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Review of Methodology for Establishing Rates for Yield Exclusion

AgRisk Management

Ames, IA

February 6, 2015

Executive Summary

A new method for establishing rates for coverage levels above 85% need to be 

established because of the new requirement to allow farmers to drop low yields 

from their APH yield calculation. The dropping of low yields increases effective 

coverage levels above nominal coverage levels. RMA wants to determine insurance 

rates for nominally selected coverage levels that represent effective coverage level 

risk. The two procedures proposed to determine effective coverage level rates 

include linear interpolation of rates when effective coverage levels are less than 

85% and linear extrapolation of rates when coverage levels exceed 85%.

I used Monte Carlo methods to determine the direction and magnitude of errors in 

rates that are likely using the proposed interpolation and extrapolation methods. 

Because premium rates increase at an increasing rate for effective coverage levels 

below 85%, the proposed interpolation methods lead to an upward bias in rates. 

That is, interpolated rates are greater than actuarially fair rates. However, because 

the interpolation points are so close together, linear interpolation does an adequate 

job approximating the actual nonlinear relationship. Thus I conclude that the 

proposed method of using linear interpolation does a good job approximating the 

nonlinear relationship between premium rates and effective coverage levels. 

The proposed linear extrapolation method is based on the change in premium rates 

at two points—the 80% and 85% premium rates—and uses this change in premium 

rate to determine premium rates for all other effective coverage levels that exceed 

85%. This method is appropriate and will lead to small approximation errors only 

when the relationship between premium rates and effective coverage levels above 

85% is linear. But the general relationship is nonlinear. For most any well-behaved 

and commonly used yield distribution, and for most risk levels, the relationship is 

convex (rates increase at an increasing rate). The lower the risk level, as 

represented by the 65% premium rate, the greater the convexity. This convexity 

implies that linear extrapolation will underestimate actuarially fair premium rates. 

Most Corn Belt corn and soybean farmers have low 65% premium rates. And most 

of the book of business for the crop insurance program is represented by Corn Belt 

corn and soybean farmers. Thus the proposed rule will result in a large 

underestimation of premium rates for effective coverage levels that exceed 85% in 

the most important crop insurance region, in terms of liability and premium.

For higher risk farmers, the relationship between premium rate and effective 

coverage level is less convex, approximately linear, or even concave (rates increase 
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at a decreasing rate), depending on the level of risk, the effective coverage level, and 

the form of the yield distribution. Simulations show that if the relationship is 

concave, which can occur when the yield distribution is skewed sharply to the right, 

then the proposed extrapolation method can lead to a large overestimation of 

premium rates. Otherwise, extrapolation errors are likely to be lower in magnitude 

than the errors made for low risk farmers.

Exacerbating the problems with underestimation of premium rates for low risk 

farmers is the likelihood of increased moral hazard for very high effective coverage 

levels. The likelihood that a farmer will take less care of a crop when it is insured for 

more than it is expected to be worth is likely much greater than when a farmer has a 

significant insurance deductible. Thus the underestimation of premium rates will 

likely be larger than estimated here. 

An alternative method of determining premium rates for greater-than-85% 

coverage levels is to take advantage of the yield draws from the truncated normal 

distribution that are used by RMA to determine revenue insurance rates. They will 

give a much better approximation of how premium rates should behave for 

coverage levels above 85% for low risk farmers than the proposed extrapolation 

method. For high risk farmers the use of the draws may be less appropriate because 

the truncated normal distribution does not allow for a concave relationship between 

premium rate and coverage levels for reasonable effective coverage levels.
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Research Report

(1) Description of the methodology used by the expert reviewer.

The method used in this review was to carefully think about and use Monte Carlo 

simulations to determine the likely relationship between premium rates and 

effective coverage levels. The likely relationship was then used to determine 

whether the methods proposed by RMA are appropriate for calculation for premium 

rates when yields are dropped from APH yield calculations.

