Securitizing Area Insurance: A Risk Management Approach

Transcription

1 ournal of Fnancal sk Management 13. Vol., No.3, 55-6 Publshed Onlne September 13 n Sces ( Securtng Area Insurance: A sk Management Approach Pasquale Luco Scando Unversty of ome Tor Vergata, ome, Italy Emal: scando@unroma.t eceved Aprl 19 th, 13; revsed May 19 th, 13; accepted May 6 th, 13 Copyrght 13 Pasquale Luco Scando. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. Ths paper eamnes the possblty of developng a rsk management nstrument by desgnng a fnancal securty whose value s lnked to the average revenue of a gven area. Ths type of program s suffcently general to be consdered for any group of busnesses that face producton uncertanty. In agrculture, t has been proposed as an alternatve to multple perl crop nsurance programs, as area yeld, revenue or ranfall nsurance n order to elmnate e ante and e post moral haard. Whle most of the lterature concentrates on the determnaton of value of the ndemnty and the payment of such an nsurance, ths paper focuses on the fact that, unlke other forms of nsurance, area nsurance can be cast n the form of a hedgng securty and, as a consequence, rather than dependng only on the demand for dversfcaton (the beta of the Captal Asset Prce Model), t makes possble a rsk shftng strategy based on the heterogenety of rsk atttudes of the economc agents operatng n a gven area. Keywords: sk; Insurance; Moral Haard; Securty Introducton Objectve of ths paper s to present and analye the characterstcs of a new type of contngent contract, amed at securtng nsurance aganst producton rsks. Whle the concept proposed has a wder applcablty, ths paper consders n partcular the case of area nsurance. Ths s an nsurance program that pays an ndemnty proportonal to an ndcator of average revenue or ncome for a gven area. In the case of agrcultural AI, for eample, the ndcator may be drectly yeld, or some other statstc, such as ranfall, that can be used as a relable predctor for the revenue shortfall, whch s the object of nsurance. The greatest advantage of AI and GI s that, unlke other nsurance programs, and provded that the subject nsured s not large enough to affect the average revenue of the area, ths type of program s almost completely free of moral haard. In the USA, eperence wth area nsurance n agrculture began n 199, when the farm bll provded funds for the Federal Crop Insurance Corporaton (FCIC) to plot-test new nsurance products. The Group sk plan was the frst of these epermental programs, ntroduced n 1993 as a plot area-yeld crop nsurance for soybeans. Ths program was followed n 1994, as a consequence of budgetary provsons n the 1993 Omnbus Budget econclaton Act, by a mandate to FCIC to offer area based nsurance on about 1 countes for barley, cotton, peanuts, gran sorghum, soybeans, and wheat. Although these contracts have not appeared to be very popular wth farmers so far, commtment of FCIC has remaned strong over the years, and the coverage of the program has been consderably etended (more than 3,) from the lmted number of farmers (less than 1,) partcpatng n the frst two years. A recent applcaton of GI nsurance s Group sk Income Protecton (GIP), an area-based revenue nsurance product that pays the nsured n the event the county average per-acre revenue falls below the nsured s trgger revenue. GIP derves from the Group sk Plan of Multple Perl Crop Insurance. The addton of a prce component to GP to form a revenue guarantee was developed by Dr. Bruce Babcock and Dr. Dermot Hayes, Professors of Economcs, Iowa State Unversty. GIP s lnked to the Chcago Board of Trade (CBOT) negotatons, snce ts epected prce s defned as the smple average of the last fve fnal daly settlement prces n February on the CBOT December corn futures contract and the nearby November soybean futures contract for the current crop year. Harvest prce, on the other hand, s defned as the smple average of the fnal closng daly settlement prces n November on the CBOT nearby December corn futures contract and n October on the nearby CBOT November soybean futures contract for the current crop year. A GIP ndemnty payment wll occur f the county revenue s less than the producer s trgger revenue based on the selected coverage level. In ths paper, I eamne the possblty of desgnng a rsk management nstrument through a fnancal securty whose