Risk Modeling Concepts Relating to the Design and Rating of Agricultural Insurance Contracts

Transcription

1 Publc Dsclosure Authorzed Publc Dsclosure Authorzed Publc Dsclosure Authorzed Publc Dsclosure Authorzed DOCUMENT OF THE WORLD BANK Rsk Modelng Concepts Relatng to the Desgn and Ratng of Agrcultural Insurance Contracts Barry K. Goodwn North Carolna State Unversty Olver Mahul The World Bank Abstract Goodwn and Mahul dentfy the key ssues and concerns that arse n the desgn and ratng of crop yeld nsurance plans, wth a partcular emphass on producton rsk modelng. The authors show how the avalablty of data shapes the nsurance scheme and the ratemakng procedures. Relyng on the U.S. experence and recent developments n statstcs and econometrcs, they revew rsk modelng concepts and provde techncal gudelnes n the development of crop nsurance plans. Fnally, they show how these rsk modelng technques can be extended to prce rsk n order to develop crop revenue nsurance schemes. World Bank Polcy Research Workng Paper 3392, September 2004 The Polcy Research Workng Paper Seres dssemnates the fndngs of work n progress to encourage the exchange of deas about development ssues. An objectve of the seres s to get the fndngs out quckly, even f the presentatons are less than fully polshed. The papers carry the names of the authors and should be cted accordngly. The fndngs, nterpretatons, and conclusons expressed n ths paper are entrely those of the authors. They do not necessarly represent the vew of the World Bank, ts Executve Drectors, or the countres they represent. Polcy Research Workng Papers are avalable onlne at Acknowledgements: We are grateful to Jerry Capro and Rodney Lester for ther helpful comments and suggestons.

2 Executve summary Crop nsurance s one mechansm for the management of the rsks assocated wth random yeld shocks once all cost-effectve rsk mtgaton strateges have been mplemented. Many developng and developed countres have well-establshed publcprvate crop nsurance programs. However, ther overall fnancal experence has been very expensve for the taxpayers and, n many nstances, far from popular wth farmers. The poor fnancal performance s usually due to the nablty or the unwllngness of the government to charge adequate premums to farmers. Beyond the nteractons wth the government, whch s tempted to use crop nsurance programs as a socal vehcle for subsdy to the agrcultural sector, the lack of sound ratemakng procedures s one of the key reasons for ther poor fnancal performance n several developng countres, lke Inda. Snce the late 1990s, due to dwndlng government subsdes to agrcultural producers n developng countres, there has been a renewed nterest among development lenders and polcy makers n usng agrcultural nsurance as an mportant rsk management tool. The fnancal vablty of agrcultural nsurance programs reles on the desgn and the ratng of actuarally sound nsurance contracts n order to gve prvate nsurance companes ncentves to offer market-based products and/or to assess the fscal consequences of government subsdy programs. In Inda, for example, the government s lookng for nternatonal techncal assstance to devse and mplement an actuarally sound ratemakng process for the Natonal Agrcultural Insurance Scheme. The purpose of ths paper s to provde Bank staff and polcy makers nvolved n agrcultural nsurance programs n developng countres wth an overvew of the latest developments n the modelng of crop yeld rsk and, to some extent, crop revenue rsk, and to dscuss how these modelng concepts affect the desgn and the ratng of crop nsurance. The overarchng concern wth such models s the dervaton of the parameters needed to adequately measure rsk and thus desgn and rate vable crop nsurance contracts. Though the procedure commonly used to measure yeld rsk (.e., collectng data and estmatng parameters of the yeld dstrbuton) seems straghtforward, a number of mportant concerns are germane to the process. Indeed, these concerns underle actuaral scence and underwrtng consderatons. Ths paper draws some lessons from the U.S. experence. The U.S. crop nsurance program s a jont effort of the Federal government and prvate ndustry. The nsurance scheme, known as Multple Perl Crop nsurance (MPCI), has been offered to the U.S. farmers for more than three decades. In crop year 2003, the MPCI program provded coverage on 217 mllon acres (almost 76%) of U.S. cropland, nsured USD 40.6 bllon n crops, generated a total premum of USD 3.4 bllon (of whch USD 2.0 bllon are premum subsdes) and dstrbuted USD 3.2 bllon of ndemnty payments. The Rsk Management Agency of the U.S. Department of Agrculture has developed a well accepted ratemakng procedure for the MPCI program. Ths paper lays out recommendatons on the desgn and ratng of vable agrcultural nsurance polces. It provdes Bank staff and polcymakers wth two sets of gudng prncples for the development of vable agrcultural nsurance programs n developng countres. 1

