RevenueRisk,CropInsuranceandForwardContracting

CoryG.Walters

DepartmentofAgriculturalEconomics,UniversityofKentucky

cgwalters@uky.edu

RichardPreston

PrestonFarms,HardinCountyKentucky

rkpreston@windstream.net

SelectedPaperpreparedforpresentationattheAgricultural&AppliedEconomicsAssociations2013CropInsuranceandtheFarmBillSymposium,Louisville,KY,October89,2013

Copyright2013byWaltersandPreston.Allrightsreserved.Readersmaymakeverbatimcopiesofthisdocumentfornoncommercialpurposesbyanymeans,providedthatthiscopyrightnoticeappearsonallsuchcopies.

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RevenueRisk,CropInsuranceandForwardContracting

Introduction

Overthepasttwentyyears,theUSgovernmenthasdramaticallyenhancedtheFederalCropInsurance

program.Subsidiesandthenumberinsurancecontractswereexpandedthroughboththe1996

AgriculturalReformActandthe2000AgriculturalRiskProtectionAct.Asaresult,insuredacres

increasedfrom100millionto279millionbetween1994and2012(USDARMANational,2012).The

RiskManagementAgency(RMA),whooverseestheFederalCropInsuranceprogram,providesenrolled

producerswithriskmanagementbenefitsthroughamyriadofinsurancecontracts.Withsome

insurancecontractsprovidingmoreriskmanagementbenefitsthanothers.Whatisunclearishow

muchdifferentinsurancecontractsreduceriskandmoreimportantly,riskreductionwhenprivaterisk

managementtools,suchasfuturesmarkets,areusedinconjunctionwithcropinsurance.Theresult

couldpossiblybeincreasedrisktotheproduceriftheydonotunderstandtheimpactofprivatehedging

tools(i.e.,futuresmarkets)onrevenuewhenapublictool(i.e.,cropinsurance)ispresent.

Ofallavailableinsurancecontractchoices,theintroductionofrevenueinsurance,whichadds

priceprotection,createsadirectlinkagewithfuturesmarkets.Thiscontract,calledrevenueprotection,

isthemostpopularrepresenting175millionofthe279millioninsuredacresin2012(USDARMA

National,2012).Enrolledacreageevidencesuggestsproducerprefersincludingfuturespricesinto

theircropinsurancerevenueguarantee.1Producerswitharevenueprotectionmayrespondby

reducing(orevenquitting)participationincommodityfuturesmarketseitherdirectlyorindirectly

throughcashforwardcontractsbecauseinsurancehascoveredthisrisksourceforthem.Thisstudyis

intendedtohelpfarmersunderstandhowrevenueinsuranceandfuturestradingcomplementeach

othertoimprovetheirriskmanagementplan.Weuseaportfolioapproachinmakingriskmanagement1Additionally,participatinginfuturesmarketsviaasubsidizedinsuranceproductmakespriceprotectioncostcheaperthandirectfuturesmarketparticipation.

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decisionsdistributingincomesourcesthroughavarietyoftoolstounderstandtheriskmanagement

linkagesbetweenthesepublicandprivatetools.

Inthispaperweexamineriskfromtheproducerspointofview.Atthebeginningoftheyear

producersarefacedwithuncertainpricesandyields.Producershavetoolsavailabletohelptamethis

uncertainty.Differentfarmershavedifferentobjectivesanddifferentresourcesandarefacedwitha

widerangeofinsurancechoicesandforwardpricingopportunities.Consequencesofdifferentdecisions

areexaminedtohelpindividualproducerscraftriskmanagementplansthatbestsupporttheirindividual

resources,values,andobjectives.Wefocusonthelongtermobjectiveoffarmsurvivalandtheshort

termobjectiveofproducingapositivenetincomeeachyear.Thesearegoalscommontomanyfamily

farmersacrosstheUS.

Weinvestigatetheriskofnotmeetingtheseobjectivesbymodelingpriceexpectationthrough

theuseofcommodityfuturespricesestimatedbythemarketsviewofriskdemonstratedbyusing

pricesofactualoptionstradedviewedwithintheframeworkoftheBlackScolesMertenmodelof

futuresmarketbehaviorandmodelingyieldexpectationbyanalyzinghistoricalfarmyieldsaidedand

constrainedbyexperienceandcommonsense.Futuresmarketsexisttoprovidethebestinformation

onwhatfallpriceswillbeinthespring.Producersareattractedtofuturesmarketsbecausethey

provideaguaranteedpriceinthespringforfalldelivery.Producersareattractedtocropinsurancefor

riskmanagementbenefitsfromyieldandpriceprotection(underarevenueprotectionpolicy)andthe

presenceofthesubsidy.Weuseproducerlevelhistoricalyieldriskadjustedforconsensuslongerterm

risktocapturefarmspecificrisks,risksthatcoulddeviatesubstantiallyfromaggregationtothosebased

oncounty,state,ornationallevelaggregatedata(MillerandKahl,1987).

Futurespricehedgingintroducesanotherrisk.Soldcropmustbeproduced;therefore,crop

yieldriskincreases.Sellingmorecropthanproducedresultsinbuyingbackpresoldfuturespositions.

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Buyingbackpresoldpositionscanbefinanciallyandemotionallypainful.Havingtherightcrop

insurancepolicyinplacecanreducethispainsubstantiallybecauseitreducesyieldrisk.Consequently,

forwardcontractingandcropinsurancecomplementeachother.Butfromtheproducerspointofview

thequestionishowmuchdocropinsuranceandcomplementeachother?Cropinsurancewithharvest

priceprotectionprovidesapartialhedgeforshortfuturespositions.However,theimpactonthe

amountofavailableworkingcapitalwhenanadverseoutcomearisesisnotfullyunderstood.We

developaportfolioapproachthatcandirectlyestimate(withinthelimitsofthemodelanditsdata

input)thelevelofworkingcapitalonhandforfarmsurvival.

