A Multivariate Analysis of Intercompany Loss Triangles

• DOI: http://doi.org/10.1111/jori.12102
• Date: 01 June 2017
• Author: Peng Shi
• Published date: 01 June 2017

©2015 The Journal of Risk and Insurance. Vol.84, No. 2, 717–737 (2017).

DOI: 10.1111/jori.12102

A Multivariate Analysis of

Intercompany Loss Triangles

Peng Shi

Abstract

The prediction of insurance liabilities often requires aggregating experience

of loss payment from multiple insurers. The resulting data set of intercom-

pany loss triangles displays a multilevel structure of claim development

where a portfolio consists of a group of insurers, each insurer several lines

of business, and each line various cohorts of claims. In this article, we pro-

pose a Bayesian hierarchical model to analyze intercompany claim triangles.

A copula regression is employed to join multiple triangles of each insurer,

and a hierarchical structure is specified on major parameters to allow for

information pooling across insurers. Numerical analysis is performed for an

insurance portfolio of multivariate loss triangles from the National Asso-

ciation of Insurance Commissioners. We show that prediction is improved

through borrowing strength within and between insurers based on training

and holdout observations.

Introduction

Loss reserves represent the best estimate of an insurer’s outstanding loss payments,

and they are often the largest liability on a property–casualty insurer’s balance sheet.

The primary goal of valuation actuaries is to set proper reserves to fund losses that

have been incurred but not yet developed. Improper reserving, either underreserving

or overreserving, is detrimental to a company’s financial position.

Because claim payments develop over time, and different obligations are incurred

from year to year, losses are often organized in a triangular fashion, known as a run-

off triangle, to emphasize the longitudinal and censored nature of the data. There is a

vast literature on alternative methods of modeling loss triangles, and we refer to the

Peng Shi is at the Department of Actuarial Science, Risk Management, and Insurance, School

of Business, University of Wisconsin–Madison, 975 University Avenue, Madison, WI 53706.

Shi can be contacted via e-mail: pshi@bus.wisc.edu. We wish to express our gratitude to the

reviewers for the valuable comments that have helped improve the quality of the work tremen-

dously. Thanks also go to Editor Professor Keith Crocker for his patience and encouragement

in the revision process. We acknowledge the financial support from the Centers of Actuarial

Excellence (CAE) Research Grant from the Society of Actuaries.

717

718 The Journal of Risk and Insurance

two monographs of Taylor(2000) and W ¨

uthrich and Merz (2008) for a comprehensive

review.Current literature focuses more on the multivariate loss reserving method and

emphasizes the implication of dependence between business lines on the reserving

variability. Existing studies fit into two groups. One group relies on the distribution-

free setup (see Braun, 2004; Hess et al., 2006; Merz and W¨

uthrich, 2008; Zhang, 2010;

Merz and W ¨

uthrich, 2009, among others). The other group uses more parametric

models. (see, e.g., Shi and Frees, 2011; de Jong, 2012; Zhang and Dukic, 2013; and Shi,

2014.

Current studies on multivariate loss reserving methods, though emphasizing the de-

pendency among business lines, have been focusing on single company experiences.

However, the prediction of insurance liabilities often requires aggregating the expe-

riences of loss payment from multiple insurers. First, different insurers sometimes

share similar claim payment patterns for the same line of business; thus, by combin-

ing experiences from other companies, an insurer borrows strength in the prediction

of its own outstanding claims. In addition, an insurer can use a model of several

companies to understand and compare experience with its rivals. Second, regulators

and professional associations benefit from an analysis of experiences of multiple in-

surers. For example, an industry-wide study helps regulators identify overall under-

or overreserving problems that cause inefficiency or even failure of the insurance

market. Another group who deals with intercompany claims data is reinsurers. A

reinsurer’s portfolio usually consists of a group of insurers and multiple lines from

each. A single model representing the experience of the group of companies will aid

both rate-making and reversing practices.

Motivated by the above observations, the goal of this study is to promote a multivari-

ate loss reserving modeling framework to analyze intercompany claim experiences.

The resulting pooled data of intercompany loss triangles display a multilevel struc-

ture of claim development, where a portfolio consists of a group of insurers, each

insurer several lines of business, and each line various cohorts of claims. To accom-

modate this unique data structure, we propose a Bayesian hierarchical model with

several features: within triangles, both parametric and semiparametric formulations

are considered for modeling the process of loss development over time; the associa-

tion among triangles of each individual firm is accommodated through a parametric

copula; a hierarchical structure is specified on major parameters to allow information

pooling across insurers.

One advantage of Bayesian methods is that one not only has the point estimate of

future outcomes, but also obtains the predictive distribution of these future outcomes

conditional on the observations to date. This is critical to insurers in the loss-reserving

context because valuation actuaries are more interested in a reasonable reserve range

than a point estimate. Wenote that Bayesian methods are not new to the loss reserving

literature, with the earliest efforts found in 1990s (see, e.g. Jewell, 1989, 1990; Verrall,

1990). With the development of the Markov chain Monte Carlo (MCMC) method,

Bayesian methods have found more applications in reserving for a single line of busi-

ness. Recent examples include Antonio and Beirlant (2008), de Alba and Nieto-Barajas

(2008), Peters et al. (2009), Meyers (2009), Merz and W ¨

uthrich (2010), and Zhang et al.

(2012). Despite the rapidly growing literature, Bayesian studies on multiple lines of