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Insurance premium calculation using credibility analysis: an example from livestock mortality insurance.

ABSTRACT

A major problem facing livestock producers is animal mortality risk. Livestock mortality insurance is still at the initial stages, and premium computation approaches are still relatively new and will require more research. This study seeks to provide a first step for developing a better understanding of livestock insurance as a solution to mortality risk, as it explores improved methods for livestock mortality insurance modeling procedures, and premium computation, using credibility analysis. The purpose of this study is to develop improved estimates for livestock mortality insurance premiums for Canada under a credibility framework. We illustrate our approach through one example using livestock data from 1999 to 2007.

INTRODUCTION

Highly contagious animal diseases, such as classical swine fever, and foot and mouth disease, can have significant economic consequences. For example, the outbreak of Classical Swine Fever in the Netherlands in 1997/1998 resulted in total financial losses estimated at US$2.3 billion (Meuwissen et al., 1999). The epidemic of foot-and-mouth disease in the United Kingdom in the spring and summer of 2001 cost $16 billion to the agricultural and support industries. In the United States, Livestock Risk Protection (LRP) insurance and Livestock Gross Margin (LGM) insurance were first offered in the summer of 2002, but were pulled from the market in December 2003 following discovery of Bovine spongiform encephalopathy (BSE), and were resumed in October 2004, with substantial program modifications (Babcock, 2004).

While crop insurance premium computation is fairly straightforward, livestock mortality insurance is still at the initial stages, and premium computation approaches are still relatively new, evolving, and require more research. The purpose of this study is to develop improved estimates for livestock mortality insurance premiums for Canada under a credibility framework.

BACKGROUND

Livestock mortality insurance differs from other agricultural insurance such as crop insurance in a number of ways. First, it is more complex than crop insurance, with large loss events or possible contagious epidemic diseases that can spread quickly across borders and create large losses. For "large-event" livestock diseases such as BSE, avian flu, and foot and mouth disease, it is difficult to know the frequency and intensity of such events. Second, for livestock disease, government policy (disease surveillance, regulation, monitoring, and reporting) and insurance policy requirements (discounts or surcharges) can play a role in preventing disease and lowering insurance costs in the long term. Third, livestock losses in one production period may be followed by losses in the next production period. These consequential losses are often because a barn contains disease. When the animals die, or must destroyed due to disease, sometimes the producer cannot quickly aquire new breeding stock for rebuilding an animal herd, and this may indicate a need for business interruption insurance. Fourth, livestock insurance involves insuring different stages of animal production and different premium rates, in contrast to crop insurance. Fifth, livestock insurance is more likely to encounter moral hazard, the possibility that producers may take less care of animals if they know they are insured. In a worst case scenario, a producer could attempt to save on feed costs and other costs by allowing animals to die and collecting on the insurance policy. Moral hazard is potentially more of a problem with livestock than crops. This because it may be relatively easier for a livestock producer to allow their livestock to die in order to collect an indemnity payout from the insurer, because livestock requires more daily management and care (e.g., feeding, vaccinations, etc., compared to crops). Crops on the other hand, depend heavily on nature (e.g., weather) rather than primarily producer management. Also, livestock are often held within a barn, and so improper management or fraud is less transparent, whereas crops are produced outdoors in an open area where questionable management practices or fraudulent behavior can be more easily discovered. Sixth, it may be difficult to verify cause of death for mortality insurance, and not all causes of death may be covered by mortality insurance. For example, electricity may fail in a barn killing a number of animals due to ventilation fan failure, and mortality insurance could be assumed not to cover this event. However, a producer might claim that it was social brutality (animal biting) that killed the animals, and this could cause a loss measurement dispute problem. Seventh, since livestock insurance is relatively new, it could take considerable time before producers would be comfortable purchasing it. This could result in lower insurance use by livestock producers, and increased costs for administration.

CREDIBILITY THEORY

Credibility theory is a set of quantitative tools that allows an insurer to adjust future premiums based on past experience on a risk, or group of risks. The first of these quantitative methods is the limited fluctuation credibility theory that was suggested in workers compensation insurance by Mowbray (1914). In general, the credibility estimate can be written as a weighted average of two known quantities (Whitney, 1918). See the Encyclopedia of Actuarial Science (2004) for more information.

