Evolutionary dynamics of collective index insurance


Index-based insurances offer promising opportunities for climate-risk investments in developing countries. Indeed, contracts conditional on, e.g., weather or livestock indexes can be cheaper to set up than conventional indemnity-based insurances, while offering a safety net to vulnerable households, allowing them to eventually escape poverty traps. Moreover, transaction costs by insurance companies may be additionally reduced if contracts, instead of arranged with single households, are endorsed by collectives of households that bear the responsibility of managing the division of the insurance coverage by its members whenever the index is surpassed, allowing for additional flexibility in what concerns risk-sharing and also allowing insurance companies to avoid the costs associated with moral hazard. Here we resort to a population dynamics framework to investigate under which conditions household collectives may find collective index insurances attractive, when compared with individual index insurances. We assume risk sharing among the participants of each collective, and model collective action in terms of an N-person threshold game. Compared to less affordable individual index insurances, we show how collective index insurances lead to a coordination problem in which the adoption of index insurances may become the optimal decision, spreading index insurance coverage to the entire population. We further investigate the role of risk-averse and risk-prone behaviors, as well as the role of partial correlation between insurance coverage and actual loss of crops, and in which way these affect the original coordination thresholds.

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  1. Alaton P, Djehiche B, Stillberger D (2002) On modelling and pricing weather derivatives. Appl Math Finance 9(1):1–20

    Article  MATH  Google Scholar 

  2. Arrow K (1963) Aspects of the theory of risk bearing, YRJO Jahnsson lectures, also in 1971. Essays in the theory of risk bearing. Markham, Chicago

  3. Barnett BJ, Black JR, Hu Y, Skees JR (2005) Is area yield insurance competitive with farm yield insurance? J Agric Resour Econ 30(2):285–301

    Google Scholar 

  4. Barrett CB, McPeak JG (2006) Poverty traps and safety nets. Springer, Berlin

  5. Carter MR, Galarza F, Boucher S (2007) Underwriting area-based yield insurance to crowd-in credit supply and demand. Sav Dev 31(3):335–362

    Google Scholar 

  6. Clarke DJ (2011) A theory of rational demand for index insurance. University of Oxford, Discussion paper Series, Number, Department of Economics 572

  7. Clarke D, Kalani G (2011) Microinsurance decisions: evidence from Ethiopia, University of Oxford, PhD Thesis

  8. Clarke D, Das N, de Nicola F, Hill RV, Kumar N, Mehta P (2012a) The value of customized insurance for farmers in rural Bangladesh. Research paper, International Food Policy Research Institute (IFPRI)

  9. Clarke DJO, Mahul K, Rao N, Verma N (2012b) Weather based crop insurance in India. World Bank Policy Research, Working Paper 5985

  10. Cole S, Giné X, Tobacman J, Townsend R, Topalova P, Vickery J (2013) Barriers to household risk management: evidence from India. Am Econ J Appl Econ 5(1):104

    Article  Google Scholar 

  11. De Bock O, Gelade W (2012) The demand for microinsurance: a literature review. ILO Microinsurance Innovation Facility Research Paper, Number 26

  12. De Janvry A, Dequiedt V, Sadoulet E (2014) The demand for insurance against common shocks. J Dev Econ 106:227–238

    Article  Google Scholar 

  13. Deng X, Barnett BJ, Vedenov DV (2007) Is there a viable market for Area-Based Crop Insurance? Am J Agric Econ 89(2):508–519

    Article  Google Scholar 

  14. Dercon S, Hill RV, Clarke D, Outes-Leon I, Seyoum Taffesse A (2014) Offering rainfall insurance to informal insurance groups: evidence from a field experiment in Ethiopia. J Dev Econ 106:132–143

    Article  Google Scholar 

  15. Dixit AK, Levin SA, Rubenstein DI (2013) Reciprocal insurance among Kenyan pastoralists. Theor Ecol 6(2):173–187

    Article  Google Scholar 

  16. Gaurav S, Cole S, Tobacman J (2011) Marketing complex financial products in emerging markets: evidence from rainfall insurance in India. J Mark Res 48(SPL):S150-S162

  17. Giné X, Townsend R, Vickery J (2007) Statistical analysis of rainfall insurance payouts in southern India. Am J Agric Econ 89(5):1248–1254

    Article  Google Scholar 

  18. Hazell PB (1992) The appropriate role of agricultural insurance in developing countries. J Int Dev 4(6):567–581

    Article  Google Scholar 

  19. Hess U, Skees J, Stoppa A, Barnett B, Nash J (2005) Managing agricultural production risk: innovations in developing countries. Agriculture and Rural Development (ARD) Department Report (32727-GLB)

  20. Hossain S (2013) Problems and prospects of weather index based crop insurance for rural farmers in Bangladesh. Dev Ctry Stud 3(12):208–220

    MathSciNet  Google Scholar 

  21. Keswell M, Carter MR (2014) Poverty and land redistribution. J Dev Econ 110:250–261

    Article  Google Scholar 

  22. Norton M, Osgood D, Madajewicz M, Holthaus E, Peterson N, Diro R, Mullally C, Teh T-L, Gebremichael M (2014) Evidence of demand for index insurance: experimental games and commercial transactions in Ethiopia. J Dev Stud 50(5):1–19

    Article  Google Scholar 

  23. Osgood DE, McLaurin M, Carriquiry M, Mishra A, Fiondella F, Hansen JW, Peterson N, Ward MN (2007) Designing weather insurance contracts for farmers in Malawi, Tanzania and Kenya: final report to the Commodity Risk Management Group. Agriculture and Rural Development, World Bank