3 The frst set of gudng prncples refers to some key ssues that should be addressed n devsng agrcultural nsurance contracts. Insurance contract desgn s drven by the data avalablty. Indvdual farm data are almost always mssng or unrelable. Consequently, aggregate data can often be used ether to develop ndvdual nsurance coverage, where the ndvdual ndemnty depends on the ndvdual loss, or ndex-based coverage, where the ndvdual ndemnty s based on some ndces correlated wth the ndvdual loss (e.g., area yelds, weather parameters). Insurance coverage should be offered at the whole farm level n order to stablze the farmer s overall agrcultural revenue. Ths global coverage focuses on losses that cannot be mtgated through dversfcaton and avods fraudulent clams (e.g., yeld shftng across felds). All-perl coverage should be subject to suffcently hgh deductbles and consurance and to effectve loss adjustment process n order to effcently control moral hazard problems. Mult-year contracts should be promoted because they protect nsurers aganst nter-temporal adverse selecton (.e., farmers seek nsurance only when they expect a bad year) and they offer more fnancal stablty to the emergng agrcultural nsurance market. Crop revenue nsurance, coverng both yeld and prce downsde varatons, should be consdered as an extenson of crop yeld nsurance. The second set of gudng prncples pertans to crop rsk modelng, whch s a key step n nsurance contract desgn and ratemakng procedures, partcularly when hstorcal loss data are lmted or mssng. Its man task s to get nformaton about the stochastc nature of random yelds. Ultmately, the objectve s to derve the estmated crop yeld stochastc dstrbuton functon. The modelng approach (e.g., parametrc or non-parametrc statstcal methods) and the underlyng assumptons (e.g., famly of stochastc dstrbutons) are of fundamental mportance as they affect drectly the estmated yelds, the estmated losses and thus the nsurance premum rates. Non-parametrc methods should be preferred when large datasets are avalable because they mpose no mnmal structure on the estmated dstrbuton. On the contrary, parametrc methods are more preferable when the data set s small. Crop yeld dstrbutons are usually skewed to the rght, wth yelds close to the maxmum yeld observed more frequently than very low yelds. The normalty assumpton s thus usually rejected. When hstorcal losses are avalable, ratng procedures should be based on hstorcal loss cost ratos. The ratemakng methodology should rely on several steps, such as loss adjustment to a common level of coverage, dervaton of unloaded base rates, base rate loadng, cappng rate changes and updatng. 2

4 When hstorcal losses are not avalable, the ratng methodology should rely on smulated losses derved from statstcal crop yeld models. Crop revenue nsurance ratng requres nnovatve procedures that are avalable through recent experence on the U.S. crop nsurance market. The gudng prncples outlned n ths paper am at devsng vable crop nsurance products. They are based on a sound actuaral approach. Experence shows that such a rgorous approach may be mssng n government-sponsored crop nsurance programs, because the hgh socal welfare content mposed by the government may not be compatble wth these nsurance gudelnes. For example, premums rates derved under these actuaral prncples may be consdered as too hgh to meet socal objectves. As a consequence, a prerequste n the mplementaton of these gudng prncples s to make a clear dstncton between socal nsurance, whch cannot be acheved through marketbased products, and market-based nsurance, whch should rely on the above prncples. 3

5 1. Introducton The management of crop producton rsks s an ssue of fundamental mportance to agrcultural economes. Because of the random nature of producton condtons (e.g., weather, pests, dseases), agrcultural producers face an array of rsks that may nfluence ther level of output per acre from year to year. Management of such yeld rsks has long been an mportant ssue for producers as well as for polcy makers. Crop nsurance s one mechansm for the management of the rsks assocated wth random yeld shocks once all cost-effectve rsk mtgaton strateges have been mplemented. Many countres have well-establshed publc and/or prvate crop nsurance programs, although most, f not all, publc mult-perl schemes are not sustanable wthout heavy government subsdes. In the Unted States, for example, federally subsdzed crop nsurance has been n exstence for over 60 years. 1 Such nsurance s currently avalable for over 75 dfferent crops and lvestock products, wth premum subsdes representng on average 60% of total premums n 2003 (RMA 2004). 2 In ts most fundamental form, a crop nsurance plan wll pay producers an ndemnty n the event that ther yelds fall below a pre-determned level. Constructon of such a seemngly smple contract requres representaton of a number of mportant parameters. Accurate measurement of such parameters may be qute complex, especally n cases where lmted knowledge of the rsks or levels of protecton beng provded are avalable. The challenges assocated wth accurately measurng the parameters that determne lablty, premums, ndemntes and other components of a crop nsurance plan are often complcated and may requre the applcaton of rather complex actuaral methods, models, and assumptons n order to desgn and rate vable nsurance contracts. The purpose of ths paper s to provde Bank staff and polcy makers nvolved n crop nsurance programs n developng countres wth an overvew of the latest developments n the modelng of crop yeld rsk and, to some extent, crop revenue rsk, and to dscuss how these modelng concepts affect the desgn and the ratng of crop nsurance. The overarchng concern wth such models s the dervaton of the parameters needed to adequately measure rsk and thus desgn and rate vable crop nsurance contracts. Though the procedure commonly used to measure yeld rsk (.e., collectng data and estmatng parameters of the yeld dstrbuton) seems straghtforward, a number of mportant concerns are germane to the process. Indeed, these concerns underle actuaral scence and underwrtng consderatons. Based on the latest developments n agrcultural rsk modelng and on the U.S. experence, ths paper revews each of these concerns and dentfes possble solutons and recommendatons that may be commonly used to address each ssue. Ths paper s organzed as follows. Practcal consderatons n devsng crop nsurance schemes are presented n Secton 2. Secton 3 descrbes how producton rsk can be modeled usng parametrc and non-parametrc methods. Secton 4 examnes how rsk 1 Goodwn and Smth (1995) revew the hstory and operaton of the U.S. crop nsurance program. The European Commsson (2001) provdes a descrpton of crop nsurance programs n European countres, Canada and Japan. FAO (1991) descrbes several crop nsurance programs n developng countres (e.g., Chle, Cyprus, Maurtus, Phlppnes). 2 Subsdes are appled to a base premum rate of 1.075, as requred by law. 4

