Including Risk Part 1 Non Adaptive. Bruce A. McCarl

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1 Including Ris Part Non Adaptive Bruce A. McCarl Specialist in Applied Optimization Professor of Agricultural Economics, Texas A&M Principal, McCarl and Associates agecon.tamu.edu/faculty/mccarl B.A. McCarl, May 00 Including Ris Part Non Adaptive (firmris) page

2 Including Non Adaptive Ris Many face pervasive uncertainty. Prices, weather, labor, and other factors induce large changes in revenue, yields, woring rates and resources available. When incorporating ris into the program models there are three big issues. What is the nature of ris? a. What parameters of the model are uncertain? and b. How do we describe their distribution?. When during the model time horizon are ris outcomes revealed? Are there times when the model should reflect that the producer has received information about uncertain events and will mae adaptive decisions?. How do we model the decision maers behavioral reaction to ris? Is expected profit maximization not to the proper obective but rather some degree of aversion to the variation caused by ris? B.A. McCarl, May 00 Including Ris Part Non Adaptive (firmris) page

3 Including Non Adaptive Ris Our ris treatment will be limited and somewhat specialized because of time constraints (see newboo.pdf chapter and probab.pdf for more extensive treatment) The treatment will be limited in several principal ways. Will specialize in ris in obective function coefficients. Will not discuss how to form such probability distributions except through a few casual remars (see probab.pdf for more extensive treatment). We will only cover the expected value variance formulation of model obective function alterations to ris (this is the two main one used in the literature -- see newboo.pdf chapter for more extensive treatment). We will treat non adaptive behavior here. In a later section adaptive behavior using discrete stochastic programming or stochastic programming with recourse B.A. McCarl, May 00 Including Ris Part Non Adaptive (firmris) page

4 Including Non Adaptive Ris Why model ris Why not ust solve for all values of risy parameters Curses of dimensionality and certainty Dimensionality Number of possible plans ( possible values for 5 parameters 5 = ) Certainty Each plan would be certain of data so we would have different things we could do What would we do? General Ris Modeling Aim Generate a plan which is Robust in the face of the Uncertainty Not best performer necessarily in any setting, but a good performer across many or most of the uncertainty spectrum B.A. McCarl, May 00 Including Ris Part Non Adaptive (firmris) page

5 Including Non Adaptive Ris Ris entry into a programming model Maximize C Subect to A # b $ 0 Obective function returns - C Variability in prices Variability in production quantities Variability in costs Variability in maret sales Resource usages - A Variability in raw input quality Variability in woring conditions Variability in intermediate product yields Variability in product requirements Resource endowments - b Variability in demand firm faces Variability in resources available Variability in woring conditions B.A. McCarl, May 00 Including Ris Part Non Adaptive (firmris) page 5

6 Including Non Adaptive Ris Forms of assumed reaction to ris Non-Recourse or non adaptive decision maing Decisions made now consequences felt later No additional decisions made between now and when consequences felt Example Buy stoc now mae no decisions for one year Recourse or adaptive decision-maing Decisions made now consequences arise over time Later time during model additional decisions made. In this later decision period Decision maer nows what happened between first decision and now. Decision maers cannot revise prior actions but can adust current decisions ie current decisions can be employed to mae adustments in the face of realized events -- phenomena called irreversibility and recourse Example Buy stoc now, review decisions quarterly possibly selling and buying other stocs B.A. McCarl, May 00 Including Ris Part Non Adaptive (firmris) page 6

7 Including Non Adaptive Ris Decision Maer reaction to ris Expected Value Maximization max c s. t. A b o Conservative - Fat or thin coefficients where ~ c =c Ris Discount max s. t. ~ a = a Ris Discount ~ ~ b =b Ris Discount c ~ A ~ b ~ o E- V Maximize E(income) - RAP * Variance(income) Expected utility Maximize Sum(p,Probability(p)*U[Wealth(p)]) S.T. Wealth(p)=InitWealth Income(p) for all p Income(p)=C(p)* for all p Safety First based Maximize Sum(p,Probability(p)*Income(p)) S.T. Income(p)=C(p)* for all p Income(p)$safety for all p B.A. McCarl, May 00 Including Ris Part Non Adaptive (firmris) page 7

8 Including Non Adaptive Ris First Ris Model Marowitz mean-variance portfolio choice formulation Given Problem max sum(invest, moneyinvest(invest)*avgreturn(invest)) s.t sum(invest,moneyinvest(invest)*price(invest))# funds Marowitz observed not all money is invested in the highest valued stoc Inconsistent with LP formulation Why? Not a basic solution Marowitz posed the hypothesis that average returns and the variance of returns were important B.A. McCarl, May 00 Including Ris Part Non Adaptive (firmris) page 8

