tails curtailed:accounting for nonlinear dependence …. j. agr... · a common theme in ﬁnancial...

TAILS CURTAILED:ACCOUNTING FOR NONLINEAR

DEPENDENCE IN PRICING MARGIN INSURANCE FOR

DAIRY FARMERS

MARIN BOZIC, JOHN NEWTON, CAMERON S. THRAEN, AND BRIAN W. GOULD

Livestock Gross Margin Insurance for Dairy Cattle (LGM-Dairy) is a risk management toolfor protecting milk income over feed cost margins. In this article, we examine the assumptionsunderpinning the method used to determine LGM-Dairy premiums. Analysis of the milk–feeddependence structure is conducted using copula methods, a rich set of tools that allow modelers tocapture nonlinearities in dependence among variables of interest. We find a significant relationshipbetween milk and feed prices that increases with time-to-maturity and severity of negative priceshocks. Extremal, or tail, dependence is the propensity of dependence to concentrate in the tailsof a distribution. A common theme in financial and actuarial applications and in agriculturalcrop revenue insurance is that tail dependence increases the risk to the underwriter and resultsin higher insurance premiums. We present, to our knowledge, the first case in which tail depen-dence may actually reduce actuarially fair premiums for an agricultural risk insurance product.We examine hedging effectiveness with LGM-Dairy and show that, even in the absence of basisor production risk, hedging horizon plays an important role in the ability of this tool to smoothfarm income over feed cost margins over time. Rating methodology that accounts for tail depen-dence between milk and feed prices extends the optimal hedging horizon and increases hedgingeffectiveness of the LGM-Dairy program.

JEL codes: G13, Q13, Q18.

The objective of this article is to provide anevaluation of the Livestock Gross MarginInsurance for Dairy Cattle (LGM-Dairy),which is a pilot insurance program admin-istered by the USDA’s Risk ManagementAgency (RMA). LGM-Dairy was created in2008 to provide dairy producers with indi-vidualized protection against catastrophicfinancial losses. LGM-Dairy allows dairyfarmers to insure an income over feed cost(IOFC) margin, which is defined as gross

Marin Bozic is an assistant professor in the Department ofApplied Economics at the University of Minnesota–TwinCities. John Newton is a clinical assistant professor in theDepartment of Agricultural and Consumer Economics at theUniversity of Illinois Urbana-Champaign. Cameron S. Thraenis an associate professor in the Department of Agricultural,Environmental, and Development Economics at The OhioState University. Brian W. Gould is professor in the Depart-ment of Agricultural and Applied Economics at the Universityof Wisconsin–Madison.

The authors are grateful to David Hennessy, two anony-mous referees, as well as various conference and seminarparticipants for valuable insights that have substantiallyimproved this work. Financial support from Minnesota MilkProducers Assocation and the Ohio Agricultural Research andDevelopment Center is gratefully acknowledged.

milk revenue less the declared cost of pur-chased livestock feed (Gould and Cabrera2011; Risk Management Agency 2005; Thraen2012). Various authors have examined thedesign and use of livestock revenue insur-ance products for margin risk management(Burdine and Maynard 2012; Hart, Babcock,and Hayes 2001; Liu 2005; Novakovic 2012;Turvey 2003; Valvekar, Cabrera, and Gould2010; Valvekar et al. 2011). However, thisanalysis is the first to address nonlineari-ties in dependence between milk and feedprices and to evaluate the performance ofalternative LGM-Dairy rating methods thatincorporate flexible dependence structures.Analysis of the milk–feed dependence struc-ture is conducted using copula methods,which include a rich set of tools that allowmodelers to capture nonlinearities in depen-dence among variables of interest. Sklardemonstrated that, for any multivariable dis-tribution function, the dependence structurecan be modeled separately from univariablemarginal distributions. The dependence struc-ture, called a copula, is itself a multivariable

Amer. J. Agr. Econ. 1–19; doi: 10.1093/ajae/aau033© The Author (2014). Published by Oxford University Press on behalf of the Agricultural and Applied EconomicsAssociation. This is an Open Access article distributed under the terms of the Creative Commons AttributionNon-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/), which permits non-commercialre-use, distribution, and reproduction in any medium, provided the original work is properly cited. Forcommercial re-use, please contact [email protected]

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distribution function with uniform marginaldistributions (Genest and Favre 2007; Nelsen2006; Woodard et al. 2011).

Extremal, or tail, dependence is thepropensity of dependence to concentratein the tails of a distribution (Joe 1997). Acommon theme in financial and actuarialapplications and in agricultural crop revenueinsurance is that tail dependence increasesthe risk to the underwriter, thus increas-ing insurance premiums (Donnelly andEmbrechts 2010; Goodwin 2012; Kouskyand Cooke 2009; Liu and Miranda 2010).Our analysis is the first, to our knowledge, topresent a case in which tail dependence mayactually reduce actuarially fair premiums foran agricultural risk insurance product.

Analysis of the effectiveness of hedgingthat uses this product complements the anal-ysis of the LGM-Dairy rating method. Awide body of literature has examined opti-mal hedge ratios and measures of hedgingeffectiveness (Chen, Lee, and Shrestha 2003;Garcia and Leuthold 2004; Lien and Tse2002; Williams 2001). More recent work hasfocused on hedging downside risk (Chen,Lee, and Shrestha 2001; Kim, Brorsen, andAnderson 2010; Mattos, Garcia, and Nelson2008; Power and Vedenov 2010; Turvey andNayak 2003). However, there has been lim-ited analysis of the importance of hedginghorizon for hedging effectiveness (Chen,Lee, and Shrestha 2004; Ederington 1979;Geppert 1995; Malliaris and Urrutia 1991).We show that, even in the absence of basis orproduction risk, the hedging horizon plays animportant role in the ability of LGM-Dairyto smooth farm IOFC margins over time.For a dairy farm that buys all of its livestockfeed needs, we find that a rating method thataccounts for tail dependence between milkand feed prices extends the optimal hedginghorizon and increases hedging effectiveness.

We proceed with a discussion of the LGM-Dairy rating method, followed by an analysisof assumptions about the milk–feed depen-dence structure. We find that the Gaussiancopula fails to adequately represent theobserved data. We also find strong evidenceof lower tail dependence between distantmilk and corn futures contracts. We proposealternatives to the official LGM-Dairy ratingmethod and analyze cost implications for avariety of insurance policy profiles. Next, wepresent an analysis of hedging effectiveness.We conclude by discussing implications ofthe research findings for evaluating other

dairy margin risk insurance products that areoffered by the private sector and the federalgovernment in the 2014 Farm Bill.

