The Use of Copula Functions in Pricing Weather Contracts for the California Wine Industry Don Cyr (contact author) Dean, Goodman School of Business Brock University St. Catharines ON, Canada dcyr@brocku,ca Robert Eyler Professor and Chair of Economics Sonoma State University Rohnert Park, California, USA Michael Visser Assistant Professor of Economics Sonoma State University Rhonert Park, California, USA 1
Introduction One of the primary impacts of climate change is the increased volatility of weather in many parts of the world, representing a significant risk for many economic endeavors. Since the introduction of weather contracts in the late 90 s, and their growing availability in the over-thecounter market, interest in and use of weather derivatives (contracts) for hedging weather volatility has increased in recent years. However, they have not been widely adopted in some sectors such as agriculture, even though weather is known to have significant consequences for quality and yield (Chicago Mercantile Exchange, 2009). Viticulture, for example, represents a high-value agricultural sector in California, of significant economic importance, that faces a myriad of weather-related risks. Many such risks could potentially be managed through weather contracts, but it appears that little use has been made of them to date (Gedeon, 2008). Unfortunately the lack of consensus with regards to a pricing model for weather contracts and their ultimate effectiveness for hedging purposes, particularly in agriculture, has mitigated their adoption. The basic issue with respect to weather contract pricing is that weather variables are not traded assets and therefore the traditional pricing approach based on riskless arbitrage arguments is not theoretically justifiable. Consequently, the most common approach to the pricing of weather contracts remains that employed in the insurance industry known as burn rate analysis and index modelling, involving the estimation of the probability distribution of the weather variable from historical data (Jewson and Brix, 2005). Not unrelated to the pricing debate is uncertainty regarding the stochastic process identified for a weather variable. Given that the traditional option pricing framework requires the specification of a continuous time process, a number of authors have attempted to identify stochastic processes for daily temperatures (Dischel, 1998; Cambell and Diebold, 2005; Schiller et al., 2010; and Benth and Benth, 2012) and in some cases daily precipitation (Odening et al., 2007). Weather contracts, however, are frequently based on cumulative weather measures (indices) over a month or season and increasingly it has been shown that using stochastic processes identified from daily data, to infer monthly or seasonal measures, results in cumulative errors and misestimation of variance, which can reduce the impact of the modelling benefits provided by daily data (Campbell and Diebold, 2005; Geman and Leonardi, 2005; Schiller et al., 2012). Unfortunately not much research has focused on identifying stochastic processes for cumulative weather variables, for use in index modelling, although it has been suggested (Geman and Leonardi 2005) this may be a more appropriate approach. In addition the interest in forecasting cumulative measures such as seasonal weather has increased significantly in the meteorological field, with some success (Troccoli, 2010). The stochastic process and distributional properties of cumulative weather indices is also an important consideration in attempting to model the complex relationship that may exist between weather variables and agricultural output or input measures. This relationship, if not modelled appropriately, can result in significant basis risk that limits the hedging effectiveness of weather contracts (Woodward and Garcia, 2008). Recent research into this issue has focused on the application of copula functions to identify and characterize multivariate distributions 2
representing weather and agricultural variables that account for varying correlations (tail dependence) across the range of the marginal distributions (Vedenov and Barnett, 2004; Barnett and Mahul, 2007; Scholzel and Friederichs, 2008; Skees, 2008; Vedenov, 2008; Liu and Miranda, 2010; Xu et al., 2010: Bokusheva, 2011; Okhrin et al, 2012) ) Although the identification of an appropriate copula function does not require knowledge of the specific marginal distributions, simulation for the purposes of testing hedging effectiveness does. The value of identifying a stochastic process for cumulative weather variables therefore, aids in the determination of appropriate marginal distribution assumptions for copula function based simulation. In summary, the focus of this paper is to examine the stochastic process of a cumulative weather variable critical to the California viticulture industry over an appropriate time period, and the use of copula function technology in modelling the pricing of a weather contract as a case study. Using, as a basis, growing degree days (GDD s) from three viticulture pricing districts in California, the current study examines the stochastic process of seasonal GDD s and employs copula function technology to specify a multivariate distribution between GDD indices and grape prices. Monte Carlo simulation is then used to price and test the effectiveness of a weather contract for hedging purposes. Data The data consists of seasonal cumulative GDD s for three regions of viticulture in California, for the years 1960 through 2010. In particular the study employs measures from 11 different weather stations relating to the grape pricing districts of Napa and Sonoma in the north, and the more southern district of Santa Barbara. In addition to GDD data, the weighted average grape harvest prices from 1976 through 2010 for Chardonnay, Cabernet Franc, Cabernet Sauvignon, Merlot and Zinfandel varietals, were obtained for the three districts from the annual USDA crush report. Methodology The first stage of the study employed standard time series with outlier intervention analysis to examine the nature of both the GDD and grape price data. The purpose was to determine whether there were any time series structural changes evident, that may ultimately result in non-stationarity in terms of the multivariate distribution identified for GDD and grape prices. The time series analysis provided evidence of significant structural changes in the GDD data, roughly associated with the period of time after the late 1970 s. In particular the time series analysis indicated, on average, an increased number of statistically significant time series level shifts (increases) in the GDD data along with an increased number of time series outliers for the three regions. These increases in GDD levels and volatility are likely attributable to climate change. Alternatively, grape prices exhibited basic non stationarity and consequently were detrended. It noted however that the simple linear correlation of GDD and adjusted grape prices significantly decreased for the period of 1990 through 2010 versus the period of 1976 through 1990. 3
