Sumaria Systems, Inc. 

Authored by: Barry K. Goodwin, Ardian Harri, Roderick 

M. Rejesus, Keith H. Coble 

Actuarial Review for Price Volatility Factor Methodology 

Responses to Review Comments 
 

 

 

 

1 Actuarial Review for Price Volatility Factor Methodology 12/27/2014 

Actuarial Review for Price Volatility Factor 

Methodology 

Responses to Review Comments 

We greatly appreciate the thoughtful comments and suggestions offered by the reviewers. We 

are also grateful to respond. In the discussions that follow, we attempt to respond to each issue 

raised in the reviews. In many cases, we believe that the specific questions and concerns relate 

to issues that fall outside of the scope of the Tasks and Work Requirements of this particular 

project. Many of these issues were addressed in an earlier report by Sumaria Systems.1 More 

specifically, many of the concerns raised in the reviews pertain to how a measure of price 

volatility is used in the rating system rather than in how the price volatility factor is calculated. 

We refer to this earlier report in many cases. 

Before proceeding to responses to individual items, it is helpful to briefly review the “Specific 

Tasks and Work Requirements” contained in the original solicitation (Number D13PS00570). 

2.3 Specific Tasks and Work Requirements: Task Order against the IDIQ 

2.3.1 Review of price volatility factor calculation methodology Draft Report (FFP) 

Price volatility factors are necessary to the development of premium rates for crop 

insurance programs offering revenue coverage based on commodity futures markets. 

Price volatility factors are currently based on the average implied volatilities for closeto-the-money puts and calls during the last five trading days of the projected pricemonitoring period for the given commodity (as determined by Barchart.com). The 

market estimates reflect an annualized value that is further adjusted by the square root of 

the percent of the year under which the insurance contract is in place (i.e. the square root 

of the percent of a calendar year between the projected price and harvest price 

discovery). However, imbedded within the option premium, and therefore implied 

volatility valuation, are certain assumptions or costs that may not be appropriate within 

the crop insurance setting. 

The contractor shall perform the following tasks: 

1. Analyze the data and assumptions and assess the adequacy of RMA’s existing 

method for establishing price volatility factors for COMBO (see brief outline 

at: www.rma.usda.gov/pubs/2011/volatilitymethodology.pdf). 

2. Review various comments/suggestions RMA has received regarding the 

determination and use of implied volatilities in establishing premium rates. 

3. Review existing literature regarding implied volatility determination. 

4. Compare and contrast two or more alternative methods for calculating implied 

volatility to the method used by RMA. 

1

 See “A Comprehensive Review of the RMA APH and COMBO Rating Methodology: Final Report,” (April 2010), 

which is available online at http://www.rma.usda.gov/pubs/2009/comprehensivereview.pdf. 
 

2 Actuarial Review for Price Volatility Factor Methodology | 12/27/2014 

5. Review the use of the price volatility factor within the revenue rate simulation 

model underlying COMBO, including an evaluation of the underlying 

price/yield correlations assumed and the interacting effect price volatility and 

price/yield correlation have on revenue rates. 

6. Provide a draft technical report outlining all findings and recommendations. 

The Contractor shall produce a draft report that provides the analysis, results, and 

recommendations. 
3 Actuarial Review for Price Volatility Factor Methodology 12/27/2014 

Responses to NCIS Review Comments 

General Comments 

Many of the NCIS review comments argue in favor of an alternative retrospective approach to 

measuring price volatility. An argument is made that the Black-Scholes (BS) volatility has not 

tended to provide an accurate representation of the market’s estimate of the volatility associated 

with future prices. The fact that the market’s “best estimate,” as it is reflected in the BS 

volatility, appears to lack precision when compared to ex-post realized prices is taken to suggest 

that the approach is conceptually flawed. Analyses of the relationships between price 

movements between planting and harvest and the BS implied volatilities are used to argue that 

there is very little correlation between the ex-ante forecast of the BS volatility and the ex-post 

realized price differences. An analysis of alternative distributional assumptions is applied to 

bolster this argument. 

We believe these assertions are flawed on several grounds. First, as the original report has 

maintained, the relevant issues pertain not only to the accuracy of the predicted variability but 

also on suggestions for alternative approaches that are superior on intuitive, theoretical, or 

empirical grounds. Because options are a fairly recent phenomenon and in light of the annual 

nature of crop production (one observation per year), the amount of data available for an 

empirical analysis is necessarily limited. In particular, we have a total of 25 observations that 

allow a direct comparison between the options implied BS volatility and the actual price 

changes. Thus, the lack of precision regarding how well the BS volatility reflects actual price 

changes may have much to do with the relatively limited data available. On conceptual grounds, 

the BS is bolstered by a fundamental economic assumption of the rationality of agents and the 

efficiency of markets. 

