Expert Review Of

Proposed Rating Methodology for APH Yield Exclusion

BPA No.: AG-645S-B-14-0026

Call Order AG-645S-K-14-0014

Prepared for

Federal Crop Insurance Corporation

Risk Management Agency

Prepared by

Bickerstaff, Whatley, Ryan & Burkhalter, Inc.

1701 N. Collins, Suite 226A

Richardson, TX 75080

February 6, 2015 David R. Bickerstaff, FCAS MAAA

Consulting Actuary
1

Executive Summary

RMA has proposed a methodology to make adjustments in the APH rates to cover a new option to 

exclude any recorded or appraised yield for any crop year in which the county yield is “at least 50 

percent below the simple average during the previous 10 consecutive crop years.” This new option 

was mandated in the 2014 Farm Bill.

As described in the example provided by RMA, the exclusion of one or more years’ yields will 

artificially increase the APH and the effective coverage level will be increased in proportion to the 

ratio of the average yield with the exclusion to the ratio before the exclusion. In the illustration, 

the APH after the exclusion is 108 compared to 100 before the exclusion, thereby increasing a .75 

coverage level to an effective level of .81 (.75 x 108/100). The actual expected (mean) yield for 

the producer, which underlies the rate per guarantee, remains at 100.

The RMA proposal to reflect this increase in the effective coverage level in the rates is to do a 

straight line interpolation of the existing coverage level relativities for effective coverage levels 

less than the current maximum (.85). For effective coverage levels greater than .85 the relativities 

will be extrapolated straight line by extending the slope between the current .85 and .80 out the 

appropriate intervals.

By employing our table of calculated loss costs by Coefficient of Variation (CV), which translates 

one-to-one into a loss cost, by coverage levels, we extended the table out to include the effective 

coverage levels of .95 and 1.05. For the most part, the straight line extrapolation procedure resulted 

in inadequate rates for these higher coverage levels. The higher the effective coverage level and/or 

the lower the CV, the more inadequate the extrapolated loss costs will be. 

As an alternative to the straight line extrapolation procedure, we are suggesting using Stirling’s

Interpolation Formula, which is not overly complicated to implement and produces vastly more 

accurate results.

In conclusion, while the proposed Yield Exclusion program has a great deal of merit and should 

be welcomed by APH insureds, it is highly recommended that RMA rethink the extrapolation 

process for coverage level relativities for the resulting very high effective coverage levels, 

especially those with relatively low base rates. 
2

Expert Review of the Proposed Rating Methodology for APH Yield Exclusion

I was engaged on December 22, 2014 by the Federal Crop Insurance Corporation (FCIC) of the 

U.S. Department of Agriculture (USDA) to prepare a review of a proposed methodology to adjust 

rates to address the option of producers to exclude yields of certain years in determining their 

Actual Production History (APH) guarantees for a prospective policy year. The 2014 Farm Bill 

directed the Risk Management Agency (RMA) to provide producers the option to exclude any 

recorded or appraised yield for any crop year in which the county yield is “at least 50 percent 

below the simple average during the previous 10 consecutive crop years.” The Act further 

mandated that the premiums shall be adjusted to reflect the risk associated with the exclusion of 

any designated years’ yields. RMA has proposed a methodology to make the necessary 

adjustments in the APH rates to cover this new option and a description of this proposed scheme 

has been provided its reviewers for an analysis and comments. I have concluded my review and I 

am pleased to present herein the results of my analysis.

Our Approach to this Review

Our approach to this review can essentially be broken down into five phases:

1. Review in some detail the recent report1to RMA providing a review of the proposed 

rating enhancements for the Yield Exclusion (YE) feature.

2. Review of the RMA-supplied monograph, The Actual Production History Yield 

Exclusion: Overview of Premium Rating. This document summarized the proposed 

scheme to account for expected loss cost increases attributable to the YE provision. 

3. Construct a table of expected APH loss costs (per guarantee) by the Coefficient of 

Variation (CV) of the yield probability distribution, further subdivided by coverage 

levels. In addition to the customary coverage levels currently available (.50, .65, ..., 

.85), the table will include effective coverage levels in excess of the .85 level, the 

highest currently available. 

4. Determine if the proposed straight line extrapolation of coverage level relativities 

greater than .85 will produce reasonable and adequate premiums.

5. If the answer to (4) is no – or at least doubtful – investigate alternative extrapolation 

schemes which would (a) result in reasonable rates for higher effective coverage levels 

and (b) not be overly complicated to implement..

