Rate Making for Farm-Level CropInsurance: Implications forAdverse SelectionJerry R. Skees and Michael R. Reed

This research identifies two problems in the new Federal Crop Insurance that maycause adverse selection: (a) the relationship between rate making and expected yieldsfor individual farmers, and (b) the bias introduced in coverage protection when trendsare not used to establish expected yields. A theoretical investigation using thenormality assumption demonstrates the potential severity of these problems, andempirical results from farm-level data lend further support. As crop insurance changesto individualized methods of protection, these issues will be particularly important fordeveloping rates.

Key words: adverse selection, crop insurance, crop yields, normality, risks.

Until recently, Federal Crop Insurance (FCl)used geographic area average yields as thebasis of protection and rate making for allfarmers in the specified area. Thus, farmerswith lower expected yields than the area aver­age could purchase more protection thanfarmers with yields above the average. Thiscaused adverse selection over time-farmerswith higher expected yields (than average)opted out, and farmers with lower expectedyields purchased crop insurance-thus in­creasing indemnity payments relative to pre­miums paid.

One of the more significant changes madesince the 1980 Federal Crop Insurance Act is amove toward developing coverage based uponindividual expected yields (first the IndividualYield Coverage program and now the ActualProduction History-APH-program). Underthe APH program, FCI uses either ten years ofcertified farm data or a combination of limitedfarm data and an area indexing scheme to es-

Jerry R. Skees is an assistant professor, and Michael R. Reed is anassociate professor of agricultural economics, both at the Univer­sity of Kentucky.

This research is a product of a four-state (Michigan, Kentucky,Minnesota, and Kansas) project on Federal Crop Insurance. Theinvestigation reported in this paper (No. 86-1-26) was also sup­ported by the Kentucky Agricultural Experiment Station and ispublished with approval of the Director.

The authors wish to thank Bob King and Roy Black for theirassistance.

Review was coordinated by Bruce Gardner, associate editor.

tablish an individual yield level. Thus, FCI hastaken a significant step toward eliminating ad­verse selection problems associated with us­ing area average yields.

However, at least two features of the FCIprogram have the potential to cause adverseselection: (a) the manner in which FCI ratesare developed for farms with different APHyields and (b) the manner in which the APHyield is established. 1 This research investi­gates how these two features impact adverseselection.

Conceptual Framework

Adverse selection occurs when farmers withhigher relative risk have opportunities to pur­chase insurance at the same cost as farmerswith lower relative risk. If farmers are able torecognize these differences and participationreflects such recognition, FCI tends to attractfarmers with relatively high risk. Over time,

I FCI uses a two-step process to determine insurance premiumsfrom FCI rates. The first step is to calculate an area average insur­ance rate. These rates facilitate the integration of different priceprotection levels in premium calculations. Equations (8) and (9)provide a more complete formulation for differences in FCI ratesand premiums. Throughout this manuscript FCI rates are actualrates charged by PCI-the procedure for developing these ratesover time is not reviewed in the paper-and theoretical rates rep­resent rates where premium payments equal indemnity paymentsover time.

Copyright 1986 American Agricultural Economics Association

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654 August 1986 Amer. J. Agr. Econ.

For most commodities, FCI provides protec­tion based on a percentage of APH yields (i.e.,50%, 65%, and 75%). In the transition fromarea average yields to more individualizedyields, FCI set premium charges based ontheir rate for the area in which the individualfarm was located. These premium chargeswere the same regardless of whether a farmerparticipated in the area coverage option orthe individualized program, though the yieldguarantee differed. An implicit assumptionof this procedure is that relative yield risk(coefficient of variation) is constant acrossfarms with different expected yields.

If farms with higher expected yields haveless risk in percentage terms, those farmersreceive less protection but pay the same pre­mium (i.e., multiplying a percent by a small

Using work by Botts and Boles and thepolynomial function for integration of a nor­mally distributed density function produces amore specific formulation for estimating ex­pected losses from a truncated normal distri­bution:

Rate Setting and Expected Yields:Simulation Results

(3) Z= [_1_]. e -1/2.[ (EYs~ Yg

)ryJ2; ,

T = 1

1 + b( EYS~ Yg)'

(5) P = Z(atT + a2T2 + a3T3),

(6) EL = P(Yg - EY) + Z· SD,

where b = .33267, at = .4361836, a2 =- .1201676, and a3 = .937298 (Abramowitzand Stegun). Other variables include: EY, ex­pected value; SD, standard deviation; P, prob­ability of collecting in any given year; and thevariables defined above or through inter­mediate calculations (Z and T).