Investigatingtherelationshipbetweencropinsuranceandfuturesmarketsonfarmincomerisk

provideschallenges.Areasonableyielddistribution,anapproximatepricedistribution,andthe

dependencebetweenyieldandpricemustbedetermined.Theprimaryissuefacingthemodelerishow

torepresenttheproducersunderlyingjointdistributionbetweenyieldsandpriceswhich

comprehensivelycombinesallthisinformation.Often,thisnegativerelationshipbetweenyieldsand

pricesisreferredtoasthenaturalhedgeandisrelevantindecisionmakingunderuncertainty(Finger,

2012).Farmlocationinfluencesthesizeoftheprice/yieldjointdistribution,makingitdifficulttoidentify

theoptimalcropinsurancecontractandhedgingamount.Additionally,widerangingstatementsabout

whatproducersshoulddobecomesirrelevantiftheprice/yieldjointdistributionisnttakenintoaccount

ormisunderstood.Wedescribetheprice/yielddependencestructurethroughacopula.Wethen

incorporatethecopulaintoaMonteCarlosimulationmodeltoanalyzerevenueriskfacedbyaKentucky

farm.

Anumberofvaluableresultscomeoutofthisstudy.Alloftheresultshelpshedlightonthe

producerscomplexdecisionenvironment.First,weanalyzetailrisk(i.e.,adverseunexpected

outcomes)inprobabilityspace.Second,weincorporateacopulaintothemodeltoaccountforthe

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inversedependencebetweenpriceandyields.Thisisanimportantfeaturebecausethestrengthofthe

dependenceisthefoundationformakingriskmanagementdecisions.Third,thefinancialbenefitsof

combiningcropinsuranceandforwardcontractingcanbeeasilycalculated.Fourth,amountofworking

capitalneededtoensurefarmsurvivalcanbemeasured,in$,throughdifferentcombinationsofcrop

insuranceandforwardcontracts.Thisallowsustocalculatethebenefitsofcropinsuranceduring

adverseoutcomesaloneaswellaswhenforwardcontractsareused.

PriorLiterature

Analyzingpriceriskimpactsproducerrevenueisnotanewendeavor(e.g.HansonandLadd,1991and

Coyle1992).McKinnon,1967foundthathedgingdependsupontheproducersprice/yieldcorrelation.

Whatisrelativelynewisanalyzingimpactsofcropinsuranceandhedgingonfarmrevenue.Dhuyvetter

andKastens(1997)foundthataverageincomewassimilaracrossinsurancetypesbutrevenueinsurance

resultedintheleastincomevariability.Cobleet.al.,(2000)demonstratedthatrevenueinsurancetends

toresultinslightlylowerhedgingdemandwhencomparedtoyieldinsuranceusingthesamecoverage

level.AfollowupstudybyCobleet.al.,(2004),thatintroducedLoanDeficiencyPaymentsintothe

model,foundthatgovernmentprogrammelevelsprofoundlyalterproducersoptionalhedgingstrategy.

Mahul(2003)foundthatrevenueinsurancetendstoresultinlowerdemandforfuturescontracts.

Wanget.al.,(2004)foundthatfutureshedgingisnegativelyaffectedbyrevenueprotection.

Theimportanceofaccountingforyieldandpricedependenceinmeasuringriskisimportant.

Forexample,TejadaandGoodwin,(2008)foundthatforratingrevenueinsurancebyincorporatingthe

price/yieldjointdistribution(viaacopula)resultedinasmallerprobabilityofpayoutthanpresent

methodsbeingused.

Ourextensionofpreviousworkliesinthewayweincorporateproduceravailableinformation

onyieldsandpricesinthefaceofuncertainty.Producersusehistorical,trendadjusted,yieldsasthe

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expectationfornextyearsyield.Theyunderstandyieldvariabilityexistsfromtheironfarmyielddata

andfromresultsoftheirneighborsyields.Manyproducersalsorememberthewidespreadcropfailures

intheearly1980s,andtheirparentsrecollectionofthe50s.Someevenrememberthestoriesofthe

30shandeddownbytheirgrandparents.Thisaccumulatedlocalknowledgehaspushedfarmersto

embraceriskmanagementtoolssuchascropinsuranceandlimitedforwardcontractingtoprotect

revenueagainstadverseimpacts.Forprices,producerslookforwardintime,viafuturesmarkets,

specificallynewcropcontractsforthemostrelevantpriceinformation.Consequentlyweincorporate

historicalyieldrealizationsandforwardlookingnewcropfuturesmarketpricesinassessingrevenue

risk.

ModelingProducerRisk

Traditionallyriskevaluationhasbeenthroughtheexpectedutilityframeworkwhereproducersare

presumedtoberiskaverse,expectedutilitymaximizers(Chavas,2004).Wewritetheexpectedutility

functionas

(1) U E f V ,

whereUisutility,E ,andV isthevarianceofprofit.Thevariancetermrepresentsuncertaintystemmingfromunknownyieldsandunknownprices.Usingthisprofitstructuretheinsuranceselection

andhedgingselectionaffectboth E ,andV .Futureshedgingreducespriceriskbutincreasesyieldrisk.Yieldprotection,aspecificcropinsurancepolicytype,impactsonlyyieldrisk.Revenueprotection,

anotherspecificcropinsurancepolicytype,impactsbothpriceriskandyieldrisk,butnotinasimple

linearadditivefashion.Riskmaybereducedwhenincorporatingbothcropinsuranceandhedging.The

amountofriskprotectiondependsupontherelationshipbetweenactualyieldandprice.