1. Ineffectiveness of BS Volatility in Reflecting Price Risk 

Response to Correlation Comment. A Pearson (linear) correlation of the BS volatilities and 

realized price changes is presented in the review. It is noted that the linear correlation 

coefficients are generally close to zero. Of course, linear correlation is a narrow approach to 

measuring dependence. Monotonic transformations of the relevant variables, which by 

definition do not change ordering, can have drastic effects on Pearson correlation coefficients. 

An approach that may offer advantages involves rank correlation. We repeated the correlation 

analysis of squared realized returns (represented by the log of the ratio of harvest to planting 

prices) and the Bridge/CRB monthly average implied volatilities. Note that use of rank 

correlation lessens the reliance of the results on the particular metric used to represent price 

differences. For example, using the absolute value of returns yields the same correlation 

structure as squared returns. 
4 Actuarial Review for Price Volatility Factor Methodology | 12/27/2014 

As we have noted, the BS volatility reflects the market’s collective judgment regarding the 

uncertainty of future prices and it is difficult to draw strong conclusions from its actual ex-post 

performance in such a small sample. However, the Spearman correlation coefficients do indeed 

demonstrate a correspondence between the BS implied volatilities and the squared returns (the 

squared logarithm of the ratio of harvest to planting prices). Table 1 presents the Spearman rank 

correlation matrix for corn and soybeans, the same commodities evaluated in the NCIS review. 

The relationship is statistically significant at the α = .10 level with a rank correlation of 0.36 for 

corn. In the case of soybeans, the correlation coefficient is 0.30 and narrowly misses being 

statistically different from zero (with a p-value of 0.14). While we would caution against 

making definite conclusions about this relationship with such a small sample, it certainly is not 

the case that there is “virtually no correlation.” 

Response to comment on frequent realization of low probability events. Probabilities listed in the 

column titled “Implied Probability %” in Table 2 of the NCIS document are derived using the 

normal probability density function. An example of the normal density for a random variable X 

with mean μ and variance 2 σ is depicted in the Figure 1 below. 

Probabilities calculated for different years can be compared with each-other even though the 

mean μ and variance 2 σ change every year as one could easily transform the normal random 

variable X with mean μ and variance 2 σ to a standard normal random variable Z with mean 0 and 

variance 1. 

As depicted in Figure 1 below, actual values x of the random variable X that are two standard 

deviations away from the mean, μ − 2 2 σ ≥≥+ x μ σ , have a 5 percent probability of 

occurrence. Said differently, there is a 5 percent probability of x values being two standard 

deviations away from the mean or 2.5 percent probability of x ≤ μ − 2σ and 2.5 percent 

probability of x ≥ + μ 2σ . 

We updated the harvest price in Table 2 of the NCIS comments, and its corresponding 

probability, for 2014. The harvest price is $3.49 and the implied probability is 8.5%. The 

original results of Table 2 in the NCIS document with the updated values for 2014 are presented 

in Table 2 below. The implied probability values in the NCIS Table 2 and Table 2 below give 

the probability of P P ≤ actual where P is corn price and Pactual is the observed corn harvest price. 

For example in 2014, 3.49$ / bu. P actual = and the probability that the price of corn would be less 

than or equal to $3.49/bu. is 8.5 percent. For 2012, 7.50$ / bu. P actual = and the probability that 

the price of corn would be less than or equal to $7.50/bu. is 91.2 percent, or said differently, the 

probability that the price of corn would be greater than or equal to $7.50/bu. is 8.8 percent. 

The connection between the interpretation of the implied probability values in the NCIS Table 2 

and Table 2 below, as well as the graphical presentation in Figure 1, is illustrated by the 

following two examples. If 2 P actual = − μ σ then the probability that the price of corn would be 
5 Actuarial Review for Price Volatility Factor Methodology 12/27/2014 

less than or equal to Pactual is 2.5 percent. Similarly, if 2 Pactual = μ + σ then the probability that 

the price of corn would be less than or equal to Pactual is 97.5 percent, or said differently the 

probability that the price of corn would be greater than or equal to Pactual is 2.5 percent. 

Looking at the values of the implied probability in the NCIS Table 2 and Table 2 below, it is 

obvious that none of the probability values is less than 2.5 percent or greater than 97.5 percent. 