value s lnked to the average revenue of a gven area. Ths type of program s suffcently general to be consdered for any group of busnesses that face producton uncertanty. In agrculture, t has been proposed as an alternatve to multple perl crop nsurance programs, as area yeld, revenue or ranfall nsurance (Mranda, 1991, 1999; Skees et al., 1997) n order to elmnate e ante and e post moral haard. Wthn the same contet, Mahul (1999) and Mahul et al. (1) have shown how to determne the optmum value of the ndemnty and the payment of such an nsurance. However, I focus on the fact that, unlke other forms of nsurance, area nsurance (AI) can be cast n the form of a hedgng securty and, as a consequence, rather than dependng only on the demand for dversfcaton (the beta of the Captal Asset Prce Model), t makes possble a rsk shftng Copyrght 13 Sces. 55

2 strategy based on the heterogenety of rsk atttudes of the economc agents operatng n a gven area. The Compettve Market Model Assume that a group of farmers of a gven area are engaged n the producton of several crops,.e. corn, wheat, soybeans, etc., and that the ncome resultng from producton s uncertan, because t depends on weather, pest attacks, prce fluctuatons and other random events. The -th farmer chooses the number of acres allocated to each crop by mamng the epected value of utlty, whch s defned as functon ncreasng n the farmer s net revenue (postve frst dervatve). The utlty functon s also supposed to be such that ts ncrease declnes wth ncome ncreases (negatve second dervatve). Formally, ndcatng wth U the utlty nde for the -th farmer, wth y yj the vector of total ncome generated by the j-th crop, wth the vector j of the number of acres allocated to the j-th crop, and wth the vector j of the stochastc net revenue per ha of the j-th crop, we can formulate the decson problem of the -th farmer as follows: ma EU y ; subject to: y, D b (1) where E s the epectaton operator, prmes denote transposes, D dkj s a matr of nput-output coeffcents, measurng the quantty of the kth - factor requred to produce one unt of the j-th crop, and b bk s a vector of resource constrants for the -th farm k 1,,, K. Takng a second order Taylor seres appromaton of the utlty functon around the mean values of the random varables j, or, alternatvely, assumng that the pj s are dstrbuted accordng to a two parameter dstrbuton (e.g. the normal), we obtan the famlar E-V utlty functon:.5 E, U du dy U U p U, () y yy where p y denotes the frst dervatve (.e. the margnal utlty) of each farmer wth respect to total ncome from all crops, U d d yy U y s the correspondng second dervatve, and jm (3) s the varance-covarance matr of crop ncomes per ha of the -th farm pj. Dvdng both sdes of () by U, we can re-wrte the problem n (1) as follows: ma EV p,5, subject to: D b. where V U U and y Uyy Uy equals Pratt coeffcent of absolute rsk averson. To solve ths problem, form the Lagrangean: where L p.5 D b (4) k s a vector of Lagrange multplers. The Kuhn-Tucker condtons for the soluton of (3) can be wrtten as follows: K (5) p d or, m1,,, m jm j km k m j1 k1 dkjj bk or k, k 1,,, K (6) j1 Epresson (5) states the well known condton of optmalty requrng that for each crop to whch a non ero area s allocated, epected revenue per ha be equal to the rsk premum plus the cost per ha of all nputs evaluated at the shadow prces k. Each of these shadow prces, accordng to condton (6), on the other hand, are non ero only f the correspondng resource constrant s bndng. Let s ntroduce now the possblty of acqurng (gong long or short) a securty, whch yelds to long holders a random payoff I gs, where g s a postve constant, for S, and requres a gven payment I c for S for long holders. For short holder, the stuaton would be reve rsed, as they would receve a payoff c from the long holders for S and would pay them I gs for S. S s thus the trgger level for the payoff, n 1 j n j 1 j1 j 1 j1 s random revenue per ha n the area (.e. for all farmers consdered). Denotng wth G the dstrbuton functon of area revenue per ha, the epected value of the premum s d 1 EI g G c G. We can reformulate the -th farmer mamaton problem as follows: ma EV ma.5, q p q EI (7) qi q Cov, I subject to: D b ; j 1,,, n; j 1,,, The objectve functon for the -th farm n (7) s now formed of four parts: 1) the epected value of net farm revenue from crop producton, ) the