6 modelng can be used to develop crop nsurance contracts. Ratemakng procedures for exstng nsurance schemes and new schemes are detaled n Secton 5. Secton 6 examnes how to extend coverage to address prce rsk through crop revenue nsurance. Fnally, the key ssues are summarzed n the conclusons. 2. Practcal Consderatons n Devsng Crop Insurance The theory underlyng the constructon of nsurance contracts s generally straghtforward. However, there are a number of ssues that are relevant to any practcal applcaton of these prncples. Indeed, t s attenton to such ssues that makes up much of actuaral scences. In ths secton, we dscuss these ssues and varous approaches used n the practcal development of crop nsurance tools Data Avalablty The feasblty of any partcular crop nsurance plan s often nfluenced by the avalablty of yeld data. Detaled tme seres of ndvdual farm data of the sort needed n measurng yeld rsks are usually rare, even n cases where farm records are mantaned. In the case of the U.S. crop nsurance program, only 4-10 years of yeld data are used n determnng the yeld guarantees. Seldom, f ever, do substantally rch yeld records exst at the ndvdual farm level that would permt the detaled ratng of crop nsurance contracts at the ndvdual level. Ths suggests that the development of rates and levels of protecton for ndvdual farmers may be dffcult and that developers wll lkely need to gve consderaton to the use of aggregate data. The use of such aggregate data n devsng crop nsurance contracts may follow two dstnct avenues: ndvdual nsurance coverage, and ndex-based nsurance coverage. 3 Indvdual nsurance coverage One may wsh to use aggregate data to make nferences about the rsks facng ndvdual farmers. Ths s very common n the development of ndvdual plans. For example, under the U.S. crop nsurance program, twenty years of the loss cost experence at the county level s used to obtan a measure of the typcal level of rsk for farmers n the county. 4 Rates at the ndvdual level are adjusted n accordance wth how the ndvdual farm s average yeld compares to the county average. Other approaches have been used to combne ndvdual farm yeld data wth aggregate data. The developers of the U.S. Income Protecton (IP) program, a revenue nsurance scheme, combne county yeld data wth ndvdual farm yelds to make use of the longer county seres and thus mprove the amount of yeld nformaton used n developng the IP contracts. In the case of the IP program, an attempt s made to dstngush yeld varablty specfc to the farm from yeld varablty experenced by the entre county. Under provsons of the U.S. federal crop nsurance program, farmers lackng a yeld hstory are assgned a yeld that s based upon the county average yeld (n 3 A conceptual dscusson on the role of nformaton n nsurance contract desgn s provded n the Appendx. 4 Loss cost s gven by the rato of ndemntes to total lablty and represents a measure of the expected loss rate whch, n turn, s a measure of the actuarally far premum rate. The ratemakng procedures are dscussed n Secton 6. 5

7 some cases, a proporton of the average yeld s used). Other approaches to combnng aggregate and dsaggregate data are possble. Most approaches adopt a measure of the relatonshp between farm level yelds and yelds at a more aggregate level such as the county. For example, one may use regresson methods to estmate the relatonshp between a lmted number of farm-level yelds and county yelds and then use ths relatonshp along wth a longer seres of county levels to nfer a pseudo seres of farmlevel yelds. In a related way, aggregate county yelds are often used to approxmate mssng farm-level yelds. In the U.S. crop nsurance program, farms that lack a yeld hstory from whch to construct a measure of ther expected farm-level yeld are assgned a proporton of the county average yeld n place of the mssng data. For example, f a farmer can prove that they have never produced the crop before, they are elgble for 100% of the county average. Farmers that are not new producers but that are unable to furnsh yeld hstores are assgned 60% of the county average yeld. Research by the Rsk Management Agency (RMA) suggests that ndvdual average yelds below the county average tend to be more varable than those above the county average. Ths nverse relatonshp s ncorporated nto the ratng of ndvdual contracts by consderng an ndex formed by the rato of the farm average yeld to the county average yeld. Ths accounts for the fact that county average yelds are varable across countes and thus adjustments to rates are made on the bass of how an ndvdual farm may compare to the county average. Hence, rates are hgher for growers wth below average yelds and lower for growers wth above average yelds. Index-based nsurance coverage An alternatve approach to devsng crop nsurance plans when ndvdual data are lackng s to base coverage on some aggregate ndex that conveys nformaton about (e.g., that are correlated wth) the ndvdual losses. Examples nclude area-yeld nsurance, whch bases coverage and ndemntes on aggregate yelds of a gven geographcal area, and weather-based nsurance, whch s based on an ndex reflectng the weather experence (e.g., ranfall) at a partcular weather staton. These ndex-based plans have substantal advantages n that aggregate data are generally more relable and longer hstores of aggregate experence are more lkely to be avalable. A potental dsadvantage assocated wth such plans relates to the fact that ndemnty payments may not be perfectly correlated wth the ndvdual loss and thus there may be yeld bass rsk. Consequently, an ndvdual farmer may suffer a loss and not be elgble for an ndemnty payment. The degree of correlaton between an ndvdual farmer s yelds and the ndex that forms the bass of the area-wde plan s a key factor determnng the extent to whch an ndvdual s yeld rsks are covered by the nsurance plan. Ths rases a substantal dstncton between actuaral soundness and effectveness of the plan for protectng aganst ndvdual rsks. An ndex for whch relable data are abundantly avalable may allow one to construct very accurate measures of rsk and thus accurate premum rates. In terms of actuaral soundness, beng able to accurately measure the ndex and model ts uncertanty s the prmary concern. However, the effectveness of the ndex-based nsurance plan and ts lkely acceptance by farmers wll be determned by the extent to whch the ndex reflects ther ndvdual rsks. For a farmer wth yelds that are unrelated to the ndex, the ndex-based plan wll provde lttle protecton for ther 6

