Wishart‐gamma random effects models with applications to nonlife insurance

• Published date: 01 June 2021
• Author: Michel Denuit,Yang Lu
• Date: 01 June 2021
• DOI: http://doi.org/10.1111/jori.12327

J Risk Insur. 2021;88:443–481. wileyonlinelibrary.com/journal/JORI

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443

Received: 9 March 2020

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Accepted: 30 August 2020

DOI: 10.1111/jori.12327

ORIGINAL ARTICLE

Wishart‐gamma random effects models with

applications to nonlife insurance

Michel Denuit

1

|Yang Lu

2,3

1

Institute of Statistics, Biostatistics and

Actuarial Science (ISBA), Louvain Institute

of Data Analysis and Modeling‐LIDAM,

Université Catholique Louvain, Louvain‐la‐

Neuve, Belgium

2

Department of Economics (CEPN),

University Sorbonne Paris Nord (formerly

University of Paris 13), Villetaneuse, France

3

Department of Mathematics and Statistics,

Concordia University, Montreal, Quebec,

Canada

Correspondence

Yang Lu, University Sorbonne Paris Nord

(formerly University of Paris 13),

Department of Economics (CEPN),

Villetaneuse, France.

Email: yang.lu@univ-paris13.fr

Funding information

Agence Nationale de la Recherche,

Grant/Award Number: Labex MME‐DII;

Centre National de la Recherche

Scientifique, Grant/Award Numbers: 2019,

2020 Delegation grants

Abstract

Random effects are particularly useful in insurance

studies, to capture residual heterogeneity or to induce

cross‐sectional and/or serial dependence, opening

hence the door to many applications including ex-

perience rating and microreserving. However, their

nonobservability often makes existing models com-

putationally cumbersome in a multivariate context.

In this paper, it is shown that the multivariate ex-

tension to the Gamma distribution based on Wishart

distributions for random symmetric positive‐definite

matrices (considering diagonal terms) is particularly

tractable and convenient to model correlated random

effects in multivariate frequency, severity and dura-

tion models. Three applications are discussed to

demonstrate the versatility of the approach:

(a) frequency‐based experience rating with several

policies or guarantees per policyholder, (b) experi-

ence rating accounting for the correlation between

claim frequency and severity components, and (c)

joint modeling and forecasting of the time‐to‐

payment and amount of payment in microlevel re-

serving, when both are subject to censoring.

KEYWORDS

credibility, fractional calculus, frailty model, Laplace transform,

multivariate Gamma distribution, predictive (Bayesian)

ratemaking, symbolic calculation

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© 2020 American Risk and Insurance Association

1|INTRODUCTION AND MOTIVATION

In insurance ratemaking, there is typically a lot of information that the insurer cannot access.

This may be due to regulatory or budget constraints, for instance. These hidden features cause

overdispersion and induce both cross‐sectional correlation across the basket of products sold to

the same policyholders, and serial correlation across different periods. In a longitudinal, or

panel data context, these dependencies can be exploited to revise premiums based on past

claims history.

The standard actuarial approach to account for this asymmetric information consists in

including random effects in the frequency and/or severity component of the claim distribution.

Once the model has been fitted to available data, the distribution of the random effects can be

revised based on past observations. This predictive distribution can then be used to compute

predictive, or experience premiums depending on the claims reported by each policyholder

during the past coverage periods.

The main difficulty inherent to random effects models is the need to integrate out the

random effects for predictive premium calculation. This often requires advanced numerical

procedures (Gauss‐Hermite quadrature approximation or Monte‐Carlo integration, for in-

stance), unless the analysis is performed with the so‐called conjugate families (such as

Gamma‐distributed random effects for Poisson counts, for instance). While many conjugate

familiesareavailableintheunivariatecase,thechoiceismuchmorerestrictedinhigher

dimensions. The present paper aims to provide risk analysts with a new multivariate

Gamma distribution for the random effects that is tractable for many parent distributions

commonly used in insurance, like the Poisson, Gamma, Weibull or time‐changed

Exponential models. Meanwhile, the parametrization by a scalar shape coefficient and a

matrix scale coefficient appears to be quite flexible. This new family is derived from the

Wishart distribution, by taking the marginal distribution of its diagonal terms. We will term

it the Wishart‐Gamma distribution in the paper.