Introduction to LGM-Dairy

With the introduction of a market-orienteddairy policy in the late 1980s, U.S. farm-levelmilk prices have exhibited increased volatilityas supply and demand shocks are no longercompensated for by government purchases ofexcess dairy products. However, despite twodecades of increased revenue volatility, openinterest in dairy futures and options contractsstill accounts for less than 10% of milk mar-ketings. This suggests that dairy producershave used accumulated equity as a buffer tosmooth net farm income across consecutivemarketing years. More recently, high volatil-ity of livestock feed prices has necessitateda shift in focus from milk revenue to IOFCmargin risk management. A dairy producer’sability to borrow against farm equity, com-bined with the need to protect IOFC marginsrather than just milk revenue, implies that aninsurance product of high interest to dairyproducers would be an affordable insurancepolicy that protects against rare but deep andprolonged IOFC margin slumps.

The LGM-Dairy insurance product allowsdairy farm operators to purchase insuranceto protect against decreases in their grossmargin, which is defined as the differencebetween milk revenue and purchased feedcosts (Gould and Cabrera 2011). Under thisinsurance policy, an indemnity at the endof the coverage period is the difference, ifpositive, between the total guaranteed grossmargin determined at the purchase of theinsurance contract and the total actual grossmargins realized over the desired coverageperiod. Unlike dairy and grain futures andoptions contracts, which protect against milkand feed price shocks in a specific month,LGM-Dairy protects against a decline inaverage IOFC margins over a period of upto ten months.1 LGM-Dairy offers the abil-ity to insure average margins, rather than asequence of monthly margins, as one would

1 LGM-Dairy insurance rules do not allow coverage for themonth immediately following the purchase date. The maximumten-month coverage period protects average margins over thesecond through eleventh month following the month when thepolicy is purchased.

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Bozic et al. Tails Curtailed 3

do with the purchase of milk puts and live-stock feed calls. This renders LGM-Dairya more affordable risk management toolcompared with exchange-traded instruments.

Let t denote the date of the LGM-Dairycontract purchase, and let i = 1, . . . , 10 denotethe ith month insurable under the LGM-Dairy contract. Let pM

i denote expected milkprices and Mi the insured milk marketingsin each of up to ten insurable months. Sim-ilarly, let pC

i and pSBMi , respectively, denote

expected corn and soybean meal prices, andlet Ci and SBMi denote declared amounts ofcorn and soybean meal, respectively. Finally,let D denote the gross margin deductible,which is the threshold decline in expectedgross margin at which LGM-Dairy paysindemnities. All decision variables can bestacked into an input vector It , where

It = [M1, . . . , M10, C1, . . . , C10,(1)

SBM1, . . . , SBM10, D]′.

Given the above, the gross margin guaranteeGt is calculated as:

Gt(It) =10∑

i=1

(pM

i − D)

Mi −10∑

i=1

pCi Ci(2)

−10∑

i=1

pSBMi SBMi.

The realized gross margin AT is calculated atthe LGM-Dairy contract expiry date T as

AT (It) =10∑

i=1

sMi Mi −

10∑i=1

sCi Ci(3)

−10∑

i=1

sSBMi SBMi

where sMi , sC

i , and sSBMi are realized milk,

corn, and soybean meal prices, respectively,for insurable month i under a contract sold attime t.

As stated in the LGM-Dairy ratingmethodology, expected milk and feed pricesare based on the three-day average of classIII milk, corn, and soybean meal futuresprices, before and including the contract

purchase date.2 Similarly, terminal prices arecalculated as the average of futures prices forthe last three trading days before associatedfutures contracts expiry. For those months forwhich corn or soybean meal futures are nottraded, the associated prices are defined asweighted averages of futures prices, which areobtained from surrounding months for whichfutures contracts do trade.

An actuarially fair insurance price sets thepolicy premium equal to expected indemnity.The LGM-Dairy premium at time t, Lt(It),depends on the declared milk marketings,declared feed amounts, and gross margindeductible3:

(4) Lt(It) = Et [max(Gt(It) − AT (It), 0)] .

To calculate equation (4), the joint distribu-tion function of actual LGM-Dairy pricesmust be identified. The joint distributionfunction has two fundamental structures,its marginal distributions FM,i, FC,i, andFSBM,i where i = 1, . . . , 10, and the assumeddependence structure Z. According to Sklar(1959), these two structures can be modeledseparately as

F(sM

i , sCi , sSBM

i

)(5)

= Z(FM,i

(sM

i

), FC,i

(sC

i

), FSBM,i

(sSBM

i

)).

For purposes of LGM-Dairy premium esti-mation, the marginal distributions FM,i, FC,i,and FSBM,i are assumed log-normal, with theirmoments obtained from associated futuresprices and options premiums. The functionZ is a copula that “couples” these marginaldistributions in such a way that it fully con-tains the dependence structure reflected inthe joint distribution function. Given themaximum contract length of ten months,the terminal price joint distribution will bebased on information from the third throughthe twelfth nearby class III milk futuresand option prices, the first six nearby cornfutures and option prices, and the first eightnearby soybean meal futures and optionprices. The joint distribution function willthus have twenty-four degrees of freedom

2 Under the Federal Milk Marketing Order System, class IIImilk is the milk used in cheese (except cottage) manufacturing.We use the term milk throughout this article to refer to class IIImilk.

3 Full insurance premium costs include administrative andoverhead fees and a 3% surcharge paid to the Federal CropInsurance Corporation.

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(i.e., twenty-four nonconstrained marginaldistributions).

In the official LGM-Dairy rating method,copula methods are not explicitly mentioned.Instead, the method developed by Iman andConover (1982) is used to couple terminalprice marginal distributions. The purposeof the dependence structure is to accountfor the correlation of price shocks. The offi-cial LGM-Dairy rating method representsthis dependence structure through a cor-relation matrix of rank-transformed priceshocks.

We define a price deviate xd as the dif-ference between realized and expectedLGM-Dairy prices:

(6) xd ≡ sd − pd d = 1, . . . , 24

where d = 1, . . . , 10 corresponds to milk,d = 11, . . . , 16 corresponds to corn, andd = 17, . . . , 24 corresponds to soybean mealprices. Here, we include only those insur-able months that directly correspond tocommodity futures trading months. LetX = (x1, . . . , x24)

′ be a random vector of pricedeviates. Given a sample {Xj}j=1,...,n, the

pseudo-observations Uj =(

Uj,1, . . . , Uj,24

)′

are defined as

(7) Uj,d ≡ nn + 1

Fd(Xj,d) d = (1, . . . , 24)

with Fd(x) denoting the univariate empiricaldistribution function,

Fd(x) = 1n

n∑j=1

1{Xj,d ≤ x}(8)

d = (1, . . . , 24)

where 1{·} is the indicator function.Given that Fd(x) returns the number ofobservations that are less than x, the pseudo-observations correspond to ranks of thedata because Uj,d = Rj,d/(n + 1), whereRj,d = (rank of Xj,d in {X1,d, . . . , Xn,d}).