 

1 Coble, K.H.,Knight, T.O., Miller, M.F.,Barnett, B.J., Review of Adjustment in Actual Production History to Establish 

Insurable Yields: Determination of Actuarially Sound Premium Rates, 2014
3

The Proposed Premium Rating Methodology for YE

To illustrate how YE can affect APH, the RMA-supplied monograph described the proposed

methodology with a hypothetical example of a corn producer’s yield experience over a 10-year 

period, reproduced below. Both the 2011 and 2012 years are eligible for exclusion. The 

preliminary step in this procedure is, for years 2011-13, substituting 60% of the county T-yield 

(the “yield plug”). Then, the 10-year average yield, which becomes the approved yield, is 100 

bushels. Then, after excluding the 2011 and 2012 years, the average yield after YE becomes 108.

Year

County

T-yield

60% of

T-yield

Grower

APH

APH

w/60% 

Substitutio

n

APH w/ 

Excluded

2005 96 58 110 110 Yields110

2006 104 62 100 100 100

2007 104 62 88 88 88

2008 104 62 118 118 118

2009 106 64 111 111 111

2010 106 64 110 110 110

2011 119 71 55 71 Excluded

2012 119 71 20 71 Excluded

2013 119 71 57 71 71

2014 119 71 153 153 153

 Approved Yield 92 100 108

For a given coverage level selection by the producer – say .75 – the adjusted guarantee becomes 

81 bushels (75 x 108/100). This in effect increases his effective coverage level to .81 from the 

original .75 in relation to the approved yield of 100. The appropriate premium rate would then be 

a function of an interpolated coverage level relativity for a .81 coverage level. The proposed 

methodology calls for a straight line interpolation between the relativities for the .80 and .85 

coverage levels.

The validity of the above described rating methodology depends on a very critical assumption and 

to fully appreciate how critical this assumption is we need to go back to APH 101. For the producer 

in question there exists an implied expected (mean) yield for the season in question. There also 

exists a probability distribution around this mean. So, for this distribution we have a mean (the 

expected yield) and a coefficient of variation (CV), or the standard deviation divided by the mean. 

For a given coverage level, the CV dictates the loss cost per guarantee, which in turn translates to 

a gross rate. So, here is the critical assumption: even after excluding years in the yield-averaging 

process, the expected yield and the mean of the distribution remains at 100 bushels. The YE 

process simply artificially jacks up the guarantee to an elevated effective coverage level, requiring 
4

the calculation of a higher coverage level relativity, by interpolation. The rates for this particular 

commodity in this county are, after all, based on actual loss experience which includes those years 

which are artificially excluded in this YE program, so it is reasonable to assume that the expected 

yield (and CV) for any producer would remain the same as before the exclusion is applied.

APH Loss Costs by Coverage Level by Underlying CV

The example described above deals with an interpolated coverage level relativity between two

existing coverage levels available to the producer (.80 and .85). I think it is safe to assume that, 

due to the proximity of the current available coverage levels (intervals of .05), a straight line 

interpolation formula would be reasonable to estimate relativities for coverage levels less than .85 

that are not exact multiples of .05. It is entirely conceivable, however, that the YE procedure could 

lead to effective coverage levels well in excess of the .85, which is the highest now offered. 

Effective coverage levels even exceeding 1.00 would not be that unlikely, it would seem. In this 

event, the proposed procedure would simply use a straight line extrapolation of the coverage level 

relativities using the slope between the two highest levels offered (.80 and .85). The key question 

here, and the key focal point of this review, is whether or not this straight line extrapolation will 

result in adequate rates for the higher effective coverage levels.

At this point, it is informative to revisit the theoretical APH loss costs per guarantee by coverage 

level by CV, using a table and accompanying graph we have used for some previous studies. In 

Exhibit 1 we have tabulated loss costs by CV for a selected group of coverage levels. For this 

expanded table we have included coverage levels .95 and 1.05. The probability distribution we 

have used is the beta distribution. Even though there is no overwhelming consensus that this 

distribution is the most representative of reality, I have found in reviewing literature over the past 

10 years that the beta is assumed to provide a better fit than either the normal or any other 

distribution, although corroborating data at the producer level seems to be hard to come by. The 

beta distribution requires parameters representing the maximum and minimum values and we have 

selected the mean plus two standard deviations as the max and zero as the min. 

The loss cost data in Exhibit 1 is depicted graphically in Exhibit 2. We have used this graphical 

depiction in previous studies but for this review we have expanded it to include two higher 

coverage levels: .95 and 1.05. One might find it curious that for each of the coverage levels less 

than 1.00 the loss cost approaches zero as the CV approaches zero. A CV of zero (theoretically) 

means that whatever the expected (mean) yield is, it is a certainty, with no variability whatsoever. 

Therefore, no losses. With a coverage level of 1.05, however, with a CV of zero the loss cost will 

be .05/1.05 (.04762), recalling that the expected yield is 100 and we are indemnifying any result 

under 105. A CV of zero is, of course, a strictly theoretical exercise but it does help in understand 

how this graph – and the underlying process – works. 