9 Including Non Adaptive Ris E-V Model Statistical Bacground Given a linear obective function Z = c c Where, are decision variables c, c are uncertain parameters Then Z is distributed with mean and variance Z = σ c c Z = s s s a Defining terms s ii is the variance of obective function coefficient of i, which is calculated using s i = (c i - c i ) /N where s i c i c i is the th observation on the obective value of i and N the number of observations. for i is the covariance of the obective function coefficients between i and, calculated by the formula s i = (c i -c i )(c -c )/N. Note s i = s i. is the mean value of the obective function coefficient associated with i, calculated by c i = c i /N. (Assuming an equally liely probability of occurrence.) B.A. McCarl, May 00 Including Ris Part Non Adaptive (firmris) page 9

10 Including Non Adaptive Ris E-V Model Statistical Bacground In matrix terms the mean and variance of Z are Z = c σ Z = where in the two by two case Z = E = [ c c ] s σ Z = s Marowitz Formulation ' S Z [ ] s s Min s. t. σ Z E = K Freund Formulation Min E φ σ Z or Min E - RAP * Variance Why Use Freund reuse of RAP and Transferability B.A. McCarl, May 00 Including Ris Part Non Adaptive (firmris) page 0

11 Including Non Adaptive Ris E-V Model Commonly Used Formulation Min E - RAP * Variance Max s. t. c φ s funds 0 for all Where E = expected value of risy c times choice of x Var = sum of var time x squared minus twice covariance times x s N = ris aversion parameter B.A. McCarl, May 00 Including Ris Part Non Adaptive (firmris) page

12 Including Non Adaptive Ris EV Example Data Data for E-V Example -- Returns by Stoc and Event ----Stoc Returns by Stoc and Event---- Stoc Stoc Stoc Stoc Event Event Event 8 6 Event Event Event6 0 5 Event7-6 Event Event9 7 5 Event Stoc Stoc Stoc Stoc Price Mean Returns and Variance Parameters for Stoc Example Stoc Stoc Stoc Stoc Mean Returns Variance-Covariance Matrix Stoc Stoc Stoc Stoc Stoc Stoc Stoc Stoc B.A. McCarl, May 00 Including Ris Part Non Adaptive (firmris) page

13 Including Non Adaptive Ris EV Model Example Max E - N F = E - RAP * Variance Max s. t. c φ s funds 0 for all or for the example Max s. t. [ ] φ [ ] or Max ϕ s. t B.A. McCarl, May 00 Including Ris Part Non Adaptive (firmris) page