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Insteadofdefiningriskinageneralframework,werankdecisionsbyhowwellandhowlikely

theyhelptheproducermeethis/herspecificobjectives(Clemen,1996).Wedefinespecificrisksasthe

riskofnotmeetingspecificobjectives.Forexample,iftheobjectiveistomaximizeincomethen

decisionsaremeasuredbysimplycalculatingtheexpectedvalueofincome.Iftheproducersobjective

istomaximizeincomewhileminimizinguncertaintyboththemeanandthevarianceofthedistributions

couldbeexaminedtofindthedecisionthatbestsupportsthatobjectiveusingthemaximumexpected

utilityapproachgiveninequation(1).Hereweconcentrateonthespecificriskoffarmsurvivalby

examiningtheshapeoftheincomeprobabilitydistributions.Thisissimilartodevelopingasafetyfirst

setofpreferenceswheredecisionmakerstrytominimizethechanceoffallingbelowagiven

subsistencelevel(Chavas,2004).Farmsurvivalisdownsiderisk.Downsideriskliesinthetailofthe

incomedistribution.Producersdecisionswhichsupporttheobjectiveoffarmsurvivalmakedecisions

tominimizetheirexposuretodownsideriskwhilemaintainingapositiveexpectedincome.

Wemodelvarianceasfarmsurvivalorhavinga1%chanceofanincomeshortfall.Witha1%

chanceoffailure,theproducerthensetsasideXdollarsofworkingcapitaltocoverthe1%incomeat

risklevel.Bysettingbacktheminimumtocoverthe1%probabilityshortfallthereexistsa99%chance

ofsucceedingeachyear.2Wechose1%becauseitisaextremelyrare(1in100year)eventandbecause

workingcapitalneededtocovera1%chancewillcoverotherlowprobabilityevents,suchasthose

comingfrom5%and10%incomeshortfalls.

ConstructingFarmIncomeDistributions

Conceptually,wemodelfarmincomeriskbylookingatthefarmyieldandpricecombinationintheeyes

ofuncertainty.WeconstructaMonteCarlosimulationmodeltoanalyzerisksfacedbyaKYgrainfarm

bycalculatingadistributionofincomeoutcomesgivenreasonableinputdistributionsforpricesand

2Assumingindependentweatherthechanceofsuccesstosurviveeachyearbecomes.99^n.Forn=20yearstheproducerwhoworriesaboutprotecting1%incomeshortfallhasan82%chanceofsuccess.

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yieldscombinedwithanestimateoftheprice/yieldcoupling.Riskcanbemeasuredbystudyingthe

shapeoftheresultingincomedistributions.

Peracreprofit,withoutinsuranceorhedging,isgivenby

(2) p y c z y

Whereprepresentsoutputprice,yisactualyield,cisthefixedcostofproduction,andthefunctionz y representstheperbushelharvestcost.PurchasinginsuranceaddsthetermI r, r , z u z toequation(2),whereI . representstheinsuranceindemnityorpaymentfromtheinsurancecompanytotheproducer,rrepresentsactualrevenue(y p , wherep representstheinsuranceharvestprice;r isguaranteedrevenue,calculatedas y p CL ,wherey representsthe

yieldguarantee,p representstheinsurancebasepriceandCLrepresentsthepercentinsured.3Hedging,throughtheuseoffuturesmarkets,addsthetermp y ,wherep representsthehedgedproductionpriceandy representsthenumberofbushelshedged.Includinghedgingreducesthenumberofbushelstobesoldatharvest,therefore,wereplaceyieldinequation(2)withy whichistheremainderofy y .Withunknownyields,itispossiblefory tobegreaterthany.Whengiventheoptionofpurchasinginsuranceandhedgingtheproducersprofitperacreis

(3) p y y c z y I r, r , z u z p y

Afterthefinalyieldandpricearerealized,itwouldbeeasytoplugthevaluesintoequation(3)

anddeterminewhichdecisionchoicemaximizesincome.Ofcourse,insurancecontractselectionand

forwardcontractingdecisionsmustbemadebeforep and yareknown.Inthespring,bothp and y areuncertain.Itisusefultothinkofp and yasrandomvariableswithonlytheirjointprobability,,P( , ,knowninMarch.Thejointprobability, P( , ,measuresthechancethataparticular(pi,yi)3Underarevenueprotectionpolicy intherevenueguaranteecalculationcouldbereplacedwith if

.Forayieldprotectionpolicy isreplacedwith intheactualrevenuecalculation.

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combinationwilloccurinDecember.Associatedwitheach(pi,yi)pairandjthdecisionisanincomei,jgeneratedbyequation(2).

WecanuseaMonteCarloapproachtoobtainanestimateoftheincomeprobability

distribution.MonteCarlomethodsarerootedinthesimilaritybetweenprobabilityandvolume

(Glasserman,2004).4OnceweknowthejointprobabilityweconductaMonteCarlosimulationby

randomlychoosingpandymanytimesinafashionthatreproducestheshapeofP( , byinsuringeach Psegmenthasanequalchanceofbeingvisited(Vose,2008).Equation(3)mapseachpair

ofp, yintoanincome.Thedistribution(volume)ofincomesproducedreflectstheprobabilityofthatincomeoccurring.Risktodifferentobjectivescanbestudiedbyexaminingtheshapeoftheoutcome

volume.Eachchoiceisgoingtoproduceadifferentshapefortheincomedistribution.Theincome

resultsarecollectedinavectorofoutcomeswitheachdecisionlabelingadifferentcomponent.

ModelingtheDistributions

Thefirststepistoconstructthejointprobabilitydistribution,P( , ).Wedividetheproblemintobuildingamarginaldistributionforyield,y,constructinganestimateforthemarginaldistributionfor

price,p,anddevelopinganalgorithmtomodelthedependencebetweenyandp.

YieldDistributions

4Thinkofasimpleonevariablehistogram.Theprobabilityofaparticulareventoccurringisrelatedtothearea,whichrepresentsvolumein2D,occupiedbythecorrespondinghistogramdividedbythetotalareaofallpossibleevents.Thiscanbeextendedtohigherdimensions.AthreedimensionalhistogramrepresentationofP( , could be constructed by tossing all events that fall between pi and pi+1 and yi and yi+1 into a bin labeled P( , IwiththeheightofP( , IgivenbythenumberofeventswhicharecollectedinP( , I.NormalizingthevolumeofP( , Ibydividingthenumberofeventslandingineachbinbythetotalnumberofeventsturnsthevolumeintoajointprobability, P( , .