The Spearman correlation matrix of theset of pseudo-observations {Uj}j=1,...,n is theconditional correlation matrix of the ter-minal LGM-Dairy milk and feed prices. Inthe RMA rating method, correlation amongfeed price shocks is assumed to be monthspecific. Thus, twelve correlation matrices arecalculated, each corresponding to a particular

LGM-Dairy sales event during the year. Assuch, only one observation of futures priceshocks per year can be used. The sampleused for feed prices spans 1978–2005 andresults in n = 28 for equations (7) and (8)for corn and soybean meal. When the LGM-Dairy rating method was first developed in2006, only eight years of milk futures datawere available (1998–2005). To ensure pos-itive definiteness of the correlation matrix,LGM-Dairy developers restricted the milk–feed correlation submatrix to zero. Furtherassumptions were imposed on milk–milkcorrelations, with rank correlation restrictedto a function of the distance between con-tract months only (Risk Management Agency2005).

Terminal price marginal distributions arecoupled into a joint distribution functionby using Spearman’s rank correlation coef-ficients of historical price deviates and theprocedure developed by Iman and Conover(1982). Although they do not explicitlyaddress the copula issues, Iman and Conover(1982) chose some features of their methodprimarily to induce elliptical patterns inpairwise plots of input variables. Mildenhall(2006) further demonstrated that the Iman–Conover procedure is essentially the same asusing the Gaussian copula. It follows that theLGM-Dairy rating method is, in effect, basedon a joint distribution function that can berepresented as a set of log-normal marginaldistributions coupled with a Gaussian copula.

Examining the LGM-Dairy DependenceStructure Assumptions

In this section, we examine the official LGM-Dairy rating method assumption that shocksto milk futures prices are not correlatedwith the shocks to corn and soybean mealfutures. The creators of LGM-Dairy claimthat, as of 2005, there was no appreciablecorrelation between milk and feed priceshocks. Consistent with this claim, we findthat the correlation between the monthlyU.S. average milk and corn prices receivedby farmers from 1990 to 2005 was only 0.07.In contrast, the correlation between milk andcorn prices from 2006 to 2013 was 0.57. Thelatter period was characterized by growingdemand for corn for biofuels, which causedan increase in corn prices and a decrease ingrain inventories. Lower stocks-to-use ratios

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have made grain markets more sensitive toboth supply and demand shocks, resultingin increased volatility of corn prices (Wright2014). Increased concentration in the dairyindustry may explain why milk and feed mar-kets have become more tightly integrated(Shields 2010). The percentage of U.S. milkproduction attributed to farms with morethan one thousand cows grew from 35.4%in 2005 to 50.6% in 2012. Large farms tendto purchase a higher percentage of theirfeed requirements and are generally moreprice sensitive than more integrated dairyoperations (Adelaja 1991; Tauer 1998). Bozic,Kanter, and Gould (2012) find that long-run milk supply elasticity, with respect tofeed prices, has substantially increased since2007.

Furthermore, the correlation between milkand feed price shocks may be substantiallystronger for more distant futures contracts.Economic theory of competitive marketswith free entry predicts zero long-term eco-nomic profits. Consistent with this prediction,Bozic et al. (2012) find that the term struc-ture of IOFC margins has exhibited stronglymean-reverting behavior over the past fifteenyears. Because dairy farm profits dependprimarily on IOFC margins, it follows thatany shock to feed markets that is perceivedas persistent will be more fully transmittedto deferred milk futures prices. On the otherhand, milk supply is known to be inelasticin the short run because of fixed productionfactors (Chavas and Klemme 1986). There-fore, we expect rank correlation of milk andfeed price deviates for distant LGM-Dairyinsurable months to be positive and higherthan the correlation for nearby months.

The strength of dependence between cornand milk price shocks may also be contingenton the state of the world. Extreme and rareevents may render milk and feed marketsmore tightly integrated than in a typical envi-ronment. For example, demand shocks tomilk and feed markets could be less corre-lated in normal economic times and morecorrelated during periods of large macroe-conomic instability and decreased economicactivity.

To examine the milk–corn price shockdependence structure, Figure 1(a) containsthe scatterplot for milk and corn price shocksat a four to six–months horizon, with datacovering March 1998 through April 2013.Figure 1(b) shows the scatterplot for priceshocks using a one-year horizon. Rank

correlation between the fourth nearby milkfutures and second nearby corn futuresis only 0.22. In contrast, rank correlationbetween the twelfth nearby milk futures andfifth nearby corn futures is 0.48. The scatter-plot in figure 1(b) suggests that corn priceshocks with absolute values less than $1 perbushel have barely any predictive poweron the direction of milk prices. However,dramatic corn price surges are much morelikely to be accompanied by discernibleshocks to milk prices. Of particular interestare the realizations of shocks in the lower leftcorner of the one-year horizon scatterplot.We find that the eight most adverse shocksto corn prices have been matched by theeight most adverse shocks to milk prices.Codependence in the lower tail is found tobe much stronger than it is in the center ofthe diagram. Therefore, measures of depen-dence that cannot account for state-specificcorrelation strength are not likely to accountfor weakly dependent markets, which aremore strongly integrated in extreme circum-stances. These scatterplots strongly suggestthat the assumption of zero-order correla-tion between corn and milk prices cannotbe supported. Thus, a more robust examina-tion of milk–corn price shock dependence isrequired.

To examine further whether milk–feedcorrelations are indeed positive and increas-ing in time to maturity, we calculate a fullSpearman’s rank correlation matrix by usingfutures data from 1998 through 2013. Theupper triangle of table 1 lists the selectedpairs of price shock correlations. The fullcorrelation matrix can be found in the sup-plementary appendix online. We find thatmilk–feed rank correlations increase withtime to maturity, as predicted by economictheory, with a maximum milk–corn corre-lation coefficient of 0.48 achieved for thetwelfth nearby milk and fifth nearby corncontract.

