Revenue Risk, Crop Insurance and Forward...

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Revenue Risk, Crop Insurance and Forward Contracting      Cory G. Walters Department of Agricultural Economics, University of Kentucky [email protected]   Richard Preston Preston Farms, Hardin County Kentucky [email protected]      Selected Paper prepared for presentation at the Agricultural & Applied Economics Association’s 2013 Crop Insurance and the Farm Bill Symposium, Louisville, KY, October 89, 2013      Copyright 2013 by Walters and Preston.  All rights reserved.  Readers may make verbatim copies of this document for noncommercial purposes by any means, provided that this copyright notice appears on all such copies.  

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  • RevenueRisk,CropInsuranceandForwardContracting

    CoryG.Walters

    DepartmentofAgriculturalEconomics,UniversityofKentucky

    [email protected]

    RichardPreston

    PrestonFarms,HardinCountyKentucky

    [email protected]

    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.

  • 16

    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

  • 27

  • 28

    Figure2.DecemberFuturesPriceCumulativeDistributionFunction

  • 29

    Figure3.FarmYieldCumulativeDistributionFunction

  • 30

    Figure4.FarmRevenueRiskCDFwithandwithoutInsurance.

  • 31

    Figure5.FarmRevenueRiskCDFHedgingwithandwithoutCropInsurance.

  • 32

    Figure6.RevenueRiskCDF,Hedging(50%)andInsuranceacrossDifferentCoverageLevels

  • 33

    Figure7.IncomeandTailRiskEfficientFrontierunderDifferentPortfolios.

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  • 34

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