This means that none of the observed harvest prices is two or more standard deviations away 

from the mean – an occurrence with a 5 percent probability. Similarly, none of the probability 

values is less than 5 percent or greater than 95 percent. This means that none of the observed 

harvest prices is 1.645 or more standard deviations away from the mean – an occurrence with a 

10 percent probability. 

Another way to look at the historical price values presented in the NCIS Table 2 and Table 2 

below is as follows. Under the assumptions that price follows a lognormal distribution (and the 

natural logarithm of the price follows a normal distribution) one would expect that in 5 out of 

100 years the natural logarithm of the observed price would be either two standard deviations 

below its mean or two standard deviations above the mean. Similarly, one would expect that in 

10 out of 100 years the natural logarithm of the observed price would be either 1.645 standard 

deviations below its mean or 1.645 standard deviations above the mean. 

Given that there are only 25 years presented in the NCIS Table 2 and Table 2 below, one would 

expect that in at least one of the 25 years the natural logarithm of the observed price would be 

either two standard deviations below its mean or two standard deviations above the mean. 

Similarly, one would expect that in at least 2 of the 25 years the natural logarithm of the 

observed price would be either 1.645 standard deviations below its mean or 1.645 standard 

deviations above the mean. As noted above there are no such occurrences. This means that the 

volatility factor used in deriving premium rates has been too high. As an exercise, we lowered 

the volatility factor for all years to the level required to produce at least one occurrence with a 5 

percent probability and at least two occurrences with a 10 percent probability. Our results show 

that the volatility factor should be reduced by 15 percent to produce at least two occurrences with 

a 10 percent probability and by 20 percent to produce at least one occurrence with a 5 percent 

probability. 

One important conclusion resulting from the analysis presented above is that if there are any 

actuarial issues related to the methodology used to derive premium rates which cause premium 

rates to be lower than they should be, the issue is not the volatility factor used to derive the 

premium rates. 

Summary Response. The NCIS review criticizes the BS volatility on a number of grounds having 

to do with a retrospective, ex-post evaluation of the relationship between the implied volatility 

and price changes. We demonstrate that alternative considerations of this relationship, made 

using rank correlation, do reflect the expected relationship, albeit weakly in some cases. We 
6 Actuarial Review for Price Volatility Factor Methodology | 12/27/2014 

believe the sample is too short to place a heavy emphasis on these results. We would also note 

that any method for measuring price volatility that is retrospective necessarily makes 

unreasonable assumptions regarding the stationarity of the price distribution. This is discussed in 

detail below. Finally, the analysis presented in the review stops short of providing a specific 

recommendation regarding an alternative to the BS implied volatility. 

2. Lack of Analysis on Premium Rates and Premiums 

The analysis presented in the NCIS document argues that the volatility review stops short of 

providing an effective analysis of how the volatility shapes revenue premium rates. The review 

demonstrates the sensitivity of revenue protection premium rates to changes in volatility. We 

most certainly agree that the volatility is a very important factor in determining the price of 

revenue coverage. Item 5 of the Task Work Order requested that the contractor “Review the use 

of the price volatility factor within the revenue rate simulation model underlying COMBO, 

including an evaluation of the underlying price/yield correlations assumed and the interacting 

effect price volatility and price/yield correlation have on revenue rates.” This review is 

contained on pages 23-36 of the volatility report. Likewise, much of this material was 

extensively reviewed in the 2010 Combo rating review. 

Our review identified a minor error in the manner in which the volatility is used in the “Cost 

Estimator” to determine premiums. This issue was pointed out in previous correspondence with 

NCIS and we agree that a minor change is needed to address this issue. 

The 2010 report made a number of recommendations for future study that we believe remain 

important, though outside of the scope of this particular review. We noted there (pages 106-110) 

that a minor change in the manner in which rate relativities is applied has conceptual advantages. 

We also recommended the use of a direct analytical calculation of rates rather than dependence 

on a relatively small number of “pseudo-random” draws. We also suggested further study of the 

price-yield correlations. We agree with the NCIS document that this remains an important topic 

for additional evaluation. Our focus in this review was on the determination and measurement of 

expected price volatility and thus a detailed assessment of the price and yield correlations was 

beyond the scope of the study. 