epected gan from holdng (long or short) the securty, 3) the rsk premum for crop producton, 4) the rsk premum for holdng the securty, 5) the conjont rsk premum to hold a portfolo wth both crops and the securty. Note that the -th farmer decdes the number of hectares j to allocate to the j-th crop j 1,,, and the number of (ha equvalents) q of securtes to hold. Note also that q n the case of long holdng (or buyng) and q n the case of short holdng (or sellng). The formulaton n (7) descrbes the plannng problem of a rsk averse farmer (assumng ). Ths ag ent s offered the opportunty to go long (.e. to buy) or short ( to sell) on an nsurance-lke securty based on the average revenue accrung from a set of actvtes (e.g. agrculture) wthn a gven area or otherwse defned group of agents. In practce, rather than average ncome, whch s dffcult to observe drectly, the securty, n echange for a premum c pad by the long agent (respectvely, pad to the short agent) n the years where an observable varable (for eample, ranfall or any other nde that can be easly observed and s correlated wth area revenue) s above a trgger level, pays the amount h n the bad years. Denotng wth F the dstrbuton functon for, assum- 56 Copyrght 13 Sces.

3 ng s non-negatve, we can compute the epected value of I as follows: d 1 EI h F c F (8) The Kuhn-Tucker condtons for the soluton of problem (7) may be wrtten by addng to the constrant n (7) the followng: p qcov I D (9) I EI q Cov, I, (1) where s a vector of shadow prces. Epresson (9) states that for all non ero actvtes epected revenues per ha should equal m argnal costs, where these are gven by rsk premums, nclusve of nsurance, and by resource opportunty costs evaluated at shadow prces. Epresson (1), on ts part, states that the nsurance net payoff should tself equal ts rsk premum. Indcatng wth the sgn the vectors and the submatrces correspondng to non ero actvtes, and applyng the varance and covarance defntons, we can wrte: p q I D q Cov, EI Cov, I I (11) wh ere p s a,1 vector of revenues per ha for non ero farm actvty levels j, j 1,,,, Cov I ; s the covarance between I and, and s the vector of coeff- cents obtaned by projectng lnearly the dfferences between the -th farm revenue per ha and the correspondng mean onto the dfference betwee n the statstcal ndcator of total revenue per ha of the area nvolved n the scheme and the correspondng mean accordng to the so called regressablty assumpton (Bennnga et al., 1984): u beng a (vector valued) random varable ndependent of y. Accordng to ths assumpton, for each crop, the dfference between farm revenue per ha and ts mean can be decomposed lnearly nto two parts: one proportonal to the dfference be- tween the current value and the mean value of an ndcator of revenue per ha of the entre area nvolved n the scheme, and an ndependent, dosyncratc random component, representng non dversfable rsk. Each component of the vector s a coeffcent j smlar to the wdely known coeffcent of the Captal Asset Prcng Model (CAPM) and measures the senstvty of the revenue per ha of the j-th crop to movements n area revenues per ha. Gong back to the mamng condtons, epresson (11) states that prces of all non ero actvtes should equal margnal costs at the optmum, whle epresson (1) ndcates that the quantty of nsurance acqured by the -th frm equals the net value of the nsurance, ncludng ts utlty from rsk dversfcaton. Ths result can be seen more clearly, droppng the astersks for smplcty, by wrtng t as follows: where q p E u (13) d 1 F h F c F (14) But Cov I, I 1, I F Cov, I so that the correlaton between area revenue and the nsurance s Cov I, F. I Ths brngs us to state the followng proposton: Proposton 1. The amount of the securty purchased (sold) by each farmer s proportonal to the rato between her epected gan and her rsk premum (the subjectve beneft-cost rato ) plus the agent s relatve gan from dversfcaton (her objectve beneft-cost rato). Dvdng (14)) by j where s the,1 sum vector, we obtan the epresson for the amount of coverage, defned as the rato of the quantty purchased or sold of the securty n ha equvalents to total cultvated has of the ndvdual farm: q d 1 F h F c F 1, (15) where denotes the relatve coeffcent of rsk averson of the -th agent and s the rato between the beta of the ndvdual crop and the land cultvated for the -th agent. Corollary 1. The amount of