Extrapolated Relativities for Coverage Levels Greater than 0.85

In Exhibits 1 and 2 we have tabulated loss costs per guarantee rather than relativities, but it is clear 

that they are, by definition, proportional, as the relativities are simply the loss costs at a specific 

coverage level divided by the loss cost at the base (0.65) coverage level. In Exhibit 3 we have 
5

then compared the theoretical loss costs at coverage levels .95 and 1.05 with the loss cost estimates 

derived from the proposed straight line extrapolation. Columns (a) through (f) are the loss costs 

derived in Exhibit 1. The extrapolated values for coverage level .95 are shown in column (g). As 

shown in the notes, the extrapolation is accomplished using the difference between the loss costs 

at .85 and .80 coverage levels and extending that slope the appropriate number of intervals. 

Similarly, the extrapolated values for the 1.05 coverage level are shown in column (i). In columns 

(h) and (j) we calculated the ratio of the actual loss costs to the extrapolated values, and these ratios 

are shown in red. The pattern that emerges is that, as the CV’s (which translate to loss costs and 

then to rates) decrease, the error increases and as the effective coverage level increases the error 

also increases. Only with CV’s in excess of, say, .40, can one reasonably conclude that the errors 

are not material. 

Alternative Extrapolation Methods

Given what appears to be an inadequacy in the extrapolated loss cost relativities for higher 

effective coverage levels, we have investigated alternative methods of extrapolation. One 

promising candidate is the Stirling Interpolation Formula, which has fairly wide use and does not 

appear to be that challenging to implement. It can be described as follows:
6

1 1

1 1

Define the following three points:

(a) ( , )

(b) ( , )

(c) ( , )

The x's represent the coverage levels and the y's represent the corresponding coverage level rel

i i

i i

i i

x y

x y

x y

 

 

1 1

2

1 1 1

ativities.

Also:

Let 

 [ .05]

and

 coverage level in question for which interpolated or extrapolated relativity is to be determined

and

then

( ) ( 2

2 2

i i i i

x

i

x i i i i i

h x x x x h

x y

x x

u

h

u u

y y y y y y

 

  

    







     1)

Example (using actual loss costs for CV=.20 instead of relativities):

(a) (.75, 0.01907)

(b) (.80, 0.02684)

(c) (.85, 0.03673)

then find extrapolated loss cost ( ) for c

i

x

y

y

 

2

overage level ( ) = 1.02

.05

1.02 .80 4.4

.05

4.4 4.4 .02684 (.03673 .01907) (.03673 2 0.02684 .01907)

2 2

.08621

i

x

x

x

h

x x

u

h

y

y



 

  

      



In columns (k) through (p) in Exhibit 3 we have calculated the extrapolated loss costs using 

Stirling’s formula and the results are much closer to actual loss costs than the straight line 

extrapolation. For the very low CV’s (say, less than .15) there is still a fairly significant error, but 

it is much less than the results with the straight line extrapolation. I am pretty sure, though, that 

CV’s less than .15 are rare.
7

Questions and Answers Prescribed by RMA

(Note: the “Procedure for the Submission and Review of New and Revised Crop Insurance 

Policies” includes topics which address Protection of Producers’ Interests, Administrative Burden, 

Marketability, and Other Review Areas, in addition to Actuarial Soundness. For this review, 

however, the only item called upon to be reviewed is Actuarial Soundness)

(1) Actuarial soundness.

(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 rate methodology is approved?

Obviously, there currently exists no actual loss data for effective coverage levels greater than 0.85. 

Coverage Level relativities have been calculated and promulgated for coverage levels up to .85. 

The relativities are a function of (a) the base rate at the .65 coverage level and (b) the coverage 

level itself. If the loss costs in Exhibit 1 were translated to relativities (dividing each by its 

respective loss cost at the .65 level) then the resulting pattern would closely resemble the relativity 

tables as modified about four years ago. Accordingly, expanding my table in Exhibit 1 to higher 

coverage levels should provide a reasonable estimate of the associated loss costs at these levels. 

So, the absence of actual loss experience at these higher levels should not be a deal breaker. If I 

am writing automobile collision coverage with $250 and $500 deductible options and wish to add 

a $1,000 deductible option, I would have the underlying data from which to properly price the 

$1,000 deductible coverage. With this in mind, adequate data is currently available and will 

continue to be available for this program embellishment. Vulnerability to tampering would be, at 

most, a remote possibility.

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

The key assumption is, again, that the expected (mean) yield for the producer remains the same as 

before the excluded years, and I believe this is a reasonable assumption..