A recent study by Englund et al. (2009) uses a multivariate generalization of the recursive credibility estimator of Sundt (1981) where the risk parameter itself is modeled as an autoregressive process. In the Bayesian credibility theory, the Bayesian premium is obtained from the predictive distribution function. Jewell (1974) shows that if the likelihood function is of exponential family and the prior is conjugate, the Bayesian premium and the credibility premium are equal. See Sundt (1986), Halliwell (1996), Greig (1999), and Buhlmann and Gisler (2005) for more comprehensive surveys on credibility theory. The focus of this article is on Bayesian credibility theory, and the procedure for the estimation is outlined in detail using Canadian livestock data as an example.

DATA

Hogs are typically produced in three stages, and the total time to raise one hog is about 25 weeks. Stage 1, called the "prewean" stage, is from 0 to 3 weeks, where the hog ends at about 11 pounds. The second stage, called the "wean/weaner/weanling" stage is from 3 to 9 weeks, where the hog ends at about 50 lbs. Finally, the third stage of production, called the "finisher/market/feeder" stage, is from 10 to 25 weeks, where the hog ends at about 260 lbs. The focus of this study is on Stage 3 finishing pigs, where approximately 139,000,000 pigs are examined over monthly periods from 1999 to 2007. For a number of periods between 1999 and 2005, limited amounts of data were available.

The (stage) mortality rate was computed from a North America-wide sample (including Canada, Central United States, Midwest United States, Southeast United States, and West United States), and was found to be 5.93 percent for Stage 3. The stage mortality rates for Stage 1 were found to be 13 percent and for Stage 2, 4.2 percent. For example, for Stage 3, if the producer started with 100 pigs, then at the end of 16 weeks when the pigs were ready for market, about 6 pigs (5.93) would have died, leaving 94. A Canadian sample showed a mortality rate of 3.83 percent. The North America wide sample was also studied in order to get a larger sample beyond Canada, and it also provided a more conservative estimate of risk (e.g., a higher mortality rate than the 3.83 percent). For example, diseases could occur in the future in Canada with higher mortality rates than were observed in the 3.83 percent Canadian-specific data, bringing it to converge at the higher mortality rate given by the North American data. Therefore, the higher 5.93 percent North America-wide data used in this study may allow for this scenario.

BAYESIAN FRAMEWORK FOR MORTALITY AND CATASTROPHE INSURANCE

In this section, we build a Bayesian model with fully specified priors to accommodate catastrophe insurance. Canadian Food Inspection Agency (CFIA) reportable diseases are among the most serious of all diseases to affect animals, and tend to be highly transmissible and spread quickly, and so producers are required to report them to the CFIA. While not frequent due to the many precautions taken by producers and regulators, such diseases have potential to cause high animal loss and impact many producers, and result in "epidemics" that have potential for economic catastrophe for the animal sector. These diseases usually involve the animal dying, or else welfare slaughter, and quarantines, in order to quickly stamp out the disease. A few of these diseases could also potentially spread to the human population. Border closure may result due to export bans, as importing countries may fear that the disease could enter their country. Examples of CFIA reportable diseases are avian flu, BSE, and foot and mouth disease.

Let [n.sub.ijk] be the number of animals from the ith period, i = 1, ..., 1, jth producer, j = 1, ..., J, and kth barn, k = 1, ..., [K.sub.ij]. Let [X.sub.ijk] be the number of dead animals from [n.sub.ijk] within the given stage. In the Bayesian credibility model, we assume that each animal in [n.sub.ijk] will die within the stage with probability [Q.sub.j] = [q.sub.j] for all causes. Each producer has its own (stage) mortality rate but the mortality rates remain the same for all farms owned by the same producer. We also assume that the expected mortality rate will not change over time unless the occurrence of a catastrophe event. Assuming independent mortality, [X.sub.ijk] follows a binomial distribution with parameters [n.sub.ijk] and [Q.sub.j] = [q.sub.j]. That is,

[X.sub.ijk]|[Q.sub.j] = [q.sub.j] ~ Bin([n.sub.ijk], [q.sub.j]), i = 1, ..., j = 1, 2, ..., J, k = 1, ..., [K.sub.ij].

The prior distribution of [Q.sub.j]s are assumed to be independent with parameters [a.sub.j], and [b.sub.j] (can take all ([a.sub.j], [b.sub.j]) equal). That is,

[Q.sub.j] ~ Be([a.sub.j], [b.sub.j]), j = 1, ..., J.