The rest of the research report is organized as follows. First an overview of what 

determines the relationship between premium rates and coverage levels for yield 

insurance is provided. Next, a comparison of an “ideal” method based on knowledge 

of the yield distribution to RMA’s proposed methods that are based on linear 

interpolation and extrapolation is made. The report then provides a discussion of 

the relevance of this review’s findings and provides a suggestion for an alternative 

method that could be used if desired. Answers to the review questions are then 

provided.

Overview of Relationship between Coverage Levels and Premium Rates

Actuarially fair premiums for yield insurance when plotted against the coverage 

level will take the same general shape of put option premiums on futures prices 

when plotted against the strike price. When the strike price is far below the futures 

price, the option premium will increase at an increasing rate with increases in strike 

prices. Once the strike price approaches the futures price then the option premium 

increases linearly. Eventually the option premium will increase at a decreasing rate. 

The reason why option premiums increase at an increasing rate when the strike 

price is low in relation to the futures price is that initially the probability that the 

option will be exercised is quite low. As the strike price increases from low levels 

the probability increases at an increasing rate because the probability distribution 

of price increases rapidly as the strike price moves towards the mean of the 

distribution. Once the strike price approaches the mode of the price distribution the 

probability of exercising the option no longer increases rapidly. This is why the 

premium eventually becomes linear in the strike price. Eventually the probability of 

exercising the option increases at a decreasing rate so the option premium increases 

at a decreasing rate also. As with option premiums so too with yield insurance 

premiums, assuming that yields follow single-peaked probability density functions, 

which will be assumed throughout this review. This means that actuarially fair yield 

insurance premiums will increase at an increasing rate for low coverage levels but 

will eventually become linear at very high coverage levels, and then increase at a 

decreasing rate.

The proposed method of RMA to linearly interpolate premiums rates for coverage 

levels between 50% and 85% means that the interpolation will overestimate 

premiums because at these coverage levels almost all yield insurance premiums 
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increase at an increasing rate with coverage levels. The upward bias will be small 

when the rate relationship is approximately linear between any two coverage levels. 

For coverage levels above 85% the proposed method to extrapolate premiums rates 

based on the increase in rate between 80% and 85% will underestimate premium 

rates if the premiums are still increasing at an increasing rate between these two 

coverage levels. The method will be largely unbiased for coverage levels not too 

much about 85% if the relationship between 80% and 85% is linear. When the 

relationship between 80% and 85% is linear, then because the rate of increase in 

the actuarially fair premium rate will start to decline, the extrapolation method will 

lead to an overestimate of premium rate for a very high coverage level.

To clarify these concepts, Figure 1 shows how actuarially fair premiums increase 

when yields follow a beta distribution with two 65% premium rates: 0.0082 and 

0.10. These two rates represent, respectively, low risk and high risk crops. Figure 2 

shows the coverage level differentials (CLD) for these two representative 

distributions which is simply the ratio of a premium rate to the 65% coverage level 

premium rate. 

Figure 1. Simulated Premium Rates for Two Beta Distributions
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Figure 2. Coverage Level Differentials for High and Low Risk Yield 

Distributions

From these two figures it is straightforward to see the ramifications of extrapolating 

premium rates for coverage levels above 85% based on the rate of increase of 

premiums rates between 80% and 85%. In Figure 1 it is apparent that premium

rates are increasing at an increasing rate for the low risk yield distribution at the 

85% coverage level. This implies that linear extrapolation will lead to an 

underestimate of what premium rates should be for effective coverage levels greater 

than 85%. It is apparent that premium rates are increasing at a decreasing rate for 

the high risk yield distribution. This implies that linear extrapolation will

overestimate premium rates for greater than 85% effective coverage levels.

