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**Agricultural insurance ratemaking: development of a new premium principle.**
*(English)*
Zbl 1429.91286

Summary: Determining the appropriate premium to charge for the underlying risk is central to delivering a sustainable agricultural insurance program. Though this is fundamental to all types of insurance, in agriculture this is a particularly challenging task given systemic risk, information asymmetry, and a number of multifaceted factors pertaining to the loss experience data, including scarcity and credibility. The objective of this article is to formally introduce premium principles to the agricultural insurance literature, with a focus on a new premium principle approach based on the multivariate weighted distribution. The multivariate weighted premium principle (MWPP) formalizes the reweighting of historical loss experience using auxiliary factors in order to refine the agricultural insurance pricing. These auxiliary factors may reflect systemic risk and include material information, such as economic and market conditions, weather, soil, etc. In the empirical study, a unique reinsurance data set from the province of Manitoba, Canada, is used to evaluate a number of potential premium principles. With the flexibility of the MWPP, the empirical results indicate that the MWPP approach can be a viable premium principle for pricing agricultural insurance. Furthermore, the MWPP redistributes premium rates and assigns increased loadings to higher risk layers, helping reinsurers manage their reserves and achieve improved sustainability in the long term.

### MSC:

91G05 | Actuarial mathematics |

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\textit{W. Zhu} et al., N. Am. Actuar. J. 23, No. 4, 512--534 (2019; Zbl 1429.91286)

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### References:

[1] | Agriculture and Agri-Food Canada, Evaluation of the AgriInsurance, private sector risk management partnerships and wildlife compensation programs (2012), Ottawa, Canada: Agriculture and Agri-Food Canada |

[2] | Anderson, T. W., On the distribution of the two-sample cramér-von mises criterion, The Annals of Mathematical Statistics, 33, 3, 1148-59 (1962) · Zbl 0116.37601 |

[3] | Anderson, T. W.; Darling., D. A., Asymptotic theory of certain “goodness-of-fit” criteria based on stochastic processes, The Annals of Mathematical Statistics, 23, 2, 193-212 (1952) · Zbl 0048.11301 |

[4] | Assa, H., On optimal reinsurance policy with distortion risk measures and premiums, Insurance: Mathematics and Economics, 61, 70-75 (2015) · Zbl 1314.91132 |

[5] | Babcock, B. A.; Hart, C. E.; Hayes, D. J., Actuarial fairness of crop insurance rates with constant rate relativities, American Journal of Agricultural Economics, 86, 3, 563-75 (2004) |

[6] | Barnett, B. J.; Mahul., O., Weather index insurance for agriculture and rural areas in lower-income countries, American Journal of Agricultural Economics, 89, 5, 1241-47 (2007) |

[7] | Borman, J. I.; Goodwin, B. K.; Coble, K. H.; Knight, T. O.; Rejesus, R., Accounting for short samples and heterogeneous experience in rating crop insurance, Agricultural Finance Review, 73, 1, 88-101 (2013) |

[8] | Brockett, P. L.; Goldens, L. L.; Wen, M.-M.; Yang, C. C., Pricing weather derivatives using the indifference pricing approach, North American Actuarial Journal, 13, 3, 303-15 (2009) |

[9] | Bühlmann, H., An economic premium principle, ASTIN Bulletin, 11, 52-60 (1980) |

[10] | Bühlmann, H.; Delbaen, F.; Embrechts, P.; Shiryaev, A. N., No-arbitrage, change of measure and conditional Esscher transforms, CWI Quarterly, 9, 4, 291-317 (1996) · Zbl 0943.91037 |

[11] | Chambers, R. G., Insurability and moral hazard in agricultural insurance markets, American Journal of Agricultural Economics, 71, 3, 604-16 (1989) |

[12] | Coble, K. H.; Harri, A.; Anderson, J. D.; Ker, A. P.; Goodwin, B., Review of County Yield Trending Procedures and Related Topics (2008), USDA Risk Management Agency, Washington, DC |

[13] | Coble, K. H.; Knight, T.; Miller, M.; Goodwin, B.; Rejesus, R.; Boyles, R., Estimating structural change in U.S. crop insurance experience, Agricultural Finance Review, 73, 1, 74-87 (2013) |

[14] | Coble, K. H.; Miller, M. F.; Rejesus, R. M.; Boyles, R.; Knight, T. O.; Goodwin, B. K., Methodology analysis for weighting of historical experience (2011), USDA Risk Management Agency |

[15] | Deprez, O.; Gerber., H. U., On convex principles of premium calculation, Insurance: Mathematics and Economics, 4, 3, 179-89 (1985) · Zbl 0579.62090 |

[16] | Duncan, J.; Myers., R. J., Crop insurance under catastrophic risk, American Journal of Agricultural Economics, 82, 4 (2000) |

[17] | Fischer, G.; Shah, M.; Tubiello, F. N.; Van Velhuizen, H., Socio-economic and climate change impacts on agriculture: An integrated assessment, 1990-2080, Philosophical Transactions of the Royal Society B: Biological Sciences, 360, 1463, 2067-83 (2005) |