Let [X.sub.ijk] be the number of dead animals from [n.sub.ijk] within the given stage. Let [M.sub.i] be the catastrophic indicator for the ith month, i = 1, 2,.... That is, [M.sub.i] = 0 if there is no catastrophe event within the month, and [M.sub.i] = 1 otherwise. We assume that [M.sub.i]s are independent and identically distributed Bernoulli random variables given the parameter [Q.sub.M] = [q.sub.M]. That is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let [L.sub.ijk] be the catastrophic indicator for the ith month, jth producer, and kth barn. If [M.sub.i] = 0, then [L.sub.ijk] = 0 with probability 1, for all j = 1, ..., J and k = 1, ..., [K.sub.ij]. If [M.sub.i], then [L.sub.ijk] are assumed to be independent and identically distributed Bernoulli random variables with parameter [Q.sub.L] = [q.sub.L]. That is,

([L.sub.ijk] |[M.sub.i] = 1) ~ Bin(1, [q.sub.L]), i = 1, 2, ..., j = 1, ..., J, k = 1, ..., [K.sub.ij].

When the catastrophe occurs in a given month, all animals in the same barn attacked by the given federal reported disease will be slaughtered. Thus, if [L.sub.ijk] = 1, then [X.sub.ijk] = [n.sub.ijk] with probability one. If [L.sub.ijk] = 0, [X.sub.ijk] follows a binomial distribution with parameters

[n.sub.ijk] and [Q.sub.j] = [q.sub.j]. That is,

([X.sub.ijk] | [L.sub.ijk] = 0) ~ Bin([n.sub.ijk], [q.sub.j]), i = 1, ..., j = 1, 2, ..., J = k = 1, ..., [K.sub.ij].

The prior distributions of [Q.sub.L] and [Q.sub.M] are assumed to be independent following the beta distribution. That is,

[Q.sub.L] ~ Be([a.sub.L], [b.sub.L]) and [Q.sub.M] ~ Be([a.sub.M], [b.sub.M]).

Let Q = ([Q.sub.1], ..., [Q.sub.J], [Q.sub.L], [Q.sub.M]) be the parameter vector and assume that the parameters are mutually independent.

Given the latent variables, the conditional likelihood of observations is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The posterior distribution of latent variables is as prior, only with updated parameters:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

RISK ASSESSMENT IN MONTH I + 1

The total claim amount in month I + 1 (drop subscript I + 1) out of [n.sub.jk], j = 1,2, ..., J, k = 1, ..., [K.sub.ij], is

X = [[summation].sub.j][[summation].sub.k][(1 - [ML.sub.jk])[X.sub.jk] + [ML.sub.jk] [n.sub.jk]] = n.. - [[summation].sub.j][[summation].sub.k] (1 - [ML.sub.jk])([n.sub.jk] - [X.sub.jk]).

The expected value of the total claim is

E[X] = n.. - (1 - E[[Q.sub.M]]E[[Q.sub.L]]) [[summation].sub.j][n.sub.j]. (1 - E[[Q.sub.j]]) (1)

and the variance is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2)

The expected value of powers of [Q.sub.M], [Q.sub.L], and [Q.sub.j] are to be taken in the posterior distribution, under which they are independent beta variables. Note that we have used the assumption that variables in different years are conditionally independent for given latent variables. See the Appendix for a detailed proof. The total expected claim amount (1) is the sum of the individual premiums in (3). The variance in (2) is useful in the context of solvency control, under which a capital requirement is made for the entire portfolio.

For a single barn, the claim amount for producer j and barn k (drop subscript I + 1) is

(1 - [ML.sub.jk])[X.sub.jk] + [ML.sub.jk] [n.sub.jk].

When there is no deductible, the credibility estimate (posterior mean) is

[n.sub.jk] - (1 - E[[Q.sub.M]E[[Q.sub.L]) [n.sub.jk] (1 - E[[Q.sub.j]]). (3)

Let P be the insured price per animal and [d.sub.j] be the deductible in terms of number of animals in a given barn. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be the total net premium for the/th producer kth barn with deductible [d.sub.j] and coverage level [phi]. We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [[x].sub.+] = x if x [greater than or equal to] 0, 0 otherwise. Here, the total liability for the jth producer, the kth barn, is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and the premium rate per dollar of liability is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Again, all the expectations above are to be taken in their posterior distributions.