Figure 3 shows the actual coverage level differentials and the extrapolated 

differentials for the low risk yield distribution. Figure 4 shows the same graph for 

the high yield distributions. These two figures show that errors in both cases can be 

quite high. The RMA method extrapolation method underestimates premium rates 

for this low risk yield distribution and overestimates premium rates for this high 

risk yield distribution. The degree of overestimation for the high risk distribution 

may be capped by RMA’s limitation that the increase in premium cannot exceed the 

increase in liability. Figure 5 shows the percent error in premium rate for the two 

distributions. For the high risk distribution the error is small until the effective 

coverage level increases above 100%. The percentage effort for the low risk 

distribution is significant in percentage terms at all coverage levels. However, the 
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absolute magnitude of the error is smaller than the percentage error for the low risk 

distribution for coverage levels close to 85% as shown in Figure 6.

Figure 3. Actual and Extrapolated Coverage Level Differentials for the Low 

Risk Yield Distribution
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Figure 4. Actual and Extrapolated Coverage Level Differentials for the Low 

Risk Yield Distribution

Figure 5. Percent Error in Premium Rate Using Proposed Extrapolation 

Method 
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Figure 6. Extrapolation Errors Expressed in Levels

A Comparison of Ideal Methods and RMA Proposed Methods

To estimate the actual errors that would be made using RMA’s proposed 

interpolation and extrapolation methods requires knowledge of the “true” yield

distribution along with an assumption that the RMA equations used to determine 

80% and 85% CDLs are consistent with this true yield distribution. However, there 

is no way of determining what the true yield distribution is, so an alternative 

method must be used. As demonstrated above the shape of the relationship between 

actuarially fair premiums and effective coverage level above 85% determines 

whether RMA’s proposed extrapolation method leads to positive or negative errors 

and the magnitude of the errors. Because the curvature of this relationship varies 

according to the shape of the yield distribution, it is not possible to develop general 

predictions of the likely extrapolation errors for all 65% premium rates. However 

insight into the possible extrapolation errors that will be likely can be estimated by 

choosing different parametric forms for the yield distribution. In this review I use 

two distributions to obtain insight. The first is the truncated normal distribution. 

The second is the beta distribution. The advantage of using the truncated normal 

distribution is that it is the distribution used by RMA to determine revenue 

insurance premium rates for coverage levels up to 85%. The disadvantage of the 

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

Coverage Level

Low risk

High risk
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truncated normal distribution is that the general shape of the non-truncated part of 

the distribution is identical for all 65% premium rates. This implies that the general 

shape of the relationship between actuarially fair premiums and effective coverage 

levels cannot vary too much. As shown in Figure 1, this is not true for the beta 

distribution, hence it is used also.

The method used to gain insight into errors for these two distributions are to

normalize all distributions so that they have a mean yield of 100 and a minimum 

yield of zero. The other parameters are chosen so that yield draws from the 

distribution generate a 65% premium rate that is equal to a target rate for analysis 

purposes. Actuarially fair premiums are calculated for each coverage level using 

Monte Carlo integration methods. The Monte Carlo premium rates for 80% and 85% 

for each distribution are used to extrapolate premiums rates for effective coverage 

levels greater than 85%. These extrapolated premium rates are then subtracted 

from the Monte Carlo-estimated actuarially fair rates to determine the direction and 

magnitude of errors. 

The parameters of the distribution for six 65% premium rates are shown in Table 1.

Non-representative very high premiums rates are included in this analysis because 

of the higher likelihood that farmers and crops with high rates will be most likely to 

drop yields from their APH calculation and subsequently have high effective 

coverage levels. The maximum yield increases as the 65% premium rate increases 

to represent the fact that low risk yield distributions are simply more stable. The 

beta distribution is left-skewed when risk is low and right-skewed when risk is high.

Ignoring the truncation point, the truncated normal distribution is symmetric.