14 Including Non Adaptive Ris GAMS Formulation (EVPORTFO.GMS) SETS STOCKS POTENTIAL INVESTMENTS / BUYSTOCK*BUYSTOCK / EVENTS EQUALLY LIKELY RETURN STATES OF NATURE /EVENT*EVENT0 / ; ALIAS (STOCKS,STOCK); PARAMETERS PRICES(STOCKS) PURCHASE PRICES OF THE STOCKS / BUYSTOCK BUYSTOCK 0 BUYSTOCK 8 BUYSTOCK 6 / ; SCALAR FUNDS TOTAL INVESTABLE FUNDS / 500 / ; TABLE RETURNS(EVENTS,STOCKS) RETURNS BY STATE OF NATURE EVENT BUYSTOCK BUYSTOCK BUYSTOCK BUYSTOCK EVENT EVENT EVENT 8 6 EVENT EVENT EVENT6 0 5 EVENT7-6 EVENT EVENT9 7 5 EVENT PARAMETERS MEAN (STOCKS) MEAN RETURNS TO (STOCKS) COVAR(STOCK,STOCKS) VARIANCE COVARIANCE MATRI; MEAN(STOCKS) = SUM(EVENTS, RETURNS(EVENTS,STOCKS) / CARD(EVENTS) ); COVAR(STOCK,STOCKS)=SUM(EVENTS,(RETURNS(EVENTS,STOCKS) - MEAN(STOCKS)) *(RETURNS(EVENTS,STOCK)- MEAN(STOCK)))/CARD(EVENTS); SCALAR RAP RISK AVERSION PARAMETER / 0.0 / ; POSITIVE VARIABLES INVEST(STOCKS) MONEY INVESTED IN EACH STOCK VARIABLE OBJ NUMBER TO BE MAIMIZED ; EQUATIONS OBJJ OBJECTIVE FUNCTION OBJJ.. OBJ =E= INVESTAV INVESTMENT FUNDS AVAILABLE ; SUM(STOCKS, MEAN(STOCKS) * INVEST(STOCKS)) - RAP*(SUM(STOCK, SUM(STOCKS, INVEST(STOCK)* COVAR(STOCK,STOCKS) * INVEST(STOCKS)))); INVESTAV.. SUM(STOCKS, PRICES(STOCKS) * INVEST(STOCKS)) =L= FUNDS ; MODEL EVPORTFOL /ALL/ ; SOLVE EVPORTFOL USING NLP MAIMIZING OBJ ; SCALAR VAR THE VARIANCE ; VAR = SUM(STOCK, SUM(STOCKS,INVEST.L(STOCK)*COVAR(STOCK,STOCKS)*INVEST.L(STOCKS))) SET RAPS RISK AVERSION PARAMETERS /R0*R5/ PARAMETER RISKAVER(RAPS) RISK AVERSION COEFICIENT BY RISK AVERSION PARAMETER/ R ,R ,R ,R ,R ,R ,R R ,R ,R ,R ,R 0.050,R ,R R ,R ,R ,R ,R ,R ,R R ,R 5.,R 0.,R 0., R5 80./ PARAMETER OUTPUT(*,RAPS) RESULTS FROM MODEL RUNS WITH VARYING RAP LOOP (RAPS,RAP=RISKAVER(RAPS); SOLVE EVPORTFOL USING NLP MAIMIZING OBJ ; VAR = SUM(STOCK, SUM(STOCKS, INVEST.L(STOCK)* COVAR(STOCK,STOCKS) * INVEST.L(STOCKS))) ; OUTPUT("RAP",RAPS)=RAP; OUTPUT(STOCKS,RAPS)=INVEST.L(STOCKS); OUTPUT("OBJ",RAPS)=OBJ.L; OUTPUT("MEAN",RAPS)=SUM(STOCKS, MEAN(STOCKS) * INVEST.L(STOCKS)); OUTPUT("VAR",RAPS) = VAR;OUTPUT("STD",RAPS)=SQRT(VAR); OUTPUT("SHADPRICE",RAPS)=INVESTAV.M; OUTPUT("IDLE",RAPS)=FUNDS-INVESTAV.L ); B.A. McCarl, May 00 Including Ris Part Non Adaptive (firmris) page

15 Including Non Adaptive Ris EV Example and Solution (EVPORTFO.GMS) Max [ ] φ [ ] s. t E-V Example Solutions for Alternative Ris Aversion Parameters RAP BUYSTOCK BUYSTOCK OBJ MEAN VAR STD SHADPRICE RAP BUYSTOCK BUYSTOCK BUYSTOCK BUYSTOCK.68 OBJ MEAN VAR STD SHADPRICE RAP BUYSTOCK BUYSTOCK BUYSTOCK BUYSTOCK OBJ MEAN VAR STD SHADPRICE IDLE FUNDS 68.0 B.A. McCarl, May 00 Including Ris Part Non Adaptive (firmris) page 5

16 Mean Income Including Non Adaptive Ris EV Example and Frontier (EVPORTFO.GMS) Frontier E-V Frontier Variance of Income RAP BUYSTOCK BUYSTOCK OBJ MEAN VAR STD SHADPRICE RAP BUYSTOCK BUYSTOCK BUYSTOCK BUYSTOCK.68 OBJ MEAN VAR STD SHADPRICE B.A. McCarl, May 00 Including Ris Part Non Adaptive (firmris) page 6

17 Max s. t. r c Including Non Adaptive Ris Alternative E-V Model S p S S E E φ( p ( S E) ) funds = 0 for all = 0 0 for all E unrestricted for all α where identifies the stoc possibilities; identifies the states of nature; is amount of stoc bought; r is the cost of buying ; c is the uncertain yield of stoc realized under state of nature when the buying ; S is the income from stocs under state of nature ; p is the probability of state of nature ; E is average income; M is a ris aversion parameter " is for EV models or ½ for E STD models B.A. McCarl, May 00 Including Ris Part Non Adaptive (firmris) page 7

18 Max s. t. r c Including Non Adaptive Ris Alternative E-V Model S p S S E E E φ( p ( S E) ) α funds = 0 for all = 0 0 for all unrestricted for all Ob Depicts maximization of expected income minus the ris aversion parameter times the variance of income First constraint limits funds available Second equation sums income by state of nature Third weights income by state of nature by probability forming expected income Last two requires investment variables to be nonnegative but allows income to be positive or negative. B.A. McCarl, May 00 Including Ris Part Non Adaptive (firmris) page 8