EXAMPLE PREMIUM CALCULATION FOR LIVESTOCK MORTALITY INSURANCE

As an example of the Bayesian credibility framework outlined above, this section illustrates the methodology using pig data obtained from AgriStat. The data consist of monthly pig data from August 2000 to May 2007, for all three production stages for pigs in Canada, and the data are anonymous, so producers are not identified, for privacy reasons. The total number of placed pigs, dead pigs, and the "stage" mortality rates are shown in Table 1. The time-series plots of the mortality rates of the three stages are in Figures 1-3. From 2000 to 2005, mortality rates were mainly from larger producers and usually recorded twice a year. After 2005, the rates were from smaller producers and were available every month. Based on the plots, there does not appear to be a trend in the mortality rate; however, heteroscedasticity is observed owing to the smaller sample sizes (number of placed pigs) in the last 2 years. The remainder of this example premium computation focuses on Stage 3, finishing stage pigs.

We first test the null hypothesis that the mortality rates are equal among producers. We have a near zero p-value resulting in the rejection of the null hypothesis. For each producer in Stage 3, the prior distribution is given such that E([Q.sub.j]) = 0.05 and [square rootof Var([Q.sub.j]) = 0.05 for j = 1, ..., J = 13. The posterior distributions [Q.sub.j] are shown in Figure 4. In ad dition, some characteristics of the mortality parameters for all Stage 3 producers are shown in Table 2. The posterior means clearly differ from producer to producer, which is consistent with the result from the likelihood ratio test. Also, as expected, larger producers tend to have smaller posterior standard deviations (less than 0.02 percent for first three producers in Table 2). Let [n.sub.I+1,j,k] = 1,000, for all j (producer) and k (barn), the Bayesian credibility premiums (covering the whole stage) can be obtained form the second column of Table 2. For example, to insure 1,000 pigs with a 100 percent coverage level without deductible, the total premium per $100 value per pig for Producer OBS 001 is $100 x 1,000 x 3.24 percent = $3,240. The predictive probability mass functions for all 13 producers are plotted in Figure 5. The marginal predictive probability mass function, assuming we do not know where the risk (producer) is from, is shown in Figure 6. The expected value is about 38 (1,000 x 3.83 percent), and the distribution is skewed to the right with a heavier tail, as expected.

The prior distribution of [Q.sub.M] is given such that E([Q.sub.M]) = 0.01 and [square root Var([Q.sub.M])] = 0.01. The prior distribution of [Q.sub.L] is given such that E([Q.sub.L]) = 0.2 (20 percent farms hit) and [square root of Var([Q.sub.L])] = 0.2. That is, our prior belief is that the CFIA federally reportable disease will occur once every 100 cycles (stages), and when it occurs, 20 percent of the farms will slaughter or be ordered to slaughter their pigs. The prior distribution of [Q.sub.J] is given such that E([Q.sub.J]) = 0.05 and [square root of Var([Q.sub.J])] = 0.05 for all j. That is, 5 percent (stage) mortality rate for all producers. Since we do not observe any slaughter information from the given data set, the posterior mean of [Q.sub.M] decreases from 0.01 to 0.008, while the distribution of [Q.sub.L] remains the same given the observed data. The unconditional probability of a farm hit (a catastrophic event causing slaughter) is (0.2)(0.008) = 0.0016 within any given stage, which is about (0.0016)(12)(16)/52 = 0.6 percent within one cycle (16 weeks) of Stage 3, finishing pigs. Let [n.sub.I+1,j,k] = 1,000, the predictive distribution [X.sub.I+1,j,k] of given [n.sub.I+1,j,k], is a mixed distribution of a degenerated distribution, where = [X.sub.I+1,j,k] = [n.sub.I+1,j,k], with probability 0.6 percent, and the beta-binomial distribution with probability 1 - 0.6 percent = 99.4 percent.

The premium rates per dollar of coverage for all Stage 3 producers are listed in Table 3. The premium rates without covering the federally reportable diseases (FRD) in the second column, and are equivalent to the posterior means in the third column of Table 2. The percentage loadings for covering FRD are in the last column. The premium rates (with FRD coverage) with deductibles from 0 to 100 for all 13 producers are plotted in Figure 7. To evaluate the impact of a deductible, we calculate the loss elimination ratio (LER) which is defined as the ratio of the decrease in the expected premium with a deductible to the expected premium without the deductible. It is interesting to note that for producers with lower premium rates, the LERs drop faster but stay at higher levels when the deductible increases (Figure 8).

For example, Producer OBS 009 has the lowest premium rate (3.29 percent), and the LER drops the fastest for deductibles up to around 35, and stays at the highest level (0.16) after 40 deductible. Higher deductible helps to lower premium and it is most effective with lower risk. At the level around 40, it eliminates most of the mortality risk but only a portion of the risks associated with the federally reportable diseases.