Table 1. Truncated Normal and Beta Parameters Used for Extrapolation Analysis

65% Premium Rate

1% 2.5% 5% 10% 20% 30%

Truncated 

Normal

 mu 100 100 99.9 99.6 96.7 88.3

 sigma 104.8 73.9 48.84 36.47 29.12 23.08

Beta

 p 6.83 4.62 3.15 1.90 1.13 0.69

 q 3.70 3.25 2.82 2.29 3.38 3.45

 Max yield 154.3 170.4 189.7 200.3 400 600

 Min yield 0 0 0 0 0 0

The first step is to find the parameters of the truncated normal distribution that 

generate a 65% premium rate equal to a target premium rate and a mean yield 

equal to 100. This mean yield is held constant at 100 for all simulations. Premium 

rates for the 65% coverage level were varied across the levels shown in Table 1. 

After the parameters of the yield distribution were determined, it is straightforward 
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to simulate actuarially fair premium rates using Monte Carlo simulation and to then 

calculate the errors that are generated when premium rates for coverage levels 

Figure 7 shows interpolation errors (in percentage terms) for the representative 

65% base premium rates using the truncated normal distribution. The interpolation 

errors for the beta distribution are not shown because they are nearly identical to 

those shown in Figure 7. The errors are the difference between actuarially fair 

premium rates between the interpolation points (65%, 70%, 75%, 80%, and 85%) 

and the interpolated premium rates. The errors will equal zero at each interpolation 

point because it is assumed that the actuarially fair premium rates between the 

interpolation points are known.

The amount of interpolation error depends on the degree of curvature of the

relationship between premium rate and coverage level. Because curvature 

decreases as 65% premium rates increase, so too does the percent error. But even at 

very low 65% premium rates, the amount of interpolation error is less than 2%. And 

2% of a very low rate is a quite low number. At a 65% premium rate of 10%, the 

amount of error drops to less than 0.2%. The reason why the error is so low is that 

even with a highly curved relationship, the interpolation points are so close together 

that a linear approximation works well. Thus I conclude that linear approximation 

of premium rates for effective coverage levels between 65% and 85% is 

appropriate.

Figure 7. Interpolation Errors Using the Truncate Normal Distribution
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The method I use to determine extrapolation errors made in estimating premium 

rates for greater than 85% coverage levels is to first assume that yields follow either 

the beta distribution or the truncated normal distribution. The increase in premium 

rate between 80% and 85% coverage levels is then calculated for each yield 

distribution, which is uniquely identified by the chosen parametric form and the 

target 65% premium rates. The change in premium rate between the 80% and 85% 

coverage levels for each distribution is then used to determine the premium rates 

for effective coverage levels greater than 85%. That is, the assumed known 

actuarially fair premium rates between 80% and 85% coverage levels are used to 

determine the slope of a line relating premium rate to coverage level. This slope is 

multiplied by the difference in coverage between the effective coverage level in 

question and the 85% coverage level. The resulting product is then added to the 

85% premium rate to determine the extrapolated premium rate. The resulting error 

for each effective coverage level is simply the difference between the extrapolated 

premium rate and the actuarially fair premium rate for each effective coverage level, 

which is determined by Monte Carlo methods for each distribution. All calculations 

were done using premium rates rather than CLDs. This is equivalent to using CDLs 

because CLDs are simply the ratio of a premium rate to the 65% premium rates. 

Actuarially fair premium rates are shown in Figures 8 and 9 for the truncate normal 

and beta yield distributions respectively.

Figure 8. Actuarially Fair Premium Rates for the Truncated Normal Yield 

Distribution

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.65 0.75 0.85 0.95 1.05 1.15 1.25 1.35 1.45 1.55 premium rate

Coverage Level

rate = .01

rate = .025

rate = .05

rate = 0.1

rate = .2

rate = .3

65% pemium 

rate
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Figure 9. Actuarially Fair Premium Rates for the Beta Yield Distribution

As can be seen, there are some significant differences between the two yield 

distributions in how actuarially fair premiums increase with coverage level. At very 

high coverage levels, the truncated normal distribution implies an approximately 

linear relationship whereas the beta distribution implies a concave relationship. An 

approximately linear relationship implies that extrapolation errors will be quite 

small. A concave relationship implies that extrapolation errors will be positive 

(extrapolation overestimates premium rates). Both distributions show a convex 

relationship between coverage level and premium rate at 65% premium rates of 

0.01 and 0.025. This suggests that extrapolation errors for low risk farmers will be 

negative. For intermediate rates the truncated normal appears to maintain a convex 

relationship for 65% premium rates of 0.05 and 0.10 whereas the beta is convex at 

the 0.05 rate and a bit concave at the 0.10 rate.