3. Concerns with the Black-Scholes Model and Alternatives Considered 

Our review considered a number of alternatives to the standard BS implied volatility that have 

been proposed in the academic literature and in commentary on RMA’s approach to measuring 

volatility. Our conclusion was that the BS tends to perform comparably to a number of these 

proposed alternatives and further that it has important advantages in terms of transparency and 

acceptance by the financial community. Of course, it is impossible to evaluate the universe of all 
 

                                 

                             

7 Actuarial Review for Price Volatility Factor Methodology 12/27/2014 

possible alternatives. The review argues that we confined our analysis only to the case of lognormality. This is not correct. We considered the volatility model of Egelkraut, Garcia, and 

Sherrick (2007).2

 This methodology calibrates a log-normal distribution across the entire range 

of strikes. This is different from assuming that the distribution of prices is necessarily lognormal across all strikes. The distinction is subtle, but rather than calibrating a log-normal 

distribution to each strike, we calibrate it to the entire range of positively traded strikes. As 

Egelkraut, Garcia, and Sherrick (2007) note, this is equivalent to approximating the risk neutral 

density of prices using a log-normal distribution. The resulting volatility measure may differ 

from a standard BS estimate across the entire range of strikes. 

Before proceeding to a wider evaluation of alternatives to the log-normal, it is important to 

address the evaluation of alternative distributions estimated using historical price changes that is 

presented in the NCIS review. Table 3 of the NCIS document presents a number of alternative 

distribution functions that have been fit to historical price differences spanning 1968-2014. We 

believe that this analysis is invalid because of the fundamental assumption that is implicit in the 

approach—that the price density is stationary across this entire period. Put differently, to our 

understanding of the analysis, the volatility estimates that are presented in Table 3 of the NCIS 

document are assumed to be constant across this entire span (or, at the very least, the parameters 

that characterize the distribution of prices is assumed to be constant across the entire period). A 

valid density can only be estimated when the data are identically and independently distributed 

(iid). Changing market conditions necessarily suggest that this pricing density is different each 

year. 

This goes to a very important distinction between ex-ante and ex-post measures of price 

volatility. Using historical data to fit a distribution or otherwise measure volatility is going to 

result in a static (or at least slowly adjusting) measure of volatility. One would wonder how the 

program would have performed between 2003 and 2008, when volatilities doubled. Had the 

volatility been fixed at a level determined by the standard deviation of annual price returns 

between 1960 and 2007, insurers and reinsurers would have had rates based upon volatility 

measures of 0.18 and 0.16 for corn and soybeans, respectively, heading into the 2008 crop year. 

Implied volatilities exceeded 0.35 in 2008, suggesting a significant potential for large payouts 

that would not be covered by premiums. Large price swings did indeed arise in the years 

following 2007 and the industry would have realized significant financial losses had revenue 

rates been based upon a static volatility measure. 

Although we disagree with the implications of the analysis presented in the NCIS report, we did 

undertake some additional analysis in the same vein. It was unclear to us exactly how the price 

differences were expressed, but we chose to use the return between planting and harvest, 

measured as the log of the ratio of prices. Without loss of generality, we shift the density to the 

2 See Egelkraut, T.M., P. Garcia, and B.J. Sherrick. 2007. “The Term Structure of Implied Forward Volatility:

Recovery and Informational Content in the Corn Options Market.” American J. of Agricultural Economics 89(1): 1‐

11.
 

                                 

           

8 Actuarial Review for Price Volatility Factor Methodology | 12/27/2014 

right by adding one (that is, we use 1+ln(PH/PP)). We fit a range of relevant distributions to the 

price differences, including the Burr and log-normal. Goodness of fit statistics, including 

Kolmogorov-Smirnov, Anderson-Darling, and Cramer-von-Mises tests are presented in Table 3 

below. In the case of corn, every statistic favors a log-normal distribution while for soybeans the 

specification tests are split between the Burr and log-normal distributions (with the test statistics 

being very close to one-another). In light of the concerns regarding the assumed static nature of 

distributions, we do not place relevance on these results. However, within the context of the 

evaluation presented in the review, these results demonstrate that the differences in goodness of 

fit are modest and that the log-normal distribution does receive strong support. Figure 2 below 

presents the fitted cumulative distribution functions for the alternatives. 

The NCIS review also recommends consideration of retrospective GARCH or ARCH type 

models as a means for representing time-varying volatility. We do not believe such models 

would be appropriate for RMA’s needs for several reasons. First, such models are typically 

applied to high frequency data. To apply the models to the case of representing price volatility 

between planting and harvest time, one would need to simulate the variance for a very long outof-sample period. For a stationary GARCH model, these long-term volatility projections 

typically revert to the unconditional mean.3 

The NCIS review also recommended consideration of alternative distributions for the risk free 

pricing density. The review specifically recommended consideration of a Burr distribution, 

which has been shown to offer considerable flexibility in terms of representing skewness and 

kurtosis. We appreciate this suggestion and agree that considering the Burr broadens the scope 

of our conclusions. Table 4 below repeats the analysis presented in Table 6.1 of the original 

report with the addition of volatilities estimated using the Burr Type 3 and Burr Type 12 

distributions. The results are again very comparable though in no case does either version of the 

Burr provides a better fit to the realized volatilities than the log-normal. Again, it is reassuring 

that the results are robust with respect to alternative distributions. 