8 yeld rsks. However, f a farmer s yelds are perfectly correlated wth the ndex, the ndex-based plan wll provde perfect coverage for ther yeld rsks. However, ndex-based plans may offer some advantages. A greater avalablty of data wll lkely allow nsurance premum rates and other parameters of the contract to be measured wth greater accuracy. Ths lmts the problems assocated wth adverse selecton. Index-based plans also lessen concerns of moral hazard. If an ndvdual farmer s so small as to be unable to affect the overall ndex, there s no potental for moral hazard snce the agent s actons cannot nfluence the lkelhood that ndemntes wll be pad. In addton, reflectng the fact that such ndex-based plans are much cheaper to admnster snce ndvdual producer records are not needed and loss adjustment need only be done at the aggregate level, ndex-based plans can generally be offered at a lower cost. It may also be the case that there s a lag n measurng the ndex, generatng a delay n settlement of clams. Under the Indan crop nsurance program, for example, the delay s close to one year. It s caused by the tme taken to carry out crop cuttng experments and, n some cases, delayed flows of funds from the state government. The delays prevent the restoraton of credt elgblty of farmers for the followng crop season (World Bank 2003). Lkewse, ndemnty payments under the U.S. area yeld plan, are not made untl the followng year due to the lag assocated wth the measurement of area yelds Specfc Perl versus Multple Perl versus All Perl Coverage An mportant dstncton n the nature of coverage offered under alternatve crop nsurance plans pertans to the types of hazards that are nsured aganst. Some forms of nsurance are specfc perl, meanng that only losses from a pre-specfed perl are nsurable. A promnent example s hal nsurance, whch wll pay only for those losses that are caused by hal damages. Fre nsurance s another example. Alternatvely, crop nsurance may provde ndemnty payments when losses are realzed as a result of multple causes of loss. As an example, crops are commonly nsured aganst losses caused only by both fre and hal. It s more common for such multple perl polces to lst causes of loss that are not nsurable. That s, all causes of loss except for a specfc lst of perls wll trgger ndemnty payments. Fnally, some agrcultural nsurance plans provde all perl coverage. For example, the U.S. federal crop nsurance program s desgned to nsure yelds of farm producers over an entre growng season on an all rsk bass. The prmary cause of loss s weather, ether for a sngle dentfable event or over an extended perod. Ths program provdes protecton aganst all losses except the followng: (1) neglgence, msmanagement, or wrongdong; (2) falure to follow recognzed good farmng practces; (3) losses caused by water contaned n any publc dam or reservor; and (4) falure to carry out good rrgaton practces for rrgated crops. A fundamental ssue that s key to the queston of whether coverage can be sngle or multple perl nvolves the extent to whch the loss adjustment process s able to adequately determne the causes of loss. As was noted n the case of the U.S. crop nsurance program, losses occurrng due to mproper actons of the producer are generally not nsurable. Such exclusons are closely related to the ssue of moral hazard. Insurance aganst the adverse actons of farmers wll naturally work aganst the soundness of the nsurance program. Montorng of nsured agents actons and adequate loss adjustment s 7

9 an ssue that arses n almost all lnes of nsurance. Adequate tranng of loss adjusters s a crtcal component needed n the development of a sound nsurance program. It s also mportant that any conflcts of nterests between loss adjusters and the nsured agents be dentfed and remeded. Actuaral consderatons must, of course, make a dstncton between measurng all perl rsk and measurng the rsks assocated wth a specfc perl. It s often argued that t s easer to measure the rsks assocated wth a specfc perl than to attempt to measure all rsks, seen and unseen, that mght be expected to cause yeld losses. Ths may be true n cases where exogenous nformaton about rsks s avalable. For example, n the case of hal, hstorcal weather records may allow actuares to accurately measure the rsks assocated wth hal damages n any partcular area. Specfc perl lnes of nsurance may also be provded aganst events that are very low probablty and non-systemc. For example, fre may be a rare event and may not affect large numbers of producers. In such cases, adequate measures of rsk may be obtaned from consderng a large, crosssectonal pool of producers. Specfc perl nsurance may also rely on hstorcal yeld data to measure rsks. In such a case, actuares must be able to dentfy the exact causes of loss assocated wth hstorcal yeld shortfalls. The methods dscussed below for detrendng yelds can be extended n a straghtforward manner to condton on non-nsured causes of loss. For example, consder a case where factors assocated wth yeld shortfalls can be segmented nto a sngle hazard that s covered (e.g., hal), denoted by X, and hazards that are not covered, denoted as Z. One would run a regresson of yelds on Z, thereby condtonng out the effects of those factors affectng yelds that are not covered by the nsurance. The resduals from such a regresson represent yeld shocks that are assocated wth the covered factor X, as well as random, unmeasured factors. Thus, the overall approach to modelng the rsks assocated wth a specfc perl from hstorcal yeld data s no dfferent than what s commonly undertaken to detrend data. The only dstncton nvolves the choce of condtonng factors ncluded n the regresson Unt Insured Agrcultural nsurance s an ntegral part of rsk management. It deals wth resdual rsks that cannot be prevented through cost-effectve preventve measures. Feld dversfcaton s a standard rsk management technque that allows farmers to reduce rsk. On the demand sde, farmers would prefer to cover ther assets at rsk by purchasng an umbrella contract that guarantees a mnmum revenue at the farm level. However, premum subsdes and nformaton asymmetres would gve them ncentves to nsure each feld, plot or crop separately. On the supply sde, such an umbrella coverage would correspond to the prmary objectve of crop nsurance,.e., stablzng the agrcultural revenue at the farm level. Indemnty payments receved on a dsaggregated set of unts are at least as large as what would be receved f all the unts were aggregated. Intutvely, f yelds are not perfectly correlated across ndvdual unts, t s lkely that shortfalls for some unts may be offset by hgher than expected yelds on others, such that the aggregate yeld does not qualfy for a payment whle the ndvdual unts would qualfy f nsured ndvdually. 8