Our arguments are supported by three case studies, which cover not only the traditional

area of application of random effects models, that is, experience ratemaking, but also the

emerging research topic of microlevel reserving. We first extend the seminal framework of

Dionne and Vanasse (1989) for a single claim count per period to allow for multiple, cor-

related claim counts per period, and demonstrate its superiority especially in high dimen-

sions. Then, we propose a random effects model for claim frequency and severity that is

both tractable and flexible enough to capture the dependence between these two compo-

nents, without relying on restrictive assumptions such as independence. Finally, for the

microlevel reserving application, we propose a new, random‐effects‐based model, which is

tailor‐made to handle the censoring issue typically encountered in such data. Through these

examples, we demonstrate why the Wishart‐Gamma approach outperforms existing

methodologies.

The first, multivariate frequency regression model for multiple claim counts per period is

particularly useful to simultaneously price all the products purchased by a given individual in a

person‐centric insurance approach, or all the products bought by individuals within a group

(a household, for instance). Adopting the Wishart‐Gamma distribution for the random effects

allows the actuary to derive closed form premium updating formulas with a clear interpretation,

similar as in the univariate, Dionne and Vanasse (1989) setting. Indeed, while the approximate

linear credibility approach, worked out in this setting by Pinquet (1998) and Englund, Guillen,

Gustafsson, Nielsen, and Nielsen (2008), is easy to implement and more robust to model

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misspecification (see Hong & Martin, 2020), it may fail to capture the nonlinearity of the pricing

formula as demonstrated by Lu (2018). The linear credibility premium may even become

negative, as documented by Pinquet (2020) who finds that the conditions for the credibility

coefficients to be positive are quite complicated and unless they are imposed ex ante, the

credibility premium can potentially be negative, rendering this approach problematic, especially

from a regulation point of view. The only tractable multivariate random effects count model we

are aware of is proposed by Badescu, Lin, Tang, and Valdez (2015), who assume a mixture of

Erlang distributions for the random effects (i.e., a mixture of Gamma distributions with integer

degrees of freedom). The downside of this approach, however, is that first, Badescu et al. (2015)

restricted their analysis to estimation and it is not clear how their approach can be used to

compute predictive premiums. Second, and perhaps more importantly, this mixture model

involves a large number of parameters, which renders its analysis cumbersome. As a com-

parison, it is also argued that the approach proposed in this paper strikes a balance between

flexibility and parsimony. At dimension

d

, it involves a shape parameter

δ

and a

d

d×

sym-

metric positive‐definite matrix parameter, allowing for flexible pairwise correlation between its

components. The number of parameters involved is thus similar to those of centered multi-

variate normal distributions that are often used to model random effects in statistical studies.

Our approach will be compared to both the linear credibility approach and numerical

approximation methods currently available for the conditional expectation based predictive

premium. First, in a low‐dimensional setting (

d

=

2

), when each policy has only two

guarantees, we consider the same Wishart‐Gamma based model, and benchmark the up-

dating formulas of the predictive premium against those coming from linear credibility. This

comparison is particularly instructive, as we are able to derive extremely simple formulas

for both the predictive and credibility premia. We analyze to which extent the functional

form of the predictive premium is different from its linear credibility counterpart, and show

that the mispricing of the credibility premium can sometimes be as high as ten percent for

reasonable values of the parameters. Then, we switch to a high‐dimensional setting (

d

=

7

),

and compare the closed form predictive premium and their numerical approximations based

on state‐of‐the‐art methods, to demonstrate its superiority both in terms of computational

cost and precision.

As a second application, a random effects model for correlated claim frequency and severity

is worked out. While the existence of such a correlation and its economic significance is well

documented in the literature (see e.g., Park, Kim, & Ahn, 2018), existing multivariate random

effects based models capable of capturing dependence between claim frequency and severity

typically suffer from computational intractability (see, e.g., Baumgartner, Gruber, & Czado,

2015; Czado & Gschlossl, 2007; Oh, Shi, & Ahn, 2020). Some recent papers try to evade this

burden by letting the number of claims enter as a covariate in the severity model, without

introducing random effect for the severity component (see, e.g., Garrido, Genest, &

Schulz, 2016; Jeong, Valdez, Ahn, & Park, 2017; Park et al., 2018). The downside of this

approach is that by doing so, the predictive power of previous claim costs is lost. Let us also

mention the related works of Frees and Wang (2005), Kramer, Brechmann, Silvestrini, and

Czado (2013), and so forth, who adopt a model without any random effect, which they call the

“frequentist”approach.

1

As Frees and Wang (2005) put it, “the frequentist perspective avoids

1

Note that this frequentist approach has also been applied to the second application of this paper, that is, dependent

frequency‐severity modeling, see for example, Lee and Shi (2019), Yang and Shi (2019). There the pros and cons of both

approaches are rather similar and therefore will be omitted.

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