Flexible modelling of multivariate risks in pricing margin protection insurance: modelling portfolio risks with mixtures of mixtures

ABSTRACT

Margin Protection Programs (MPPs) are relatively new insurance plans, introduced by USDA’s Risk Management Agency (RMA). The attractiveness of these risk management instruments lies in the fact that the financial stability of agricultural production and farming operations is more dependent on margins than solely revenues, which neglect production costs, as is the case for Revenue Protection Programs (RPPs). This article examines the structure and rating of margin protection insurance policies by considering a broad class of high-dimensional copula models that parameterize the dependence among multivariate sources of risks. A variety of copula methods, including Archimedean Copulas (ACs), Mixture Copulas (MCs), and Vine Copulas (VCs) are used to analyse the dependence structure between revenues and input costs. In terms of methodology, flexible mixtures of parametric distributions are applied to characterize marginal densities, and likewise flexible mixtures of alternative copulas are used to model dependence. This article also argues that the rating methodology that accounts for irregular and anomalous features of dependence such as asymmetry, non-linearity, non-ellipticity, and tail dependence between input prices and output prices can result in more accurate premiums, and therefore, can increase the hedging effectiveness of the MPPs and the market efficiency in the US crop insurance market.

KEYWORDS:

• Insurance
• mixture distribution
• vine copulas
• margin protection plans
• dependence modelling

JEL CLASSIFICATION:

• G22
• Q12
• Q13
• Q18

Supplementary data

Supplemental data for this article can be accessed here.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

¹ In this article, the term Margin Protection Program (MPP) is used as an umbrella term that encompasses any insurance programs providing a margin-based policy. Typically, such programs guarantee certain components (quantities and/or prices) of both output revenues as well as input costs. Two notable examples of margin protection insurance plans are Livestock Gross Margin for Cattle (LGM-Cattle), which provides producers with coverage against unanticipated, financially adverse declines in their ‘gross margin’ (i.e., market value of livestock minus feeder cattle and feed costs on cattle), and Margin Protection Program for Grains (MPP-Grains), which provides farmers with coverage against unanticipated, financially adverse declines in their area-based ‘operating margins’ (i.e., expected area revenues minus expected area input costs).

² The program is called Livestock Gross Margin insurance plan for dairy producers (for short, LGM-Dairy), which is separate from Livestock Gross Margin insurance plan available for cattle feeders (for short, LGM-Cattle).

³ As for the table of summary statistics, it is important to note that although the magnitude of the average price deviates seem to be trivial, they are in fact average ‘daily’ price deviates, and as a result, even such seemingly small deviates are absolutely considerable if one takes into account the compounding nature of price increases over time.

⁴ To examine whether extreme values tend to occur together in the same period (here, on the same day), one can take advantage of scatterplots. Tail independence in the context of multiple random variables can be seen in a scatterplot matrix by noticing that the outliers have tendency to lie along x- and y-axes. On the other hand, tail dependence occurs when outliers tend to occur together, that is, in the upper-right and lower-left corners, instead of being concentrated along the axes. In short, when the outliers lie along x- and y-axes, it is said that the scatterplot indicates tail independence, while when the outliers are concentrated in the upper-right and lower-left corners, it is said that the scatterplot implies tail dependence.

⁵ Considering the fact that the number of observations in this empirical application is quite large (2997), and that the dot points on the scatterplot matrix are rather large, clearly observing some of the above-mentioned interesting features on these scatterplots is hard or impossible.

⁶ Some argue that the SW test could be statistically significant from a normal distribution in any large samples, which is due to the fact that the test is biased by sample size. As such, they suggest that an examination of the associated Q–Q plots needs to be done for verification in addition to the SW normality test. This task was conducted in the present article before the SW normality test, verifying that the data are non-normal.

⁷ Without copulas, modellers are usually limited to using normal or student’s t distributions, which may or may not explain well the frequency of the observations at hand. This freer choice of parametric distributions for modelling marginals will help modellers attain better fits, more predictive and more realistic models, and as a consequence, actuarially fairer premiums.

⁸ The existence of any sizable tail dependence between an input price and the output price, ceteris paribus, will result in a natural hedge, and thereby, accounting for such a tail dependence tends to estimate an actuarially fair premium rate that is lower than that computed using the conventional method (i.e., computed based on the assumption of multivariate normality, which by construction has zero tail dependence). On the other hand, the existence of any sizable upper tail dependence between two input prices, ceteris paribus, will result in a higher degree of risk, and thereby, accounting for such a tail dependence tends to estimate an actuarially fair premium rate that is greater than that computed using the conventional method (i.e., computed on the basis of the assumption of multivariate normality, which has no tail dependence).