Abstract
The European Commission has introduced new risk management tools in the rural development pillar 2 of the Common Agricultural Policy. One of them consists in providing co-financing support to mutual funds compensating farmers who experience a severe drop in their income. This paper analyses this income stabilisation tool for a region in Belgium by means of a skew normal linear mixed model. Relying on the farm accountancy data network, this analysis focuses on estimating the probability that such a fund would need to intervene and, in that case, the expected amount of each farm income compensation. The predictive distribution of future incomes given past revenues trajectory is derived and used for evaluation purposes. Particular attention is paid to additional requirements that could be imposed to the income stabilisation tool.
Similar content being viewed by others
References
Mahul O, Stutley CJ (2010) Government support to agricultural insurance. The World Bank, Washington
Meuwissen M, Huirne R, Skees J (2003) Income insurance in European agriculture. EuroChoices 2:12–17
Diaz-Caneja M, Garrido A (2009) Evaluating the potential of whole-farm insurance over crop-specific insurance. Spanish J Agric Res 7:3–11
Mahul O, Wright B (2003) Designing optimal crop revenue insurance. Am J Agric Econ 85:580–589
OECD (2011) Managing risk in agriculture: policy assessment and design. OECD Publishing, Paris
European Commission (2009) Income variability and potential cost of income insurance for EU. Directorate-General for Agriculture and Rural Development, European Commission, Brussels
Coble K, Dismukes R, Thomas S (2007) Policy implications of crop yield and revenue variability at differing levels of disaggregation. In: Paper presented at the American Agricultural Economics Association Annual Meeting, Portland, Oregon, USA
Antonio K, Beirlant J (2007) Actuarial statistics with generalized linear mixed models. Insur: Math Econ 40:58–76
Greene W (2011) Econometric analysis. Prentice Hall, New Jersey
Bolance C, Guillen M, Pelican E, Vernic R (2008) Skewed bivariate models and nonparametric estimation for the CTE risk measure. Insur: Math Econ 43:386–393
Eling M (2012) Fitting insurance claims to skewed distributions: are the skew normal and skew-student good models? Insur: Math Econ 51:239–248
Pigeon M, Antonio K, Denuit M (2013) Individual loss reserving with the multivariate skew normal framework. ASTIN Bull 43:399–428
Azzalini A (1985) A class of distributions which includes the normal ones. Scand J Stat 12:171–178
Azzalini A, Dalla Valle A (1996) The multivariate skew-normal distribution. Biometrika 83:715–726
Gupta A, Chen J (2004) A class of multivariate skew normal models. Ann Inst Stat Math 56:305–315
Akdemir D, Gupta A (2010) A matrix variate skew distribution. Eur J Pure Appl Math 3:128–140
Arellano-Valle RB, Bolfarine H, Lachos V (2005) Skew-normal linear mixed models. J Data Sci 3:415–438
Lin T (2009) Maximum likelihood estimation for multivariate skew normal mixture models. J Multivar Anal 100:257–265
Arellano-Valle RB, Genton MG (2005) On fundamental skew distributions. J Multivar Anal 96:93–116
DGARNE (2011) Évolution de l’économie agricole et horticole de la Région wallonne 2009–2010, Direction Générale opérationnelle de l’Agriculture, des Ressources naturelles et de l’Environnement. Service public de Wallonie, Namur
Henry de Frahan B, Saegerman C, Denuit M, Dubuisson B, Ledoux O, Pigeon M, Vandeputte S, Weynants S (2011) Étude de la possibilité et proposition de mise en place de mécanismes assurantiels ou de mutualisation des risques dans le secteur agricole en région wallonne. Université catholique de Louvain et Université de Liège, Louvain-la-Neuve
Barndorff-Nielsen O (1977) Exponentially decreasing distributions for the logarithm of particle size. Proc R Soc Lond A 353:401–419
McNeil A, Frey R, Embrechts P (2005) Quantitative risk management: concepts, techniques and tools. Princeton University Press, Princeton
Breymann W, Dias A, Embrechts P (2003) Dependence structures for multivariate high-frequency data in finance. Quant Fin 3(1):1–14
Abramowitz M, Stegun I (1972) Handbook of mathematical functions. Dover Books on Mathematics, Dover, New York
WTO (1994) Final act of the 1986–1994 Uruguay round of trade negotiations. World Trade Organization, Geneva
Rejesus R, Coble K, Knight R, Jin Y (2006) Developing experience-based premium rate discounts in crop insurance. Am J Agric Econ 88:409–419
Turvey C (2012) Whole farm income insurance. J Risk Insur 79:515–540
Stokes J, Nayda W (2003) The pricing of revenue assurance: reply. Am J Agric Econ 85:1066–1069
Myers R, Liu Y, Hanson S (2005) How should we value agricultural insurance contracts? In: Paper presented at the American Agricultural Economics Association Annual Meeting, Providence, Rhode Island, USA
Chambers R (2007) Valuing agricultural insurance. Am J Agric Econ 89:596–606
Acknowledgments
The authors wish to thank the anonymous referees and the editor for numerous constructive comments which have improved earlier versions of the present paper. The financial support from the Direction Générale opérationnelle de l’Agriculture, des Ressources naturelles et de l’Environnement of the Belgian Ministry of Wallonia is gratefully acknowledged.