10 However, actuarally sound premum rates would be expected to fall as the nsurance unt s defned over more and more ndvdual farm unts. The fact that yeld rsk tends to fall as the unt of observaton ncreases reflects factors that are assocated wth the correlaton of yelds across unts. Spatal dversfcaton may be possble f yeld correlaton s less than perfect. Indvdual unts may be separated by a consderable dstance. Indvdual farm unts may have ther own dosyncratc factors affectng yelds. For example, the sol qualty on one unt may be much hgher than on another. Mult-feld contracts In the case of the U.S. crop nsurance program, an across-the-board dscount of 10% s gven to farmers that nsure ther ndvdual felds growng the same crop (whch are called optonal unts ) as a whole (whch s called the basc unt ). However, under the optonal unts, t may be dffcult for the nsurer to verfy the feld from whch nsured producton orgnated. To the extent that t s dffcult to montor and verfy producton from ndvdual felds, producers may have an ncentve to shft producton across ndvdual felds. For example, f two felds had an dentcal guarantee of 100 bushels per acre and each feld produced 105 bushels, no ndemnty would be pad. If, however, the producer reports that one feld generated 125 bushels and the other generated 85 bushels, ndemntes would be trggered. Ths pont nvolves actons on the part of producers that are obvously fraudulent. The extent to whch such actons actually take place n the U.S. crop nsurance program s unclear, though unpublshed research has suggested that there s evdence of such yeld swtchng n some areas and for some crops. 5 When the nterest s n addressng whole farm rsk or rsk at some aggregate level, polces based on producton across all relevant felds should be consdered. In addton, the potental for fraudulent clams through yeld shftng across felds may exst when coverage s provded at the ndvdual feld level. Ths s essentally a montorng and yeld verfcaton problem and thus ts relevance depends upon the extent to whch adjusters can verfy yelds on ndvdual felds. Mult-crop contracts Most contracts are developed for a sngle crop. However, there may be some value to provde coverage over multple crops due to the dversfcaton effect, as ths effect apples over multple felds for a same crop. If yelds across crops are not perfectly correlated, overall rsk wll be lower when multple crops are combned nto a sngle contract and, therefore, such contracts may be offered at a lower prce. For example, shortfalls n corn yelds may be offset by good soybean producton on a sngle farm. In such a case, corn nsurance would trgger an ndemnty payment whle soybeans would not. However, when total revenues from the two crops are combned, no ndemnty s trggered snce the below average corn yelds are offset by above average soybean yelds. 5 Unpublshed research at Montana State Unversty has provded evdence that may be consstent wth such swtchng. 9

11 Ratng such mult-crop plans presents a number of challenges snce one must have adequate measures of the correlaton of yelds for dfferent crops on a sngle farm. Measurement of the degree of correlaton of yelds for dfferent crops on a sngle farm s dffcult and may requre strong assumptons about the degree of correlaton. Other factors such as the dstance between felds for alternatve crops may be relevant to the degree of correlaton. The authors are aware of only two nsurance polces avalable on a mult-crop bass n the U.S. The whole farm opton s avalable for corn and soybean producers under the Revenue Assurance plan. In ths case, the revenue guarantee and ndemnty trgger are based upon the total revenue generated by both corn and soybeans. Substantal dscounts are avalable to growers that nsure ther crops as a whole farm unt. The second example s the Adjusted Gross Revenue (AGR) plan of nsurance. The coverage and ndemntes are based on the farm s adjusted gross revenue as reported on the farm s tax return form. Ths plan covers revenues from all sources, ncludng lvestock commodtes Mult-Year Contracts Most crop nsurance contracts are of a sngle year n duraton. Ths reflects the seasonal nature of crop producton and the fact that producton decsons are typcally made over an annual growng perod. The same general arguments regardng the varablty reducton that may be possble from combnng yelds across crops also apples to the case of combnng yelds over tme. If one bases coverage on the value of producton over multple years, t may be possble to lower the cost of nsurance snce poor producton n one year may be offset by above average producton n another. In lght of the fact that many farmers are credt constraned and dependent upon short term loans, mult-year coverage may not be a vable alternatve. A related ssue pertans to the potental for offerng dscounts to producers that have strong performance from year to year through a bonus/malus arrangement. Producers that do not submt a clam over a few years may reveal that they are of a lower rsk to the nsurer than was thought when they came nto the program. In such a case, t may be preferable to lower ther premum rates, thus makng use of the revealed nformaton about ther rsk. Ths smply nvolves usng ther performance nformaton to fne-tune rates n an attempt to make them more accurate. Ths ntuton also works n the opposte drecton; t may be to the nsurer s advantage to rase the rates of agents that frequently submt clams. Commercal lnes of nsurance typcally undertake such actons n an attempt to make premum rates more accurate. 3. Producton Rsk Modelng 3.1. Tme Trend Crop yelds have realzed substantal changes n recent years as mproved producton technques have been adopted. Conceptually, ths means that the data-generatng process underlyng yeld realzatons s not stable but rather changes over tme. Thus, one cannot combne yelds observed over dfferent perods of tme. For example, corn yelds observed n the 1970s are clearly not comparable to those observed today. To address the problem of structural changes n yelds observed over tme, a varety 10