coverage purchased (sold) wll be proportonal to the sum of two terms: 1) the ndvdual rato between the epected net beneft and the rsk premum, and ) the weghted average of the agent betas. Comment. The nsurance program ntroduces an element of rsk due to the varablty of area revenue. Those who go long on the securty, n fact, pay n every perod a premum c to those who go short on the same securty, unless area revenue (or ts proy) s below ts threshold level, n whch case the shorters wll have to pay the longers a compensaton. The compensaton pad by the shorters to the longers, n turn, wll be a functon of the dfference between current area revenue (or the current value of the proy used) and the threshold. Demand for the securty wll be larger, the hgher the trgger value for the payment of the ndemnty, the smaller the rsk premum that each farmer s wllng to pay to hold the securty, the lower the premum to be pad to the short holders and the hgher the correlaton between the performance of the buyer and the average performance n the area. On the other hand, supply for the securty wll be larger, the larger the premum 1 Iy FIc F Z 1 E E F Var Cov, d, 1 EyF Ey c F E Copyrght 13 Sces. 57

4 pad to shorters, the lower the rsk premum and the larger, n absolute value, the negatve correlaton between the suppler s revenue per ha and area revenue per ha. Note also that n order to hold the securty, t s not necessary that the agent s a producer,.e. the dversfcaton component may be ero and the quantty of the securty purchased (sold) may stll be postve. In ths case, the agent holdng (long or short) the securty wll act as a pure speculator. Equaton (15) can be nterpreted as the demand for holdng long the securty (for q ) and the demand for holdng t short q.snce each unt of the securty promses to yeld a net epected revenue of r 1 EI F E h c F to long holders and EI to short holders, the demand elastcty wth respect to the payment to be made s c 1 F EI F for long holdng and F E h EI F r for short holdng. In equlbrum, demand for nsurance should equal supply so that the sum of ecess demand n q. Summng over all farms the terms n (15) and solvng for c, we can thus fnd the equlbrum value of the premum for the nsurance polcy: F F X c Eh (16) 1 n where. n 1 1 n 1 s the harmonc mean of the ndvdual rsk averson coeffcents, X n s the correspondng average relatve rsk averson coeffcent, evaluated at the average level of cultvated land n the area, E h E h t for, and I have used the property:, n 1 X, X I 1 j1 s total land cultvated n the area and Cov, s the lne ar regresson coeffcent of the ndcator wth respect to av erage area revenue per ha. Proposton. The payment for the short holders of the securty s equvalent to an nsurance prem um. The level of the premum that equlbrates demand and supply s always greater Usng agan the regressablty assumpton, we can wrte: E E v, where v s a randomly dstrbuted dsturbance. Substtutng nto (13), we obtan: p E v u j than the actuarally far premum (.e. the epected value of the payoff for the long holders) dependng on the average (area) rsk premum from holdng the securty. Comment. Gven the number of traders, n equlbrum the epected value of the payoff for both long and short holders equals the average premum for rsk that the farmers nvolved n the scheme are wllng to pay. Therefore, the premum to be pad (respectvely, to pay) from those who buy (to those who sell) the securty wll equal the sum of the epected ndemnty and of the average (n the harmonc mean sense) subjectve rsk. The premum wll be greater than ts actuarally far level of a loadng factor reflectng average rsk averson (n the harmonc mean sense), where the average s taken over both long and short securty holders. An ncrease n the rsk averson of any agent, n other words, ncreases the premum (.e. the take of the short holders) ndependently on whether ths concerns somebody who would buy or sell the securty. Corollary. The equlbrum epected level of the equvalent premum (the payment to short holders) equals the epected utlty gan of the average (n the harmonc mean sense) agent. Comment. Epresson (17) shows that, n order to be feasbly supported by the farmers of a gven area, the contract proposed must be far n the sense that the epected charge should equal the epected utlty gan of a representatve farmer. Such an agent s defned as one, whose