As a measure of dependence, correlation isappropriate only in the context of multivari-able normal or elliptical models, and it maynot adequately capture the apparent propen-sity of dairy and feed markets to be moreclosely integrated in extreme market environ-ments (McNeil, Frey, and Embrechts 2010).Such flexibility can be achieved by usingcopula methods that account for extremal,or tail, dependence. In the case of bivari-able copulas, tail dependence relates to theamount of dependence in the upper-quadrant

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Figure 1. Milk versus corn futures price Shocks (a) Futures price shocks at the 4–6 monthshorizon (b) Futures price shocks at the 1-year horizon

or lower-quadrant tail of a bivariable dis-tribution (Joe 1997). Extremal dependenceis measured by a coefficient of lower taildependence, defined as

(9) λL = limu→1

Pr (U1 < u | U2 < u)

and a coefficient of upper tail dependence,defined as

(10) λU = limu→1

Pr (U1 > u | U2 > u) .

A Gaussian (normal) copula always exhibitszero tail dependence, no matter how strongthe correlation coefficient is (Joe 1997).We use formal statistical tests to examinewhether the Gaussian copula can be consid-ered an appropriate approach to modelingdependence between milk and feed priceshocks. We use the Cramér-von Mises test,based on Kendall’s transforms as suggestedby Genest and Rivest (1993) and Wang andWells (2000). Goodness-of-fit tests for dis-tributions generally proceed by designinga test statistic that summarizes the dis-tance between the empirical distribution

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function and the hypothesized cumulativedistribution function. Let Fθ be the hypoth-esized cumulative distribution function andFn the empirical distribution function. TheCramér-von Mises statistic (Sn) is given by

(11) Sn =∫+∞

−∞[Fn(x) − Fθ(x)]dFθ(x).

In the test developed by Genest and Rivest(1993), a bivariable copula is first reducedto a one-dimensional distribution function,known as Kendall’s transform, with empiri-cal and hypothesized versions of Kendall’stransforms used instead of Fn and Fθ inequation (11).

We perform this test for each of the 276pairs of marginal distributions. In the lowertriangle of table 1, we present the resultingp values. To preserve space, only a subsetof p values is shown. The full table can befound in the supplementary appendix online.At a 95% confidence level, we find that theGaussian copula is rejected for 99 of 276 pair-wise comparisons. Considering the sevenththrough twelfth nearby milk contracts andthe third through sixth nearby corn contracts,we find that a Gaussian copula is rejected fornine of twenty-four pairs at a 95% confidencelevel and for seventeen of twenty-four pairsat a 90% confidence level. In the supplemen-tary online appendix we further examinethe nature of dependence between distantmilk and corn futures prices, and we find theGaussian copula to be inferior to most ofthe commonly used, bivariable, parametriccopulas that exhibit lower tail dependence.

These findings strongly suggest that theLGM-Dairy rating method should be revisedto allow a positive dependence structurebetween milk and feed prices. Revisionsshould use the method flexible enough toallow strength of the relationship betweenmilk and feed prices to increase with timeto maturity and with the severity of negativeprice shocks.

Alternative LGM-Dairy Rating Methods

We propose two new LGM-Dairy ratingmethods. Our first method, Full Correlation,allows for nonzero milk–feed correlationsand is flexible enough to allow correla-tion coefficients to differ based on time tomaturity. The second method, Empirical

Copula, goes further and allows dependenceamong futures price deviates to be strongerin extreme events.

The Full Correlation Method

The official RMA rating method requirestwelve large correlation matrices to becalculated and as many as 887 correlationcoefficients to be estimated, despite forcingall milk–feed correlation coefficients to zero.Furthermore, because correlation matricesare month-specific, only one observation of afutures price shock can be collected per cal-endar year. Under this approach, a minimumof twenty-four years of data is necessary for atwenty-four-variable correlation matrix to bepositive definite. Currently, only fifteen yearsof class III milk futures data exist. There-fore, if correlation matrices are allowed tobe LGM-Dairy sales month specific, it wouldnot be possible to relax the zero milk–feedcorrelation restriction.

Given our previous discussion, we showthat the previously described zero correla-tion restriction is not supported by theory orrecent data. Therefore, for the first alterna-tive rating method, we propose a simplifyingassumption regarding the correlation struc-ture. In this assumption, each correlationcoefficient is stipulated as dependent onlyon time-to-maturity horizons for each pairof futures price deviates. This nearby-basedapproach is flexible enough to allow cor-relation coefficients to depend on time tomaturity in addition to distance betweencontract months. At the same time, this mod-ification greatly simplifies the estimationburden. Under the current premium deter-mination method with LGM-Dairy salesmonth–specific correlations, allowing fornonzero milk–feed correlation requires esti-mating more than two thousand correlationcoefficients in twelve separate correlationmatrices. Under the Full Correlation method,a single 24 × 24 correlation matrix is used.

The Empirical Copula Method

Rüschendorf (1976) and Deheuvels (1979)introduced the Empirical Copula concept.4Given the joint distribution function inequation (5), the copula Z is a uniquelydefined distribution function with domain

4 In this section, we follow the notation of Blumentritt andGrothe (2013).

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Table 1. Conditional Rank Correlation Matrix (Upper Triangle) and Results of Cramér-von Mises Test of the Gaussian Copula (LowerTriangle)

Conditional Spearman’s Rank Correlation

Class III Milk Futures Corn Futures Soybean Meal Futures

M7 M8 M9 M10 M11 M12 C3 C4 C5 C6 S4 S5 S6 S7 S8

Cra

mér

-Von

Mis

esTe

stfo

rG

auss

ian

Cop

ula

(pV

alue

s)

Cla

ssII

IM

ilkF

utur

es

M7 — 0.92 0.79 0.68 0.62 0.55 0.28 0.31 0.24 0.20 0.09 0.12 0.08 0.02 0.02M8 0.30 — 0.91 0.77 0.67 0.60 0.29 0.33 0.27 0.22 0.14 0.17 0.13 0.06 0.04M9 0.01 0.32 — 0.91 0.78 0.67 0.31 0.35 0.33 0.24 0.17 0.21 0.18 0.12 0.08M10 0.04 0.07 0.04 — 0.91 0.78 0.33 0.37 0.37 0.28 0.19 0.22 0.22 0.17 0.13M11 0.16 0.04 0.07 0.47 — 0.90 0.37 0.39 0.42 0.33 0.21 0.25 0.25 0.23 0.20M12 0.25 0.14 0.02 0.03 0.18 — 0.38 0.41 0.48 0.40 0.25 0.32 0.31 0.29 0.30

Cor

nF

utur

es

C3 0.15 0.05 0.09 0.06 0.08 0.21 — 0.76 0.58 0.40 0.53 0.44 0.32 0.24 0.20C4 0.25 0.11 0.10 0.13 0.00 0.04 0.11 — 0.77 0.60 0.51 0.54 0.52 0.44 0.38C5 0.00 0.04 0.07 0.17 0.07 0.03 0.11 0.09 — 0.83 0.37 0.47 0.56 0.60 0.60C6 0.02 0.06 0.03 0.09 0.06 0.01 0.02 0.01 0.12 — 0.23 0.33 0.43 0.53 0.63