Figures 10 and 11 present the extrapolation errors that correspond to Figures 8 and 

9 respectively. As expected, both distributions imply large extrapolation errors for 

yield distributions calibrated to low 65% premium rates. The reason is that the 

relationship between coverage level and premium rate is convex at the 85% 

coverage level when risk is low so linear extrapolation leads to underestimates of 

what premium rates should be. The truncated normal maintains this convex 

relationship even for higher risk yield distributions. Thus, as shown in Figure 10, a 

truncated normal distribution implies that extrapolation errors will only be negative 

and they will be large for effective coverage levels above 90% unless the 65% 

premium rate is quite high. In contrast a beta yield distribution implies that at high 

0

0.1

0.2

0.3

0.4

0.5

0.6

0.65 0.75 0.85 0.95 1.05 1.15 1.25 1.35 1.45 1.55 premium rate

Coverage Level

rate = .01

rate = .025

rate = .05

rate = 0.1

rate = .2

rate = .3

65% pemium 

rate
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premium rates extrapolation errors will be positive. With the yield distribution 

calibrated to a 65% premium of 0.10, the beta distribution implies nearly a linear 

relationship so at that rate the extrapolation errors will be low.

Figure 10. Extrapolation Errors for the Truncated Normal Yield Distribution

Figure 11. Extrapolation Errors for the Beta Distribution 

-50%

-45%

-40%

-35%

-30%

-25%

-20%

-15%

-10%

-5%

0%

0.65 0.75 0.85 0.95 1.05 1.15 1.25 1.35 1.45 1.55 Extrapolation error

Coverage Level

rate = .01

rate = .025

rate = .05

rate = 0.1

rate = .2

rate = .3

65% pemium 

rate

-40%

-35%

-30%

-25%

-20%

-15%

-10%

-5%

0%

5%

10%

0.65 0.75 0.85 0.95 1.05 1.15 1.25 1.35 1.45 1.55 Extrapolation error

Coverage Level

rate = .01

rate = .025

rate = .05

rate = 0.1

rate = .2

rate = .3

65% pemium 

rate
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Discussion 

The analysis presented here demonstrates that substantial extrapolation errors are 

likely if the relationship between actuarially fair premium rates and coverage levels 

above 85% is either convex or concave. If the relationship is convex, as it will be for 

low-risk farmers, then the errors will be negative, leading to underestimation of 

premium rates using the proposed extrapolation method. When the relationship is 

concave, as it could be at very high coverage levels or if the probability density 

function is skewed to the right, then the errors will be positive, leading to 

overestimation of rates. 

The foregoing conclusions assumes that the current coverage level differentials used 

by RMA to determine 80% and 85% premium rates are consistent with the 

actuarially fair premiums that were used in this analysis. This assumption is, at best, 

only approximately true because the equations used to estimate the 80% and 85% 

coverage level differentials are not estimated using Monte Carlo methods from a 

well-specified probability distribution of yields. Rather they are estimated 

econometrically from loss cost data. Thus it is not really possible to determine 

exactly how large an error will be made using the proposed extrapolation methods. 

However, we know that the actuarially fair value of a yield insurance contract is 

simply the average indemnity payoff from the contract, conditional on the 

indemnity being positive, multiplied by the probability that an indemnity will be 

received. Whether this actuarially fair value is convex, concave, or linear with 

respect to coverage level depends on the change in the probability of collecting an 

indemnity and the change in the average size of an indemnity if one is received. 