absolute (or relatve computed at the mean) rsk averson coeffcent s the harmonc mean of the rsk averson coeffcents of all other farmers supportng the scheme. In other words, n order for the scheme to be sustanable, the representatve agent should be unable to gan (or lose) on average from purchasng or sellng the securty. Ths noton of farness does not concde wth the usual actuaral noton, snce a rsk loadng factor s added to the actuarally far premum. Substtutng (16) nto (15), we obtan: X q n (17) Usng the defnton of relatve rsk averson, we can also wrte: q H (18) where and are both coeffcents of relatve rsk averson respectvely at the overall average and average revenue level for the -th agent. Proposton 3. The equlbrum holdng level for the -th farmer of an area nsurance securty s ndependent of both the se of the premum and the trgger level for the ndemnty. It equals the dfference between a measure of the demand for dversfcaton (the average beta of the farmer) and the recprocal of a measure of relatve rsk averson (the rato of average to the farmer s own rsk averson coeffcent). Comment. In equlbrum, the epected value of the payoff equals the average rsk premum of the partcpants to the scheme (Proposton ). Thus, whle the epected payoff for long and short holders wll vary wth the trgger level of the ndemnty, the equlbrum level of the amount of the securty bought and sold by each farmer wll not. Proposton 3 translates nto the smple rule: whatever the ndemnty epected, buy (sell) the securty for a share equal to your average beta mnus the rato of average to your rsk averson coeffcent. The rea- 58 Copyrght 13 Sces.

5 son for ths s that the demand for the securty depends n the frst nstance on the degree to whch t wll dversfy the portfolo of the productve actvtes of the farmer. Because carryng the securty mples an addtonal rsk, ths degree, whch can be measured by the weghted beta, has to be corrected wth the rato between average and ndvdual rsk averson. The larger ths rato, the larger the rsk that the farmer s wllng to carry wth respect to the average, thus requrng less nsurance. Demand (and supply) for the securty wll be ero f all farms are dentcal (all betas are equal to one) and all rsk averson coeffcents are also equal. In general, however, the amount of the securty demanded (offered) wll be larger (smaller) the larger (the smaller) the correlatons nvolved (between the nde and area revenue and between own and area revenue), and the smaller (the larger) the rato between average and own rsk averson. For eample, f the -th agent rsk averson s twce the harmo-, optmum coverage wll be 1. Note, n partcular, that even f all farmers revenues are postvely correlated wth area revenues, t wll pay for some of them to go short on the securty. A suffcent heterogenety n rsk averson, n other words, wll ensure that a market for the hedgng securty may develop even n the absence of negatve correlatons across farms. For the same reason, n equlbrum, we may epect pure speculators to hold short postons. From Equaton (5), by totally dfferentatng wth respect to nc mean, and 1 q, we can derve for the m-th crop: 1 d D F for all m dq (19) Provded that matr n the square parenthess s full, t wll also be postvely defnte snce the frst term s the varancecovarance matr, whle the second s the matr gven by the product of the nput-output coeffcents by the shadow prces. Ths matr wll be sngular f the number of factors s smaller than the number of products, but ts sum wth a full varance covarance matr wll also be full. Moreover, by the propertes of shadow prces at the optmum, an ncrease n crop producton may not result n a decrease n any shadow prce k (.e. for all k,, j). j Thus, we can state the followng proposton: Proposton 4. In equlbrum, for long holders of the securty, producton of the crops whose revenues per ha are postvely (negatvely) correlated wth area revenue per ha ncreases (decreases). For short holders, producton of the crops whose revenues per ha are n egatvely (postvely) correlated wth area revenue ncreases (decreases). Comment. The ntroducton of area nsurance determnes a dfferentaton n farm portfolo holdngs, snce farms now hold, n addton to ther cultvated plots, a securty whose performance s negatvely correlated wth average performance over all farms and crops. If the m-th crop revenue per ha s postvely correlated wth