12 of methods for detrendng yeld data have been adopted. Generally, one uses least squares regresson technques to account for determnstc trends that have moved yelds up over tme. Devatons from ths trend can then be added to current yelds to produce a seres of yelds that can be compared over tme. Trends may be lnear or nonlnear (the latter s more common) and a varety of estmaton approaches have been used to account for such trends. These unvarate methods nclude autoregressve-movng average models, splnes, and local nonparametrc smoothng technques. 6 A closely related ssue that one must address when detrendng yelds nvolves the second moment of the detrended yeld data. In partcular, s the varance of the yelds constant or does t vary as the level of yelds changes? Goodwn and Ker (1998) evaluate ths ssue usng parametrc and nonparametrc tests for heteroscedastcty. Ther results ndcate that the standard devaton of yelds tended to be proportonal to the average yeld. Ths suggests a somewhat dfferent approach to normalzng yelds. Assume that the followng functonal relatonshp s approprate for detrendng a temporal seres of crop yelds and has been estmated for a seres of yelds rangng from t=1970,, 2003: y = X β + e, t t t where X t represents some lnear or nonlnear functon of tme. Estmates of such a relatonshp wll yeld trend-predcted yelds (y t ) and devatons from the trend (e t ). If one wants to normalze yelds to a 2003 level and one beleves the magntude of the errors s not affected by the level of yelds, one would add all of the resduals to the 2003 yeld predcton, such that: (normalzed yeld) t = y e t. However, f one beleved that the devatons from the trend tended to be proportonal to the level of yelds, one mght consder constructng normalzed yelds as: (normalzed yeld) t = y 2003 (1+e t /y t ). Both approaches have been adopted n the lterature though the latter probably has more emprcal support. One must balance the complexty of emprcal technques that may be more approprate aganst practcal consderatons assocated wth modelng what s often a very large collecton of yeld trends. 7 We have not assumed that yeld trends are lnear n nature. In many cases, the exact functonal form for trend effects s unknown and thus nonlnear functons ncludng hgher-ordered polynomals, splnes, and nonparametrc regresson procedures may be used to represent tme trends. In the case of the U.S. group rsk (GRP) program, a lnear 6 For a detaled dscusson of ths lterature and references to specfc applcatons, see Goodwn and Ker (2002). 7 A number of other subtle ssues underle the normalzaton of yelds. Departures from normalty, whch often underle consderatons of modelng yeld rsk, may suggest that least squares technques are less than optmal snce least squares tends to make detrended yelds (resduals from the OLS regresson) more symmetrc than the populaton error dstrbuton. Ths ssue s dscussed n detal by Goodwn and Ker (1998), who conclude that a quadratc loss functon of the sort that underles least squares regresson may be merted on economc grounds. One could also argue that any suspected heteroscedastcty should be modeled n the detrendng equaton drectly rather than accountng for t after estmaton. 11

13 splne wth a sngle knot pont s used to detrend county-level yelds. Of course, t s always possble to mprove the ft of any regresson by addng addtonal terms (or n the case of nonparametrc regresson, by lowerng the bandwdth). Put dfferently, devatons from trend can always be made smaller by ncreasng the order of a polynomal regresson of yelds on tme. One must take care to avod overfttng when estmatng trends and other determnstc factors. One approach to guardng aganst overfttng when detrendng yeld data s to choose the functon on the bass of out-of-sample predctve power rather than n-sample performance. One mght ft the regresson model on one porton of the sample and predct the values of the other porton. Alternatvely, one mght drop out each observaton, ft the model, and predct the omtted observatons, an approach known as cross-valdaton. Semparametrc and nonparametrc methods are well-known for ther overfttng problems and thus care should be used when modelng trend or other determnstc factors Spatal Correlaton Varatons n crop yelds tend to be drven by factors that typcally affect a large area, such as weather and pest nfestatons. Ths rases the ssue of systemc (.e. covarate) rsk or, put dfferently, spatal correlaton of yelds. 8 The fact that average yelds tend to be subject to substantal correlaton across space suggests that standard appeals to normalty based upon central lmt theorem results are lkely to be nvald. Goodwn (2001) demonstrates that yeld correlaton tends to de off slowly over space. Hs results also suggest that the degree of correlaton may be state-dependent,.e., yelds tend to be more hghly correlated over space as extreme yeld events (e.g., drought) occur. The ssue of spatal yeld correlaton has many other mportant mplcatons for the constructon of yeld nsurance contracts. Systemc rsk may suggest that the potental for catastrophc losses s hgher than what one mght reveal n a short sample of data. The degree to whch rsks n a large nsurance pool can be dversfed s dmnshed as yelds become more spatally correlated. In addton to spatal correlaton, t s possble that the effects of exogenous factors that underle realzed yelds may persst beyond a sngle producton cycle. For example, the effects of a severe drought on crop yelds n one year may persst nto the next year as sol mosture levels reman below average. Lkewse, the outbreak of a dsease or pest nfestaton may affect yelds over multple growng seasons, suggestng that low yelds ths year may be followed by low yelds agan next year. The potental for such persstence s an agronomc ssue and s lkely to be crop- and regon-specfc. To the extent that the effects of weather or other exogenous factors persst across years, t may be necessary to nclude forms of autoregressve or movng average effects n any detrendng equaton. In dong so, care must be exercsed n usng the results snce one may want predctons of future yelds to account for any such effects but may not want to condton out the effects of such factors when detrendng yelds, snce devatons from trend for ndemnfable reasons should not be accounted for when modelng devatons from trend. Evdence regardng the extent to whch yeld shortfalls may persst beyond a sngle growng cycle s unclear, though some authors have obtaned results that suggest such persstence may ndeed occur (Goodwn and Ker 1998). 8 Ths correlaton tends to decrease as there are elevaton varatons (e.g., hlls, valleys). 12

