# Optimal relativities in a modified Bonus-Malus system with
long memory transition rules and frequency-severity dependence

Jae Youn Ahn<EMAIL_ADDRESS>Eric C.K. Cheung<EMAIL_ADDRESS>Rosy Oh<EMAIL_ADDRESS>Jae-Kyung Woo∗<EMAIL_ADDRESS>Department of
Statistics, Ewha Womans University, Seodaemun-Gu, Seoul 03760, Korea.  School
of Risk and Actuarial Studies, Australian School of Business, University of
New South Wales, Australia.  Department of Industrial and Systems Engineering,
Korea Advanced Institute of Science and Technology (KAIST), Daejeon 34141,
Korea.

###### Abstract

In the classical Bonus-Malus System (BMS) in automobile insurance, the premium
for the next year is adjusted according to the policyholder’s claim history
(particularly frequency) in the previous year. Some variations of the
classical BMS have been considered by taking more of driver’s claim experience
into account to better assess individual’s risk. Nevertheless, we note that in
practice it is common for a BMS to adopt transition rules according to the
claim history for the past multiple years in countries such as Belgium, Italy,
Korea, and Singapore. In this paper, we revisit a modified BMS which was
briefly introduced in Lemaire, (1995) and Pitrebois et al., 2003a .
Specifically, such a BMS extends the number of Bonus-Malus (BM) levels due to
an additional component in the transition rules representing the number of
consecutive claim-free years. With the extended BM levels granting more
reasonable bonus to careful drivers, this paper investigates the transition
rules in a more rigorous manner, and provides the optimal BM relativities
under various statistical model assumptions including the frequency random
effect model and the dependent collective risk model. Also, numerical analysis
of a real data set is provided to compare the classical BMS and our proposed
BMS.

###### keywords:

Bonus-Malus System, Extended Bonus-Malus Levels , Optimal Relativity,
Frequency-severity Dependence, A Priori Classification, A Posteriori
Ratemaking.

##  1 Introduction

From Lemaire, (1998), as far as automobile insurance is concerned, insurers in
the U.S. and Canada mainly use many of a priori variables (observable factors)
to classify the risks of the policyholders while some other countries in
Europe, Asia, Latin America and Africa use also other a posteriori rating. The
common a posteriori ratemaking process can be done via Bonus-Malus System
(BMS), and such a posteriori ratemaking possibly resolves residual
heterogeneity still remaining within risk classes. As in Section 1 of Lemaire,
(1998), although BMS is not commonly used in North America, the situation can
change later since regulatory authorities may prohibit insurers from using
certain classification variables, and moreover a posteriori rating is known to
be an efficient way to assess individual risks. Based on the policyholder’s
claim experience, such system discounts premium as a reward of claim-free case
(bonus) and charges additional premium in the presence of accidents (malus).
This a posteriori ratemaking mechanism takes on various forms across countries
but it generally consists of three major components: Bonus-Malus (BM) levels;
transition rules; and BM relativities. Each new policyholder is classified
into one of the finite number of risk classes depending on his/her observable
risk characteristics, and he/she is also assigned an initial BM level. After
each policy period, the BM level moves up or down according to the
prespecified transition rules penalizing based on the number of reported
claims and rewarding those without claim. In a typical BMS, claim experience
in the previous policy period is usually utilized to determine the transition
rule. However, for better separation of consistently and temporarily good (or
bad) risks, we consider to capture claim experience in the multiple previous
policy periods in the design of the transition rule like in Belgium. Then, the
premium is renewed at a rate which is the product of the BM relativity and the
base premium. Here the BM relativity is determined by the BM level which plays
a role in the premium adjustment coefficient. In this way, the actual premium
charged to policyholders in each BM level could match more closely to their
risks reflected in the claim number record for the past few policy periods and
consequently the claim costs are fairly shared. In the classical BMS, the base
premium is often calculated as a product of the mean frequency and the mean
severity due to the assumed independence between these two random variables,
e.g. Pitrebois et al., 2003b ; Denuit et al., (2007); Tan et al., (2015).
However, the independence assumption is often criticized since some recent
insurance reports demonstrate the existence of significant dependence between
claim frequency and severity. In the actuarial literature, various statistical
models were developed in attempt to capture certain dependency structures
between frequency and severity of claims. These include, for example, copula
models (Czado et al.,, 2012; Frees et al.,, 2016), shared or bivariate random
effect models (Hernández-Bastida et al.,, 2009; Baumgartner et al.,, 2015; Oh
et al., 2020b, ), and two-part models (Shi et al.,, 2015; Garrido et al.,,
2016; Park et al.,, 2018; Valdez et al.,, 2020).