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Itisdifficulttogetalongenoughtimeseriesoffarmlevelyielddatathatadequatelydescribesfarm

yieldrisk.Asaresultresearchersoftenusecountyleveldatatodescribefarmyieldvariation.

Aggregationtothecountylevelcouldaverageoutpoororexcellentoutcomes.Inthisstudy,todescribe

farmlevelyieldvariabilityweuseyearlyfarmaverageyieldforasingleproducer.Thisproducerhas

beeninbusinesssince1979.

Sincethefarmstartedproducingin1979through2011yieldsincreasedsteadily.Todetermine

theamountofyieldriskwefirstremovetheyieldtrend.Failuretoremovethetrendwillresultinan

overestimateoffarmyieldrisk.Tocalculatethetrend,weregresscurrentyieldonthepreviousyield

usingLinearLeastSquaresusinghistoricalfarmdatafrom1979to2011.Theresidualratiodistribution,

Rf,whichweuseasthesamplingdistribution,iscalculatedas

(4) Rf

whereY representsyieldattimet,YT representstheestimatedyieldtrendattimet,andRfrepresentstheresidualratio.Wecanexpresstheunknown,nextyearsyieldas

(5) Rf

WeassumeRfcanbeusedasastationarysamplingdistribution,i.e.,thatthedistributionofratiosof

differencefromthetrenddividedbythetrenddonotchangeovertime.

The33yearsofyielddataareasampledrawnfromapopulationdistributionwhichisassumed

toexist.Figure(1)showstheRcdistributions,forHardincountyKY,Rfshowsthehistogramsforthe

farmunderstudy,RsandRnshowthehistogramsforthestateandnationallevels,respectively.The

normaldistributionhasshowndoesapoorjobofrepresentingyielddataashasbeenreportedby

severalauthors(Hennessy,2009;Sherricketal.,2004;Atwood,Shaik,andWatts,2002).Boththefarm

andcountydistributionsareskewedtotheleftsuggestingthatanormaldistributionwouldunder

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estimatetheprobabilityoflowyieldevents.Effortstofitthefarmdatatoseveralcommonfunctional

formsproposedintheliteratureincludingtheBetafunctionandWeibullfunctionfailedtoadequately

representthelowyieldtail(Sherricketal.,2004).WewantourRfdistributiontoadequatelydescribe

thechanceofanymajoryieldimpactingevent;therefore,wedevelopanempiricaldistributionby

plottingthecumulativedistributionanddrawingasmoothlinethroughthepointswiththeadditional

constraintthatthelinemustpassthroughapointwith1/30probabilitythattheyieldwillbeatleastas

badasitwasin1983.Thisconstraintwasaddedtomodeltheextremelowyieldeventswhichhave

occurredinthe30s,50s,and80swhichisaboutonceevery30years.Thisconstraintleadstoafatlow

yieldtail.ThisfattailbehaviorwaspredictedbeforethemajorKentuckydroughteventof2012.The

majoryieldshortfallwhichoccurredin2012iseasilyexplainedwithinthemodelforRfproposedforthis

farm.

PriceDistributions

WeareinterestedinthechangeinDecemberfuturespricefromMarch1toDecember1.Webuildan

approximatelystationarysamplingdistributionbytakingtheLnoftheDecemberfuturespriceon1

dividedbytheDecemberfuturespriceonMarch1andwedesignatethisdistributionasLnPh.

AssumingthedataresultedfromsamplingofaLogNormalpricedistribution,wecanusethisdatasetto

estimatetheparametersofLnPhwhichleadstoanestimateofaLogNormalhistoricaldistributionof

prices.Inthenextsectionweinvestigatethecouplingbetweenourpricesamplingdistribution,LnPh ,

andouryieldsamplingdistribution,Rf.

Insteadofusing LnPh directlyasasamplingdistribution,wechosetousethenewcropfutures

contractmarketinformationonMarch1toproduceaforwardlookingdistributiontoaccountforthe

possibleincreaseordecreaseinpricevolatility.Optionmarketpriceswereusedtoestimatetheprice

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probabilitydistribution.5Ourgoalistofocusonpricechange,butweconstructLnPbbyusingaBlack

ScholesMertonstochasticmodeltorelateastartingpointonMarch1toadistributiononDecember1.

Inordertodeterminethepriceprobabilitydistributionwefirstcalculatetheimpliedvolatility.

Weusedtheaverageofpremiumspaidfortheatthemoneycallandtheatthemoneyputfornew

cropfuturescontractoptionpricesandtheoneyearLIBORratetoestimateimpliedvolatilityusing

DerivaGemsoftware(Hull,2012).ThisimpliedvolatilitywasusedinBlacksmodelforvaluingoption

contractsatexpiration(Hull,2012).BlacksmodelassumesthatEuropeanoptionscanbeusedtomodel

Americanstyleoptions.6Blacksmodelprovidesalognormalpriceprobabilitydistribution.We

estimatethepriceprobabilitydistributioninthespringonMarch1st,justafterthecropinsurancebase

wasset.7Decembercornoptionpricesexpireapproximatelyoneweekbeforefuturescontractsenter

thedeliveryperiod.

JointdependenceandCopulas

Webaseouranalysisoffarmincomeriskonthejointdistributionbetweenyieldsandprices.Yieldsand

pricesarerelatedthroughsupply,loweryieldsreducesupplywhichholdingeverythingelseconstant

wouldresultinhigherprices.Thestrengthoftheinversepriceyieldrelationshipdependsupon

producerlocationrelativetotheprimarycropgrowingarea.8Forexample,becauseIowaistheprimary

corngrowingstate,averylargedeclineforanIowaproducersfarmyieldwouldlikelybeassociatedwith

lesscornmarketsupply,which,holdingeverythingelseconstant,wouldpushcornpriceshigher.As

producerlocationincreasesawayfromtheprimarycorngrowingareatheinversepriceyield

5Foradiscussionofprobabilitypricedistributionsusingthisapproachseehttps://www.msu.edu/~hilker/crnfut.htm6Europeanstyleoptions,oncepurchased,mustbehelduntilexpirationvs.Americanstyleoptionsthatcanbecontinuouslytradedanytimebeforeexpiration.7ForcorninKentuckythespringbasepriceistheaverageofDecembercornfuturesmarketduringthemonthofFebruary.8Irrigationwouldalsoplayarolebutisnotincludedinthisanalysis.