area revenue, two effects wll ensue: 1) a reducton of producton costs, due to the possblty of epandng ts producton and coverng ts rsk by gong long on the securty, and ) an ncrease n costs due to the fact that nsurance ncurporates a rsk premum dependng on average rsk averson and on the varance of area revenue. For long holders, the frst effect wll domnate the second one, whle the opposte wll occur for short holders. By a basc property of mathematcal program- mng, snce the ntroducton of a new dmenson of optmaton s equvalent to the removal of a constrant, we have that D q D q,.e. the shadow cost of the resources decreases as a consequence of the ntroducton of the securty. Ths mples a hgher frst term on the rght hand sde of Equaton (19). For q, and m (.e. the m-th crop s postvely correlated wth area revenue, producton of the m-th crop wll thus unequvocally ncrease, snce shadow costs are lower and the rsk premum to hold the new securty s also lower. In other words, farmers can now hedge the crops that are postvely correlated wth area revenue by gong long on a securty whch s also negatvely correlated wth area revenue. If q, and m, we have a smlar result. In other words, farmers who grow crops that are negatvely correlated wth area revenue can now hedge them by gong short on a securty whch s also negatvely correlated wth area revenue. For the same reasons, we wll see a reducton of the producton of long holders (short holders) crops negatvely (postvely) correlated wth area revenue. These crops, n fact, do not offer any further opnty to ncrease earnngs because of the ntroducton of the portu new securty. Concluson In ths paper I have shown that area nsurance may be molded n the form of a securty or a contngent clam, whose payoff depends on the value taken by a statstc of a gven populaton. Ths statstc, whch may be the value of average or medan ncome, revenue or yelds, or any statstc correlated wth t (e.g. ranfall), dsplays a value that changes wth the state of nature. When the statstc s below a crtcal level, the people who have gone short on the securty wll pay a premum, whle when t s above the crtcal level, they wll collect a payment from those who have gone long. Actuaral group or area nsurance may be consdered a specal case where only one nsurer goes short on the equvalent securty, by commttng herself to pay the premum n the bad states, n echange for the payment n the good states. The rewards that the securty promses to long and short holders possess some smple propertes. Frst, the amount of the securty purchased (sold) by each agent s proportonal to the rato between her epected ndemnty and her rsk premum (the subjectve beneft-cost rato ) plus the agent s relatve gan from dversfcaton (hs objectve beneft-cost rato). Second, gven an epected reward for long holders, the securty pay off for short holders that equlbrates demand and supply s greater than the actuarally far premum and depends postvely on the average (area) rsk premum from buyng or sellng nsurance. Thrd, the equlbrum epected level of the premum charged for the securty equals the epected utlty gan of the average (n the harmonc mean sense) agent. Fourth, the optmal level of nsurance demanded or suppled n equlbrum wll be proportonal to the rato of the beta of each agent to her share of total actvty level mnus the rato of average relatve rsk averson to the agent s own rsk averson coeffcent. EFEENCES Bennnga, S., Eldor,., & Zlcha, I. (1984). The optmal hedge rato n unbased futures markets. ournal of Futures Markets, 4, Feder, G., ust,. E., & Schmt, A. (198). Futures markets and the theory of the frm under prce uncertanty. Quarterly ournal of Eco- Copyrght 13 Sces. 59

6 nomcs, Mahul, O. (1999). Optmum area yeld crop nsurance. The Amercan ournal of Agrcultural Economcs, 81, Mahul, O., & Stutley, C. (1). Government support to agrcultural nsurance: Challenges and optons for developng countres. World Bank Publcatons. M randa, M. L. (1991). Area yeld crop nsurance reconsdered. Amercan ournal of Agrcultural Economcs, 73, Skees,.., Black,.., & Barnett, B.. (1997). Desgnng and ratng an area yeld crop nsurance contract. Amercan ournal of Agrcultural Economcs, 79, Vargas Hll,., Kumar, N., & Hoddnott,. (1). Adopton of weather-nde nsurance: Learnng from wllngness to pay among a panel of households n rural Ethopa. Mmeo: IFPI. 6 Copyrght 13 Sces.