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

recognized as appropriate? Do they provide credible, relevant results?

Not applicable.

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

Not applicable.
8

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

reasonable reserve?

Using the proposed straight line extrapolation, there is a clear inadequacy in the resulting rate 

relativities as you increase the effective coverage level and as you decrease the underlying base 

rate. See Exhibit 3. 

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

As detailed in Exhibit 3, the straight line extrapolation falls short for the higher effective

coverage levels. Using Stirling’s Interpolation Formula gives much better results.

Conclusion

While the proposed Yield Exclusion program has a great deal of merit and should be welcomed 

by APH insureds, it is highly recommended that RMA rethink the extrapolation process for 

coverage level relativities for the resulting very high effective coverage levels, especially those 

with relatively low base rates. 
9

CURRICULUM VITAE

Name of Reviewer: David R. Bickerstaff, FCAS, MAAA

Name of Firm: BICKERSTAFF, WHATLEY, RYAN & BURKHALTER, INC.

 Richardson, Texas

Profession: Consulting Actuary

Membership in Professional Societies: • Fellow, Casualty Actuarial Society

 • American Academy of Actuaries

Education: University of Mississippi – 1960 – B.A. Mathematics

Key Qualifications:

Mr. Bickerstaff’s experience in crop insurance ratemaking dates back more than 30 years. In the 

early 1970’s, as Vice-President and Actuary at Southern Farm Bureau Casualty Insurance 

Company in Jackson, Mississippi, he performed periodic studies on crop hail programs 

underwritten in Texas (cotton, soybeans) and South Carolina (tobacco). In 1983, as a Principal 

with Milliman & Robertson (Pasadena, California, office), he was one of four co-authors of a 

comprehensive analysis and review of the procedures used by the Federal Crop Insurance 

Corporation (FCIC) of Kansas City, Missouri. Mr. Bickerstaff was the principal author of two of 

the nine separate reports resulting from that engagement: “Analyzing the Effects of Unitizing on 

Loss Costs and Related Issues” and “Analysis of Procedures to Determine Rating Areas and 

Classifications.” He was also a major contributor to a third report in that series, “Analysis of Area 

Average Yield and Individual Yield Programs.” 

Since 2002, Mr. Bickerstaff has been engaged by the Risk Management Agency (RMA), the 

operational wing of FCIC, to perform actuarial reviews of proposed new crop insurance products. 

In 2002-2003 he reviewed a Livestock Risk Protection product. In 2005, he reviewed the Pasture, 

Rangeland, and Forage Rainfall Index and NDVI Index pilot programs. In 2006, he reviewed the 

Experience Based Producer Discount Program. In 2007, reviews were performed for the 

Apiculture Vegetation Index and the Apiculture Rainfall Index programs. In 2008, he reviewed 

three other proposals, one involving a Crop Technology Discount Program, another dealing with 

Trout and Catfish farms, and a third involving a new Named Peril Weather Program. In 2009 he 

reviewed an Actual Revenue History program for navel oranges and a Maximum Indemnity Factor 

amendment to the existing Apiculture program. In 2010 he contributed to the report “Feasibility 

Research Report for Insuring Honey Bees”, prepared for RMA by AgForce of Perryton, Texas. 

Then, in later 2010, he prepared a review of a proposed County ACRE program. In 2011 Mr. 

Bickerstaff provided a review for a proposed APH product for Camelina. In 2012 he prepared 

reviews for a Pulse Crop program and a proposed Downed Rice Endorsement. In 2013 he prepared 

a review for a proposed ARPI program for rice. In 2014 he was part of the AgForce, Inc., team to 
10

prepare the rating for a new Drought Monitor Endorsement to the existing Pasture, Rangeland, 

and Forage Rain Index program. 

In 2001, Mr. Bickerstaff was engaged by Agriservices, LLC, of Blackfoot, Idaho, to provide the 

actuarial input for a proposed Whole Farm Revenue Protection plan. The plan will be proposed as 

a pilot in a selected group of counties in Idaho. The model simulates (a) the yields for various 

crops produced at the grower level and (b) the harvest prices for these crops, incorporating the 

correlation of both the yields and the prices among the crops at the grower level. To determine the 

Whole Farm loss cost and the whole farm premium, the model samples 100,000 random scenarios. 

Mr. Bickerstaff was a speaker at the November 2006 CAS Meeting in San Francisco on the subject 

of GRP programs.

Employment Record:

1990 – present Bickerstaff, Whatley, Ryan & Burkhalter, Inc., Indian Wells, California and 

Richardson, TX (since 2009). Former names: Bickerstaff & Associates; Bickerstaff & Whatley, 

Inc.