Copula function technology, although most fully developed in the field of mathematical statistics early on by Sklar (1959), did not gain much attention in terms of applications in finance until the seminal work of Li (2000) in pricing default risk in the mortgage-backed security market. Although much maligned in terms of the 2008 financial crisis (Salmon, 2009) copula functions provide for a practical way of characterizing a multivariate distribution, independent of the specification of the univariate marginal distributions, which can capture basic non-linear relationships, also known as tail dependence, over the range of the distribution. As noted above, the applications of copula function technology in characterizing the non linear relationships between weather and agricultural variables has increased in recent years. Results In general preliminary results indicate that the best fitting marginal distributions for seasonal GDD s appears to be that of a Weibull, while grape prices are best characterized by the lognormal distribution. In terms of copula functions the results would appear to indicate that the Gaussian copula is the best characterization of the multivariate distribution between GDD and grape prices for all five varietals. This analysis was based upon the average GDD s measured from several weather stations in each pricing district. However, a copula function analysis of GDDs between weather stations indicates some non-gaussian copula forms, with possible tail dependence. As in Liu and Miranda (2010) this has critical implications for suppliers of weather contract written on average GDD s. In addition, recent indications (Heimfarth et al., 2012) are that any benefits of hedging, based upon aggregated data, as in the case considered, must be tempered when considering the benefits to individual producers. References Barnett, B.J., and Mahul, O. (2007). Weather index insurance for agriculture and rural areas in lower-income countries. American Journal of Agricultural Economics 89(5): 1241-1247. Benth, J.S. and Benth, F.S. (2012). A critical view on temperature modelling for application in weather derivatives markets. Energy Economics 34(2):592-602. Bokusheva, R. (2011). Measuring dependence in joint distributions of yield and weather variables, Agricultural Finance Review 71(1): 120-141 Campbell, S. D. and Diebold, F. X. (2005). Weather forecasting for weather derivatives. Journal of the American Statistical Association 100(469): 6 16. Chicago Mercantile Exchange Group/Storm Exchange (2009). Best Practices for the Agricultural Community (Webniar Series). Accessed on September 19 th 2009 at http://www.cmegroup.com/education/events/forms/storm_webinar_series.html. Dischel, B. (1998). At last: A model for weather risk. Energy & Power Risk Management 11(3): 20 21. 4
Gedeon, J. (2008). Wine industry is slow to warm up to weather derivatives: experts say various factors account for hesitation. Wine Business Monthly, 06/15/2008. Heimfarth, L.E., R. Finger, and O. Musshoff, (2012). Hedging weather risk on aggregated and individual farm-level: pitfalls of aggregation biases on the evaluation of weather index-based insurance. Agricultural Finance Review, 72(3): 471 487. Jewson, S. and Brix, A. (2005). Weather Derivative Valuation: The Meteorological, Statistical, Financial and Mathematical Foundations, Cambridge University Press. Li, D.X. (2000). On default correlation: a copula function approach, Journal of Fixed Income 9, 43-54. Liu, P. and Miranda, M.J. (2010).Tail dependence among agricultural insurance indices: the case of Iowa county-level rainfall. Paper presented at the Agricultural & Applied Economics Association 2010 AAEA, CAES, & WAEA Joint Annual Meeting, Denver, Colorado, July 25-27, 2010. Odening, M., Mubhoff, O. and Hanson, S. (2007). Analysis of rainfall derivatives using daily precipitation models; opportunities and pitfalls. Agricultural Finance Review, 67, 135-156. Okhrin, O., Odening, M. and Xu, W. (2012). Systemic Weather Risk and Crop Insurance: The Case of China. Journal of Risk and Insurance. doi: 10.1111/j.1539-6975.2012.01476.x Salmon, F. (2009). Recipe for disaster: the formula that killed Wall Street, Wired, February 23, 2009. Schiller, F., Seidler, G. And Wimmer, M. (2012). Temperature models for pricing weather derivatives, Quantitative Finance, 2(3): 489-500. Scholzel, C. And Friederichs, P. (2008). Multivariate non-normally distributed random variables in climate research introduction to the copula approach. Nonlinear Processes in Geophysics, 15, 761-772. Skees, J.R. (2008). Challenges for use of index-based weather insurance in lower income countries. Agricultural Finance Review 68(1): 197-217. Sklar, A. (1959). Fonctions de répartition à n dimensions et leurs marges, Publ. Inst. Statist. Univ. Paris 8: 229 231. Troccoli, A. (2010). Seasonal climate forecasting: A review, Meteorological Applications 17(3): 251 268. Vedenov, D.V. (2008). Application of copulas to estimation of joint crop yield distributions. Contributed paper at the Agricultural and Applied Economics Association 2008 meetings, Orlando, USA, July. Vedenov, D.V., Barnett, B.J. (2004). Efficiency of weather derivatives as primary crop insurance instruments. Journal of Agriculture and Resource Economics, 29; 387-403. Woodward, J.D. and Garcia, P. (2008). Basis risk and weather hedging effectiveness, Agricultural Finance Review, 68, 99-117. 5
Xu, W., Filler, G., Odening, M., and Okhrin, O. (2010). On the systemic nature of weather risk. Agricultural Finance Review, 70(2), 267 284. 6