The focus of this paper is two-fold. First, we revisit the extension of the BM
levels (Lemaire,, 1995; Pitrebois et al., 2003a, ) and define our long memory
transition rules in a more formal setting. Second, under the proposed
transition rules, we consider the optimal relativities not only in a
frequency-only model but also in a dependent collective risk model that allows
for dependence between frequency and severity. As an example of the dependent
collective model, we consider the copula-based bivariate random effect model
as in Oh et al., 2020b , among many choices of shared or bivariate random
effect models. While the policyholder’s BM level goes down if there is no
claim in the past year in the classical BMS, the BM level in our model moves
to a lower one when he/she does not have any claim history consecutively for
years equivalent to the “period of penalty” in this modified BMS. Similar case
was discussed in Chapter 17 of Lemaire, (1985) and Section 5 of Pitrebois et
al., 2003a  where the transition is made based on the number of consecutive
claim-free years. However, the optimal relativities in this paper are obtained
by solving different forms of optimization problems. Under two models, namely
the frequency random effect model and the bivariate random effect model, the
optimal relativities and the hypothetical mean square error (HMSE) are
studied. The rest of the paper is organized as follows. In Section 2, the
description of two random effect models and the optimal relativity results
under the classical BMS rules are given. In Section 3, a period of the penalty
(denoted as “$pen$”) is introduced in the transition rules such that the BM
level goes down only when there is no claim for the last consecutive $(1+pen)$
years. Under the modified transition rules, an “augmented” BM level is newly
defined so that its transition process has the Markov property. In turn, the
optimal relativities are provided under the two different models. Finally, in
Section 4, a data set from the Wisconsin Local Government Property Insurance
Fund (e.g. Frees et al., (2016)) is utilized to illustrate the impact of the
long memory transition rules on our BMS compared to the classical BMS.

##  2 Model descriptions and basic results

For the $i$-th policyholder in the $t$-th policy year, we let $N_{it}$ be the
number of claims and $\boldsymbol{Y}_{it}=(Y_{it1},\ldots,Y_{itN_{it}})$ be
the vector of associated claim amounts, where $Y_{itj}$ is the $j$-th claim
amount. (Note that $\boldsymbol{Y}_{it}$ is undefined when $N_{it}=0$.) Also,
the aggregate claim amount and average claim amount are defined as

$S_{it}:=\begin{cases}\sum_{j=1}^{N_{it}}Y_{itj},&N_{it}>0\\\
0,&N_{it}=0\end{cases}\quad\quad\hbox{and}\quad\quad
M_{it}:=\begin{cases}\frac{1}{N_{it}}\sum_{j=1}^{N_{it}}Y_{itj},&N_{it}>0,\\\
\hbox{0},&N_{it}=0,\end{cases}$ (1)

respectively. Clearly, these two quantities are directly linked as
$S_{it}=N_{it}M_{it}$.

In the insurance ratemarking process, a priori ratemaking results in risk
classification of the (new) policyholder whose past claim history is
insufficient for the ratemaking purposes. Using the observed risk
characteristics of the policyholders (collected in the row vector
$\boldsymbol{X}_{i}$ for the $i$-th policyholder), their risk classes are
determined and the a priori premiums are also obtained. Then, the residual
heterogeneity is explained by capturing the unobserved risk characteristics of
policyholders (denoted as $\Theta_{i}$ for the $i$-th policyholder) in a
posteriori ratemaking process and consequently their premium amounts are
adjusted based on the claim history. Here, $\boldsymbol{X}_{i}$ and
$\Theta_{i}$ do not depend on time. In general, the frequency part may be
considered enough to construct the BMS under the assumption of independence
between frequency and severity. However, recent BMS research in Oh et al.,
2020b  reveals that the dependence between frequency and severity plays a
critical role in the determination of optimal relativities in case of
significant dependence. In this regards, the superscripts [1] and [2]
representing frequency and severity components respectively are introduced to
variables and their corresponding realizations. When there is only the
frequency component, we drop the notation [1] for convenience. To model the
frequency and the severity of claims, the techniques of Generalized Linear
Model (GLM) in De Jong and Heller, (2008) with the assumption of exponential
dispersion family for the random components will be applied (see A for the
descriptions).

