Abstract
It is well known that the hedging effectiveness of weather derivatives is interfered by the existence of geographical basis risk, i.e., the deviation of weather conditions at different locations. In this paper, we explore how geographical basis risk of rainfall based derivatives can be reduced by regional diversification. Minimizing geographical basis risk requires knowledge of the joint distribution of rainfall at different locations. For that purpose, we estimate a daily multi-site rainfall model from which optimal portfolio weights are derived. We find that this method allows to reduce geographical basis risk more efficiently than simpler approaches as, for example, inverse distance weighting.
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Notes
A different loss function could be applied here. In particular, one could use the semi-variance or another downside risk measure that implies an asymmetric loss function (see McNeil et al. (2005) and Ang et al. (2006) for more details about downside risk measures). This would reflect the buyer’s asymmetric preference towards positive and negative deviations between realized losses and insurance payoffs. From the seller’s point of view, however, this preference is reversed. Hence, we prefer to use the symmetric quadratic loss function, which is also the most prevalent loss function (Lee 2008).
Climate change may cause inter-year shifts of the model parameters, i.e. of the transition probabilities and rainfall intensity parameters. However, we refrain from taking this into account for two reasons: First, the complexity of the daily rainfall model would further increase and could hardly be handled. Second, the observation period of the training phase consists of 28 years and it would be difficult to estimate effects of climate change during this rather short period. Moreover, in a recent study, Wang et al. (2013) do not find clear empirical evidence for a change in rainfall risk in Germany in the last decades.
Instead of using standard normally distributed random numbers, a uniform random variable \(u_{t,k}\) could be compared with \(p_t^{01/11}(k)\) (rain if \(u_{t,k}\le p_{t,k}^{01/11}\)) in Eq. (8). However, the notation with the standard normal cumulative distribution function \(\Phi \) was chosen because it is more practical for the multi-site modelling.
The correlations \(\sigma (k,l)\) cannot be derived directly from the data as only the correlations between the historical occurrence processes are observable, \(\xi ^0(k,l)=\text {Corr}\left[ X^0_{t,k},X^0_{t,l}\right] \). The correlation of the random variables \(\sigma (k,l)=\text {Corr}\left[ \epsilon _{t,k},\epsilon _{t,l}\right] \) can be obtained via trial and error by guessing \(\sigma (k,l)\) for every pair of locations and comparing the resulting \(\xi (k,l)\) with the historical \(\xi ^0(k,l)\). By using an adequate algorithm, this procedure can be repeated until \(\xi (k,l)\) and \(\xi ^0(k,l)\) match.
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Another way of sorting the neighbours is the correlation of the historical payoffs of the put option. The RMSE for the daily rainfall, however, almost does not change if the neighbours are sorted by correlation instead of distance.
The neighbours are again sorted by distance. For the payoff from the put option, the difference between sorting the neighbours by distance or by correlation is larger than for the daily rainfall. However, the way of sorting does not change the overall message so that we will continue with sorting by distance only.
Bentler and Chou (1987) set a minimal ratio of the sample size to the number of free parameters of 5:1.
References
Ang, A., Chen, J., & Xing, Y. (2006). Downside risk. Review of Financial Studies, 19(4), 1191–1239.
Bentler, P. M., & Chou, C.-P. (1987). Practical issues in structural modeling. Sociological Methods Research, 16, 78–117.
Berg, E., & Schmitz, B. (2008). Weather-based instruments in the context of whole-farm risk management. Agricultural Finance Review, 68(1), 119–133.
Buishand, T. A., & Brandsma, T. (2001). Multisite simulation of daily precipitation and temperature in the Rhine Basin by nearest-neighbor resampling. Water Resources Research, 37(11), 2761–2776.
Cao, M., Li, A., & Wei, J. (2004). Precipitation modeling and contract valuation: A frontier in weather derivatives. The Journal of Alternative Investments, 7(2), 93–99.
Cheng, G., & Roberts, M. C. (2004). Weather derivatives in the presence of index and geographical basis risk: Hedging dairy profit risk. In Proceedings of the NCR-134 Conference on Applied Commodity Price Analysis, Forecasting, and Market Risk Management, St. Louis, MO.
Cowden, J. R., Watkins, D. W, Jr., & Mihelcic, J. R. (2008). Stochastic rainfall modeling in West Africa: Parsimonious approaches for domestic rainwater harvesting assessment. Journal of Hydrology, 361(1–2), 64–77.
Diaz-Caneja, M. B., Conte, C. G., Pinilla, F. J. G., Stroblmair, J., Catenaro, R., & Dittmann, C. (2009). Risk management and agricultural insurance schemes in Europe. JRC Reference Report, EU-23943, EN-2009, European Commission.
Diebold, F. X., & Mariano, R. S. (1995). Comparing predictive accuracy. Journal of Business & Economic Statistics, 13, 134–144.
Dischel, R. S. (2002). Climate risk and the weather market. London: Risk Books.