Soyb

ean

Mea

lF

utur

es

S4 0.00 0.21 0.25 0.25 0.59 0.61 0.08 0.24 0.02 0.00 — 0.83 0.68 0.56 0.45S5 0.00 0.00 0.00 0.02 0.09 0.21 0.04 0.15 0.04 0.00 0.91 — 0.85 0.72 0.57S6 0.10 0.08 0.13 0.13 0.54 0.38 0.03 0.17 0.08 0.07 0.51 0.41 — 0.87 0.73S7 0.49 0.34 0.32 0.06 0.00 0.15 0.26 0.04 0.10 0.11 0.69 0.27 0.35 — 0.84S8 0.36 0.88 0.46 0.66 0.43 0.10 0.40 0.31 0.36 0.15 0.44 0.41 0.03 0.05 —

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[0, 1]24 and uniformly distributed margins.Function Z can be estimated nonparametri-cally from a sample of futures price deviates{Xj}j=1,...,n. We use futures price data fromJanuary 1998 through April 2013. Afterdropping months with missing data, thetotal sample size is n = 157.5 Using pseudo-observations described in equation (7), theempirical copula is estimated as

Z(u) = 1n

n∑j=1

24∏d=1

1{Ud,j ≤ ui

},(12)

u ∈ [0, 1]24.

The Empirical Copula can be furthersmoothed using kernel functions as explainedby Fermanian, Radulovic, and Wegkamp(2004). However, the high dimensionality ofthe LGM-Dairy empirical copula renders thisapproach infeasible. If no smoothing is done,simulating data based on the empirical cop-ula is a simple bootstrap that is implementedby drawing rows, with replacement, from amatrix of pseudo-observations (Blumentrittand Grothe 2013). Elements of the drawnrow-vector are then used as quantiles ofmarginal distributions, which in the case ofLGM-Dairy are defined parametrically.

Alternatively, if pseudo-observations areused directly as quantiles, then marginal dis-tributions can be discretized to points. Toremain consistent with the original RMArating method, we discretize futures andoptions-based marginal distributions to fivethousand points, corresponding to quan-tiles (1/5001)…(5000/5001). This is a muchfiner grid than allowed by the sample sizeif the direct method is used. To allow fora fine discretization of marginal distribu-tions, we use a modified bootstrap method.Instead of bootstrapping from a matrixof pseudo-observations U, we draw froma matrix of price deviate ranks, obtainedas R = U(n + 1). We first determine thedimensions of an auxiliary matrix B to ben�5000/n × 24, where �5000/n denotesthe largest integer not greater than 5000/n.Matrix B is first populated by integers thatare uniformly drawn between 1 and 5,000and then sorted one column at a time.

5 As implied by our introduction, the class III milk futures andoptions contract markets were relatively thin during the initialtrading years.

Matrix B can be split into n blocks of size�5000/n × 24.

The next step is similar to regular empir-ical copula bootstrapping, as we randomlydraw a row (with replacement) of the rankmatrix R. However, for each column of theselected row j, each rank Rj,d determinesnot the quantile of the dth marginal distri-bution, but the Rj,d − th block in the dth

column of matrix B. The specific quantileis finally determined by dividing a randomdraw from rows n(Rj,d − 1) + 1 to nRj,d ofthe Bd column vector by 5,001. The fivethousand bootstrap rounds then result in a5000 × 24 matrix of quantiles from marginaldistributions. Distribution of quantiles ineach column is uniform, as is needed for anycopula. This method allows marginal distri-butions to be sampled using a finer grid thanis allowed by the sample size that underpinsthe empirical copula, but it does not arti-ficially augment the information availableto estimate the empirical copula. Finally,although the Empirical Copula method isrich enough to capture nonlinearities independence between included variables, itis simple enough to be implementable forLGM-Dairy insurance purposes.

LGM-Dairy Premiums and Indemnitiesunder Alternative Rating Methods

The impacts on insurance premiums ste-mming from the previously described mod-ifications to the LGM-Dairy rating methodare likely to be more important for someLGM-Dairy insurance policy profiles thanthey are for others. The Full Correlationmethod is not likely to reduce premiums forhigh-feed insurance policy profiles that pro-tect margins only in nearby months, whereasinsurance policies that protect distant monthsare more likely to be more affordable underthis method than they are under the cur-rent rating methodology. Likewise, theEmpirical Copula method that allows forstronger dependence between milk and feedin extreme circumstances is likely irrelevantfor an insurance policy profile that declaresminimal amounts of corn and soybean meal.For an insurance policy profile with declaredhigh feed usage, there may be a substantialdecrease in the insurance premium.

To examine the impacts of alternative rat-ing methods on LGM-Dairy premiums, wedefine two representative insurance policy

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profiles: Feed Grower and Feed Buyer. Thesetwo policy designs capture two opposing endsof a spectrum of production systems usedin the U.S. dairy sector. Some operationspursue an integrated production model, inwhich livestock feed needs are grown on thefarm. These farmers enjoy a natural hedgeagainst increases in feed prices. Instead ofincurring feed hedging costs, they managethe cost of raising their own feed. An alter-native approach is to outsource all feedproduction to third parties and use equitycapital to support farm expansion and man-agement of a larger dairy herd. This strategyis especially well suited when feed pricesare stable because it allows the producerto exploit economies of scale. However, abusiness model based on buying all livestockfeed lacks the resiliency of an integratedfeed–milk production model. Unless feedcosts are hedged, this strategy can lead toinsolvency during prolonged periods ofincreased feed price.

We define the Feed Grower policy pro-file such that the insured corn and soybeanmeal amounts correspond to the minimumfeed amounts that must be declared, perLGM-Dairy insurance rules: 0.13 bushelsof corn and 0.00081 tons of soybean mealper hundredweight of milk.6 As for the FeedBuyer policy profile, a briefing paper by theNational Milk Producers Federation (2010)provides a reasonable approximation of thefeed ration for a typical dairy farm that buysall of its livestock needs. In 2010, the NationalMilk Producers Federation assembled adairy-policy working group to construct arepresentative feed ration using corn, soy-bean meal, and alfalfa hay. The nutritionspecialists in this working group suggesteda formulated dairy ration, which the U.S.Congress later modified and adopted. Thisdairy ration, which is the foundation of thenew federal dairy policy, includes 1.0728bushels of corn, 0.00735 tons of soybeanmeal, and 0.0137 tons of alfalfa hay per hun-dredweight of milk (National Milk ProducersFederation 2010; U.S. Congress 2014).