  24. Pacheco JM, Santos FC, Souza MO, Skyrms B (2009) Evolutionary dynamics of collective action in N-person stag hunt dilemmas. Proc R Soc B 276(1655):315–321

    Article  Google Scholar 

  25. Santos FC, Pacheco JM (2011) Risk of collective failure provides an escape from the tragedy of the commons. Proc Natl Acad Sci USA 108(26):10421–10425

    Article  Google Scholar 

  26. Souza MO, Pacheco JM, Santos FC (2009) Evolution of cooperation under N-person snowdrift games. J Theor Biol 260(4):581–588

    Article  MathSciNet  Google Scholar 

  27. Vasconcelos VV, Santos FC, Pacheco JM, Levin SA (2014) Climate policies under wealth inequality. Proc Natl Acad Sci USA 111(6):2212–2216

    Article  Google Scholar 

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The authors would like to dedicate this paper to Mats Gyllenberg on the occasion of his 60th birthday. We also thank Avinash Dixit for his insightful comments. J.M.P. and F.C.S. acknowledge financial support from FCT-Portugal through Grants EXPL/EEI-SII/2556/2013, PTDC/MAT-STA/3358/2014, and PTDC/EEI-SII/5081/2014, and by multi-annual funding of CBMA-UM and INESC-ID (under the Projects UID/BIA/04050/2013 and UID/CEC/50021/2013) provided by FCT-Portugal. S.A.L. acknowledges financial support from National Science Foundation Grants EF-1137894, GEO-1211972 and Project “Green Growth Based on Marine Resources: Ecological and Socio-Economic Constraints (GreenMAR)”, funded by Nordforsk.

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Correspondence to Simon A. Levin.

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Appendix: When actual loss and insurance coverage do not fully overlap

Appendix: When actual loss and insurance coverage do not fully overlap

Here we follow (Clarke 2011) and investigate the effect of introducing two different probabilities instead of the risk r employed in the main text: p, the probability that harvest is lost; q, the probability that any index insurance is activated (as defined previously already in terms of the risk r).

Depending on which one of the four possible states we are in, namely :

  • (0, 0)—(no Loss, no insurance activated)

  • (0, I)—(no Loss, insurance activated)

  • (L, 0)—(Loss, no insurance activated)

  • (LI)—(Loss, insurance activated)

we can write down the following payoff tables for Cs and Ds:

State Probability Payoff
(0,0) \(p_{0,0}\) b
(0,I) \(p_{0,1}\) b
(L,0) \(p_{1,0}\) 0
(L,I) \(p_{1,1}\) 0
State Probability Payoff
   \(n_C \ge M\) \(n_C <M\)
(0,0) \(p_{0,0}\) \(b-c_c \) \(\gamma (b-c)+(1-\gamma )b\)
(0, I) \(p_{0,1}\) \(b(1+\alpha )-c_c\) \(\gamma \left[ {b(1+\alpha )-c} \right] +(1-\gamma )b\)
(L, 0) \(p_{1,0}\) \(-c_c \) \(\gamma (-c)\)
(LI) \(p_{1,1}\) \(\alpha b-c_c \) \(\gamma (\alpha b-c)\)

In the simplest scenario, we may assume that the probability p of occurrence of a loss is statistically independent from the probability q that the index I of the II is activated. Then we can write for the probabilities of occurrence of each of the 4 states:

$$\begin{aligned} p_{0,0}= & {} (1-p)\times (1-q)\\ p_{0,1}= & {} (1-p)\times q\\ p_{1,0}= & {} p\times (1-q)\\ p_{1,1}= & {} p\times q \end{aligned}$$

Such a statistical independence is unlikely, however. Thus, the joint probability distribution will not, in general, be the product of the 2 marginal probability distributions. Following (Clarke 2011), we investigate a symmetric joint probability distribution, defined in terms of a new parameter u, which corresponds to the probability that an individual will incur a Loss but without the II be activated (\(p_{1,0}\)). In terms of u, the probabilities now read:

$$\begin{aligned} \begin{aligned} p_{0,0}&= 1-q-u&p_{0,1}&= q-p+u \\ p_{1,0}&= u&p_{1,1}&= p-u \\ \end{aligned} \end{aligned}$$

Again, we can write for the fitness difference between Cs and Ds

$$\begin{aligned} f_C (x)-f_D (x) =\Delta +\left[ {\Omega -\Delta } \right] B_{N,M} (x) \end{aligned}$$

with \(\Delta =\gamma c\big (\frac{p_{0,1} +p_{1,1} }{m}-1\big )\)and \(\Omega -\Delta =c \big (\gamma +\frac{p_{0,1} +p_{1,1} }{m}-s_C \big )\).

The expressions for \(\Delta \) and \(\Omega -\Delta \)depend only on the combination \(p_{0,1} +p_{1,1} =q\), despite the fact that, individually, both \(p_{0,1} \) and \(p_{1,1}\) depend on p, q and u. Since \(p_{0,1} +p_{1,1} =q\), irrespective of whether the p and q distributions are independent or not, this means that the results we obtain for the original model also hold for this “extended model” provided that the original risk r is replaced by the probability q.

\(\square \)

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Pacheco, J.M., Santos, F.C. & Levin, S.A. Evolutionary dynamics of collective index insurance. J. Math. Biol. 72, 997–1010 (2016). https://doi.org/10.1007/s00285-015-0939-3

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  • Index insurance
  • Collective action
  • Evolutionary game theory
  • Non-linear returns

Mathematics Subject Classification

  • 91-XX
  • 91A22
  • 91A06
  • 91A40
  • 91B18
  • 91B30