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relationshipwouldweakenorpossiblydisappear.CornyieldsinWashingtonStatewouldhavelittleto

noimpactoncornfuturespricessinceWashingtonStateissofarremovedfromtheprimarycorn

growingregion.9Duetothepresenceofaninversepriceyieldrelationshipitisimportanttomodelthis

asajointdistributionthroughtheuseofcopulas.Copulasallowstwofunctions,whowethinkare

related,tobetiedtogether.Copulasarearelativelynewmethodinansweringriskmanagement

questions(Embrechs(2002),Embrechs(2003),Kole(2007),Goodwin(2012)).Animportantfactorwhen

usingcopulasisthecopulaselection.Koleetal.(2007)foundthatriskswereoverorunderstated

dependinguponthetypeofcopulaused.

CopulasarethecombinationofmultiplejointdensitiesintoonejointdistributionthroughSklars

theorem(Sklar,1973).Thecopuladescribesthedependencebetweentherandomvariablesthrough

mappingfromtheindividualdistributionfunctionstothejointdistributionfunction

(6) F x , , x C F x , . , F x

wherex , , x representsanhcontinuousrandomvariableswithacumulativedistributionfunctionF x , , x ,marginaldistributionfunctionsofF x , . , F x andcopulafunctionC.10

Forourmodelweareconcernedidentifyingacopuladisplayingtaildependence.TheClayton

copula,fromtheArchimedeancopulafamily,displaystaildependence.TheClaytoncopulatakesthe

form

(7) C x , x , x x 1 /

where,thedependenceparameter,isrestrictedtobegreaterthanzero.Archimedeancopulashavethespecialpropertythatallowsthemtobeconstructedfromagenerator,,(Nelson)

9Itwouldlikelyimpactregionalcornbasis.10ForathoroughbackgroundoncopulamodelingseeTrivediandZimmer(2005).

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(8) , [1](( 2))

withtheClaytongenerator(Nelson,2006)givenby

(9)(t)=(1/)(t1)

Asapproacheszerothemarginaldistributionsbecomeindependent.Asincreasesinsizethemarginaldistributionsbecomemoredependent.TheClaytoncopulawith>0accountsforpositivedependence,notnegativedependenceandourhistoricaldatasuggestsverystrongnegative

dependence;lowyieldsimplyamuchstrongchanceforhigherprices.FortheClaytoncopulatowork

withnegativedependencewetransformthedataintopositivedependencebychangingthesampling

distributionforpricesfromLnPbpriceratioto1 LnPbpriceratio.Thehistoricaldataforthisfarmdoes

notsuggestthathighyieldsimplyastrongchanceoflowerprices.WefoundthattheClaytoncopula

exhibitsthistypeofrelationship;strongtaildependenceononesideofthedistributionandweak

dependenceontheotherside.

AnArchimedeancopulacanberelatedtoKendalltau,,whichisaglobalmeasureof

dependencebetweentworandomvariables(MariandKotz,2001)by

(10)=14 01(/')

UsingthegeneratorfortheClaytoncopula(Nelson,2006andVose,2008)relatestheparameter,,to

asshowninequation(11)

(11)=2/(1)

Weestimateusingequation(11)andhistoricalfarmyieldsandpricestocalculateKendaltaubetweenRfand1 LnPb..

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TorunthefinalMonteCarlomodelseveralstepsweretakentoincorporatetheCopula.The

firstissueisgivenanyx howtogetthecorresponding x .WeusetheLaplacetransformmethodbecauseitisstraightforwardwiththeClaytoncopulasinceitsgeneratoristheinverseLaplacetransform

oftheGammafunctionwitharguments1/and1(Nelson,2006)andcomputationallyefficient

(Chrisman,2013).With x , x )pairswethenfindyieldsandpricesbytakingtheinverseoftheirrespectiveCDF.Because1 LnPbwasassumedtohaveanormaldistributionwetaketheinverseofthe

normaldistributionusingthelibraryroutineprovidedbyANALYTICAdecisionsoftware(LuminaDecision

Software,2013).Thesamplingdistribution, Rf,connectedtoyieldisempirical.Atwodimensionalarray

withdimensions(30000,2)isbuiltcontainingx inonecolumnandCDFarrayforRfintheother.Becausex andtheCDFsdomainsare[0,1]andtheCDFisaincreasingfunctionon[0,1]theinversecanbenumericallyestimatedbyusingatablelookupmethod.WeusealibraryroutineinAnalyticawhich

usescubicspinestointerpolatebetweentheelementsofthearray.

Model

ToanalyzerevenueriskmodelwerunaMonteCarloprobabilitysimulationwith30,000iterations.

MonteCarloprobabilitysimulationallowstheprobabilitycalculationof,farmrevenue.Fromthiswecan

assessthelevelofincomeriskandimpactfromriskmanagementtools(cropinsuranceandfutures

hedging).RiskmanagementvariablesincludedwereCropinsurance(noinsuranceandrevenue

protection,RP,andrevenueprotectionwithharvestpriceexclusion,RPHPE,atdifferentcoverage

levels),hedging(futuresmarket,nohedgingto110%ofexpectedproductionhedgedin10%

increments).Forcropinsuranceunitchoicewelimitouranalysistoonlyenterpriseunitsbecausethe

produceristryingtomanagefarmincomeandfieldsenteringandexitingproductionovertimemakesit

difficulttoobtainfieldlevelyieldprobabilitydistributions.Welimithedgingtothefuturesmarketviaa

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hedgingaccountatabrokeragefirm.Theproduceriscompletelyresponsibleformarginpayments.11

GrainistobevaluedonDecember1atharvestusingthehistoricalbasisonDecember1.Historicalbasis

isestimatedastheaverageoverthe20yearsattheelevatortheproducertraditionallydeliversto.We

useANALYTICA,decisionsupportsoftwarespecializinginriskanalysisusingMonteCarlomethods,to

analyzeourincomeriskmodel.