SUMMARY

A major problem facing Canadian livestock producers, such as pig producers, poultry producers, and cattle producers, is animal mortality risk. This study seeks to provide a first step for developing a better understanding of livestock insurance as a solution to mortality risk, as it explores improved methods for livestock mortality insurance modeling procedures, and premium computation, using Bayesian credibility analysis. This analysis is of interest as there are currently no widely accepted methods for modeling and computing livestock insurance premiums. This stems from the fact that while livestock insurance has been introduced in a limited fashion in some areas of Europe, and Nova Scotia in Canada, it is relatively new, and is not commercially widespread across developed countries. Therefore, the objective of this study is to provide an overview and discussion of livestock mortality insurance, and implications for modeling, and premium estimation.

It is straightforward to incorporate CFIA reportable diseases into credibility premium calculations. In our premium computations, data are from a North American sample and premiums are adjusted to include a possible CFIA reportable disease that might occur once every 100 cycles on average, impacting 20 percent of producers, and resulting in 100 percent mortality of those producers' animals (animal death due either to the disease or welfare slaughter, or slaughter as ordered by CFIA to prevent the disease from spreading). In the case of CFIA reportable diseases, mortality insurance would pay the producer indemnities for the portion of CFIA reportable disease losses that were not covered by the CFIA. In our study based on the 13 Canadian producers, the premium loading with the FRD coverage ranges from 6 percent to 21 percent, with an average around 12 percent (see Table 3). Larger producers tend to have higher premium loadings owing to the scale of the slaughter.

The credibility approach (referred to as greatest accuracy credibility theory) provides a more accurate method for computing premiums (Klugman, Panjer, and Willmot, 2008). It can assist in overcoming the problem of adverse selection, where low-risk producers in the sample are over-charged for premiums, and high risk producers are undercharged. The adverse selection problem results in the lower risk producers refusing to buy insurance, and the insurer left only with high-risk producers, possibly causing producer participation rates to fall, administration costs per producer to rise, and the insurance to fail.

Informal mortality estimates based on large pig operations in Canada indicate around 4 percent for Stage 3, finishing production. However, smaller producers may have higher pig mortality rates, and larger producers may have lower mortality, as larger operators may possibly benefit due to improved disease management procedures and barns that have stricter disease prevention. However, more analysis and study of the data would be needed in order to gain more information, and compute premiums more accurately. Questions as to why the North American data as a whole have higher mortality than the Canadian region and the U.S. Midwest region data should be studied further. Possible differences between larger and smaller producers, and implications for different premium rates between the two groups should be studied further as well.

Further research is also needed on livestock reinsurance, given the high risk of livestock. Also, overcoming any moral hazard problems is important, as moral hazard may be one of the main reasons why livestock insurance has not yet been widely adopted. While moral hazard is not included in this analysis, it could be accounted for by adding a premium loading. In summary, livestock insurance premium computation is relatively new, and this study provides further analytical research for improved premium estimates by applying credibility analysis.

APPENDIX

Given Q, the first two conditional moments of X are

E[X|Q] = n.. - (1 - [Q.sub.M][Q.sub.L]) [summation over (j)] [n.sub.j]. (1 - [Q.sub.j]) (A1)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A2)

The details leading to (A2) are as follows (recall that a binary variable like M or 1 - M [L.sub.jk] is equal to its square):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The second term in the last expression is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

From these calculations one gathers (A2). Using (Al) and (A2) we calculate the expected value and variance of the total claim amount:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

DOI: 10.1111/jori.12024

REFERENCES

Bruce A., 2004, Implications of Extending Crop Insurance to Livestock, Agricultural Outlook Forum 2004 (32995), United States Department of Agriculture, Agricultural Outlook Forum.

Biihlmann, H., and A. Gisler, 2005, A Course in Credibility Theory and Its Applications (Berlin, Germany: Springer-Verlag).

Englund, M., J. Gustafsson, J. P. Nielsen, and F. Thuring, 2009, Multidimensional Credibility With Time Effects: An Application to Commercial Business Lines, Journal of Risk and Insurance, 76(2): 443-453.

Greig, R., 1999, Random Effects Linear Statistical Models and Biihlmann-Straub Credibility, CAS Forum, (Winter): 387-404.