Equations (3)-(6) are first used with hy­pothetical data to simulate expected losses.Next, data for com and soybean farms in Il­linois and Kentucky are used with equations(3)-(6). The empirical results are comparedwith results that disregard the normality as­sumption and use trend-adjusted yield data toexamine what losses would have been overthe time series.

(1) CV = SDIEY.

Both SD (standard deviation) and EY (ex­pected per-acre yield) vary by farm. Thesevalues determine the relative risk and the the­oretical rate for individual farms. Therefore,both farm-level expected values and a mea­sure of farm-level variability are fundamentalto an individual farmer's decision to purchasecrop insurance. Farmers must be able to eval­uate their expected yield, standard deviation,willingness to accept risk, and the cost of theinsurance in making the decision to buy.Those farmers who are risk averse (otherfarmers should not be interested in actuariallysound insurance) compare what they believeis their expected returns from insurance towhat they must pay for insurance. Thus, farm­ers' subjective assessment of their relativerisk is critical.

It is possible to estimate expected indem­nity payments once expected yield and stan­dard deviation are known by assuming aknown parent distribution. Botts and Bolesexplored techniques for developing expectedcrop losses from a normal distribution ofyields. The distribution must be integrated inthe region below the coverage level:

(2) EL = f:: (Yg - Y)f(Y)dy,

where EL is expected losses in bushels, Yg isthe yield guarantee (APH yield multiplied bythe percentage level of protection), Y is theactual yield, andf(Y) is the probability densityfunction for yields.

FCI insurance rates increase to reflect the rel­ative risk of the class of farmers that are par­ticipating in FCI, compounding the problem.Setting insurance rates to reflect relative riskof different farmers is necessary to forestallthis adverse selection.

Theoretical rates are a function of expectedlosses and yield guarantee levels. If yields arenormally distributed, knowledge of the firsttwo moments (expected value-EY-andstandard deviation-SD) is sufficient to deter- (4)mine theoretical rates. Further, since yieldguarantees in FCI are based upon percentagesof expected yields (FCI has not really cap­tured expected yields in their APH program asis developed below) expected losses and theo­retical rates are a function of relative risk, asmore fully developed below.

Relative risk can be measured by the co­efficient of variation (CV):

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Skees and Reed Crop Insurance and Adversity Selection 655

where PR is premium and Pg is price protec­tion. Again, until the 1985 crop year, the yieldguarantee used to calculate premium was thearea average value. Thus, premiums were the

where R is the theoretical rate, EL is the ex­pected loss from equation (6) and Yg is theyield guarantee level. Since FCI does notknow what EL is for each farm, the calculationof their premium is based upon their rate(which assumes an EL), yield guarantee, andprice protection:

number yields a smaller number than multiply­ing the percent by a large number). For ex­ample, a farmer who chooses 75% protectionfor an expected com yield of 80bushels wouldreceive indemnity payments for yields below60 bushels (20bushels below his average). Hisneighbor, who has an expected yield of 120bushels, would receive indemnity paymentsfor yields below 90 bushels (30 bushels belowhis average). If the farmer with the 120-bushelaverage has a standard deviation for yieldwhich is 50% higher than his neighbor, hewould have the same coefficient of variationas the farmer with the 8o-bushel expectedyield. Both farmers would have the same rela­tive risk and should be charged the same pre­mium. However, if both farmers have similaryield dispersions (similar standard deviations)and different expected values, charging thesame premium for a fixed level of protection(e.g., percentage of average yield) creates anadverse selection problem-farmers with rela­tively high expected yields and potentiallylower relative risk will not participate in FCI.The empirical question involves the relation­ship between expected yields and measures ofvariation (e.g., standard deviation). This de­termines whether farmers with relativelyhigher expected yields also have lower rela­tive risk.

Equations (3)-(6) and (7) and (8) can beused to illustrate the potential differences intheoretical rates and premiums as expectedyields vary. Expected yield losses under vari­ous insurance program designs can be con­verted to theoretical rates and premiums. Atheoretical premium should simply equalthe expected loss multiplied by the level ofprice protection. Theoretical rates can be de­veloped from the expected loss information:

(7)

(8)

R = (ELIYg ) 100,

PR = Y . P . Rll00 = EL . Pg g s:

same regardless of the individual farm yield.In 1985, FCI began to offer discounts for somecrops as expected yields increase.