###  2.1 Random effect models

For the unobserved risk characteristics in the a posteriori ratemating, let us
describe two different models which take into account the randomness of
frequency only or both frequency and severity. For convenience, we assume that
there are $\mathcal{K}$ risk classes predetermined at the moment of when the
contracts begin. We start with the descriptions of the model for the frequency
part only.

######  Model 1 (Frequency model with random effect).

First, the weight of the $k$-th risk class is defined as

$w_{k}:=\mathbb{P}(\boldsymbol{X}=\boldsymbol{x}_{k}),\qquad
k=1,\ldots,\mathcal{K},$ (2)

where $\boldsymbol{X}$ is the row vector of the observed risk characteristics
of a randomly picked person from the population of policyholders, and
$\boldsymbol{x}_{k}$ is a vector of the observed risk characteristics for the
$k$-th risk class. Note that $w_{k}$ can be regarded as the fraction of
policyholders with observed risk characteristics $\boldsymbol{x}_{k}$. The a
priori premium for the $i$-th policyholder is determined as
$\Lambda_{i}:=\eta^{-1}(\boldsymbol{X}_{i}\boldsymbol{\beta})$, where
$\eta(\cdot)$ is a link function, $\boldsymbol{X}_{i}$ is the row vector of
observed risk characteristics of this $i$-th policyholder, and
$\boldsymbol{\beta}$ is a column vector of regression coefficients in the
count regression model estimated from the past data concerning the number of
reported claims. Incorporating unobserved risk component for the
frequency111In practice, it is very hard to figure out all the possible
combinations of the observed risk characteristics for the model. Hence,
introduction of unobserved risk characteristics $\Theta_{i}$ is essential, and
this provides theoretical and practical justifications for the a posteriori
risk classification., the conditional distribution of the number of claims
$N_{it}$ for the $i$-th policyholder in the $t$-th policy year given the
observed risk characteristics $\boldsymbol{X}_{i}=\boldsymbol{x}_{i}$ and
unobserved risk characteristics $\Theta_{i}=\theta_{i}$ along with the
distribution of $\Theta_{i}$ is provided in B.

The following model is an extension of two-part model (Frees and Valdez,,
1998; Garrido et al.,, 2016) by introducing a bivariate random effect to model
various types of dependence such as:

  * 1.

dependence among frequencies;

  * 2.

dependence among severities; and

  * 3.

dependence between frequency and severity.

######  Model 2 (Collective risk model with bivariate random effect (Oh et
al., 2020b, )).

Recall that the superscripts [1] and [2] represent the frequency and the
severity components respectively. As an extension of Model 1 above, the random
component of the claim severity is also taken into account for a priori and a
posteriori ratemaking processes. In contrast to (2), the weight of the $k$-th
risk class is defined as

$w_{k}:=\mathbb{P}({\boldsymbol{X}^{[1]}=\boldsymbol{x}_{k}^{[1]},\boldsymbol{X}^{[2]}=\boldsymbol{x}_{k}^{[2]}}),\qquad
k=1,\ldots,\mathcal{K},$

where $(\boldsymbol{X}^{[1]},\boldsymbol{X}^{[2]})$ is the pair of row vectors
containing the observed risk characteristics relevant to the frequency and
severity of claims for a randomly picked person from the population of
policyholders, and $(\boldsymbol{x}_{k}^{[1]},\boldsymbol{x}_{k}^{[2]})$ is
the corresponding pair for the $k$-th risk class. The a priori premium for the
$i$-th policyholder can be obtained from the information of the vector
$(\Lambda_{i}^{[1]},\Lambda_{i}^{[2]})$, which is in turn determined by the
observed risks characteristics as