Halliwell, L., 1996, Statistical Models and Credibility, CAS Forum, (Winter): 61-152.

Jewell, W. S., 1974, Credible Means Are Exact Bayesian for Simple Exponential Families, ASTIN Bulletin, 8: 77-90.

Klugman, S. A., H. P. Panjer, and G. E. Willmot, 2008, Loss Models: From Data to Decisions, 3rd edition (Hoboken, NJ: John Wiley & Sons, Inc.).

Meuwissen, M. P. M., S. H. Horst, R. B. M. Huirne, and A. A. Dijkhuizen, 1999, A Model to Estimate the Financial Consequences of Classical Swine Fever Outbreaks: Principles and Outcomes, Preventive Veterinary Medicine, 42: 249-270.

Mowbray, A., 1914, How Extensive a Payroll Is Necessary to Give Dependable Pure Premium? Proceedings of the Casualty Actuarial Society, 1: 24-30.

Sundt, B., 1981, Recursive Credibility Estimation, Scandinavian Actuarial Journal, 3-22.

Sundt, B., 1986, The Special Issue on Credibility Theory, Insurance: Abstracts and Reviews, 2.

Teugels, J., and B. Sundt, eds., 2004. Encyclopedia of Actuarial Science (Hoboken, NJ: John Wiley & Sons, Inc.).

Whitney, A. W. (1918), The Theory of Experience Rating, Proceedings of the Casualty Actuarial Society, 4: 274-292.

Jeffrey Pai is the Warren Professor in the Warren Centre for Actuarial Studies and Research, Asper School of Business, University of Manitoba. The author can be contacted via e-mail: jeffrey.pai@ad.umanitoba.ca. Milton Boyd is a Professor in the Department of Agri-Business & Agricultural Economics, and an Adjunct Professor at the Asper School of Business, University of Manitoba. The author can be contacted via e-mail: milton.boyd@ad.umanitoba.ca. Lysa Porth is an Assistant Professor in the Warren Centre for Actuarial Studies and Research, Asper School of Business, University of Manitoba. The author can be contacted via e-mail: lysa.porth@ad.umanitoba.ca. We thank Sam Cox, the editors, and an anonymous referee for their valuable suggestions and comments. We are also grateful to Agriculture and Agri-Food Canada (AAFC) for the research funding.

TABLE 1
Stage Mortality Rates (Canada 2000-2007)

                         Total        Total        Total        Stage
                        No. of       No. of        No. of     Mortality
Stage                  Producers   Placed Pigs   Dead Pigs      Rate

Prewean (3 weeks)         15       12,902,370    1,433,483     11.11%
Nursery (6 weeks)         11       10,979,235      241,065      2.20%
Finishing (16 weeks)      13        9,510,204      364,584      3.83%

TABLE 2
Posterior Mean and SD (Standard Deviation)
of [Q.sub.j] (Producers' Mortality Rate)

Producer   Posterior Mean   Posterior SD

OBS 001        3.24%           0.009%
OBS 002        3.59%           0.013%
OBS 003        4.70%           0.019%
OBS 004        3.98%           0.021%
OBS 005        3.88%           0.023%
OBS 006        4.30%           0.027%
OBS 007        6.34%           0.040%
OBS 008        5.52%           0.081%
OBS 009        2.72%           0.058%
OBS 010        6.49%           0.171%
OBS Oil        4.28%           0.169%
OBS 012        8.45%           0.237%
OBS 013        6.40%           0.389%

TABLE 3
Premium Rates and FRD Loadings

               Premium Rates

           Without FRD   With FRD
Producer    Coverage     Coverage   FRD Loading

OBS 001       3.24%       3.80%         17%
OBS 002       3.59%       4.15%         16%
OBS 003       4.79%       5.26%         12%
OBS 004       3.98%       4.54%         14%
OBS 005       3.88%       4.44%         14%
OBS 006       4.30%       4.86%         13%
OBS 007       6.34%       6.89%         9%
OBS 008       5.52%       6.07%         10%
OBS 009       2.72%       3.29%         21%
OBS 010       6.49%       7.03%         8%
OBS Oil       4.28%       4.84%         13%
OBS 012       8.45%       8.98%         6%
OBS 013       6.40%       6.94%         9%

Note: FRD, federally reportable diseases.
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Author:Pai, Jeffrey; Boyd, Milton; Porth, Lysa
Publication:Journal of Risk and Insurance
Article Type:Abstract
Geographic Code:1CANA
Date:Jun 1, 2015
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