Finally, the NCIS review raises some valid questions regarding a potential inconsistency 

between the use of monthly averages to calculate expected prices and the use of only five days in 

calculating volatilities. We agree that this issue merits additional consideration on the part of 

RMA. The use of such averages always attempts to balance the effects of random variation 

(which is diminished by averaging) and the accuracy of projections. 

4. Adjustments for Trade Volume Weighting and Volatility Transformation 

In the volatility report, we argued that care must be exercised when dealing with thinly traded 

options. We recommended dropping option quotes with zero trades and a consideration of 

3 See, for example, Christoffersen, P.F., and F.X. Diebold. 2000. “How Relevant Is Volatility Forecasting for Financial

Risk Management.” Review of Economics and Statistics 82:12–22.
9 Actuarial Review for Price Volatility Factor Methodology 12/27/2014 

volume-weighting in the calculation of volatilities. In that the Barchart methods use near-themoney option quotes, which are the most fluidly traded instruments, we believe that the issue of 

trade volume weighting is not critical for the current methods. However, any alternative 

analytical method that might be adopted for estimating volatilities should give careful 

consideration to volume. It appears we are largely in agreement with NCIS on this issue. 

5. Need for Assessment of Price/Yield Correlations 

The NCIS review argues that the issue of parameterizing price/yield correlation deserves greater 

attention. In accordance with the Task Order Statement of Work, we did review the price/yield 

correlation issue. On page 28 of our report, we concluded that “Our suggestion is that RMA 

consider updating price-yield correlations both spatially and using newer data. Further we 

suggest RMA follow the recent developments related to applying copulas for modeling revenue. 

However, we make no stronger recommendation at this time.” On page 109 of the 2010 “Combo 

Report,” we also noted that a number of questions regarding assumed price/yield correlations 

remain relevant. We recommended that "RMA evaluate estimating price-yield correlations at a 

level below the state level as there may be clear reason to allow correlation to vary across 

production regions in a state.” We believe these recommendations remain relevant. A more 

detailed study of the proper parameterization of price/yield dependencies is clearly outside of the 

scope of the volatility study, but we agree it merits additional investigation. 

Technical Considerations and Corrections 

1. RMA’s current implied volatility estimation method. 

The review argues that RMA’s methods assume constant volatility across the growing season. 

We disagree. As the review correctly notes, the volatility is a forecast made at a point in time. 

This forecast will of course be updated by the market, but the important point is that rates must 

be determined prior to the growing season. 

Questions are also raised regarding the suitability of 500 random draws. As we noted above, we 

have recommended further evaluation of this issue in the 2010 review. We agree that 500 

random draws is a small number and further that direct analytical solutions to the determination 

of rates may be preferred to simulations run within the ADM software. 

2. SR’s alternatives to the current method. 

All of the evaluations undertaken in this study used daily closing settlement quotes for the 

relevant options. 

3. SR’s evaluation statistics. 
 

 

 

 

10 Actuarial Review for Price Volatility Factor Methodology | 12/27/2014 

The NICS report argues that modest differences may arise due to the use of the Black (1976) 

implied volatility formula and a slight variant presented by Hull. We provide below an 

explanation for the discrepancy. It is not correct to state that Black’s formula assumes nonrandomness in the evolution of the commodity spot prices as Black’s models uses futures rather 

than spot prices as the underlying price. The fact that futures prices in a risk-neutral world have 

zero drift accounts for the discrepancy between the two formulas, BSM and Black’s. A more 

technical explanation follows. 

In the formulas below, c is the call option price, S0 is the price of an asset (other than a futures), 

F0 is the futures price, K is the strike price, N(.) is the standard normal CDF, T is the term of the 

option and r is the constant risk free interest rate. 

BSM’s call option formula: 

0 1 2 

0 

1 

2 1 

( ) ( ) 

2 ln( ) ln( ) T/ 2 where 

rT c S N d K e N d 

S rT K d 

T 

d d T 

σ

σ

σ

− = −

+ − + =

= −

Black’s call option formula: 

0 1 2 

0 

1 

2 1 

( ) ( ) 

2 ln( ) ln( ) T/ 2 where 

rT rT c e F N d e K N d 

F K d 

T 

d d T 

σ

σ

σ

− − = −

− + =

= −

To highlight the well-known difference between the two call pricing formulas we will display the 

first part of each formula side-by-side: 

BSM’s formula: 0 1 2 ( ) ( ) rT c S N d K e N d − = −

Black’s formula: 0 1 2 ( ) ( ) rT rT c e F N d e K N d − − = − . 