There are no exchange-traded futuresor options contracts with alfalfa hay. TheLGM-Dairy rules do not allow hay pricesto be hedged directly, although suggestedconversion coefficients to corn and soymeal

6 Similar to hedging with exchange-traded contracts, producerscan decide to protect only a share of their expected milk marketingsand livestock feed purchases.

equivalents are provided to those dairy pro-ducers that wish to cross-hedge hay pricerisk with corn and soybean meal. To keepour analysis straightforward, we considera policy that insures only the partial rationthat includes corn and soybean meal coef-ficients, as previously suggested. We do notcross-hedge alfalfa with corn and soybeanmeal.

In addition to choosing feed amounts perhundredweight of milk, the purchasers of aLGM-Dairy policy must decide on declaredmilk marketings in each of the ten insurablemonths. For this analysis, we consider a setof rolling three-month strategies. We assumea new LGM-Dairy policy is purchased everymonth and that one-third of expected milkmarketings are declared for insurable monthsi, i + 1, and i + 2. By letting i vary from 1through 8, we obtain eight different hedgingapproaches that differ only in the horizonused for protecting the IOFC margin underthis three-month strategy. If i = 1, then thefirst three insurable months are protectedwith the LGM-Dairy policy, which resultsin a hedging horizon of 30 to 120 days. Incontrast, if i = 8, then the eighth, ninth, andtenth insurable months are protected, with ahedging horizon that spans 240 to 330 daysafter contract purchase date. Increasing thedistance of the hedging horizon increases theLGM-Dairy policy premium, as the marginrisk is increasing in time-to-maturity, ceterisparibus.

The third and final contract choice involvesthe deductible level. The chosen deductiblelevel indicates the buyer’s risk aversion. Sim-ilar to more traditional insurance products,choosing $0.00 per hundredweight (cwt)deductible results in a relatively high policypremium with a relatively high probabil-ity of an indemnity compared with policeswith higher deductibles. Choosing a highdeductible level (the maximum is $2.00/cwt)implies that LGM-Dairy is mainly used toprotect against catastrophic downside mar-gin risks. Because the primary objectiveof our analysis is to examine the effects ofmilk–feed tail dependence on the cost ofcatastrophic dairy margin insurance, ini-tially we use $2.00/cwt as the deductible inour analyses. A sensitivity analysis of theimpacts under alternative deductible levels isundertaken next.

Table 2 presents results of the analy-sis of LGM-Dairy premium costs undercurrent RMA and two alternative rating

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methods. We consider Feed Grower and FeedBuyer insurance policy profiles with hedginghorizons that cover a three-month period,starting from 30 to 240 days after the contractpurchase. We assume that the LGM-dairycontract is available and purchased everymonth from January 2007 through December2012. Table 2 presents six-year average percwt indemnities and policy premiums. As inequation (4) and unlike expected indemni-ties, realized indemnities are a function ofguaranteed and realized margins only. Theydo not depend on the rating method used.For the Feed Grower policy, average indem-nities with a $2.00/cwt deductible variedfrom $0.10/cwt for the 30 to 120 days hedginghorizon up to $0.60/cwt for the 240 to 330days horizon. Similarly, for the Feed Buyerprofile, average indemnities increase from$0.10 for policies covering nearby monthsto $0.35/cwt for the most distant horizonconsidered.

Because of the 2008–2009 recession anda major drought that occurred in 2012, therealized IOFC margin volatility exceededthe expected volatility from 2007 to 2012.As such, we expect fairly priced insurance toresult in indemnities that exceed the premi-ums paid over the stated period. Under thecurrent RMA rating method, premiums areindeed much lower than indemnities for theFeed Grower policy profile, with net bene-fits (i.e., indemnity received less premiumpaid) increasing from $0.05/cwt for the 30to 120 days hedging horizon to $0.35/cwt forthe 240 to 330 days horizon. In contrast, netbenefits for the Feed Buyer profile exhibit adifferent picture. Net benefits are negativefor all hedging horizons, with losses averagingas high as −$0.23/cwt for the most distanthedging horizon. In contrast with supposedactuarially fair premiums under the currentprogram procedures, we find that even dur-ing the period when shocks are higher thanexpected, the original RMA rating methodfails to provide positive net benefits to thispolicy profile. This finding reinforces theargument that the rating method needs to beupdated.

Adopting the Full Correlation or theEmpirical Copula rating methodology doesnot appreciably change premiums underthe Feed Grower scenario. In contrast, sub-stantial premium effects are found for theFeed Buyer profile. Under the Full Corre-lation method, premiums are up to 45%lower than those generated using the actual

RMA method. Using the Empirical Cop-ula method reduces premiums even further,with savings averaging $0.08/cwt relative tothe Full Correlation method for all hedginghorizons starting more than five months intothe future. Under Full Correlation, premi-ums are close to realized indemnities. Underthe Empirical Copula method, premiumsincrease net payouts relative to the Full Cor-relation method by more than three timesfor the most distant hedging horizon, from$0.03/cwt to $0.10/cwt.

Per current LGM-Dairy insurance pol-icy rules, users cannot choose a deductiblelevel higher than $2.00/cwt. To put thesedeductible levels in perspective, considerthat the new federal dairy policy, which isbased on lessons learned from the LGM-Dairy pilot program, allows margin coveragelevels that are up to $4.00/cwt below thehistorical average margins. We expect the taildependence impact on policy premiums toincrease in the level of deductible. In figure 2,we further explore the savings under theEmpirical Copula method, relative to the FullCorrelation method. We allow deductiblelevels to vary from $0.00/cwt to $3.00/cwt.We find that allowing for tail dependenceusing the Empirical Copula rating methodresults in substantial premium reductions forthe Feed Buyer policy. Compared with theFull Correlation method, the premiums fora hedging horizon of 240 to 330 days underthe Empirical Copula method are lower by4.8% at a $1.00/cwt deductible, by 21.6% ata $2.00/cwt deductible, and by 49.7% underat a $3.00/cwt deductible. In contrast, underthe Feed Grower profile, allowing for taildependence does not produce any premiumreductions. To the contrary, premiums actu-ally increase up to 12.4%. We conclude that,although explicitly accounting for tail depen-dence between milk and feed prices reducespremiums, tail dependence between consec-utive class III milk futures prices increasespremiums for those policies with minimalfeed amounts.

