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An average model approach to experience based premium rates discounts: an application to Spanish agricultural insurance

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Abstract

We address some issues in agricultural insurance, describing drawbacks of the bonus-malus system (BMS) methodology used in Spain and many other EU countries. We develop an alternative experience based premium rate discount system taking into account the adverse years when high losses caused by extreme weather events happen. Our contribution consists of a two-step methodology. Firstly, we use tobit or Tweedie regressions to calculate yearly correction rates. Secondly, we calculate the mean of the correction rates. This average model acts as a buffer against adverse year losses. We compare three alternatives: our two resulting average models and the BMS operating in the Spanish line of business exemplified—table grapes.

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Notes

  1. In the US there are several practical implementations of the idea of experience-based ratemaking. One reviewer let us know that US rating procedures ‘fine-tune’ base rates according to historical average yields. Since a higher average yield leads to a rate decrease, farms with higher average yields will have lower relative risk. Also, the Risk Management Agency (RMA) has implemented a biotech discount, lowering the premiums to those farmers that use modern technologies. In addition, discounts are also offered to producers taking coverage at higher levels of aggregation, see [14].

  2. See [5] for an alternative Bayesian approach to weighting the experience in order to deal with extreme weather events.

  3. The Tweedie distribution depends on a parameter p, known as the ‘index parameter’, and the different values of this parameter allow to obtain as particular cases some important continuous, discrete and mixed distributions. For example, the Normal, Poisson and Gamma are particular cases of a Tweedie distribution with p = 0, 1 and 2, respectively. When 1 < p < 2, the Tweedie becomes a Poisson-Gamma compound distribution. This is precisely the interesting case for us, because it is a mixed distribution with a discrete mass of probability assigned to zero (interpreted as the probability of no claims) and a positive continuous part (interpreted as the density of the claim amounts). For information about the Tweedie distribution see Chapter 12 of [8], and for its ratemaking applications, see [13]. Regarding the Tobit model, it can also be considered as a mixture of a claim/no claim binary distribution and a truncated normal distribution giving the density of the positive claim amounts, see [16]. Although this decomposition makes sense in insurance problems, it is rarely found in the actuarial literature.

  4. The Spanish “Consorcio de Compensación de Seguros” (CCS), is a public business entity attached to the Spanish Ministry of Economy. The CCS relies on its own capital, without any dependence of public funds. From the agricultural insurance perspective, it develops three main lines of activity, playing a key role as a system compulsory reinsurer, monitoring loss adjustments and participating in the coinsurance scheme. The involvement of the CCS in the Spanish system of agricultural insurance is crucial since it decreases the impact of losses making possible risk underwritings.

  5. For more information, see Sect. 14. Bonificaciones y Recargos in the document Seguro de Explotaciones de Uva de Mesa: condiciones especiales (in Spanish) https://agroseguro.es/fileadmin/propietario/Productos/AGRICOLAS/321%20UVA%20DE%20MESA/PLAN_2018/CES-321-18-1.0.pdf

  6. The selection of \(n=5\) years for the return period has been based on the fact that we can find both favourable and unfavourable years within this period, as we can check in Tables 1, 2 and 3. Rejesus et al. also choose a period of 5 years for the analysis, based on different considerations: “The choice of five years is arbitrary but reflects a balance between a longer period (that would have greater statistical power) and choosing a shorter period (that would make it possible for more producers to qualify but would have lower statistical power)” (see [21], pg. 414).

  7. It is important to highlight that the empirical factors \({F}^{(-t)}, t\in \{1,\dots ,5\}\), were always significant in all the Tobit and Tweedie regressions. For the numerical calculations, we used the R packages ‘AER’ (https://cran.r-project.org/web/packages/AER/AER.pdf) and ‘tweedie’ (https://cran.r-project.org/web/packages/tweedie/tweedie.pdf).

  8. The outputs of the statistical procedures are available to the readers upon request.

  9. The values of the mse have been calculated using the Leave One Out Cross Validation (LOOCV) method.

  10. In order to reach the financial equilibrium, we should multiply the Mean Tobit Premiums by \(\frac{9,818}{\mathrm{9,828}}\), and the Mean Tweedie Premiums by \(\frac{\mathrm{9,818}}{\mathrm{9,636}}\).

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Acknowledgements

This work was supported by Agroseguro under UCM contract number 437-2016. The authors are very thankful to Pablo Ezquerra and Felix Novoa from AGROSEGURO, for their helpful comments and suggestions.

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Authors and Affiliations

  1. Department of Financial and Actuarial Economics & Statistics, Universidad Complutense de Madrid, Madrid, Spain

    José L. Vilar-Zanón & Antonio Heras

  2. Agroseguro S.A., Madrid, Spain

    Estela de Frutos

  3. Departamento de Economía Financiera y Actuarial y Estadística, Universidad Complutense de Madrid, Campus de Somosaguas. Facultad de Ciencias Económicas y Empresariales, 28223, Pozuelo de Alarcón, Spain

    José L. Vilar-Zanón

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  1. José L. Vilar-Zanón
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Correspondence to José L. Vilar-Zanón.

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Vilar-Zanón, J.L., Heras, A. & de Frutos, E. An average model approach to experience based premium rates discounts: an application to Spanish agricultural insurance. Eur. Actuar. J. 10, 361–375 (2020). https://doi.org/10.1007/s13385-020-00234-1

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