 

Position: Chairman

Types of Activities: Wide assortment of medical professional liability engagements, having served 

physician and hospital clients in every area of the U.S. Has also had extensive experience in 

personal lines pricing and loss reserves, and in the development of risk models and actuarial 

information systems. Clients have included insurance companies, managing general agents, 

captives, self-insurance trusts, risk retention groups, trade associations, law firms, and regulatory 

bodies. Highlights of other past assignments include:

 Actuarial expert reviewer for Risk Management Agency/Federal Crop Insurance 

Corporation/U.S. Department of Agriculture

 Actuarial advisor for northwest USA carrier covering agri-business risks, assisting in that 

carrier's exclusive design of a whole-farm revenue protection policy for selected crops.

 Participant in major study of Federal Crop Insurance Corporation in mid-80's

 Extensive Expert Testimony in insurance-related litigation and rate hearings

 Consultant to the Insurance Council of Texas (formerly Texas Automobile Insurance 

Service Office and Texas Insurance Organization) since 1989, annually preparing 

proposals for private passenger benchmark rates.

 Consultant to large workers’ compensation self insurance trust for restaurant owners in 

California
11

 Designed pricing models for vehicle service contract and GAP products for carrier in 

western states

 Developed Surety Bond product for one of the largest bail bondsman in Nevada

 Assisted in the development of a Patent Infringement Liability product

 Consultant to General Accounting Office (U.S. Government) for Medical Malpractice 

study

 Designed and implemented Actuarial Information System for major Southeastern personal 

lines carrier

 Consultant to major HMO in Southern California

 Extensive Workers' Compensation studies for large Midwestern state insurance department

 Consultant in 1980’s to nation's largest extended automobile warranty carrier 

1974 – 1990 Milliman & Robertson, Inc., Pasadena, California (now known as Milliman USA)

Position: Consulting Actuary; Principal

1965 – 1974 Southern Farm Bureau Casualty Insurance Company, Jackson, MS

Position: Vice President and Actuary

1963 – 1965 State Farm Mutual Insurance Company, Bloomington, IL

Position: Senior Actuarial Assistant

1960 – 1963 United States Navy

Rank: Lieutenant Junior Grade

Awards and Publications:

Mr. Bickerstaff has served on the Board of Directors of the Casualty Actuarial Society (1975-78). 

In 1972 he received the Woodward-Fondiller Prize for his paper in the Society's Proceedings. He 

has also made frequent contributions to the C.A.S. Call Paper program and the annual loss reserve 

and ratemaking seminars sponsored by the Society.

He is the author of the following published papers:

"Automobile Collision Deductibles and Repair Cost Groups: The Lognormal Model" in 

Proceedings of the Casualty Actuarial Society, 1972.

"Hospital Self-Insurance Funding: A Monte Carlo Approach", Casualty Actuarial Society Forum,

Fall 1989.
12

"Evaluating Contingent Premium Liabilities for Excess-of-Loss Swing Plans", C.A.S. Discussion 

Paper Program, 1988.

"How Your Free Tail Liability Policy is Funded," Medical Practice Management, 

September/October, 2000.
APH Loss Cost per Guarantee by Coverage Level by Yield CV 

Beta Distribution (max = mean + 2 standard deviations, min = 0)

