management Solution NPV Rik PDF f (ξ) Ω = 900 ENPV = 908 Ω = 900 Ω = 1100 ENPV = 1074 Ω increae Ω = 1100 PP PP ENPV =1140 Ω PP NPV ( (M$) ξ, M$ ) 41

42 Proce Planning with Inventory PROBLEM DESCRIPTION: A B D D MODEL: The ma balance i modified uch that now a certain level of inventory for raw material and product i allowed A torage cot i included in the objective OBJECTIVES: Maximize Expected Net Preent Value Minimize Financial Rik 42

43 Example with Inventory SP Solution Period year Chemical Chemical Chemical 1 Chemical kton 1.94 kton Chemical kton Proce Proce 1 Proce Proce 2 2 Proce 1 Proce kton Chemical Chemical 2 2 kton Chemical Chemical 6 Chemical Chemical kton kton Chemical Chemical kton 3.40 Proce kton Chemical Chemical Proce 3 Proce 5 Proce Chemical 4 Chemical

44 Example with Inventory Solution with Min DRik (Ω=900) Period year Chemical kton Chemical 5 kton Chemical Chemical Proce 1 Proce Proce Proce Chemical 1 kton 0.30 Chemical Chemical 2 kton 3.64 Proce 4 Proce Chemical kton Chemical kton Chemical Chemical Proce Proce 5 Chemical 8 Chemical 8 Proce 3 Proce Proce Chemical 3 Chemical Chemical Chemical kton kton Chemical kton 4.11 kton 44

45 Example with Inventory - Solution Rik DRik PP Ω olution = 900 ENPV = 980 E[NPV ] = 1140 M$ Without Inventory PPI ENPV = 1140 PPI olution With PPI Inventory E[NPV ] = 1237 M$ ENPV = DRik Ω = 1400 ENPV = NPV NPV (M$) (M$) 45

46 Downide Expected Profit Definition: Ω DENPV ( x, pω) = ξ f ( x, ξ ) dξ = Ω Rik( x, Ω) DRik( x, Ω) Rik 1.0 CEP (M$) Ω = 900 Ω = 1100 PP PP olution E[NPV ] = 1140 M$ Ω = E[NPV ] = 908 M$ NPV (M$) Rik Up to 50% of rik (confidence?) the lower ENPV olution ha higher profit expectation. 46

47 Value at Rik Definition: VaR i given by the difference between the mean value of the profit and the profit value correponding to the p-quantile. Profit PDF Ω x fixed 0.10 VaR( x, pω) Area = Rik (x,ω) 0.00 Profit ξ E[ Profit( x)] VaR( x, p Ω ) = E[ Profit( x)] Rik 1 ( x, Ω) VaR=z p σ for ymmetric ditribution (Portfolio optimization) 47

48 COMPUTATIONAL APPROACHES Sampling Average Approximation Method: Solve M time the problem uing only N cenario. If multiple olution are obtained, ue the firt tage variable to olve the problem with a large number of cenario N >>N N to determine the optimum. Generalized Bender Decompoition Algorithm Firt Stage variable are complicating variable. Thi leave a primal over econd tage variable, which i decompoable. 48

49 Example Generate a model to: Evaluate a large network of natural ga upplierto-market tranportation alternative Identify the mot profitable alternative() Manage financial rik

50 Network of Alternative Supplier Autralia Indoneia Iran Kazakhtan Malayia Qatar Ruia Tranportation Method Pipeline LNG CNG GTL Ammonia Methanol Market China India Japan S. Korea Thailand United State 50

51 Network of Alternative 51

52 Reult Rik Management (Downide Rik): Malayia GTL Thailand China DR-200 Mala-GTL Ship: 4 & 2 ENPV:4.570 DR@ 4: DR@ 3.5: Indo-GTL 200 Ship: & 3 ENPV:4.633 DR@ 4: DR@ 3.5:

53 Value at Rik (VaR( VaR): VaR i the expected lo for a certain confidence level uually et at 5% VaR =ENPV p-quantile Opportunity Value (OV) or Upper Potential (UP): OV = (1-p)-quantile ENPV 53