Härdle, W. K., & Osipenko, M. (2011). Pricing Chinese rain: A multi-site multi-period equilibrium pricing model for rainfall derivatives. SFB 649 Discussion Paper 2011–055.
Hughes, J. P., Guttorp, P., & Charles, S. P. (1999). A non-homogeneous hidden Markov model for precipitation occurrence. Applied Statistics, 48(1), 15–30.
Lazo, J. K., Lawson, M., Larsen, P. H., & Waldman, D. M. (2011). US economic sensitivity to weather variability. Bulletin of the American Meteorological Society, 92(6), 709–720.
Lee, T.-H. (2008). Loss functions in time series forecasting. In W. A. Darity Jr. (Ed.), International encyclopedia of the social sciences (2nd ed., Vol. 9). Detroit: Macmillan Thomson Gale Publishers.
Li, Z., Brissette, F., & Chen, J. (2012). Finding the most appropriate precipitation probability distribution for stochastic weather generation and hydrological modelling in Nordic watersheds. Hydrological Processes. doi:10.1002/hyp.9499.
López Cabrera, B., Odening, M., & Ritter, M. (2013). Pricing rainfall futures at the CME. Journal of Banking and Finance, 37, 4286–4298. doi:10.1016/j.jbankfin.2013.07.042.
McNeil, A. J., Frey, R., & Embrechts, P. (2005). Quantitative risk management: Concepts, techniques, and tools. New Jersey: Princeton University Press.
Mehrotra, R., Srikanthan, R., & Sharma, A. (2006). A comparison of three stochastic multi-site precipitation occurrence generators. Journal of Hydrology, 331(1–2), 280–292.
Miranda, M. J., & Gonzalez-Vega, C. (2011). Systemic risk, index insurance, and optimal management of agricultural loan portfolios in developing countries. American Journal of Agricultural Economics, 93(2), 399–406.
Mußhoff, O., Odening, M., & Xu, W. (2011). Management of climate risks in agriculture—will weather derivatives permeate? Applied Economics, 43(9), 1067–1077.
Odening, M., Muhoff, O., & Xu, W. (2007). Analysis of rainfall derivatives using daily precipitation models: Opportunities and pitfalls. Agricultural Finance Review, 67(1), 135–156.
Roldán, J., & Woolhiser, D. A. (1982). Stochastic daily precipitation models: 1. A comparison of occurrence processes. Water Resources Research, 18(5), 1451–1459.
Salsón, S., & Garcia-Bartual, R. (2003). A space-time rainfall generator for highly convective Mediterranean rainstorms. Natural Hazards and Earth System Sciences, 3, 103–114.
Shepard, D. (1968). A two-dimensional interpolation function for irregularly-spaced data. In Proceedings of the 1968 ACM National Conference (pp. 517–524).
Vedenov, D. V., & Barnett, B. J. (2004). Efficiency of weather derivatives as primary crop insurance instruments. Journal of Agricultural and Resource Economics, 29(3), 387–403.
Wang, W., Bobojonov, I., Härdle, W. K., & Odening, M. (2013). Testing for increasing weather risk. Stochastic Environmental Research and Risk Assessment, 27, 1565–1574.
Wilks, D. S. (1998). Multisite generalization of a daily stochastic precipitation generation model. Journal of Hydrology, 210(1–4), 178–191.
Woodard, J. D., & Garcia, P. (2008). Basis risk and weather hedging effectiveness. Agricultural Finance Review, 68(1), 99–117.
Woolhiser, D. A., & Pegram, G. G. S. (1979). Maximum likelihood estimation of Fourier coefficients to describe seasonal variations of parameters in stochastic daily precipitation models. Journal of Applied Meteorology, 18(1), 34–42.
Woolhiser, D. A., & Roldán, J. (1982). Stochastic daily precipitation models: 2. A comparison of distributions of amounts. Water Resources Research, 18(5), 1461–1468.
Xu, W., Filler, G., Odening, M., & Okhrin, O. (2010). On the systemic nature of weather risk. Agricultural Finance Review, 70(2), 267–284.
Acknowledgments
The financial support from the German Research Foundation via the CRC 649 ‘Economic Risk’, Humboldt-University Berlin, is gratefully acknowledged. Moreover, the authors would like to thank the participants of the CRC 649 Conference 2011 for their helpful comments and discussions on this topic.
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Appendix
Appendix
Empirical and estimated rainfall probabilities for Koßdorf (left) and Nordhausen (right), data 1973–2000; FS Fourier series
Estimated parameters for the mixed exponential distributions for Koßdorf (left) and Nordhausen (right), data 1973–2000; FS Fourier series
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Ritter, M., Mußhoff, O. & Odening, M. Minimizing Geographical Basis Risk of Weather Derivatives Using A Multi-Site Rainfall Model. Comput Econ 44, 67–86 (2014). https://doi.org/10.1007/s10614-013-9410-y
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DOI: https://doi.org/10.1007/s10614-013-9410-y