Exhibit 1

Yield

CV

Coverage Level

0.50 0.65 0.75 0.85 0.95 1.05

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

0.00000 0.00000 0.00000 0.00000 0.00000 0.04762

0.00000 0.00000 0.00000 0.00000 0.00001 0.04766

0.00000 0.00000 0.00000 0.00000 0.00035 0.04770

0.00000 0.00000 0.00000 0.00001 0.00157 0.04774

0.00000 0.00000 0.00000 0.00007 0.00359 0.04836

0.00000 0.00000 0.00001 0.00032 0.00618 0.04985

0.00000 0.00000 0.00004 0.00084 0.00918 0.05201

0.00000 0.00001 0.00014 0.00170 0.01245 0.05464

0.00000 0.00003 0.00036 0.00291 0.01592 0.05757

0.00000 0.00008 0.00073 0.00447 0.01953 0.06074

0.00000 0.00019 0.00129 0.00635 0.02326 0.06407

0.11

0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.20

0.00001 0.00037 0.00207 0.00852 0.02708 0.06752

0.00003 0.00067 0.00308 0.01094 0.03097 0.07106

0.00008 0.00109 0.00433 0.01360 0.03493 0.07469

0.00015 0.00165 0.00580 0.01645 0.03893 0.07837

0.00026 0.00238 0.00751 0.01949 0.04299 0.08211

0.00043 0.00329 0.00943 0.02268 0.04708 0.08590

0.00068 0.00437 0.01156 0.02602 0.05121 0.08972

0.00100 0.00564 0.01388 0.02948 0.05537 0.09358

0.00142 0.00709 0.01639 0.03305 0.05956 0.09747

0.00195 0.00872 0.01907 0.03673 0.06378 0.10139

0.21

0.22

0.23

0.24

0.25

0.26

0.27

0.28

0.29

0.30

0.00260 0.01054 0.02191 0.04050 0.06803 0.10534

0.00337 0.01253 0.02490 0.04435 0.07229 0.10931

0.00428 0.01469 0.02803 0.04828 0.07658 0.11330

0.00533 0.01701 0.03129 0.05229 0.08089 0.11731

0.00652 0.01950 0.03467 0.05635 0.08522 0.12135

0.00786 0.02213 0.03816 0.06047 0.08957 0.12540

0.00935 0.02492 0.04176 0.06465 0.09393 0.12947

0.01099 0.02784 0.04546 0.06888 0.09832 0.13356

0.01278 0.03090 0.04925 0.07316 0.10272 0.13766

0.01471 0.03408 0.05313 0.07748 0.10713 0.14178

0.31

0.32

0.33

0.34

0.35

0.36

0.37

0.38

0.39

0.40

0.01680 0.03739 0.05709 0.08184 0.11156 0.14592

0.01903 0.04081 0.06113 0.08624 0.11601 0.15007

0.02140 0.04433 0.06524 0.09067 0.12047 0.15423

0.02392 0.04797 0.06941 0.09514 0.12494 0.15841

0.02657 0.05170 0.07365 0.09964 0.12942 0.16260

0.02935 0.05553 0.07795 0.10417 0.13392 0.16680

0.03226 0.05944 0.08230 0.10873 0.13843 0.17102

0.03531 0.06344 0.08671 0.11331 0.14296 0.17524

0.03847 0.06752 0.09116 0.11792 0.14749 0.17948

0.04176 0.07168 0.09567 0.12256 0.15203 0.18373

0.41

0.42

0.43

0.44

0.45

0.46

0.47

0.48

0.49

0.50

0.04515 0.07591 0.10021 0.12721 0.15659 0.18799

0.04866 0.08022 0.10480 0.13189 0.16115 0.19226

0.05228 0.08458 0.10943 0.13658 0.16573 0.19654

0.05600 0.08901 0.11410 0.14130 0.17031 0.20083

0.05982 0.09350 0.11880 0.14603 0.17490 0.20513

0.06374 0.09804 0.12353 0.15078 0.17950 0.20944

0.06775 0.10264 0.12830 0.15555 0.18411 0.21375

0.07185 0.10729 0.13309 0.16032 0.18873 0.21808

0.07604 0.11198 0.13792 0.16512 0.19335 0.22241

0.08030 0.11672 0.14277 0.16992 0.19798 0.22674

0.51

0.52

0.53

0.54

0.55

0.56

0.57

0.58

0.59

0.60

0.08464 0.12151 0.14764 0.17474 0.20262 0.23109

0.08906 0.12633 0.15253 0.17957 0.20726 0.23544

0.09355 0.13119 0.15745 0.18441 0.21190 0.23980

0.09811 0.13609 0.16238 0.18925 0.21655 0.24416

0.10273 0.14102 0.16734 0.19411 0.22121 0.24852

0.10741 0.14598 0.17231 0.19897 0.22587 0.25289

0.11215 0.15097 0.17729 0.20384 0.23053 0.25727

0.11695 0.15599 0.18229 0.20872 0.23519 0.26164

0.12180 0.16103 0.18731 0.21360 0.23986 0.26602

0.12669 0.16609 0.19233 0.21848 0.24452 0.27040
1 Loss Cost

Exhibit 2

APH Loss Cost per Guarantee by Coverage Level by Yield CV 

Beta DIstribution

0.30

0.25

0.20

0.15

0.10

0.05

0.00

0 0.1 0.2 0.3 0.4 0.5 0.6

Yield CV

CovLev 50

CovLev 65

CovLev 75

CovLev 85

CovLev 95 CovLev 105
2

Exhibit 3

Actual Loss costs for APH Effective coverage levels Gre ater that .85 Compared

to Extrapolations

(a) (b ) (c) (d) (e) (f) (g) (h) (i) (j) (k) (m) (n) (p)