As noted, the difference between the two formulas is that the asset price in the first term of the 

BSM’s formula, 0 S is not discounted, while the futures price in the first term in the Black’s 

formula, F0 is discounted at the risk free rate. 

The reason for 0 S not being discounted is that different risky assets (stocks, stock indexes, 

exchange rates, etc.) in a risk-neutral world would return the risk free rate, r. Thus, their value at 

time T in a risk-neutral world would be equal to 0 

rT 

T S e S = . Discount this future value to the 

present and you get: 0 0 0 ( ) rT rT rT 

T S e S e e S S − − = = = . 
 

 

11 Actuarial Review for Price Volatility Factor Methodology 12/27/2014 

The reason for F0 being discounted is the well-known fact that futures in a risk-neutral world 

would return zero. In other words, the expected value of futures at time T in a risk-neutral world 

is 0 ( ) EF F T = . Discount this future value to the present and you get: 0 

rT e F − . 

Even in light of the discussion above, note that we do not use Black’s formula directly in our 

calculation of the BS volatility4

 but instead use embedded procedures in SAS. In particular, the 

implied volatility is given by 

proc fcmp outlib=sasuser.funcs.options; 

 function blkschc(strike_price, term, p_, rate, volty); 

 return(blkshclprc(strike_price, term, p_, rate, volty)); 

endsub; 

 function bsvoltyc(settlement, strike_price,term, p_, rate); 

 array opts[5] initial abconv relconv maxiter status 

(.3 .001 1.0e-6 500 -1); 

 iv = solve("blkschc", opts, settlement, strike_price, term, p_, rate, .); 

 return(iv); 

endsub; 

run; 

This is technically correct and consistent with Hull. 

The NCIS review does point out that the calibration of a log-normal distribution across the entire 

range of relevant strikes uses Black’s original formula. However, as we have noted above, this 

does not directly correspond to the BS volatility but rather is the case of a log-normal 

approximation to an unknown pricing density. 

The NCIS report notes that the commodity symbols appearing in SAS seem confusing. We 

agree but would note that these are the correct symbols used by CRB/Bridge. We verified this 

with Bridge. The specific codes are: CZO = soybeans, LO = crude oil, PY = corn, RRC = rough 

rice, WZ = Chicago wheat. 

The mean absolute and squared error statistics presented are defined as given in the table 

captions. In particular, they are the average absolute and squared differences of the estimated 

volatility and the realized volatility. Likewise, when comparing price differences, these are 

again given in terms of differences in absolute prices (cents per bushel). 

The sample period encompasses all available options for the relevant options. The data generally 

begin in the 1985-1990 period, depending on the contract. 

4 We updated our results of table 6.1 in the original report (presented in Table 4) to include the results derived using 

the Black formula. There are only trivial changes between the results based on the Black and BSM formula. 
 

 

12 Actuarial Review for Price Volatility Factor Methodology | 12/27/2014 

The NCIS document also argues that the volatility study does not test for autocorrelation in the 

realized volatility. We agree, but see no relevance to our conclusions. There is but a single 

realized volatility for each year and the comparisons are all made to this realized volatility. 

The NCIS review further notes that the volatility study does not test the significance of the 

differences in alternative volatility measures. Table 5 below has added t-tests of the significance 

of the differences for each alternative volatility measure. Given the very large sample size, the 

differences are nearly always significant, but the results reinforce conclusions favoring the 

standard Black Scholes and log-normal volatility measures. 

The NCIS report argues that realized volatilities should have been adjusted for time differences. 

We disagree in that all volatilities apply to roughly the same time basis—planting to harvest. 

These could all be annualized but it would not change the conclusions. 
13 Actuarial Review for Price Volatility Factor Methodology 12/27/2014 

Table 1. Spearman Correlation Coefficients (BS Implied Volatility and ln(Harvest/Planting)2). 
14 Actuarial Review for Price Volatility Factor Methodology | 12/27/2014 

Table 2. The Implied Percentiles from the Corn Harvest Price Distribution 

Year Base Price 

($/bu.) 

Volatility 

Factor 

Harvest Price 

($/bu.) 

