Dynamic Frailty Count Process in Insurance: A Unified Framework for Estimation, Pricing, and Forecasting

• Author: Yang Lu
• Date: 01 December 2018
• Published date: 01 December 2018
• DOI: http://doi.org/10.1111/jori.12190

©2017 The Journal of Risk and Insurance. Vol.85, No. 4, 1083–1102 (2018).

DOI: 10.1111/jori.12190

Dynamic Frailty Count Process in Insurance:

A Unified Framework for Estimation, Pricing,

and Forecasting

Yang Lu

Abstract

We study count processes in insurance, in which the underlying risk fac-

tor is time varying and unobservable. The factor follows an autoregressive

gamma process, and the resulting model generalizes the static Poisson–

Gamma model and allows for closed form expression for the posterior Bayes

(linear or nonlinear) premium. Moreover,the estimation and forecasting can

be conducted within the same framework in a rather efficient way. An ex-

ample of automobile insurance pricing illustrates the ability of the model to

capture the duration dependent, nonlinear impact of past claims on future

ones and the improvement of the Bayes pricing method compared to the

linear credibility approach.

Introduction

We propose a time series model for count variables encountered in insurance, when

the underlying risk factor is time varying and unobservable. We introduce the au-

toregressive gamma process for the latent factor dynamics and show how it provides

an integrated framework for the efficient estimation, pricing, and forecasting of the

risk. One typical application of the model is automobile insurance, for which the in-

surer holds the history of individual yearly claim counts over several periods. This

information should then be taken into account in order to update the future premium

regularly, on an individual basis.

Ideally, this latter exercise, also called posterior pricing, should be conducted via the

Bayes rule. However, very often, due to the lack of analytic expression of the Bayes

premium for most models, the linear credibility method is employed to get an approx-

imation. This practice has some drawbacks. Indeed, the credibility framework is solely

based on the first two moments of the count process1and does not fully capture the dy-

namics of the latter,which is clearly non-Gaussian. Thus, the linear premium approach

induces a systematic pricing bias that has to be evaluated. Moreimportantly, the mean–

Yang Lu is at Aix-Marseille School of Economics (AMSE), Aix-Marseille University. Lu can

be contacted via e-mail: luyang000278@gmail.com. I thank Christian Gourieroux, as well as

two anonymous referees for insightful comments that greatly improvedthe presentation of the

article. All remaining errors are mine.

1Indeed, only Gaussian processes are characterized by its first two moments.

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1084 The Journal of Risk and Insurance

variance credibility approach is not adapted for a complete risk, which includes non-

linear pricing/forecasting, or simulation of the future trajectory of the count process.

In a pioneering work, Dionne and Vanasse(1989) propose a Poisson–Gamma (i.e., neg-

ative binomial) model. They show that in a Poisson model with Gamma distributed

individual random effect (or frailty, or unobserved heterogeneity), the Bayes premium

is linear in past observations and thus can be obtained in closed form. However, one

shortcoming of this approach is that all the past observations have the same weight in

the premium updating formula, or equivalently, the random effect is time invariant.

There is a growing interest on models that improve this feature(Sundt, 1988; Brouhns

et al., 2003; Purcaru and Denuit, 2003; Abdallah, Boucher,and Cossette, 2016; Le Cour-

tois, 2015). In particular, Pinquet, Guill´

en, and Bolanc´

e (2001) generalize the negative

binomial model by replacing the static frailty with a dynamic one and empirically

found evidence of serial correlation of the individual frailty. However, the model

suffers from classical computational burdens of models with dynamic unobserved

factors, and the only linear credibility premium is available.

Our article contributes to this literature by proposing a new dynamic frailty model.

The static gamma frailty is replaced by a dynamic process with Gamma marginal

distribution, called autoregressive gamma process (ARG), which generalizes Dionne

and Vanasse’s (1989) model. It belongs to the so-called affine, or compound autore-

gressive family (see Darolles, Gourieroux, and Jasiak, 2006). We show that it provides

a tractable, unified approach for estimation, linear or nonlinear pricing, as well as

forecasting. More precisely, first, its estimation does not require simulation-based

techniques. Second, we show that the exact Bayesian forecast formula allows for a

closed-form expression. Third, this model can be easily extended to higher dimen-

sions, when multiple-risk policies are introduced. We then illustrate the advantage

of this new approach via an example of pricing. In particular we show that how the

model successfully captures the nonlinear, and duration dependence, and how the

credibiilty premium induces a pricing bias with respect to the Bayes premium.

Contrary to the state-space based, or parameter-driven model, the usual count pro-

cess literature (see, e.g., Harvey and Fernandes, 1989; Gourieroux and Jasiak, 2004;

Frees and Wang, 2005; Bolanc´

e et al., 2007; Abdallah, Boucher, and Cossette, 2016) is

observation driven; that is, it specifies directly the dynamics of the observable count

variables without serially correlated latent factor. The advantage of a parameter-

driven model is that it provides a more intuitive interpretation in terms of omitted

pricing factors and can better capture some key features of the observable count data,

such as nonlinear serial dependence and overdispersion (see also Davis, Dunsmuir,

and Streett, 2003, for a discussion). Nevertheless, as state-space models are popular in

finance,2they have enjoyed a limited success in insurance. Clearly,one of the reasons

is the computational burden involved, as the Bayesian simulation methods typically

2See, for instance, the stochastic volatility model for asset prices, and the stochastic intensity

model for credit risk occurrences.