Stirling Interpolation Formula

actual loss costs by coverage level (beta) extrapolated

cov lev .95

error

actual/ extrap

extrapolated

cov lev 1.05

error

actual/ extrap

.95 coverage level 1.05 coverage level

CV 0.75 0.80 0.85 0.95 1.05 u loss cost u loss cost

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

0.00000 0.00000 0.00000 0.00000 0.04762

0.00000 0.00000 0.00000 0.00001 0.04766

0.00000 0.00000 0.00000 0.00035 0.04770

0.00000 0.00000 0.00001 0.00157 0.04774

0.00000 0.00001 0.00007 0.00359 0.04836

0.00001 0.00005 0.00032 0.00618 0.04985

0.00004 0.00020 0.00084 0.00918 0.05201

0.00014 0.00052 0.00170 0.01245 0.05464

0.00036 0.00107 0.00291 0.01592 0.05757

0.00073 0.00189 0.00447 0.01953 0.06074

0.00129 0.00298 0.00635 0.02326 0.06407 0.01310 1.775 0.01986 3.227 3.00 0.01817 5.00 0.03676

0.11

0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.20

0.00207 0.00435 0.00852 0.02708 0.06752

0.00308 0.00598 0.01094 0.03097 0.07106

0.00433 0.00788 0.01360 0.03493 0.07469

0.00580 0.01001 0.01645 0.03893 0.07837

0.00751 0.01237 0.01949 0.04299 0.08211

0.00943 0.01492 0.02268 0.04708 0.08590

0.01156 0.01766 0.02602 0.05121 0.08972

0.01388 0.02057 0.02948 0.05537 0.09358

0.01639 0.02364 0.03305 0.05956 0.09747

0.01907 0.02684 0.03673 0.06378 0.10139

0.01687 1.605 0.02522 2.678 3.00 0.02256 5.00 0.04420

0.02086 1.485 0.03078 2.309 3.00 0.02702 5.00 0.05133

0.02503 1.395 0.03647 2.048 3.00 0.03153 5.00 0.05811

0.02933 1.327 0.04221 1.857 3.00 0.03603 5.00 0.06454

0.03373 1.274 0.04797 1.712 3.00 0.04053 5.00 0.07064

0.03820 1.233 0.05372 1.599 3.00 0.04502 5.00 0.07645

0.04273 1.199 0.05944 1.510 3.00 0.04949 5.00 0.08199

0.04729 1.171 0.06510 1.438 3.00 0.05395 5.00 0.08731

0.05189 1.148 0.07072 1.378 3.00 0.05840 5.00 0.09244

0.05650 1.129 0.07627 1.329 3.00 0.06284 5.00 0.09739

0.21

0.22

0.23

0.24

0.25

0.26

0.27

0.28

0.29

0.30

0.02191 0.03018 0.04050 0.06803 0.10534

0.02490 0.03364 0.04435 0.07229 0.10931

0.02803 0.03721 0.04828 0.07658 0.11330

0.03129 0.04088 0.05229 0.08089 0.11731

0.03467 0.04464 0.05635 0.08522 0.12135

0.03816 0.04849 0.06047 0.08957 0.12540

0.04176 0.05242 0.06465 0.09393 0.12947

0.04546 0.05642 0.06888 0.09832 0.13356

0.04925 0.06050 0.07316 0.10272 0.13766

0.05313 0.06464 0.07748 0.10713 0.14178

0.06114 1.113 0.08178 1.288 3.00 0.06728 5.00 0.10225

0.06578 1.099 0.08721 1.253 3.00 0.07169 5.00 0.10691

0.07044 1.087 0.09259 1.224 3.00 0.07613 5.00 0.11156

0.07510 1.077 0.09792 1.198 3.00 0.08055 5.00 0.11609

0.07977 1.068 0.10318 1.176 3.00 0.08497 5.00 0.12053

0.08444 1.061 0.10841 1.157 3.00 0.08942 5.00 0.12499

0.08912 1.054 0.11358 1.140 3.00 0.09384 5.00 0.12933

0.09380 1.048 0.11871 1.125 3.00 0.09828 5.00 0.13364

0.09848 1.043 0.12380 1.112 3.00 0.10274 5.00 0.13799

0.10316 1.038 0.12885 1.100 3.00 0.10718 5.00 0.14224

0.31

0.32

0.33

0.34

0.35

0.36

0.37

0.38

0.39

0.40

0.05709 0.06883 0.08184 0.11156 0.14592

0.06113 0.07309 0.08624 0.11601 0.15007

0.06524 0.07739 0.09067 0.12047 0.15423

0.06941 0.08175 0.09514 0.12494 0.15841

0.07365 0.08615 0.09964 0.12942 0.16260

0.07795 0.09060 0.10417 0.13392 0.16680

0.08230 0.09509 0.10873 0.13843 0.17102

0.08671 0.09961 0.11331 0.14296 0.17524