Implied 

Probability (%) 

2014 4.62 19 3.25 (3.49)* 3.8 (8.5)* 

2013 5.65 20 4.39 12.9 

2012 5.68 22 7.5 91.2 

2011 6.01 29 6.32 62.5 

2010 3.99 26 5.46 90.9 

2009 4.04 34 3.72 47.3 

2008 5.4 29 4.13 21.5 

2007 4.06 25 3.58 35.6 

2006 2.59 23 3.03 78.9 

2005 2.32 20 2.02 28.2 

2004 2.83 21 2.05 7.5 

2003 2.42 17 2.26 37.4 

2002 2.32 18 2.52 71.4 

2001 2.46 19 2.08 22.4 

2000 2.51 21 2.04 18.9 

1999 2.4 19 2.01 20.3 

1998 2.84 20 2.19 11.6 

1997 2.73 19 2.81 60.1 

1996 3.08 20 2.84 37.3 

1995 2.57 15 3.23 94.2 

1994 2.68 17 2.16 11.1 

1993 2.4 15 2.49 62.3 

1992 2.7 18 2.09 9.5 

1991 2.59 16 2.51 45.3 

1990 2.47 16 2.3 35.2 

Note: * Numbers in parentheses are the updated values. 
15 Actuarial Review for Price Volatility Factor Methodology 12/27/2014 

Table 3. Evaluation of Alternativ 

B. 

ve Distributio 

A. Corn 

. Soybeans 

ons for [1+ ln(Pharvest/Pplanting)] 
          

     

     

           

             

           

             

       

     

               

           

           

         

     

           

             

           

             

       

     

               

           

           

         

 

16 Actuarial Review for Price Volatility Factor Methodology | 12/27/2014 

Table 4. Volatility Comparisons Including Burr 3 and Burr 12 

(Refers to Table 6.1 of Volatility Report) 

All Corn Soybeans Wheat

(n=1,251) (n=539) (n=554) (n=158)

Method Mean Absolute Error

Standard Black Model 0.0478 0.0409 0.0402 0.1007

Standard Black Scholes (Barcharts) 0.0479 0.0418 0.0389 0.1031

Log‐Normal (Restricted Mean) 0.0487 0.0417 0.0410 0.1021

Weighted Log‐Normal (Restricted Mean) 0.0496 0.0417 0.0429 0.1028

Egelkraut, Garcia, and Sherrick

(Unweighted) 0.0528 0.0467 0.0408 0.1191

Egelkraut, Garcia, and Sherrick (Weighted) 0.0532 0.0475 0.0410 0.1191

Burr Type 3 0.0539 0.0477 0.0420 0.1200

Burr Type 12 0.0523 0.0462 0.0404 0.1186

Model‐Free Volatility 0.1488 0.0456 0.2366 0.2012

Mean Squared Error

Standard Black Model 0.0045 0.0032 0.0028 0.0157

Standard Black Scholes (Barcharts) 0.0046 0.0033 0.0028 0.0162

Log‐Normal (Restricted Mean) 0.0046 0.0033 0.0029 0.0159

Weighted Log‐Normal (Restricted Mean) 0.0047 0.0033 0.0030 0.0159

Egelkraut, Garcia, and Sherrick

(Unweighted) 0.0059 0.0045 0.0035 0.0202

Egelkraut, Garcia, and Sherrick (Weighted) 0.0060 0.0045 0.0034 0.0204

Burr Type 3 0.0060 0.0045 0.0036 0.0201

Burr Type 12 0.0058 0.0043 0.0034 0.0198

Model‐Free Volatility 0.1656 0.0040 0.3514 0.0585
 

 

 

17 Actuarial Review for Price Volatility Factor Methodology 12/27/2014 

Table 5. T-Tests of Mean Errors 

(Refers to Table 6.1 of Volatility Report) 

All Corn Soybeans Wheat 

(n=1,251) (n=539) (n=554) (n=158) 

Method Mean Absolute Error 

Standard Black Scholes (Barcharts) 0.0479 0.0418 0.0389 0.1031 

Log-Normal (Restricted Mean) 0.0487 0.0417 0.0410 0.1021 

Weighted Log-Normal (Restricted Mean) 0.0496 0.0417 0.0429 0.1028 

Egelkraut, Garcia, and Sherrick (Unweighted) 0.0528 0.0467 0.0408 0.1191 

Egelkraut, Garcia, and Sherrick (Weighted) 0.0532 0.0475 0.0410 0.1191 

Model-Free Volatility 0.1488 0.0456 0.2366 0.2012 

t-test of Mean Error = 0 

Standard Black Scholes (Barcharts) -13.1420 -9.5873 -1.8972 -16.5655 

Log-Normal (Restricted Mean) -9.5978 -7.4063 1.7687 -16.2608 

Weighted Log-Normal (Restricted Mean) -9.2277 -7.7695 2.4561 -15.7173 

Egelkraut, Garcia, and Sherrick (Unweighted) -23.8621 -16.9253 -11.7365 -18.6134 

Egelkraut, Garcia, and Sherrick (Weighted) -23.5253 -17.5388 -10.7949 -18.3290 

Model-Free Volatility 3.9894 -11.5941 7.4533 -15.3835 
 

18 Actuarial Review for Price Volatility Factor Methodology | 12/27/2014 

Figure 1. Graphical presentation of the normal probability density function. 
19 Actuarial Review for Price Volatility Factor Methodology 12/27/2014 