Farmproductioncostsarecalculatedintwocomponents,coststhatareindependentofyield

andcostsdependentuponyield.Thiscalculationmethodaccuratelyreflectsfarmercosts.Giventhe

samecropinsuranceyieldguaranteeontwofieldstheproducerwouldbefinanciallybetteroffwitha

zeroyieldvs.ayieldonebushelbelowtheguarantee.12Itemsusedtocalculatecostsarethesameas

thosefoundinaLandGrantUniversityextensionbudget.Forcostsdependentuponyield,thosewere

estimatedusinghistoricalperacreharvestcosts.Costsincludedwouldbe:combinecosts,truckingcosts

(bothfromthefieldtobinandbintoelevator),anddryingcosts.Thecostperacreincludinglandcosts

wasassigned$606forthisfarmwhiletheestimatedperbushelyieldis%0.58.Theaverageyieldis143

bu/acreleadingtoatotalcostof$689peracre.Thisaveragetotalcostperacreissimilartotheaverage

costsfoundintheUniversityofKentuckynotillcornbudgets(Halich,2012).

Weanalyzefarmrevenueportfolioattwodistinctpointsintime.Thefirstpointisin

thespring,rightafterthecropinsurancebasepriceisdetermined.Thisalsocorrespondstothetime

whentheproducerismakinginvestmentdecisionsonwhichcroporhowmuchcroptoplant.The

secondpointisinthefall,justbeforeDecemberfuturesgointodelivery.Harvestiscompletedsocrop

sizehasbeendeterminedaswellascropinsuranceharvestprice.Forsimplicitythecropinsurance

11Herewedonotincludetheriskproducedbydaytodaymarginfluctuations.Insteadinthespiritofzerodrift(theperfectmarketassumption)wechargethecommissionplustheinterestontheinitialmarginrequirementincurredfromMarch1toDecember1.Marginriskisveryreal.Themodelhasbeenexpandedtoincludetheoptionofusingoverthecounterderivativessuchashedgetoarriveorcashcontracts.Weusuallypresentresultstofarmersusingthesemorespecificderivativesinordernottobeseenasencouragingriskybehavior.12Thisonlyappliestothecurrentyear.Thefollowingyearsyieldguaranteewouldbelowerforthezeroyieldproducerthantheonebushelbelowyieldguaranteeproducer.

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harvestpriceissetequaltofuturespriceonDecember1.13Thesepointsintimeareimportantbecause

oftherelationshiptospringplantinginvestmentsandhowcropinsurancefunctionstoprotectprices.

Forhedginginthisfirstordermodel,cropcanonlybepresoldatthefirstpoint(March1)inthespring.

Anyremainingunsoldbushelsmustbesoldatthesecondpoint(December1)andoversoldbushelsare

boughtbackatthesecondpoint.WedonotconsiderstoragepastDecember1becauseitisoutside

thecropinsurancepriceprotectionwindow.Anyincomeorlossincurredbystoringunpricedcroppast

December1canbecreditedtoaseparatestorageenterpriseandestimatedusingadifferentmodel;

therefore,postharvestpricedecisionareseparatedfrompreharvestdecisions.

Ourdataislimitedtooneproducerwith33yearsofexperience.Withonlyoneproducerovera

relativelyshorttimeframeourresultscouldleadtoaninaccurateassessmentoflinkagesbetweencrop

insuranceandhedging.Specifically,ourproducercouldhaveyieldriskthatisnotrepresentativeof

eitherthecountyorstateyieldrisk.Asaresultwecalculateyieldriskatthefarm(ourproducer),

county,stateandnationalleveloverthesamedataperiodasourproducer.Figure1presentsthese

histograms.14Itisthedistancebelowzeroonthexaxisthatisimportant.Aggregationfromthefarmto

thecountydoesnotchangethelevelofyieldrisk.However,aggregationfromthefarmtothestateor

evennationallevelreducestheamountofyieldrisk.Consequently,onewouldexpectsimilarresults

fromfarmorcountyanalysisbutnotfromthestateornationallevel.

Results

13ThefallpriceguaranteeforKYistheDecemberfuturesaveragedoverthemonthofOct.UndertheBlackScholesMertonassumptionsofastochasticprocessfollowinggeometricBrownianmotionthisisapathdependentquantity(Glassman,2004).Wehavemodeledthisquantityanditsdistributionhasaslightlysmallerspreadandthesameaverage(zerodrift)astheDecember1futurespricedistribution.HarvestpriceprotectionisnotaperfecthedgeforafuturespositionliftedonDecember1;however,wefeelthisapproximationwillnotsignificantlyalterourresults.14FfCnrepresentsthefarmyieldhistogram,CtCnrepresentsthecountyyieldhistogram,StCnrepresentsthestateyieldhistogram,andNtCnrepresentsthenationalyieldhistogram.

17

December2013cornfuturespricesCumulativeDistributionFunction(CDF),whichwasanalyzedon

March1,2012becausethatisthefirstdayafterthecropinsurancebasepricewasset,ispresentedin

figure2.AccordingtooptionpricesandBlacksoptionspricingmodelthereexistsatenpercentchance

offuturespricesoflessthan$4.00perbushelatexpiration.Therealsoexistsatenpercentchanceof

futurespricebeinggreaterthan$7.55perbushelatexpiration.Themedianpriceis$5.60perbushel.

Thehistorical,detrendedcornfarmyieldCDFispresentedinfigure3.Accordingtohistorical

yields,thereexistsatenpercentchancethatyieldscouldbebelow100bushelsperacreandaten

percentchanceofyieldsbeingabove170bushelsperacre.Themedianyieldis155bushelsperacre.