19 Including Non Adaptive Ris Alternative E-V Model (EVPORTF.GMS) 0 / ) /0*( = = = = = = = = = = = E I I I I I I I I I I I s t E I E Max R H S P N O E R I I I I a C S G O N N N N v O I A A A W V V V V r e v - r e v - r e g E T I T I C E E E E O r F I J J N S S S S B e F V, V, T T T T T J v S E S E S S OBJJ C = 0 INVESTAV E E E E < F 0 r D D D D = 0 5 e D D E D = 0 5 v D D E D = 0 5 e D D D = 0 5 n 5 D D E D = 0 5 u 6 D E E D = 0 5 e 7 D E D = D D E D = 0 5 r 9 D D E D = 0 5 e 0 D D D = 0 5 avrevenue A A A A A A A A A A = 0 0 B.A. McCarl, May 00 Including Ris Part Non Adaptive (firmris) page 9

20 Including Non Adaptive Ris Alternative E-V Model (EVPORTF.GMS) SETS STOCKS POTENTIAL INVESTMENTS / BUYSTOCK*BUYSTOCK / EVENTS EQUALLY LIKELY STATES OF NATURE /EVENT*EVENT0 / ;... SCALAR RAP RISK AVERSION PARAMETER / 0.0 / alpha term if is var if 0.5 is std err // n sample size; n=card(events); POSITIVE VARIABLES INVEST(STOCKS) MONEY INVESTED IN EACH STOCK VARIABLE OBJ NUMBER TO BE MAIMIZED rev(events) revenue by event avgrev average revenue; EQUATIONS OBJJ OBJECTIVE FUNCTION INVESTAV INVESTMENT FUNDS AVAILABLE revenue(events) accounts revenue by event avrevenue accounts average revenue ; OBJJ.. OBJ =E= avgrev - RAP* (SUM(events,/n*(rev(events)-avgrev)*(rev(events)-avgrev)))**alpha; INVESTAV.. SUM(STOCKS, PRICES(STOCKS) * INVEST(STOCKS)) =L= FUNDS ; revenue(events).. sum(stocs,returns(events,stocks)*invest(stocks))=e=rev(events); avrevenue.. sum(events,/n*rev(events))=e=avgrev; MODEL EVPORTFOL /ALL/ ; SOLVE EVPORTFOL USING NLP MAIMIZING OBJ ; SCALAR VAR THE VARIANCE ; VAR = (SUM(events, /n*(rev.l(events)-avgrev.l)*(rev.l(events)-avgrev.l))) ; SET RAPS RISK AVERSION PARAMETERS /R0*R5/ PARAMETER RISKAVER(RAPS) RISK AVERSION COEFICIENT BY RISK AVERSION / R ,R ,R ,R ,R ,R ,R R ,R ,R ,R ,R 0.050,R ,R R ,R ,R ,R ,R ,R ,R R ,R 5.,R 0.,R 0., R5 80./ PARAMETER OUTPUT(*,RAPS) RESULTS FROM MODEL RUNS WITH VARYING RAP LOOP (RAPS,RAP=RISKAVER(RAPS); SOLVE EVPORTFOL USING NLP MAIMIZING OBJ ; var=(sum(events,/n*(rev.l(events)-avgrev.l)*(rev.l(events)-avgrev.l))) ; OUTPUT("RAP",RAPS)=RAP; OUTPUT(STOCKS,RAPS)=INVEST.L(STOCKS); OUTPUT("OBJ",RAPS)=OBJ.L; OUTPUT("MEAN",RAPS)=avgrev.l; OUTPUT("VAR",RAPS) = VAR; OUTPUT("STD",RAPS)=SQRT(VAR); OUTPUT("SHADPRICE",RAPS)=INVESTAV.M; OUTPUT("IDLE",RAPS)=FUNDS-INVESTAV.L ); parameter graphit (*,raps,*); graphit("frontier",raps,"mean")=output("mean",raps); graphit("frontier",raps,"var")=output("std",raps)**; alpha=0.5; LOOP (RAPS,RAP=RISKAVER(RAPS); SOLVE EVPORTFOL USING NLP MAIMIZING OBJ ; var=(sum(events,/n*(rev.l(events)-avgrev.l)*(rev.l(events)-avgrev.l))) ; OUTPUT("RAP",RAPS)=RAP; OUTPUT(STOCKS,RAPS)=INVEST.L(STOCKS); OUTPUT("OBJ",RAPS)=OBJ.L; OUTPUT("MEAN",RAPS)=avgrev.l; OUTPUT("VAR",RAPS) = VAR; OUTPUT("STD",RAPS)=SQRT(VAR); OUTPUT("SHADPRICE",RAPS)=INVESTAV.M; OUTPUT("IDLE",RAPS)=FUNDS-INVESTAV.L ); DISPLAY OUTPUT; B.A. McCarl, May 00 Including Ris Part Non Adaptive (firmris) page 0

