Simultaneous borrowing of information across space and time for pricing insurance contracts: An application to rating crop insurance policies

• Published date: 01 March 2021
• DOI: http://doi.org/10.1111/jori.12312
• Date: 01 March 2021
• Author: Alan P. Ker,Yong Liu

J Risk Insur. 2021;88:231–257. wileyonlinelibrary.com/journal/JORI

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Received: 28 July 2019

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Accepted: 4 April 2020

DOI: 10.1111/jori.12312

ORIGINAL ARTICLE

Simultaneous borrowing of information across

space and time for pricing insurance contracts:

An application to rating crop insurance

policies

Yong Liu |Alan P. Ker

Department of Food Agricultural and

Resource Economics, University of Guelph,

Guelph, Ontario, Canada

Correspondence

Alan P. Ker, Department of Food,

Agricultural and Resource Economics,

University of Guelph, Guelph,

ON N1G 2W1, Canada.

Email: aker@uoguelph.ca

Abstract

Changing climate and technology can often lead to

nonstationary losses across both time and space for a

variety of insurance lines including property, cata-

strophe, health, and life. As a result, naive estimation

of premium rates using past losses will tend to be

biased. We present three successively flexible data‐

driven methodologies to nonparametrically smooth

across both space and time simultaneously, thereby

appropriately incorporating possibly nonidentically

distributed data into the rating process. We apply

these methodologies in estimating U.S. crop in-

surance premium rates. Crop insurance, with global

premiums totaling $4.1 trillion in 2018, is an inter-

esting application as losses exhibit both temporal and

spatial nonstationarity. We find significant borrowing

of information across both time and space. We also

find all three methodologies improve both the stabi-

lity and accuracy of crop insurance premium rates.

The proposed methods may be of relevance for other

lines of insurance characterized by spatial and/or

temporal nonstationary losses.

KEYWORDS

crop insurance, nonstationary losses, spatial smoothing, temporal

smoothing

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© 2020 American Risk and Insurance Association

1|INTRODUCTION

An increasingly volatile climate has led to significant economic (and insurance) losses from not

only property damage, but also business disruptions, increased medical claims, and even loss of

life.

1

For many forms of insurance, climate change, itself temporally and spatially nonstationary,

presents significant pricing difficulties (Born & Klimaszewski‐Blettner, 2013;Botzen,Vanden

Bergh, & Bouwer, 2010; Chang, Chang, & Wen, 2014; Hogarth & Kunreuther, 1989). Some have

argued that under these circumstances loss experience is ineffective at predicting future losses

and thus pricing insurance contracts (Linnerooth‐Bayer & Hochrainer‐Stigler, 2015;Mills,2012).

We suggest that while loss distributions may be changing in complex ways, this does not

necessarily suggest that the historical loss data are uninformative about future losses and thus

should be discarded. This manuscript proposes three methods to incorporate data from possibly

nonidentical distributions across both space and time simultaneously into the insurance rating

process. These methods are data‐driven and do not require any assumptions about the degree or

form of similarity between the loss distribution of interest and data from nonidentical loss

distributions.Given the nonstationary natureof losses for many lines of insurance, these methods

may be of general interest.

We consider the application of these methodologies to ratingcrop insurance contracts because

crop yields exhibit significantspatial and temporal nonstationarity. Moreover, crop insuranceis of

great public interest because it is heavily subsidized in most of the developed countries. Note in

2018, global crop insurance premium totaled $4.1 trillion. In the United States and Canada,

administrative and operating costs are fully absorbed by the government, and subsidies on crop

insurance premiums are around 60% (Glauber, 2013; Ker, Barnett, Jacques, & Tolhurst, 2017;

Rosa, 2018). In countries in the European Union, subsidies on premiums range from 30% to 70%

(e.g., 46% in Austria, 49% in Spain, 64% in Italy, and 65% in France; Bielza, Stroblmair, Gallego,

Conte, & Dittmann, 2007; Enjolras & Sentis, 2011). In Brazil, premium subsidies are almost 50%

(Lavorato & Braga, 2018), and in China subsidies range from 50% to over 80% (M. Wang, Shi, Ye,

Liu, & Zhou, 2011). These subsidies have generated significant transfers of public monies to the

agriculture production sector in each of these countries. Moreover, much of the world's food is

produced under some form of insurance (crop, livestock, weather index, etc.). With respect to the

U.S. crop insurance program, the total net cost between 2007 and 2016 was $72 billion—the

second largest outlay in the farm bill (nutrition being the largest). Of the $72 billion, 60%

($43 billion) was direct benefits to farmers (Rosa, 2018).

There is a great deal of methodological literature on estimating conditional yield distribu-

tions and corresponding crop insurance premium rates.

2

The spatial structure of yields has also

drawn attention in the literature. For examples, H. Wang and Zhang (2003) showed that

county‐level U.S. crop yields are spatially correlated with correlation dying off across distance.

Woodard, Schnitkey, Sherrick, Lozano‐Gracia, and Anselin (2012) found strong spatial de-

pendence in loss experience across counties, especially in central Corn Belt regions. Okhrin,

Odening, and Xu (2013) found loss dependence in Chinese yields depends on the distance but

not linearly. Most recently, the idea of borrowing spatially extraneous yield data to better

1

The Intergovernmental Panel on Climate Change (IPCC) estimates the mean present value of global economic da-

mages caused by a 1.5°C warming is $54 trillion.

2

Examples include parametric estimation (Atwood, Shaik, & Watts, 2003; Gallagher, 1987; Sherrick, Zanini, Schnitkey,

& Irwin, 2004; Tack, Harri, & Coble, 2012; Tolhurst & Ker, 2015) and nonparametric estimation (Goodwin & Ker, 1998;

Ker & Coble, 2003; Ker & Goodwin, 2000; Norwood, Roberts, & Lusk, 2004).

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