$\Lambda_{i}^{[1]}:=\eta_{1}^{-1}(\boldsymbol{X}^{[1]}_{i}\boldsymbol{\beta}^{[1]})\quad\hbox{and}\quad\Lambda_{i}^{[2]}:=\eta_{2}^{-1}(\boldsymbol{X}^{[2]}_{i}\boldsymbol{\beta}^{[2]}),$

where $\eta_{1}(\cdot)$ and $\eta_{2}(\cdot)$ are link functions,
$\boldsymbol{X}^{[1]}_{i}$ and $\boldsymbol{X}^{[2]}_{i}$ are the $i$-th
policyholder’s row vectors of observed risk characteristics for frequency and
severity, and $\boldsymbol{\beta}^{[1]}$ and $\boldsymbol{\beta}^{[2]}$ are
column vectors of parameters to be estimated. More importantly, the dependence
between claim frequency and severity is also considered as the corresponding
unobserved risk characteristics $(\Theta_{i}^{[1]},\Theta_{i}^{[2]})$ are
modeled via a bivariate copula. More specific distributional descriptions of
this model are provided in C.

It is instructive to note that when Model 2 is restricted to the claim
frequency only, it is conceptually equivalent to Model 1.

###  2.2 Optimal relativities

In the classical BMS framework, common transition rules are such that the BM
level is lowered by one for a claim-free year and increased by $h$ levels per
claim, leading to what is referred to as the $-1/+h$ system. Based on such
transition rules, the BM level for each policyholder evolves as a (discrete-
time) Markov Chain. Let us denote $L$ as a random variable representing the BM
level (from 0 to $z$) for a randomly picked policyholder from the population
in a stationary state (independent of time). Its distribution (that is, the
proportion of policyholders in each level) is determined by the frequency
component as

$\mathbb{P}(L=\ell)=\sum_{k=1}^{\mathcal{K}}w_{k}\int\pi_{\ell}(\lambda_{k}\theta,\psi)g(\theta){\rm
d}\theta,\qquad\ell=0,\ldots,{z},$ (3)

where $\pi_{\ell}(\lambda_{k}\theta,\psi)$ is the stationary probability that
a policyholder with expected frequency $\lambda_{k}\theta$ is in level $\ell$.
(It is understood that
$\lambda_{k}=\eta^{-1}(\boldsymbol{x}_{k}\boldsymbol{\beta})$, $g(\cdot)$ and
$\psi$ in (3) are replaced by
$\lambda^{[1]}_{k}=\eta_{1}^{-1}(\boldsymbol{x}^{[1]}_{k}\boldsymbol{\beta}^{[1]})$,
$g_{1}(\cdot)$ and $\psi^{[1]}$ respectively if one considers Model 2.) We
shall provide explanations on how to obtain the stationary probabilities at
the beginning of Section 3.2 for our more general model in relation to Example
2. The relativity associated with the BM level $\ell$ is denoted by
${\zeta}(\ell)$. The following lemmas review how to determine the optimal
relativity $\tilde{\zeta}(\ell)$ in two optimization settings under the model
assumptions in Section 2.1.

######  Lemma 1.

(Tan et al.,, 2015) Consider the optimization problem

$(\tilde{\zeta}(0),\ldots,\tilde{\zeta}(z)):=\operatorname*{arg\,min}_{({\zeta}(0),\ldots,{\zeta}(z))\in\mathbb{R}^{{z+1}}}\mathbb{E}[(\Lambda\Theta-\Lambda{\zeta}(L))^{2}]$
(4)

under the frequency-only Model 1. Because
$\mathbb{E}[N_{it}|\Lambda_{i},\Theta_{i}]=\Lambda_{i}\Theta_{i}$ (see (23))
is the “correct” premium for the $i$-th policyholder if we knew $\Theta_{i}$,
one can regard
$\Lambda\Theta=\eta^{-1}(\boldsymbol{X}\boldsymbol{\beta})\Theta$ as the
“correct” premium for a policyholder randomly picked from the population
having observed risk characteristics $\boldsymbol{X}$ and unobserved risk
characteristics $\Theta$. The optimization (4) amounts to choosing the
relativities to minimize the mean squared difference between $\Lambda\Theta$
and the actual premium charged $\Lambda{\zeta}(L)$ when a policyholder is in
BM level $L$. Then, the optimal relativities can be analytically calculated as