Figure 2. Estimated 

Actuarial Re 

d Cumulative 

B. 

eview for Pric 

Distribution 

A. Corn 

. Soybeans 

ce Volatility 

ns for [1+ ln( 

Factor Meth 

(Pharvest/Pplantin 

hodology 

ng)] 
20 Actuarial Review for Price Volatility Factor Methodology | 12/27/2014 

Responses to Munich-RE Review Comments 

1. Adequacy of the BSM as the Core of RMA’s Revenue Protection Rate Making 

The review comments of Munich-Re argue that reliance on financial asset markets (options) to 

derive measures of crop price volatility may induce biases not consistent with market 

fundamentals. We would argue that, if one accepts the efficiency of such markets, then marketbased measures of volatility should correspond to market fundamentals. We have considered a 

range of alternative volatility measures and have not been able to derive an alternative that would 

offer advantages over the current method. 

A number of arguments regarding the superiority of retrospective ex-post measures of price 

volatility have been advanced. As we have noted in our responses to the NCIS review, we 

disagree with these assertions and believe that reliance on static or slowly-adjusting volatility 

measures could result in substantial risks to the industry. We have also pointed out that 

assertions that the implied volatilities do not necessarily correspond to actual price changes are 

questionable. This is complicated by a relatively small sample but we do demonstrate 

statistically significant rank correlations among these alternatives. 

2. As a Public-Private Partnership, Alternatives to the BSM Should Be Considered 

The Munich-RE review argues that the risk exposure of insurers and reinsurers necessarily 

justifies recognition of longer-term risk factors. We do not feel that we are in a position to offer 

an opinion on these issues as the scope of our analysis was necessarily confined to the specific 

issue of measuring price volatility in rating annual revenue insurance contracts. 

Without making inappropriate inferences, it seems that a subtle argument regarding the necessity 

for loading of price risk much as is done for yield risk is being made. We do not have an opinion 

on this issue as it is clearly outside of the scope of our assignment. 

3. Proposed Adjustments to the BSM volatility 

An argument is made regarding the fact that prices and volatilities necessarily adjust to reflect 

new information during the growing season. This is true but we do not believe it is necessarily 

relevant to the manner in which the BSM is used in rating. As we have noted in the responses to 

NCIS, the volatility is a forecast made at a point in time. This forecast will of course be updated 

by the market, but the important point is that rates must be determined prior to the growing 

season. 

The Munich-RE review also argues that the BSM omits potentially relevant information 

contained in the tails of the price distribution. An argument is made that, despite the very thinly 
 

21 Actuarial Review for Price Volatility Factor Methodology 12/27/2014 

traded nature of strikes in the tails, information regarding tail behavior could be extracted. Much 

of our evaluation of alternatives was directed toward considering alternatives that use such 

information—as long as the assets were traded. For example, calibration of a log-normal across 

the entire range of strikes results in a volatility that differs from the BSM volatility for near the 

money strikes. We do not find that such alternatives would necessarily improve the precision of 

volatility estimates. 
22 Actuarial Review for Price Volatility Factor Methodology | 12/27/2014 

Responses to Rain and Hail Review Comments 

Rain and Hail provides a very thoughtful review that mainly emphasizes a need for further 

evaluation. The review notes that there is a perceived lack of correspondence between the BS 

implied volatilities and actual price changes. As we have noted in our responses to NCIS, we do 

not agree with the conclusion and further do not believe a preferred alternative has been 

identified. As we note, rank correlations between price changes and volatilities do demonstrate 

the expected relationships, albeit somewhat weakly. This is largely due to the relatively small 

sample and the fact that prediction of volatility is a challenging task. 

Rain and Hail argue that RMA should consider floors in the volatility or other changes in the 

SRA that would mitigate the exposure of the industry to price risk. We do not offer an opinion 

on these suggestions as they are issues outside of the scope of our assignment. Our evaluation 

was limited to the suitability of the BS volatility in representing price volatility and the 

consideration of potential alternatives.