Predicting Multivariate Insurance Loss Payments Under the Bayesian Copula Framework

• Author: Yanwei Zhang,Vanja Dukic
• Date: 01 December 2013
• Published date: 01 December 2013
• DOI: http://doi.org/10.1111/j.1539-6975.2012.01480.x

© The Journal of Risk and Insurance, 2013, Vol. 80, No. 4, 891–919

DOI: 10.1111/j.1539-6975.2012.01480.x

891

PREDICTING MULTIVARIATE INSURANCE LOSS PAYM E N T S

UNDER THE BAYESIAN COPULA FRAMEWORK

Yanwei Zhang

Vanja Dukic

ABSTRACT

The literature of predicting the outstanding liability for insurance compa-

nies has undergone rapid and profound changes in the past three decades,

most recently focusing on Bayesian stochastic modeling and multivariate

insurance loss payments. In this article, we introduce a novel Bayesian mul-

tivariate model based on the use of parametric copula to account for depen-

dencies between various lines of insurance claims. Wederive a full Bayesian

stochastic simulation algorithm that can estimate parameters in this class

of models. We provide an extensive discussion of this modeling framework

and give examples that deal with a wide range of topics encountered in the

multivariate loss prediction settings.

INTRODUCTION

In his 1987 article, B ¨

uhlmann made the comment that the casualty actuaries have to

master the skills of probabilistic thinking in order to deal with a variety of risky situa-

tions. Because of the complexity and diversity of risks arising in the property–casualty

insurance, probability distributions of unknown quantities are often required for the

formal solutions of decision problems. One such important example is the practice

of insurance loss reserving. Due to reasons such as late reported claims, judicial

proceedings, or schedules of benefits for employer’s liability claims, many types of

property–casualty insurance claims often have lengthy settlement periods, with lia-

bility claims often taking years or even decades to complete. In order to be able to

respond to outstanding claims, every insurance company must set aside a provision,

known as a loss reserve. The loss reserve is typically a property–casualty insurance

company’s largest balance sheet liability.Its proper prediction is therefore a matter of

vital importance to the company and the research around loss reserve prediction has

become a central subject in modern actuarial science.

The loss reserving literature has undergone rapid and profound changes in the past

three decades, where a large number of innovative applications of statistical methods

Yanwei Zhang is at CNA Insurance Company. Vanja Dukic is at the University of Colorado-

Boulder. The first author can be contacted via e-mail: Yanwei.Zhang@cna.com. The authors

thank an anonymous referee for his/her many constructive suggestions that have helped

improve the contents of this article greatly. The first author is also grateful to Jian Peng at MIT

for his generous help while programming the simulation algorithm.

892 THE JOURNAL OF RISK AND INSURANCE

have led to the proliferation of a great variety of stochastic loss reserving models. As

the body of the literature continues to grow,some new trends have been observed in

recent years. One such important trend is the increasing reliance on Bayesian meth-

ods (e.g., de Alba, 2002; Ntzoufras and Dellaportas, 2002; Antonio and Beirlant, 2008;

Zhang, Dukic, and Guszcza, 2012). Bayesian analysis has gained some of its popular-

ity due to its inherent advantages in the applied loss forecasting context. For example,

the posterior predictive distribution conveys a wealth of information well beyond the

uncertainty estimators commonly used in the traditional stochastic loss reserving

analysis. This full measure of liability risk can be further incorporated into the deter-

mination of economic capital for the purpose of capital allocation and making strategic

and tactical decisions. The second trend is developments in the realm of multivariate

methods, where the major focus lies in the creation of models that enable integration

of correlated multidimensional claim information for simultaneous statistical process-

ing in order to improve, with respect to inference efficiency and prediction and risk

assessment accuracy, upon univariate stochastic methods that could potentially fail

due to the negligence of the inter-relationship within the multidimensional data. For

example, substantial effort (e.g., Braun, 2004; Merz and W¨

uthrich, 2008; de Jong, 2012;

Shi and Frees, 2011) has been devoted to making the extension of standard stochastic

reserving models to account for correlations among various lines of insurance, with

the aim to depict a more accurate picture of the liability risks faced by the insurance

company.

Although considerable advances have been achieved in both of the above fields,

Bayesian analysis in the multivariate setting has rarely been found in the loss re-

serving literature (see Shi and Frees, 2011). It is the intention of this article to fill

such a gap by laying out a unified and flexible framework to perform multivariate

loss reserving analysis using Bayesian methods. One distinct feature of the proposed

framework is the use of parametric copulas (e.g., see Frees and Valdez, 1998) within

a Bayesian paradigm to construct multivariate joint distributions. The copula model

allows various marginal model structures or multivariate outcomes of mixed types to

be specified and modeled simultaneously,yielding a rich family of probability models

for analyzing multivariate reserving problems (see, e.g., Shi and Frees, 2011; de Jong,

2012). Associated with the model flexibility in copulas is the increasing difficulty in

parameter inference, which, in many situations, would have not been feasible without

the use of the Bayesian simulation-based methods. Moreover, the development of the

multivariate methods in the Bayesian framework also brings many other advantages

to the loss reserving context. For example, compared to existing multivariate loss

reserving methods, the Bayesian copula model accounts for all sources of uncertainty

from the data and parameters and generates predictive distributions for all quantities

of interest. Furthermore, when the collected data have an inherent multilevel struc-

ture, hierarchical models can be employed for efficient statistical inference. Although

much of the article is within the context of insurance loss reserving, we note that

the proposed Bayesian copula framework also has potential applications in the field

of economic capital evaluation (Tang and Valdez, 2009), enterprise risk management

(CAS, 2003), and dynamic financial analysis (Kaufmann, Gadmer, and Klett, 2001),

where Monte Carlo methods are often adopted to generate risk distributions and

the aggregation of all risk distributions needs to reflect correlations and portfolio

effects.