If tail dependence is a characteristic man-ifested only during a rare event, then onemust ask if any given sample contains suchan event at its true frequency. For example,our estimation uses data from 1998 through2013. The extreme realizations of milk andcorn price shocks in the lower left corner ofthe bottom panel of figure 1 are observed atthe onset of the 2008–2009 recession. Unlesssuch an event occurs once in fifteen years, on

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Table 2. Effect of Milk–Feed Dependence Assumptions on Hedging Costs, Benefits, and Effectiveness

Hedging Horizon

30–120 60–150 90–180 120–210 150–240 180–270 210–300 240–330Declared Feed Rating Method Hedging Effects Days Days Days Days Days Days Days Days

Feed Grower:Corn:0.13 bu/cwtSoymeal:0.0008 ton/cwt

Indemnity 0.10 0.19 0.29 0.38 0.46 0.52 0.57 0.60

Original Method Premium 0.06 0.11 0.16 0.20 0.24 0.28 0.32 0.35Threshold Semivariance −22% −37% −58% −76% −87% −91% −91% −86%

Full Correlation Premium 0.05 0.11 0.16 0.21 0.25 0.28 0.31 0.34Threshold Semivariance −22% −37% −58% −76% −87% −91% −91% −87%

Empirical Copula Premium 0.06 0.11 0.16 0.21 0.25 0.28 0.32 0.35Threshold Semivariance −22% −37% −57% −76% −87% −91% −91% −87%

Feed Buyer:Corn:1.0728 bu/cwtSoymeal:0.00735 ton/cwt

Indemnity 0.09 0.12 0.17 0.23 0.27 0.30 0.33 0.35

Original Method Premium 0.13 0.22 0.30 0.36 0.43 0.48 0.53 0.58Threshold Semivariance −11% −17% −37% −57% −41% −50% −52% −40%

Full Correlation Premium 0.09 0.15 0.21 0.25 0.29 0.31 0.32 0.32Threshold Semivariance −15% −20% −42% −64% −51% −62% −66% −59%

Empirical Copula Premium 0.08 0.13 0.17 0.20 0.21 0.23 0.25 0.25Threshold Semivariance −15% −22% −44% −68% −56% −67% −70% −62%

Note: Indemnity and Premium are in $/cwt. A $2.00 deductible is used for these simulations. All results are aggregated over 2007–2012. The average income-over-feed-costs (IOFC) margin over this period is $15.23 for the FeedGrower insurance profile and $8.29 for the Feed Buyer insurance profile. Both policy profiles use a deductible level of $2.00/cwt. Semivariance reduction is defined relative to the no-hedging threshold semivariance level, where

the sample threshold semivariance is calculated as∑N

i=1 min[IOFCi−(IOFC−$2.00),0]2N−1 .

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Figure 2. LGM-Dairy premiums under Empirical Copula relative to the Full Correlationmethod and under alternative deductible levels (% difference in premiums) (a) Feed Growerpolicy profile (b) Feed Buyer policy profile

average, we may oversample these extremeshocks. This is because an additional fifteenyears of data are not likely to contain anothershock of this magnitude. Oversampling a rareevent may bias upward premium reductionsfrom the Empirical Copula method relativeto the Full Correlation method. It may alsobias upward correlation coefficients, resultingin downward-biased premiums, under theFull Correlation method.

We conduct a jackknife-type sensitiv-ity analysis to illuminate the effects ofoversampling a rare event by recalculat-ing average LGM-Dairy policy premiumswith the underlying historical price deviatesmatrix. The matrix is modified so that oneyear is dropped from the sample. Given thefifteen years of data, there are fifteen setsof fourteen-year histories. Figure 3 shows

premiums under the Full Correlation methodfor a subset when the excluded year was oneof the seven most recent years. By excludingyear 2008, the premiums increase up to 36%for the Feed Buyer policy profile. The result-ing premiums are still substantially lowerthan the premiums under the official LGM-Dairy rating method. No change is found forthe Feed Grower policy profile.

To examine the influence on premiumreductions under the Empirical Copularelative to the Full Correlation method, wereduce the relative frequency of year 2008 toone-fifth of what it is in the original sample.Denote the original sample of price deviatesas W = {Xj}j=1,...,n. Denote with W−2008 theoriginal sample without 2008 data. The sen-sitivity analysis is conducted by replicatingthe sample five times, while allowing datafrom 2008 to be observed only once in the

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Figure 3. Sensitivity analysis of LGM-Dairy premiums under the Feed Buyer insurance pro-file (a) Jackknife analysis of premiums under the Empirical Copula with one year excludedfrom the sample relative to the Emprical Copula method with full sample (b) Oversamplinganalysis of premium reductions under the Empirical Copula relative to the Full Correlationrating method and with sample augmented as in equation (13)

augmented sample:

(13) W = [W′ W′

−2008 W′−2008 W′

−2008 W′−2008

]′ .

This reduction in frequency of the 2008–2009recession is consistent with the assumptionthat a shock of similar magnitude occurs only

once in seventy-five years, which is very closeto the time lapse between the Great Depres-sion of 1930s and the recession of 2008–2009.We find that the premium savings under theEmpirical Copula method relative to theFull Correlation method are lower than theyare in figure 2, but they are still substantial.

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Premium reductions decline from 21.6% to15.8% with a $2.00/cwt deductible and from49.7% to 39.4% for the $3.00/cwt deductible.Given this information, we conclude that ourinitial results regarding the importance of taildependence are indeed robust.

Effectiveness of Hedging with LGM-Dairy

Although allowing for milk–feed depen-dence may reduce premiums of LGM-Dairypolicies, it remains unclear whetherLGM-Dairy can indeed be used as aneffective risk management instrument. Forexample, anecdotal evidence suggests thatmany users of LGM-Dairy were unpleasantlysurprised at the inability of their LGM-Dairy policies to effectively smooth IOFCmargins in 2012, when high feed pricesreduced margins to near historical lows.In this section, we analyze LGM-Dairy hedg-ing effectiveness for a variety of LGM-Dairyinsurance policy profiles by incorporatingour Full Correlation and Empirical Copulamethods.

LGM-Dairy was introduced with an objec-tive to offer an affordable tool for protectionagainst catastrophic downside risks to dairymargins. This is evidenced by its Asian Bas-ket option–style design and by availablesubsidies that increase with the deductiblelevel. Hedging programs that are based onusing LGM-Dairy with high deductible levelsas a sole risk management instrument areconsistent with the “safety first” preferencesintroduced by Telser (1955). Measures of riskthat address hedging effectiveness in the faceof such safety-first preferences are based onthe α order lower partial moments:

LPMα(IOFC; b)(14)

=∫b

−∞(IOFC − b)αdF(IOFC)

where F(IOFC) is the distribution function ofthe IOFC margins. In the context of hedgingagricultural commodities, such a measurehas been used by Turvey and Nayak (2003);Mattos, Garcia, and Nelson (2008); Powerand Vedenov (2010); and Kim, Brorsen, andAnderson (2010).