[18] | Furman, E.; Landsman., Z., Tail variance premium with applications for elliptical portfolio of risks, ASTIN Bulletin, 36, 2, 433-62 (2006) · Zbl 1162.91373 |

[19] | Furman, E.; Landsman., Z., Economic capital allocations for non-negative portfolio of dependent risks, ASTIN Bulletin, 38, 2, 601-619 (2008) · Zbl 1274.91379 |

[20] | Furman, E.; Zitikis., R., Weighted premium calculation principles, Insurance: Mathematics and Economics, 42, 1, 459-65 (2008) · Zbl 1141.91509 |

[21] | Furman, E.; Zitikis., R., Weighted risk capital allocations, Insurance: Mathematics and Economics, 43, 2, 263-69 (2008) · Zbl 1189.62163 |

[22] | Furman, E.; Zitikis., R., Weighted pricing functionals with applications to insurance: An overview, North American Actuarial Journal, 13, 4, 1-14 (2009) |

[23] | Furman, E.; Zitikis., R., General Stein-type covariance decompositions with applications to insurance and finance, ASTIN Bulletin, 40, 1, 369-75 (2010) · Zbl 1191.62097 |

[24] | Gerber, H.; Shiu, S., Option pricing by Esscher transforms, Transactions of the Society of Actuaries, 46, 99-140 (1994) |

[25] | Glauber, J. W., Crop insurance reconsidered, American Journal of Agricultural Economics, 86, 5, 1179-95 (2004) |

[26] | Goodwin, B. K.; Ker., A. P., Nonparametric estimation of crop yield distributions: Implications for rating group-risk crop insurance contracts, American Journal of Agricultural Economics, 80, 139-53 (1998) |

[27] | Heilmann, W. R., Decision theoretic foundations of credibility theory, Insurance: Mathematics and Economics, 8, 1, 77-95 (1989) · Zbl 0687.62087 |

[28] | Henzea, N.; Zirkler., B., A class of invariant consistent tests for multivariate normality, Communications in Statistics - Theory and Methods, 19, 10, 3595-3617 (1990) · Zbl 0738.62068 |

[29] | Hogg, R.; Klugman, S., Loss distributions (1984), New York: Wiley, New York |

[30] | Hubalek, F.; Sgarra., C., On the Esscher transform and the minimal entropy martingale measure for exponential Lévy models, Quantitative Finance, 6, 2, 125-45 (2006) · Zbl 1099.60033 |

[31] | Kaas, R.; Van Heerwaarden, A. E.; Goovaerts, M. J., Ordering of actuarial risks (1994), Leuven, Belgium: CAIRE, Leuven, Belgium |

[32] | Kamps, U., On a class of premium principles including the Esscher principle, Scandinavian Actuarial Journal, 1998, 1, 75-80 (1998) · Zbl 1031.62505 |

[33] | Kaufmann, R. K.; Snell., S. E., A biophysical model of corn yield: Integrating climatic and social determinants, American Journal of Agriculture Economics, 79, 1, 178-90 (1997) |

[34] | Ker, A.; Goodwin., B. K., Nonparametric estimation of crop insurance rates revisited, American Journal of Agricultural Economics, 82, 2, 463-78 (2000) |

[35] | Mardia, K. V., Measures of multivariate skewness and kurtosis with applications, Biometrika, 57, 3, 519-30 (1970) · Zbl 0214.46302 |

[36] | MARSH, Canada insurance market report 2014 (2014), Marsh & McLennan Companies |

[37] | Mendelsohn, R.; Nordhaus, W. D.; Shaw, D., The impact of global warming on agriculture: A Ricardian analysis, American Economic Review, 84, 4, 753-71 (1994) |

[38] | Motha, R. P., Challenges and opportunities in agrometeorology, The impact of extreme weather events on agriculture in the United States, 397-407 (2011), Berlin: Springer, Berlin |

[39] | Nelson, C. H.; Loehman., E. T., Further toward a theory of agricultural insurance, American Journal of Agricultural Economics, 69, 3, 523-531 (1987) |

[40] | O’Connor, C., Soil matters: How the federal crop insurance program should be reformed to encourage low-risk farming methods with high-reward environmental outcomes (2013) |

[41] | Okhrin, O.; Odening, M.; Xu, W., Systemic weather risk and crop insurance: The case of China, Journal of Risk and Insurance, 80, 2, 351-72 (2013) |

[42] | Ozaki, V. A.; Goodwin, B. K.; Shirota, R., Parametric and nonparametric statistical modeling of crop yield: Implications for pricing crop insruance contracts, Applied Economics, 40, 9, 1151-64 (2008) |

[43] | Patil, G. P.; Rao, C. R.; Ratnaparkhi, M. V., On discrete weighted distributions and their use in model choice for observed data, Communications in Statistics - Theory and Methods, 15, 3, 907-18 (1986) · Zbl 0601.62022 |

[44] | Poon, J.; Lu., Y., A spatial cross-sectional credibility model with dependence among risks, North American Actuarial Journal, 19, 4, 289-310 (2015) · Zbl 1414.91225 |