Farmcornyieldriskdisplaysstrongtailriskinthatthereexistsamuchbetterchanceoflowyieldsthan

highyields.

Incomerisk,inprobabilityspace,facedbytheproducerwithandwithoutcropinsurancepolicy

(Revenueprotection,enterpriseunits,80%coveragelevelpolicy)arepresentedinfigure4.Large

differencesinincomeriskexistwhencropinsuranceispresent.Forexample,atthetenpercent

cumulativeprobabilitylevel(ora1in10event)thereexistsaapproximatelya$100peracrerevenue

increasewhencropinsurancecontractispurchased.Thedifferenceinrevenuebetweennocrop

insuranceandcropinsuranceincreases(decreases)asthecumulativeprobabilityleveldecreaes

(increaes).Bothpathscrosszeroincomeatapproximatelythesamepoint.Aresultindicating,forthis

producer,cropinsurancedoesnotenhanceincomeitmakeslossessmaller.Thechanceoflosingmoney

isapproximately25%.Asincomebecomesmorepositivethenoinsurancepathresultsinmoreincome,

withthedifferencebeingthepremium.

Includinghedging(50%expectedproductionhedged)intotheincomeriskmodelindicatesa

furtherreductionintailrisk(seeFigure5).Atthe10percentriskcumulativeprobabilitylevelthere

existsapproximatelya$130peracrerevenueincreasewhencropinsuranceispurchasedwithhedging

18

vs.hedgingalone.Moreimportantly,atthe10percentcumulativeprobabilityarea,includinghedging

withcropinsuranceprovidespositiveincomeofabout$30peracre.Hedginglowersriskoflosing

money.Withouthedgingtherewasabouta25%oflosingmoney(seefigure3).Withhedgingthe

chanceoflosingmoneydropsto17percentforthenoinsurancepathanddropsanastonishingamount

to4%whencropinsuranceisused.Noinsurancepathcrossestheinsurancepathatapproximately31%

cumulativeprobabilitylevel;abovethisthereexistssmalldifferencesinbothpaths.

Figure6presentsreductionsinincomeriskfromdifferentcoveragelevelswith50%hedging.

60%coveragelevelprovidesthemostincomeriskwhile85%coveragelevelprovidestheleast.At70%

cumulativeprobabilitylevelresultsareopposite;60%coveragelevelprovidesthemostrevenueand

85%providestheleast.Differencesareequaltopremiumamounts.

Figure7providestheefficientfrontierofdifferentportfolios.Fromtable7wecanidentifythe

setofoptimalportfoliosorportfoliosthatmaximizereturnrelativetoanacceptablerisklevel.

Portfolioswiththehighestexpectedreturnandlowesttailriskrepresenttheefficientfrontier.What

standsoutintable7isthatnopointsfromthenoinsurancepathexistontheefficientfrontierandat

allhedginglevelsriskincreasesovernohedging.Hedging,whennoinsuranceispresent,increasesthe

riskofnotmeetingthefarmsurvivalobjective(1%tailrisk).Hedgingreducesvariability,butitincreases

tailriskbecauseeachcontractedbushelnotraisedmustbeboughtbackifthefallpriceishigherthan

thecontractprice(Hedgingstarchesthetailsevenasitreducesthevariance).Negativecorrelation

betweenfarmyieldandfallpricemakesthisworse.Shapesofyieldandpricedistributionsdetermine

thisoutcome.

BothRPandRPHPEinsurancepolicytypesexistontheefficientfrontier.However,onlyoneRP

HPEpointexistsandthatiswithnohedging.RPandRPHPEwithnohedgingprovidethehighest

expectedreturnwithanimprovementinriskatthe1%levelofjustover$292peracreoverno

19

insurance.AshedgingincreasespastzeropercenttheefficientfrontierexitsonlyontheRPpath.15At

the1%probabilitylevelrevenueriskisminimizedwithRPbetween3040%ofexpectedyieldhedged.

Hedgingat30to40%ofexpectedyield,underaRPinsurancepolicy(othercontractelementsare80%

coveragelevelandenterpriseunits),reduced1%revenueriskbyanadditional$39peracre.After55%

hedgedrevenueriskincreasedsteadily.Thehighestrevenueriskwithinsuranceandhedging,at120%,

wasconsiderablylowerthannoinsuranceandnohedging.

ForRPonlythereisevidenceofnearlyidenticalrisklevelsusingdifferenthedginglevels,Figure

7.Forexample,10and70percenthedgingprovides41to44dollarsperacreofrevenuerisk,

respectively.Whilebothhedginglevelsprovidethesamerisk,expectedincomeatthe70percent

hedginglevelisnearly$14.00peracrelessthanthatfoundatthe10percenthedginglevel.Recallthat

theproducerselected80%coveragelevel.Evidencesuggeststhathavingahighcoverageleveldoesnot

implyhedgingtothatcoveragelevelforoptimalrisk/reward.Inourcase,hedgingtolevelssimilartothe

selectedcoverageisnotontheefficientfrontier,expectedincomecanbeimprovedwhileholdingrisk

constant.

Reportedinfigure8,theaverageincomefromcropinsurancewashigherthanwithout

insuranceacrossallhedginglevels.Aresultindicatingthatovertime,30,000yearsinourcase,crop

insuranceindemnitieswillbelargerthanproducerpaidpremiums.Thedifferenceistheamountofthe

subsidy.Cropinsuranceissetuptobeactuariallyfair,meaningthatovertimeindemnitiesequal

premiums.However,producersdonotpaytheentirepremium,justaportionofit.Asaresult,over

time,producerswillgetmoremoneyinindemnitiesthanpaidinpremiums.

Ourfindingssuggestthatarevenueprotectioncropinsuranceplanincombinationwithforward

contractingminimizesrevenuerisk.TheyarecountertofindingspreviouslynotedbyCoble(2000)and

15RPHPEincreasesinrevenueriskpastthe10percenthedginglevelbecauseyieldriskisgreaterthanthereductioninpricerisk.