54 UPSIDE POTENTIAL Point meaure for the upide Rik E(Profit) =3.4 E(Profit) =3.0 OV =0.75 VaR =0.75 VaR =1.75 OV = Profit 54

55 Reult Value at Rik (VaR( VaR) ) and Opportunity Value (OV): %: %: %: 1.82 Mala-GTL Ship: 4 & 2 ENPV:4.570 DR@ 4: DR@ 3.5: %: 1.49 Indo-GTL Ship: 5 & 3 ENPV:4.633 DR@ 4: DR@ 3.5: Reduction in VaR: 18.1% Reduction in OV: 18.9% 55

56 Reult Rik /Upide Potential Lo Ratio %: %: R-Area: %: 1.82 Mala-GTL Ship: 4 & 2 ENPV:4.570 DR@ 4: DR@ 3.5: %: 1.49 Indo-GTL Ship: 5 & 3 ENPV:4.633 DR@ 4: DR@ 3.5: O-Area: Rik /Upide Potential Lo Ratio:

57 Rik /Upide Potential Lo Ratio Rik Rik /Upide Potential Lo Ratio Rik(x 2,NPV) R_Area O_Area Rik(x 1,NPV) ENPV 2 ENPV NPV 10 = O_Area R_Area where: ψ = ψ + ψ = + + ψ ψ ψ = 0 Rik + dnpv dnpv if otherwie ψ 0 ψ if ψ < 0 = 0 otherwie NGC ( NPV ) Rik ( NPV ) NGC DR 57

58 Upper and Lower Bound Rik Curve Upper Bound Rik Curve (Envelope): The curve contructed by plotting the et of net preent value (NPV) for the bet deign under each cenario. Rik a) Poible b) Poible c) Impoible E) Envelope 0.2 d) Impoible

59 SAMPLING ALGORITHM Generate h() d=1 d=d+1 Outer Loop ye h=h(d) EXIT no d<=d_max Fix Firt-Stage Variable Solve Determinitic Model =1 Inner Loop no h=h() = +1 ye <=_max Solve Determinitic Model Store NPV(d,) Thi generate everal olution 59

60 Upper and Lower Bound Rik Curve Lower Bound Rik Curve: The curve contructed by plotting the et of net preent value (NPV) for the wort (of the et of bet deign) under each cenario. \d n d 1 NPV d1,1 NPV d2,1 NPV d3,1 NPV dn,1 d 2 NPV d1,2 NPV d2,2 NPV d3,2 NPV dn,2 d 3 NPV d1,3 NPV d2,3 NPV d3,3 NPV dn,3 : : : : : d n NPV d1,n NPV d2,n NPV d3,n NPV dn,n Min Min Min Min Min NPV 1 NPV 2 NPV 3 NPV n 60

61 Reult 1.0 Upper and Lower Bound Rik Curve Mala-GTL Ship: 4 & 3 ENPV:4.540 Lower Envelope ENPV: Upper Envelope ENPV:4.921 Indo-GTL Ship: 6 & 3 ENPV:

62 UPPER AND LOWER BOUNDS Rik 1 D 0.8 C Lower Bound B 0.2 A Upper Bound Thi i very ueful becaue it allow nice decompoition, That i, there i no need to olve the full tochatic problem 62

63 Concluion A probabilitic definition of Financial Rik ha been introduced in the framework of two-tage tochatic programming. Theoretical propertie of related to thi definition were explored. New formulation capable of managing financial rik have been introduced. The multi-objective nature of the model allow the deciion maker to chooe olution according to hi rik policy. The cumulative rik curve i ued a a tool for thi purpoe. The model uing the rik definition explicitly require econd-tage tage binary variable. Thi i a major limitation from a computational tandpoint. To overcome the mentioned computational difficultie, the concept t of Downide Rik wa examined, finding that there i a cloe relationhip between thi meaure and the probabilitic definition of rik. Uing downide rik lead to a model that i decompoable in cenario and that allow the ue of efficient olution algorithm. For thi reaon, it i uggeted that thi model be ued to manage financial rik. An example illutrated the performance of the model, howing how w the rik curve can be changed in relation to the olution with maximum expected e profit. 63