If standard deviation is fixed at 25, equa­tions (3)-(6) can be used to demonstrate theeffects of expected yield increases on theoreti­cal rates (since standard deviation is fixed,coefficient of variation declines as expectedvalues increase). Substantial differences existbetween theoretical rates when coefficient ofvariation declines. For the relatively low yieldof 65 bushels of com per acre (CV = .38)annual bushel loss is expected to be 3.88bushels. Doubling com yields reduces ex­pected bushel loss to 1.14bushels (i.e., at 130bushels, CV = .19). Therefore, given theseassumptions, FCI rate adjustments based onexpected yield can easily be justified. Someempirical questions remain: Are expectedyields and standard deviations independent?Is the assumption of normality valid for cropyields? These questions are addressed in thenext section.

The above analyses assumed that expectedyields and APH yields were equivalent. Infact, when a significant trend is present, ex­pected yields exceed APH yields. Ideally, FCIstrives to obtain ten years of certified farmyields for the APH program. In the transition,they accept limited data and use an indexingscheme with adjusted ASCS yield data for thearea. Although these methods can be chal­lenged, they must be accepted as a means ofattracting new farmers into FCI.

The problem that this study addresses isthat FeI does not make an attempt to adjustyield data for trends. Given ten years of data,FeI drops the high and low yield and takes asimple average to obtain the APH yield. Sincecoverage levels are tied to APH yields, thismeans that farmers with positive yield trendsare not able to purchase as much protection asis implied because APH yield is a biased esti­mate of expected yield. For example, a farmerwith an expected yield of 100 bushels per acreand a trend of two bushels per year wouldlikely have an APH of only 90 bushels basedon ten years of data (i.e., 100 - [2 x 5]).Rather than having the option to protect yieldshortfalls below 75 bushels (75% of expectedyield), the maximum protection available isfor yields below 67.5 bushels under the APHprogram. Due to the properties of a normaldistribution, such a discrepancy results in aneven larger difference in the expected losses.Again, assuming a standard deviation of 25,

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656 August 1986

results from equations (3)-(6) illustrate the de­gree to which neglecting trend adjustmentscan influence expected losses. For example, afarmer with an expected yield of 100 bushelsand an APH yield of 90 bushels (given the 2­bushel trend) would have an expected bushelloss of only 1.34 bushels versus 2.08 bushels ifthe APH yield were adjusted for trend. Thedifferences are not trivial, which indicates an­other reason for participation problems. Em­pirical results below provide further insight.

Empirical Results

The simulation results above raise a number ofissues: (a) Are farm-level expected yields andstandard deviations independent? (b) Arefarm-level yield distributions normally distrib­uted? And (c) are trends in farm-level datazero? Results of these hypothesis tests relatedirectly to the two issues addressed in this pa­per.

A time-series of farm-level yields (fromfarm analysis records on a per-planted-acrebasis) for a sample of farms in northern Illinoisand western Kentucky was used. Only farmswith twelve or more years of data were used(for the Illinois data set, twelve years, 1972­83, were available; for the Kentucky data set,twelve to twenty-two years were available).Farms that could be identified as being in afloodplain were eliminated from the sample.The sample included fifty-four farms for Il­linois corn and soybeans, sixty-five for Ken­tucky corn and forty-eight for Kentucky soy­beans.

Data were pooled by region to estimatetrends in yields, since a limited number ofyears were available on an individual farmbasis. In order to control for differences inmean yields for individual farms, a dummyvariable was used for each farm (this allowedthe intercept term to vary by farm). Chowtests on the assumption that yield trends werethe same across farms indicated that the hy­pothesis that yield trends were identical for allfarms in each region could not be rejected atthe 5% significance level. Annual trend values(in bushels per acre) and corresponding stan­dard errors (in parentheses) were 2.14 (.213)for Illinois corn, 1.62 (.118) for Kentuckycorn, .622 (.066) for Illinois soybeans, and.465 (.051) for Kentucky soybeans.

Because the Chow tests suggested that as­suming a common yield trend for all farms

Amer. J. Agr. Econ.

within a region and commodity was rea­sonable, each individual farm's data were ad­justed to 1983 technology:

(9) Y'it = Yit + b(l983 - t),

where Y'it is farm i's yield in year t adjustedfor 1983 technology and b is the trend value(reported above). These transformed data setswere used for the remaining analyses.