In this analysis, we use threshold semi-variance of IOFC margins as a measure ofrisk, where threshold semivariance is defined

as

TSV = E[min

(IOFC(15)

−(

IOFC − $2.00/cwt)

, 0)]2

where IOFC is the historical average margin.If the purpose of pursing a risk managementstrategy is to reduce threshold semivari-ance, then it follows that the effectivenessof any hedging program can be evaluatedby

(16) ν =(

1 − TSVH

TSVN

)100

where TSVH denotes threshold semivari-ance under hedging, and TSVN is the samemeasure but under the no-hedging scenario.This measure of hedging effectiveness iswell suited for evaluating how well LGM-Dairy protects against catastrophic downsidemargin risks.

Of particular interest is how the LGM-Dairy rating method influences the optimallength of the hedging horizon. Ederington(1979); Malliaris and Urrutia (1991); Geppert(1995); and Chen, Lee, and Shrestha (2001)examine the effect of hedging horizon lengthon hedging effectiveness. These studies findthat an increase in the hedging horizon gen-erally results in higher hedging effectiveness,which is explained by higher covariancebetween cash and futures prices at longerhorizons. However, a more predictable basisat longer horizons may not be the only rea-son why hedging months that are moredistant can be more effective in stabilizingnet revenues. Bozic et al. (2012) argue thatshocks to dairy IOFC margins take up to ninemonths to dissipate. The speed of the meanreversion in IOFC margins is reflected in theterm structure of milk futures prices. Conse-quently, a risk management strategy with ashort hedging horizon is inadequate to pro-tect against the full impact of major shocksinducing prolonged low-margin periods.In the aftermath of such a shock, expectedIOFC margins decline for those monthsthat are not yet hedged. Attempts to hedgenearby months may then result in guaran-teed margins that are already well below theaverage IOFC margins.

Consider a risk management programbased on LGM-Dairy insurance with theFeed Grower profile. In this program, the first

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three insurable months are always insured.This strategy results in lower premiumscompared with programs that use longerhedging horizons. However, it is ultimatelyinadequate to protect against major IOFCmargin shocks, such as the 2008–2009 reces-sion. Under this program, March, April, andMay 2009 were insured at the January 2009sales event. At that point, futures marketshad already incorporated information aboutthe decline in demand for dairy products. InJanuary 2009, the expected average IOFCmargin for March–May 2009 was as low as$9.92/cwt, which was more than $5.00/cwtbelow the long-run average of $15.23/cwt.As this example illustrates, for a hedg-ing program to be consistently effective insmoothing dairy IOFC margins, the hedginghorizon must be longer than the averagetime needed for margins to recover to thelong-run average.

In table 2, we measure the effectivenessof hedging with LGM-Dairy under the FeedGrower and Feed Buyer policy profiles. Aspredicted, hedging effectiveness for the FeedGrower policy increases with the length ofthe hedging horizon. A maximum reduc-tion in the threshold semivariance of 91%occurs when the hedged period begins sevenmonths after the LGM-Dairy sales event.Under the original RMA rating system andthe Feed Buyer policy, maximum hedgingeffectiveness occurs when the hedged periodstarts only four months after the sales event.Because the original RMA rating systemignores milk–feed dependence and milk–feeddependence increases with time to maturity,the magnitude of upward bias in LGM-Dairypremiums increases with the length of thehedging horizon.

Indemnities under the Feed Buyer profileincrease with time to maturity. This indicatesan opportunity for more effective smoothingof IOFC margins if longer hedging hori-zons are used. However, for horizons longerthan four months, upward bias in premiumsdominates and thus reduces hedging effec-tiveness. For the Full Correlation method,hedging effectiveness is nearly identical forfour-month and seven-month hedging hori-zons. Under the Empirical Copula method,we obtain maximum hedging effectivenessat seven months. In conclusion, we find thatignoring the nonlinearity of dependencebetween milk and feed prices results in neg-ative net payouts for longer horizons. It mayalso prompt users to use shorter hedging

horizons, which reduces effectiveness of theLGM-Dairy as a hedging instrument forproducers who purchase a majority of theirlivestock feed.

Conclusions

Finance and actuarial science fields usecopula methods for flexible modeling ofmultivariable dependence structures. Overthe last five years, these methods have beenadopted in the analysis of agricultural priceand revenue risk management. A commontheme in research efforts that apply cop-ula methods is that extremal dependence,where present, increases portfolio risk. In thisarticle, we demonstrate that lower-tail depen-dence exists between milk and corn futuresprices and between milk and soybean mealfutures prices. Unlike previous applicationsof copula methods, we find that such extremaldependence actually decreases the price ofLGM-Dairy margin insurance because fac-tors inducing exceptionally large declines inmilk revenue are likely to induce extremedeclines in feed prices as well.

The mean-reverting feature of dairy IOFCmargins predicts that hedging will be moreeffective if longer hedging horizons are used.To test this hypothesis, we analyze the rela-tionship between the length of the hedginghorizon and the ability of LGM-Dairy toreduce catastrophic downside risk. We findthat hedging effectiveness is substantiallyhigher for longer hedging horizons. The effectis more pronounced when the LGM-Dairyrating methodology is altered to allow formilk–feed tail dependence.

Our study has two policy implications.First, the current rating method for LGM-Dairy is called into question. The currentmethod, which is based on Gaussian copulaand data observations through 2005, failsto capture salient features of milk and feedmarkets. The current rating system should bereplaced with the Empirical Copula method,which is presented in this analysis. Second,in scoring federal dairy policy proposals thatinclude margin insurance, evaluation methodsshould rely on multivariable copula densitiesthat allow for lower tail dependence.

Whereas this work focuses on the struc-ture of dependence among futures prices,additional research on the LGM-Dairy rat-ing method should focus on examining the

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assumptions regarding univariate marginaldistributions used in the rating method. First,the rating method assumes there are nobiases in futures prices or implied volatilitiesinferred from option premiums. That assump-tion is particularly suspect for distant classIII milk futures and options contracts, whichsuffer from low liquidity. Second, the ratingmethod assumes all marginal distributionsare log-normal. The log-normality assump-tion may provide computational convenience,but this assumption is known to be inappro-priate when applied to the prices of a varietyof agricultural products (Fackler and King1990; Sherrick, Garcia, and Tirupattur 1996).The effect of departures from log-normalityon LGM-Dairy premiums needs to beexamined.

Supplementary Material

Supplementary material is available at http://oxfordjournals.org/our_journals/ajae/online.

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