Author information
Authors and Affiliations
Corresponding author
Appendix
Appendix
1.1 Preliminary results
Lemma 1
Let \({\varvec{X}} \sim N_n(\varvec{\mu }, \varvec{\varSigma })\). Then, for \(\mathbf {a}\) \((k \times 1)\) and \(\mathbf B\) \((k \times n)\), we have
Proof
Let \({\varvec{Z}} \sim N_k(\varvec{\nu }, \varvec{\varOmega })\) and
where \(\mathbf {U} \sim N_k(\varvec{\nu } - \mathbf {B}\varvec{\mu }, \varvec{\varOmega } + \mathbf {B}\varvec{\varSigma }\mathbf {B}')\).
Lemma 2
Let \({\varvec{X}} \sim N_q(\varvec{\nu }, \varvec{\varOmega })\) and \({\varvec{Y}} \sim N_p(\varvec{\mu }, \varvec{\varSigma })\). Then,
where \(\varvec{\varLambda } = \left( \varvec{\varOmega }^{-1} + \mathbf {A}'\varvec{\varSigma }^{-1}\mathbf {A}\right) ^{-1}\).
Proof
Define \(\mathbf {z} = {\varvec{y}} - \varvec{\mu } - \mathbf {A}\varvec{\nu }\) and \(\mathbf {w} = \mathbf {x} - \varvec{\nu }\), we have
and \(|\varvec{\varSigma } + \mathbf {A}\varvec{\varOmega }\mathbf {A}'|| \varvec{\varLambda }| = |\varvec{\varSigma }||\varvec{\varOmega }|\).
1.2 Proof of Proposition 1
-
(i)
We have, with \(\mathbf {x} = {\varvec{y}} - {\varvec{X}}\varvec{\beta } - \varvec{\varOmega }\mathbf {t}\),
$$\begin{aligned}&M_{{\varvec{Y}}}(\mathbf {t}) = E\! \left[ e^{\mathbf {t}'{\varvec{Y}}} \right] \\&= \int e^{\mathbf {t}'{\varvec{y}}}2^T\phi _T\left( {\varvec{y}}; {\varvec{X}}\varvec{\beta }, \varvec{\varOmega }^*\right) \varPhi _T\left( \varvec{\varDelta }'\left( \varvec{\varOmega }^*\right) ^{-1}({\varvec{y}} - {\varvec{X}}\varvec{\beta }); \mathbf {0}, \mathbf {I}_T - \varvec{\varDelta }'\left( \varvec{\varOmega }^*\right) ^{-1}\varvec{\varDelta }\right) \,d{\varvec{y}}\\&= 2^Te^{\mathbf {t}'{\varvec{X}}\varvec{\beta } + 0.5\mathbf {t}'\varvec{\varOmega }^*\mathbf {t}}\int \phi _T\left( \mathbf {x}; \mathbf {0},\varvec{\varOmega }^* \right) \varPhi _T\left( \varvec{\varDelta }'\left( \varvec{\varOmega }^*\right) ^{-1}(\mathbf {x} + \varvec{\varOmega }^*\mathbf {t}); \mathbf {0}, \mathbf {I}_T - \varvec{\varDelta }'\left( \varvec{\varOmega }^*\right) ^{-1}\varvec{\varDelta }\right) \,d\mathbf {x}\\&= 2^Te^{\mathbf {t}'{\varvec{X}}\varvec{\beta } + 0.5\mathbf {t}'\varvec{\varOmega }^*\mathbf {t}}E\! \left[ \varPhi _T\left( \varvec{\varDelta }'\left( \varvec{\varOmega }^*\right) ^{-1}({\varvec{X}} + \varvec{\varOmega }^*\mathbf {t}); \mathbf {0}, \mathbf {I}_T - \varvec{\varDelta }'\left( \varvec{\varOmega }^*\right) ^{-1}\varvec{\varDelta }\right) \right] , \end{aligned}$$where \(X \sim N_T\left( \mathbf {0}, \varvec{\varOmega }^*\right)\). By using Lemma 1, we obtain
$$\begin{aligned}&= 2^Te^{\mathbf {t}'{\varvec{X}}\varvec{\beta } + 0.5\mathbf {t}'\varvec{\varOmega }^*\mathbf {t}}\varPhi _T\left( \varvec{\varDelta }'\mathbf {t}\right) . \end{aligned}$$ -
(ii)
The proof is direct from result (i).
1.3 Proof of Proposition 2
The marginal probability density function is given by (with \(\varvec{\varOmega } = \varvec{\varSigma } + \varvec{\varDelta }\varvec{\varDelta }'\))
By using Lemma 2, we have
where \(\varvec{\varLambda } = \left( \varvec{\varPsi }^{-1} + {\varvec{Z}}'\mathbf {D}^{-1}\varvec{\varOmega }\right) ^{-1}\). So, the marginal probability density function can be written as
where \(\mathbf {W} \sim N_T\left( \varvec{\varLambda }{\varvec{Z}}'\varvec{\varOmega }^{-1}({\varvec{y}} - {\varvec{X}}\varvec{\beta }), \varvec{\varLambda }\right)\). By using Lemma 1, we obtain
Rights and permissions
About this article
Cite this article
Pigeon, M., Henry de Frahan, B. & Denuit, M. Evaluation of the EU proposed farm income stabilisation tool by skew normal linear mixed models. Eur. Actuar. J. 4, 383–409 (2014). https://doi.org/10.1007/s13385-014-0097-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13385-014-0097-9