14 3.3. Yeld Modelng: Parametrc vs Nonparametrc Methods Inherent n the desgn of an nsurance contract s a mechansm for determnng the probablty of loss and the expected level of loss when losses occur. More formally, one s generally nterested n a measure of the probablty densty functon (pdf) underlyng the event or events that trgger losses. Thus, the concept of modelng yeld rsk for the purposes of desgnng and ratng a crop nsurance contract s fully analogous to modelng the probablty dstrbuton for the crop yeld n queston. At ts most fundamental level, the task of modelng yeld rsk nvolves obtanng estmates of parameters or descrptons of patterns depctng the stochastc nature of random yelds. In formal terms, one s nterested n measurng the probablty densty or dstrbuton functon of yelds. Approaches to modelng yeld denstes and dstrbutons fall nto two broad areas. The frst nvolves estmaton of the parameters of a parametrc dstrbuton. The second general approach encompasses the wde varety of nonparametrc methods that are commonly used to measure or approxmate yeld dstrbutons. Alternatve sem-parametrc methods may combne elements of both parametrc and nonparametrc methods (by, for example, usng a hgh-ordered polynomal seres expanson to approxmate an unknown functon). In what follows, we provde an overvew of parametrc and nonparametrc methods commonly used n estmatng crop yeld denstes. Parametrc methods Parametrc approaches to the estmaton of probablty denstes generally nvolve usng an observed seres of yeld realzatons to estmate specfc parameters that descrbe a probablty densty or dstrbuton functon. Examples nclude the normal dstrbuton, a two parameter dstrbuton that s completely descrbed by ts mean and varance. Gven a set of ndependent and dentcally dstrbuted (d) yeld realzatons (a crtcal assumpton for yeld data, whch s dscussed below) one can estmate the parameters of the dstrbuton usng maxmum lkelhood or method of moments estmaton procedures. Of course, such an approach to modelng yelds assumes that one knows a pror the correct parametrc famly underlyng the yeld data generatng process. In realty, there s usually very lttle to gude one n choosng a specfc parametrc dstrbuton, though certan characterstcs of yeld dstrbutons (such as negatve skewness) tend to suggest specfc canddates. A number of specfc parametrc dstrbutons have been consdered for modelng crop yeld dstrbutons. A case can be made for the approprateness of the normal dstrbuton on the bass of the central lmt theorem (CLT), snce a degree of aggregaton s almost always present n crop yelds such that one s measurng average yelds. However, a problem arses when consderng central lmt theorem results for crop yelds because of the systemc (covarate) component. The rsks affectng agrcultural yelds are typcally of a systemc nature n that they arse from weather, pests, or other natural phenomena that tend to affect large geographc areas. The central lmt theorem, n ts most basc form, s based upon the ndependence of dentcally dstrbuted ndvdual random draws, a stuaton that s unlkely to occur for crop yelds. The spatal correlaton that affects crop yelds and results n systemc rsks suggests that smple versons of the CLT may be nvald when nvoked for crop yelds. Weakest versons of the CLT that 13

15 allow for dependence and non dentcal dstrbutons exst, though requrements that such dependence decays over space lmts ther sutablty for crop yeld modelng. Box 1. Basc statstcal measures of rsk Consder a sample of n observatons, y, for =1 to n. The mean (or average) s the value that s n 1 expected to occur from a draw from the dstrbuton,.e., µ =. y n = 1 The second moment about the mean s measured usng the varance, whch descrbes the amount n 2 1 y n = 1 of varaton about the mean exhbted by the dstrbuton, σ = ( µ ) Skewness s determned by the thrd moment about the mean, ( µ ) 1 n 3 nσ = 1 y 2 3. It s a measure of the extent to whch a dstrbuton s pulled to the rght (negatve skewness) or to the left (postve skewness). A completely symmetrc dstrbuton has zero skewness. Kurtoss refers to the mass that s present n the tals of the dstrbuton and s often defned n 4 y 3 4 nσ = 1 relatve to the normal dstrbuton (whch has a kurtoss of 3), ( µ ) values of ths metrc are sad to ndcate excess kurtoss and denstes are sad to exhbt leptokurtoss (for heavy tals) and platykurtoss (for lghter tals). Source: authors. 1. Postve Many researchers have observed that crop yelds tend to be negatvely skewed, wth yelds near the maxmum beng observed more frequently than yelds near the mnmum. In ths lght, a frequent choce for modelng yeld dstrbutons s the four-parameter beta dstrbuton. The beta dstrbuton can accommodate the skewness so commonly observed for crop yelds and can assume a varety of shapes. Ths dstrbuton does suffer from a number of shortcomngs, however. Although the dstrbuton has four parameters (two shape parameters and a maxmum and mnmum possble yeld), t s commonly appled wth only the shape parameters beng estmated. The maxmum and mnmum yelds are generally set by assumpton or n some ad hoc manner. 9 Other parametrc dstrbutons that have been used to model crop yelds nclude the Webull, the log-normal, the gamma, the logstc, versons of the Burr dstrbuton, and mxtures of parametrc dstrbutons. These parametrc dstrbutons vary n terms of ther flexblty and ablty to capture ntrnsc propertes of varous crop yelds and thus they dffer n terms of ther approprateness for modelng crop yeld denstes. For example, the log-normal dstrbuton mposes postve skewness on the dstrbuton, a characterstc that s not typcally expected for crop yelds. 9 Ker and Coble (2003) show that shape parameter estmates may be very senstve to assumptons about the maxmum and mnmum possble yelds. They also present evdence suggestng that estmaton of three or four parameter beta dstrbutons may be less effcent than nonparametrc or sem-parametrc alternatves. 14