Consider two yield distributions with the same mean but one is high risk and one 

low risk. As coverage increases from 80% to 85% to 90% coverage the change in the 

probability of receiving an indemnity increases faster for the low risk distribution 

because more of the mass of the probability distribution is near the mean than for 

the high risk distribution. The change in the average size of a payment is more 

similar for the two distributions. This implies that the degree of convexity for the 

lower risk distribution will be greater for coverage levels above 85% than for higher 

risk distributions. This result implies that unless the RMA coverage level differential 

equations greatly overestimate the actual slope in premium rates between 80% and 

85%, then for low-risk farmers, the proposed RMA extrapolation will lead to 

farmers being charged too little—perhaps significantly so—for effective coverage 

levels above 85%. For high risk farmers, the results presented here suggest that 

unless actuarially fair premium rates become concave in effective coverage levels, 

then extrapolation errors will be much smaller than for low risk farmers.

For low-risk farmers, if moral hazard issues arise (as seems likely) from offering 

effective coverage levels greater than 85% then the underestimation bias of the 

extrapolation method will be exacerbated because the actual indemnities paid out 

will exceed those implied by the Monte Carlo methods used here. Thus it seems 
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advisable to adopt methods that at a minimum, will not lead to systematic 

underestimation of premium rates for effective coverage levels greater than 85%.

One option that RMA could use as an alternative would be to adopt the proposed 

methods to interpolate premium rates for effective coverage levels that are less than 

85% and to then use the truncated normal draws for determine coverage level 

differentials for coverage levels greater than 85%. The yield draws are already 

available for rate-making purposes for revenue insurance plans of insurance. It 

would be feasible to simply use them instead of adopting a method that does not 

take into account the fact that premium rates are likely to be convex in effective 

coverage levels for the majority of farmers in the program.

Answers to Requested Items

(A)Are adequate, credible, and reliable rate-making data available? Is it likely 

that the data will continue to be available? Is the data vulnerable to 

tampering if the proposed ate methodology is approved?

Loss cost data above 85% coverage levels are not readily available hence the need to 

develop a method to determine how premium rates should increase for above-85% 

coverage levels. Simulation “data” based on assumptions about the functional form 

of yield distributions could be used instead of loss cost data. Data for losses less than 

85% coverage level are readily available.

(B)Are the explicit and implicit assumptions used in the rating process 

reasonable? 

Yes for interpolation of rates for effective coverage levels less than 85%. The linear 

assumption is appropriate as explained in my report above. 

But the linear assumption is not reasonable for extrapolating premium rates above 

85% because the relationship between premium rates and effective coverage levels 

is likely nonlinear. 

(C) Are the technical analyses (e.g., stochastic and other simulations) technically 

correct? Do they provide credible, relevant results?

The algebra provided was correctly done. No simulations were provided. 

(D) Is the data used for the analysis appropriate, reliable, and the best available?

No data were used in the analysis. Monte Carlo simulations could have revealed the 

limitations of the proposed methods.

 (F) Are the proposed premium rates likely to cover anticipated losses and a

 reasonable reserve?
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Not for low risk producers who drop yields form their APH. Perhaps for higher risk 

producers who drop yields. Because it is more likely that high risk producer will 

drop yields because most of them live in or adjacent to high risk counties, the 

probability of a high risk producer dropping yields is greater than the probability of 

a low risk producer who drops yields. Thus the aggregate amount that premiums 

fall short of anticipated losses is difficult to determine.

 (G) Is the actuarial methodology appropriate for the insured risks?

Not for determining premium rates above 85%. The relationship between rates and 

coverage levels is most likely nonlinear. The method used assumes a linear 

relationship, which is only appropriate for specific cases. 
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Short Bio of Bruce Babcock. 

Bruce Babcock is a partner in AgRisk Management. He has developed rating 

methods for crop insurance products that are widely in use today. He holds a PhD in 

Agricultural and Resource Economics from the University of California, Berkeley 

and MS and BS degrees from UC Davis. He currently holds the Cargill Chair of Energy 

Economics at Iowa State University.