[45] | Porth, L.; Pai, J.; Boyd, M., A portfolio optimization approach using combinatorics with a genetic algorithm for developing a reinsurance model, Journal of Risk and Insurance, 82, 3, 687-713 (2015) |

[46] | Porth, L., and K. S.Tan.2015. Agricultural insurance-More room to grow? The Actuary April/May:36-41. |

[47] | Porth, L.; Tan, K. S.; Weng, C., Optimal reinsurance analysis from a crop insurer’s perspective, Agricultural Finance Review, 73, 2, 310-28 (2013) |

[48] | Porth, L.; Zhu, W.; Tan, K. S., A credibility-based Erlang mixture model for pricing crop reinsurance, Agricultural Finance Review, 74, 2, 162-87 (2014) |

[49] | Prentice, I. C.; Cramer, W.; Harrison, S. P.; Leemans, R.; Monserud, R. A.; Solomon, A. M., Special paper: A global biome model based on plant physiology and dominance, soil properties and climate, Journal of Biogeography, 19, 2, 117-34 (1992) |

[50] | Rao, C. R., On discrete distributions arising out of methods of ascertainment, Sankhya: The Indian Journal of Statistics, Series A, 27, 2-4, 311-24 (1965) · Zbl 0212.21903 |

[51] | Rejesus, R. M.; Coble, K. H.; Knight, T. O.; Jin, Y., Developing experience-based premium rate discounts in crop insurance, American Journal of Agricultural Economics, 88, 2, 409-19 (2006) |

[52] | Royston, P., Estimating departure from normality, Statistics in Medicine, 10, 8, 1283-93 (1991) |

[53] | Schlenker, W.; Hanemann, W. M.; Fisher, A. C., The impact of global warming on U.S. agriculture: An econometric analysis of optimal growing conditions, Review of Economics and Statistics, 88, 1, 113-25 (2006) |

[54] | Shapiro, S. S.; Wilk., M. B., An analysis of variance test for normality (complete samples), Biometrika, 52, 3-4, 591-611 (1965) · Zbl 0134.36501 |

[55] | Shields, D. A. (2013) |

[56] | Skees, J. R.; Reed., M. R., Rate making for farm-level crop insurance: Implications for adverse selection, American Journal of Agricultural Economics, 68, 3, 653-59 (1986) |

[57] | Smirnov, N., Table for estimating the goodness of fit of empirical distributions, The Annals of Mathematical Statistics, 19, 2, 279-81 (1948) · Zbl 0031.37001 |

[58] | Swiss, Re. (2013) |

[59] | Swiss, Re., Partnering for food security in emerging markets, Sigma, 1 (2013) |

[60] | Wang, S., Insurance pricing and increased limits ratemaking by proportional hazards transforms, Insurance: Mathematics and Economics, 17, 1, 43-54 (1995) · Zbl 0837.62088 |

[61] | Wang, S., Premium calculation by transforming the layer premium density, ASTIN Bulletin, 26, 1, 71-92 (1996) |

[62] | Wang, S., Normalized exponential tilting: Pricing and measuring multivariate risks, North American Actuarial Journal, 11, 3, 89-99 (2007) |

[63] | Wang, S.; Young, V. R.; Panjer, H. H., Axiomatic characterization of insurance prices, Insurance: Mathematics and Economics, 21, 2, 173-183 (1997) · Zbl 0959.62099 |

[64] | Wirch, J. L.; Hardy., M. R., A synthesis of risk measures for capital adequacy, Insurance: Mathematics and Economics, 25, 3, 337-47 (1999) · Zbl 0951.91032 |

[65] | Woodard, J. D., A conditional distribution approach to assessing the impact of weather sample heterogeneity on yield risk estimation and crop insurance ratemaking, North American Actuarial Journal, 18, 2, 279-93 (2014) |

[66] | Woodard, J. D.; Garcia, P., Weather derivatives, spatial aggregation, and systemic risk: Implications for reinsurance hedging, Journal of Agricultural and Resource Economics, 33, 1, 34-51 (2008) |

[67] | Woodard, J. D.; Pavlista, A. D.; Schnitkey, G. D.; Burgener, P. A.; Ward, K. A., Governement insurance program design, incentive effects, and technology adoption: The case of skip-row crop insurance, American Journal of Agricultural Economics, 94, 4, 823-37 (2012) |

[68] | Woodard, J. D.; Schnitkey, G. D.; Sherrick, B. J.; Lozano-Gracia, N.; Anselin, L., A spatial econometric analysis of loss experience in the U.S. crop insurance program, Journal of Risk and Insurance, 79, 1, 261-85 (2012) |

[69] | Woodard, J. D.; Sherrick, B. J., Actuarial impacts of loss cost ratio ratemaking in U.S. crop insurance programs, Journal of Agricultural and Resource Economics, 36, 1, 211-28 (2011) |

[70] | Young, V. R., Premium principles (2014) |

[71] | Zhu, W. (2015) |

[72] | Zhuang, S. C.; Weng, C.; Tan, K. S.; Assa, H., Marginal indemnification function formulation for optimal reinsurance, Insurance: Mathematics and Economics, 67, 65-76 (2016) · Zbl 1348.91196 |

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