16 In summary, ntuton about the nature of crop producton (.e., the fact that bologcal constrants lmt the maxmum possble yeld, wth yelds close to the maxmum beng observed more frequently than very low yelds) suggests that yeld dstrbutons are lkely to be negatvely skewed. Lkewse, much of the exstng emprcal evdence about crop yeld dstrbutons has confrmed the prevalence of negatve skewness. Several dstrbutons are compatble wth negatve skewness, ncludng the beta, gamma, and Burr dstrbutons. Our recommendaton would be that such dstrbutons should be the frst choce n cases where a parametrc dstrbuton must be adopted. Nonparametrc methods An alternatve approach to modelng crop yeld dstrbutons s to adopt a nonparametrc type of estmator. The smplest estmator s the hstogram. The range of the data s dvded nto bns of a certan wdth and the yeld observatons fallng nto each bn are counted. An obvous problem wth such an approach nvolves the determnaton of the approprate bn wdth and placement of bns. Alternatve choces can have a substantal mpact on the shape of the estmated densty, especally n cases of small sample szes. In cases where one s workng wth large dataset to estmate nsurance premum rates or other parameters of an nsurance contract, a smple nonparametrc approach may nvolve smply countng the number of postve events that occur n a large sample (to gan a measure of the probablty of the event) or measurng the average level of loss (to gan a measure of the expected loss). Such a method has dstnct advantages n that no assumptons regardng the underlyng probablty dstrbuton are needed. Ths emprcal approach s commonly referred to as usng emprcal estmates of premum rates or expected loss, and s used n ratng the U.S. federal crop nsurance program. The obvous shortcomng of ths emprcal approach s ts relance on a large sample and the potental dffculty n measurng rare events. For example, usng smple bnomal probabltes, the lkelhood of observng a 1 n 100 year event one or more tmes n a sample of 100 years of data s only When one s nterested n measurng the rsks assocated wth rare, catastrophc events, usng emprcal lkelhood measures s problematc. In ths lght, such emprcal measures are often combned wth other approaches such as catastrophc loadng or rsk poolng to measure catastrophc rsks and the lkelhood of other rare events. An alternatve approach to measurng yeld rsks nvolves the applcaton of nonparametrc kernel methods (see Box 2). A shortcomng assocated wth kernel densty estmaton technques pertans to ther lack of structure relatve to parametrc methods. By ts very nature, a nonparametrc technque mposes mnmal structure on the estmated dstrbuton. If one knows the approprate parametrc structure (.e., the proper dstrbuton functon), t s more effcent from a statstcal perspectve to mpose ths structure when estmatng the dstrbuton. However, an napproprate choce of a parametrc dstrbuton may result n based estmates of the dstrbuton and thus naccurate nsurance premum rates. Ths tradeoff between effcency and bas s omnpresent n consderatons of the proper specfcaton of econometrc or statstcal models and one may be wllng to accept bases f the gans n effcency are large (and vce versa). 15

17 Box 2. Nonparametrc kernel methods The kernel densty estmator places an ndvdual kernel at each observaton from the densty of nterest. The area under these kernels at each pont s summed to gve an estmate of the densty at that pont. The area under a properly defned kernel must ntegrate to one. Many specfc functons, ncludng properly defned probablty densty functons (pdf), are sutable as kernels and research has demonstrated that the estmated densty s not senstve to the choce of kernel functon n moderately szed samples. More mportant than the specfc functon chosen for the kernel s the wdth of the kernel. Ths wdth, whch corresponds to the varance of the densty f a pdf s used for the kernel, s called the bandwdth parameter. It s fully analogous to the bn wdth chosen n the constructon of a hstogram. A varety of methods have been developed for choosng an optmal bandwdth. The smplest nvolves a rule of thumb, whch Slverman (1986) has shown to perform well as the data devate from normalty. Alternatve methods ncludng varous plug-n approaches and crossvaldaton procedures may offer advantages as data depart radcally from normalty. Nonparametrc kernel densty estmaton technques were appled to rate area-wde crop nsurance contracts by Goodwn and Ker (1998). Ker and Goodwn (2000) consdered a number of extensons to the nonparametrc approaches, ncludng methods that allowed the bandwdth parameter to vary across dfferent ponts of the densty and emprcal Bayes type methods that utlzed aggregate data to construct prors for use n the estmaton of nonparametrc denstes. Source: authors. Therefore, a tradeoff exsts between the effcency assocated wth mposng structure n the estmaton of rsk versus the potental for bases when one mposes an napproprate parametrc structure. Rarely does the analyst know a pror the correct parametrc form. Nonparametrc procedures, whle offerng substantally more flexblty, necessarly requre more observatons n order to obtan relable probablstc estmates. Precse recommendatons regardng whch approach to use are dffcult to make snce the approach s lkely to depend on the specfc characterstcs of the problem. In general, f one has strong pror knowledge of the statstcal dstrbuton that s lkely to descrbe yelds or f one s workng wth small data sets (of less than observatons), a parametrc approach s lkely to be preferred. In cases where a large amount of data s avalable and one s unsure of the approprate dstrbuton, a nonparametrc approach offers advantages n terms of ts flexblty. 4. Usng Rsk Models to Develop Crop Insurance Contracts We consder the smple case of a yeld nsurance contract that pays ndemntes at a predetermned, fxed prce f realzed yelds fall beneath some threshold that defnes a guarantee. Such a contract has characterzed the U.S. multple perl crop nsurance program over most of ts exstence. Two fundamental parameters are nherent n such an nsurance contract. Frst, one must establsh the guarantee that determnes the condtons under whch ndemnty payments wll be pad. The yeld guarantee establshes total lablty (.e., the maxmum possble ndemnty or, equvalently, the amount of ndemnty pad n the event of a total loss). Second, one must establsh the approprate prce (premum) that should be charged for the coverage offered under the contract. An error n ether parameter can be costly for the nsurer and/or the nsured. 16