20

Wang(2004)thatcropinsurancenegativelyimpactstheneedforforwardcontracting.However,our

resultsaresensitivetobothfuturespriceriskandalsoproduceryieldrisk.Twovariablesthatwhen

changedcouldleadtodifferentconclusions.

Conclusions

Inthispaper,weinvestigatedrevenueriskimpactsfromcropinsuranceandforwardcontractingfora

producergrowingcorn.Ourfindingscontributetothevalueofinsurancetotheproducerand

demonstratetheimportanceofcropinsurancewhenhedgingispresent.Thisarticlecanaidinthe

debateoftheimportanceofcropinsurance.

Revenueriskimpactsfromcropinsuranceandforwardcontractingwereanalyzedusinga

portfolioapproachontwodays,oneinthespringjustafterthecropinsurancebasepricewassetand

theotherinthefallrightbeforeDecemberfuturesgointothedeliveryperiod.Weanalyzehow

differentcropinsurancecoveragelevelsandhedginglevelsimpactrevenuerisk.Understandingthe

revenuerisklinkagesbetweencropinsuranceandhedgingwouldresultinproducersmakingbetter

informeddecisionsandpolicymakersbetterunderstandingofthevalueofcropinsurance.Whilecrop

insuranceimprovesriskmanagementoverdoingnothingthecombinationofcropinsuranceand

hedgingprovidesfurtherriskreduction.Themagnitudeofriskprotectionbenefitswasquitelarge.The

costsofriskmanagementforfinanciallybeneficialoutcomesarequitesmall.

Justpurchasingcropinsurance,withouthedging,providesaexceptionallylargedecreasein1%

tailrisk.Forourproducer,thereisa$292.2peracrereductionintailrisk.Asaresult,theproducer

canreleasethesedollarsfromworkingcapitalandapplythemtootherinvestments,makingthefarm

moreefficient.Thischangeinworkingcapitalonhandforadverseeventshasapositiveimpactonthe

farmeconomybecausethesedollarswouldbeinvested.While1%riskwasreducedfromjust

purchasinginsurancewithouthedgingexpectedincomerose.Forourproducer,purchasinginsurance

21

increasedexpectedincomeby$23.7peracre.Withpurchasinganinsuranceproductonewouldexpect

areductioninexpectedincome;however,wefindanincreaseinexpectedincome.Theanswertowhy

isclearthepresenceofthepremiumsubsidy.Producerspayaportionoftheactuariallyfairinsurance,

therebyovertime,allowingthemtogainbackmoreinindemnitythantheiroutofpocketcosts.

Resultsindicateexceptionallylargerevenueriskreductionatthe1%risklevelwhencrop

insuranceispresentandfutureshedgingwitharevenueprotectionpolicy.Ourresultsarespecificand

requirecautionwheninterpretingresults.First,resultsarespecificto2013cropinsurancebasepriceor

springfuturesprices.Changesinthesepriceswoulddirectlyimpactlevelofrevenueriskprotectionsoit

isimpossibletoinferlongtermresults.Second,resultsarespecifictofuturespricevolatility.Changesin

volatilitywoulddirectlyimpactthelevelofhedging.Third,weanalyzeonlyonecrop,corn,different

cropswouldundoubtedlyleadtodifferentrevenueriskimpacts.Fourth,wedonotidentifystatistically

significantdifferencesindifferentportfolios.Especiallycrucialwhentherearesmallchangesinriskat

differenthedginglevels.Fifth,ourmodeldoesnotallowgrainstorageoranyothertypeofhedging

tools(i.e.,hedgetoarrivecontractsoroptions).Slightimprovementintheresultsisexpectedusing

thesetools.Likelythelargestgainwouldbeinsavinginterestexpenseonmarginpaymentsbyusing

hedgetoarrivecontractsvs.futurescontractsdirectly.

Oneparticularlyinterestingresultisthatofhedgingwithoutinsurance.Resultsindicatethatrisk

increasesatalllevelsofhedging.Hedgingunderourobjectiveoffarmsurvival,the1%revenuerisklevel

inourcase,increasestailriskbecauseeachcontractedbushelmustbeboughtbackifthefallpriceis

higherthanthecontractprice.Negativecorrelationbetweenfarmyieldandfallpricemakesthisworse.

Thecombinationofexperiencinglowyields,higherpricesandhedgingcommitmentsistheworst

scenariofortheuninsuredproducer.Becauseourmodelanalyzestwopointsintime,onceinthe

springandtheotherinthefall,itcannotallowforbuyingbackhedgedbushels,atthelowestpossible

22

price,oncethereisastrongchancethatcontractedbushelswillnotbemet.Additionally,theproducer

couldrollundeliverablecontractedbushelsintothenextcropyear,afeatureourmodeldoesnot

allowfor.Itispossiblethatourmodelcontainstheworstcasefinancialscenarioofnotmeetinghedged

bushels.Lessfinancialoutcomesexistandwouldlikelybeexercised.

Resultsleadtoadditionalresearchquestions.Forinstance,revenueriskimpactsfromdifferent

cropsandregions.Furtherinvestigationintotheimpactofthenaturalhedgeonrevenueriskreduction

andoptimallevelsofhedgingwouldbeusefulforbothproducersandpolicymakers.

Acknowledgments

TheauthorsaregratefultoJerrySkeesforhelpfulmodelingsuggestions.

23

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26

Figure1Farm,County(HardinCounty),State(Kentucky),andNationalyielddistributions

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28

Figure2.DecemberFuturesPriceCumulativeDistributionFunction

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Figure3.FarmYieldCumulativeDistributionFunction

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Figure4.FarmRevenueRiskCDFwithandwithoutInsurance.

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Figure5.FarmRevenueRiskCDFHedgingwithandwithoutCropInsurance.

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Figure6.RevenueRiskCDF,Hedging(50%)andInsuranceacrossDifferentCoverageLevels

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Figure7.IncomeandTailRiskEfficientFrontierunderDifferentPortfolios.

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