In order to test the hypothesis that farm­level expected yields and standard devia­tions are independent, a simple regression wasused:

(10) SDi = C + aEYi,

where EYi is the expected yield (mean of thetime series using adjusted data) for the ithfarm and SD i is the standard deviation for theith farm. In all cases-Illinois corn and soy­beans and Kentucky corn and soybeans-thenull hypothesis that standard deviation is inde­pendent of expected yield cannot be rejectedat the 5% level of significance." A similar testwas performed on the hypothesis that co­efficient of variation declines as expectedyields increase. In all cases, there was a rela­tionship (the level of significance was less than1% for all crops and areas except Kentuckysoybeans where the level was 11%). These re­sults provide legitimacy to the simulation oftheoretical rates that assumed a constant stan­dard deviation across farms with different ex­pected yields.

A Shapiro-Wilks test (a common normalitytest for small sample sizes) was performed us­ing individual farm yield data. Using a 20%significance level, the normality assumptionwas rejected for 31% of the Illinois corn sam­ple, 50% of the Kentucky corn sample, 19% ofthe Illinois soybean sample, and 31% of theKentucky soybean sample.' Although normal­ity cannot be rejected for the majority offarms, these results suggest that the normalityassumption may not be appropriate for analyz­ing FCI. Examination of the third moment foreach individual farm suggests that, if there is asignificant skewness (small sample sizes pro-

2 The same procedures were used with the samples split into thelow half and high half of standard deviation by farm. This wasdone to assure that farmers with low expected yields and high SDswere not offsetting farmers with high expected yields and low SDsand thus resulting in no relationship between expected yield andSD. The split samples also revealed no relationship between stan­dard deviation and expected yields.

3 The 20% significance level was selected because lower levelsof significance would substantially increase the probability of atype II error.

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Skees and Reed Crop Insurance and Adversity Selection 657

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STEp· Slap 10",,01100 Iro .., dolo

Eltp·u ..-ctad ItU,1 lor ,leld ._onl"

APH - U APH ,i.ld. for,i.ld ••orool••

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INSURANCE RATE

guarantee. Results are quite consistent be­tween the two areas (figs. I and 2). Under eachof the procedures, the negative relationshipbetween expected yields and theoretical ratesis apparent. (These relationships were alsostatistically significantfor equations where ex­pected yield was used to establish protectionlevels.) When trend adjustments are made,theoretical rates are consistently greater thanwith an APH yield (which is consistent withthe simulated results). Figures 1 and 2 alsodemonstrate the influence that trend adjust­ments have on theoretical rates and expectedlosses. Either FCI must attempt to adjustAPH yields for trends or they must reducetheir rates to reflect the actual level of cover­age that is provided when trends are not ad­justed. Finally, figures 1 and 2 illustrate thedifferences obtained when using step integra­tion rather than normal integration. These re­sults support the notion that distributions arenegatively skewed.

Figures 1 and 2 yield two basic conclusions:(a) FCI rate discounts are appropriate as ex­pected yields increase; and (b), since the trendadjusted curves are consistently to the right ofthe APH curves (as expected from the theoret­ical foundations for effects of ignoring trend),trends in yields need to be accounted foreither by adjusting APH yields or by loweringFCI rates to reflect the coverage provided by

Figure 1. Theoretical rates for Illinois com(75% yield guarantee)

YIELD

145

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110

105

100

95

90

85

80

75

70

65

60

55

0

(when Y,. < Yg )

where EL is expected loss, Yg is the yieldguarantee (either developed based on ex­pected yield or APH yield), Y; is the trend ad­justed yield in year i, and n is the sample sizefor the farm. King shows that these proce­dures provide unbiased estimates of EL.

Since the simulations using equations (3)­(6) indicate that theoretical rates should de­cline at a decreasing rate, a curvilinear rela­tionship was chosen:

(12) R, = c + »r.:',where R, is the theoretical rate for the ith farm(under the four alternative assumptions dis­cussed above) and Yi -1 is the inverse of theexpected 1983 yield for the ith farm. Equation(12) was used to examine the relationship be­tween theoretical rate and expected yield forIllinois and Kentucky com under 75% yield

;=n

(II) EL = L (Yg - Y;)/n,;= 1

hibited statistical testing), it is negative. If thedistributions are negatively skewed, the ex­pected losses associated with FCI coverageincrease.

Given individual farm data, it is possible tocalculate expected losses and theoretical ratesfor alternative designs in FCI. Because thefocus of this research is on the relationshipbetween expected yields and theoretical rates,four alternative procedures for calculatingFCI rates are examined: (a) assuming normal­ity and allowing for a trend adjustment, (b)assuming normality and using APH proce­dures, (c) relaxing normality assumptions andallowingfor a trend adjustment, and (d) relax­ing normality assumptions and using APHprocedures. Equations (3)-(6) are used fornormality assumptions by using the farm-levelmean (expected 1983 value) and farm-levelstandard deviation. In the case of the APHassumption, the ten most recent years of un­adjusted (for trend) data were used by drop­ping the high and low value and calculating asimple average. (This replicates FCI's proce­dure for developing APH yields when a ten­year history of certified farm-level yields isavailable.) Normality assumptions are relaxedby use of step integration. That is, the sum ofthe difference between the yield guarantee andthe trend-adjusted yields that fall below thatlevel were divided by the sample size for eachfarm:

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658 August 1986 Amer. J. Agr. Econ.