21 Including Non Adaptive Ris Alternative E-V Model (EVPORTF.GMS) Mean Income Frontier Std Err E-V Frontier Variance of Income EV solutions are identical with earlier, E Standard Error is close R0 R R R R R5 BUYSTOCK BUYSTOCK RAP E E E OBJ MEAN VAR STD SHADPRICE R6 R7 R8 R9 R0 R BUYSTOCK BUYSTOCK RAP OBJ MEAN VAR STD SHADPRICE B.A. McCarl, May 00 Including Ris Part Non Adaptive (firmris) page

22 Including Non Adaptive Ris Dissecting the GAMS formulation Graphing parameter graphit (*,raps,*); graphit("frontier",raps,"mean")=output("mean",raps); graphit("frontier",raps,"var")=output("std",raps)**; *$include gnu_opt.gms * titles... more solves... graphit("std Err",raps,"Mean")=OUTPUT("MEAN",RAPS); graphit("std Err",raps,"Var")=OUTPUT("std",RAPS)**; $setglobal gp_title "E-V Frontier " $setglobal gp_xlabel "Variance of Income" $setglobal gp_ylabel "Mean Income" $batinclude gnupltxy graphit mean var This is done using a GNUPLOT interface originally developed by Rutherford but modified to gnupltxy as documented on the Web page agecon.tamu.edu/faculty/mccarl B.A. McCarl, May 00 Including Ris Part Non Adaptive (firmris) page

23 Including Non Adaptive Ris Modeling Support from GAMSCHK Nonlinear Models ## rev('event') SOLUTION VALUE.697 EQN Ai Ui Ai*Ui OBJJ ***-0.706E E-0 revenue('event') avrevenue TRUE REDUCED COST E ## EQU OBJJ ## OBJJ VAR Ai Ai* OBJ rev('event') ***-0.706E rev('event') *** E rev('event') *** rev('event') *** E E-0 rev('event5') *** rev('event6') *** rev('event7') *** E rev('event8') *** rev('event9') ***-0.56E rev('event0') *** avgrev *** =E= =E= RHS COEFF E00 *** mars nonl;inear terms Starting point and accuracy is an issue B.A. McCarl, May 00 Including Ris Part Non Adaptive (firmris) page

24 Including Firm Level Ris Finding a Ris Aversion Parameter E-MF is one tailed confidence interval when M= under normality confidence interval is 8% M=.96 interval is 97.5% Also E-V relation Max s. t. c ψσ A ( ) b 0 versus Max s. t. c θσ ( ) A b 0 σ ( ) σ ( ) c ψσ ( ) λa= 0 c θ λa = 0 which equates when θ ψ = σ ( ) Given we generally find θ in a range between 0 and 5 this implies 5 0 ψ σ ( ) B.A. McCarl, May 00 Including Ris Part Non Adaptive (firmris) page

25 Including Firm Level Ris Forming Probability Distributions Probability distributions state the relative frequency of occurrence of a set of mutually exclusive events. Finding Probability Distributions Based on Obective Data Desirable Characteristics ) each of the states of nature must be mutually exclusive; ) probability of occurrence of each of the states of nature must be an unbiased measure of the current probability of that state of nature occurring; ) the sum of the probabilities across the states of nature must equal one Second property is the most troubling in when using obective, historical data. trends, events B.A. McCarl, May 00 Including Ris Part Non Adaptive (firmris) page 5

26 Including Firm Level Ris B.A. McCarl, May 00 Including Ris Part Non Adaptive (firmris) page 6

27 Including Firm Level Ris 0.5 Figure 5. Probability Distribution 0. Probability Stationary Price Historic Price Real Price Price B.A. McCarl, May 00 Including Ris Part Non Adaptive (firmris) page 7

28 Including Firm Level Ris General Lessons Learned on Obective Probabilities Use obective data trends and other systematic effects can bias Use a procedure lie regression to develop values expected One may find residual terms are hetereosedastic B.A. McCarl, May 00 Including Ris Part Non Adaptive (firmris) page 8

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