0.09116 0.10417 0.11792 0.14749 0.17948

0.09567 0.10877 0.12256 0.15203 0.18373

0.10785 1.034 0.13386 1.090 3.00 0.11164 5.00 0.14649

0.11254 1.031 0.13885 1.081 3.00 0.11612 5.00 0.15077

0.11723 1.028 0.14379 1.073 3.00 0.12060 5.00 0.15501

0.12192 1.025 0.14871 1.065 3.00 0.12508 5.00 0.15921

0.12662 1.022 0.15360 1.059 3.00 0.12958 5.00 0.16345

0.13132 1.020 0.15847 1.053 3.00 0.13409 5.00 0.16770

0.13602 1.018 0.16331 1.047 3.00 0.13860 5.00 0.17191

0.14072 1.016 0.16813 1.042 3.00 0.14312 5.00 0.17612

0.14543 1.014 0.17293 1.038 3.00 0.14766 5.00 0.18037

0.15014 1.013 0.17772 1.034 3.00 0.15221 5.00 0.18461

0.41

0.42

0.43

0.44

0.45

0.46

0.47

0.48

0.49

0.50

0.10021 0.11340 0.12721 0.15659 0.18799

0.10480 0.11805 0.13189 0.16115 0.19226

0.10943 0.12274 0.13658 0.16573 0.19654

0.11410 0.12745 0.14130 0.17031 0.20083

0.11880 0.13219 0.14603 0.17490 0.20513

0.12353 0.13696 0.15078 0.17950 0.20944

0.12830 0.14174 0.15555 0.18411 0.21375

0.13309 0.14655 0.16032 0.18873 0.21808

0.13792 0.15137 0.16512 0.19335 0.22241

0.14277 0.15622 0.16992 0.19798 0.22674

0.15485 1.011 0.18248 1.030 3.00 0.15675 5.00 0.18883

0.15956 1.010 0.18723 1.027 3.00 0.16131 5.00 0.19307

0.16427 1.009 0.19196 1.024 3.00 0.16588 5.00 0.19733

0.16899 1.008 0.19668 1.021 3.00 0.17046 5.00 0.20159

0.17371 1.007 0.20139 1.019 3.00 0.17505 5.00 0.20585

0.17843 1.006 0.20608 1.016 3.00 0.17964 5.00 0.21011

0.18315 1.005 0.21076 1.014 3.00 0.18424 5.00 0.21439

0.18788 1.005 0.21543 1.012 3.00 0.18885 5.00 0.21867

0.19260 1.004 0.22009 1.011 3.00 0.19346 5.00 0.22296

0.19733 1.003 0.22474 1.009 3.00 0.19808 5.00 0.22725

0.51

0.52

0.53

0.54

0.55

0.56

0.57

0.58

0.59

0.60

0.14764 0.16108 0.17474 0.20262 0.23109

0.15253 0.16596 0.17957 0.20726 0.23544

0.15745 0.17085 0.18441 0.21190 0.23980

0.16238 0.17576 0.18925 0.21655 0.24416

0.16734 0.18068 0.19411 0.22121 0.24852

0.17231 0.18561 0.19897 0.22587 0.25289

0.17729 0.19055 0.20384 0.23053 0.25727

0.18229 0.19550 0.20872 0.23519 0.26164

0.18731 0.20045 0.21360 0.23986 0.26602

0.19233 0.20542 0.21848 0.24452 0.27040

0.20206 1.003 0.22938 1.007 3.00 0.20271 5.00 0.23155

0.20679 1.002 0.23401 1.006 3.00 0.20734 5.00 0.23586

0.21152 1.002 0.23863 1.005 3.00 0.21198 5.00 0.24017

0.21625 1.001 0.24324 1.004 3.00 0.21662 5.00 0.24449

0.22098 1.001 0.24785 1.003 3.00 0.22127 5.00 0.24881

0.22571 1.001 0.25244 1.002 3.00 0.22592 5.00 0.25314

0.23044 1.000 0.25703 1.001 3.00 0.23057 5.00 0.25747

0.23517 1.000 0.26161 1.000 3.00 0.23522 5.00 0.26181

0.23989 1.000 0.26618 0.999 3.00 0.23988 5.00 0.26615

0.24462 1.000 0.27075 0.999 3.00 0.24454 5.00 0.27049

Notes: (a) Coefficient of Variation of yield distribution, on which loss cost is (i) straight line extrapolation: (i) = (d) + 4 * [(d) - (c)]

based. (j) = (f) / (i)

(b) - (f) Loss cost per guarantee at various coverage levels, based (k) u = (.95 - .80) / .05

on beta distribution (m) = (c) + (u/2) * [ (d) - (b) ] + (u^2/2) * [ (d) + (b) - 2*(c) ]

(g) straight line extrapolation: (g) = (d) + 2 * [(d) - (c)] (n) u = (1.05 - .80)/.05

(h) = (e) / (g) (p) = (c) + (u/2) * [ (d) - (b) ] + (u^2/2) * [ (d) + (b) - 2*(c) ]
3

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