Implications for Adverse Selection

Figure 2. Theoretical rates for Kentuckycorn (75% yield guarantee)

current APH methods. These conclusions arenot sensitive to normality assumptions. How­ever, the step integration results also suggestthat FCI rate makers should be concernedwith use of normality assumptions.

Conclusions

This research has identified two problems withthe Federal Crop Insurance program that dis­courage participation and probably lead to ad­verse selection: (a) farmers with relativelyhigh expected yields can only expect verysmall and infrequent indemnity paymentswhen guarantees are tied to some measure ofexpected yields, and (b) unadjusted (for trend)APH yields reduce the expected indemnitypayments. FCI has taken steps to provideyield discounts so that farmers with higher ex­pected yields have lower rates. The results ofthis paper support such action, and the paperintroduces a procedure for developing theseyield discounts. However, the fact remainsthat farmers with high expected yields mayhave such a low probability of collection(when level of protection is based on a per­centage of yield) that they have little incentiveto participate. Congressional action would berequired to change the fashion in which yieldguarantees are developed.

Levels of protection ideally should be tiedto some measure of variability (e.g., standarddeviation). This would provide farmers withdifferent expected yields similar levels of pro­tection (assuming that standard deviation isnot a function of expected values-an as­sumption supported by this research). Clearly,such a program would be difficult to adminis­ter. However, this approach or somethingsimilar is needed if crop insurance is to pro­vide stability to the agricultural community.There is no reason that farmers with higherexpected yields have less need for insurancethan farmers with lower expected yields. In­deed, to the extent that farmers with higherexpected yields are farming better and moreexpensive land, they may be exposed to

deviations. If farmers are generally better atassessing expected values than yield variabil­ity, or the probability of crop losses below theoffered guarantee level, then FCI rates that donot decline as expected values increase aremore easily identifiable by farmers than FCIrates that are set inappropriately due to differ­ence in farm-level variability. Thus, accuracyin developing farm-level protection basedupon actual farm-level expected yields is a keyto avoiding adverse selection.

NORM- No,.. olily OMU""

STEP - Slip inllo,olion f,omfarm data

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APH- U.I. APH yield. forlilld guarant ..

4 5 6 7 8 9 10 II

INSURANCE RATE

o

This article has made a point of stressing thefact that tying coverage protection to APH (orany proxy for expected yields) leads to a di­rect relationship between theoretical rates andrelative risk. Since relative risk can be repre­sented by the coefficient of variation, standarddeviation and expected yield are critical torate making. Although this paper has focusedon the relationship between expected yieldand theoretical rates, differences in standarddeviation between farmers with the same ex­pected yield can also lead to inappropriate in­surance rates and potential adverse selection.If farmers are able to recognize when theirrates are lower than justified by their relativerisk, then adverse selection occurs. There­fore, the issue of major sources of adverseselection depends upon a farmer's subjectiveassessment of expected values and standard

YIELD

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Skees and Reed

greater financial risk than farmers with lowerexpected yields and less expensive land.

If FCI should attempt to tie protection tosome measure of variability (i.e., standard de­viation), then research is needed to explainwhy farms with the same expected yield havedifferent standard deviations. For the mostpart, this research has ignored this importantdimension of relative risk, primarily becauseof the current design of FCI. Yet, even underthe current design, expected yield does notexplain the bulk of differences in rates thatshould be charged to different farmers. Undera redesigned FCI program that would tiecoverage to SD, expected yields would beeven less important for setting rates. Futurework should focus on explaining differences in

Crop Insurance and Adversity Selection 659

standard deviation and why it varies betweenfarmers and different regions.

[Received August 1984; final revisionreceived September 1985.]

References

Abramowitz, Milton, and Irene Stegun. Handbook ofMathematical Functions. Washington DC: Govern­ment Printing Office, 1968.

Botts, Ralph R., and James N. Boles. "Use of Normal­Curve Theory in Crop Insurance Ratemaking." J.Farm Econ. 39(1957):733-40.

King, Robert. Presentation to Federal Crop Insurancefour-state research team, San Francisco CA, June1984.

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