$\tilde{\zeta}(\ell):=\frac{\mathbb{E}[\Lambda^{2}\Theta|L=\ell]}{\mathbb{E}[\Lambda^{2}|L=\ell]}=\frac{\sum_{k=1}^{\mathcal{K}}w_{k}\lambda_{k}^{2}\int\theta\pi_{\ell}(\lambda_{k}\theta,\psi)g(\theta){\rm
d}\theta}{\sum_{k=1}^{\mathcal{K}}w_{k}\lambda_{k}^{2}\int\pi_{\ell}(\lambda_{k}\theta,\psi)g(\theta){\rm
d}\theta},\qquad\ell=0,\ldots,{z}.$

######  Lemma 2.

(Oh et al., 2020b, ) Consider the optimization problem

$(\tilde{\zeta}(0),\ldots,\tilde{\zeta}(z)):=\operatorname*{arg\,min}_{({\zeta}(0),\ldots,{\zeta}(z))\in\mathbb{R}^{{z+1}}}\mathbb{E}[(\Lambda^{[1]}\Lambda^{[2]}\Theta^{[1]}\Theta^{[2]}-\Lambda^{[1]}\Lambda^{[2]}{\zeta}(L))^{2}]$
(5)

under the frequency-severity Model 2. As
$\mathbb{E}[S_{it}|\Lambda_{i}^{[1]},\Lambda_{i}^{[2]},\Theta_{i}^{[1]},\Theta_{i}^{[2]}]=\Lambda_{i}^{[1]}\Lambda_{i}^{[2]}\Theta_{i}^{[1]}\Theta_{i}^{[2]}$
(see C) is the “correct” premium for the $i$-th policyholder,
$\Lambda^{[1]}\Lambda^{[2]}\Theta^{[1]}\Theta^{[2]}=\eta_{1}^{-1}(\boldsymbol{X}^{[1]}\boldsymbol{\beta}^{[1]})\eta_{2}^{-1}(\boldsymbol{X}^{[2]}\boldsymbol{\beta}^{[2]})\Theta^{[1]}\Theta^{[2]}$
is the “correct” premium for a policyholder randomly picked from the
population having observed risk characteristics
$(\boldsymbol{X}^{[1]},\boldsymbol{X}^{[2]})$ and unobserved risk
characteristics $(\Theta^{[1]},\Theta^{[2]})$. The optimization (5) is
concerned with choosing the relativities to minimize the mean squared
difference between $\Lambda^{[1]}\Lambda^{[2]}\Theta^{[1]}\Theta^{[2]}$ and
the actual premium charged $\Lambda^{[1]}\Lambda^{[2]}{\zeta}(L)$ when a
policyholder is in BM level $L$. Then, the optimal relativities can be
analytically calculated as

$\displaystyle\tilde{\zeta}(\ell):=$
$\displaystyle~{}\frac{\mathbb{E}[(\Lambda^{[1]}\Lambda^{[2]})^{2}\Theta^{[1]}\Theta^{[2]}|L=\ell]}{\mathbb{E}[(\Lambda^{[1]}\Lambda^{[2]})^{2}|L=\ell]}$
$\displaystyle=$
$\displaystyle~{}\frac{\sum_{k=1}^{\mathcal{K}}w_{k}(\lambda_{k}^{[1]}\lambda_{k}^{[2]})^{2}\int\int\theta^{[1]}\theta^{[2]}\pi_{\ell}(\lambda_{k}^{[1]}\theta^{[1]},\psi^{[1]})h(\theta^{[1]},\theta^{[2]}){\rm
d}\theta^{[1]}{\rm
d}\theta^{[2]}}{\sum_{k=1}^{\mathcal{K}}w_{k}(\lambda_{k}^{[1]}\lambda_{k}^{[2]})^{2}\int\pi_{\ell}(\lambda_{k}^{[1]}\theta^{[1]},\psi^{[1]})g_{1}(\theta^{[1]}){\rm
d}\theta^{[1]}},\qquad\ell=0,\ldots,{z}.$

The optimal relativities in Lemma 1 and 2 can be applied to various transition
rules and BM levels $\ell$. For the calculation of optimal relativity in the
BMS literature, the Markov property of the transition rules is typically
utilized, and the BM level is often assumed to be stationary (Pitrebois et
al., 2003b, ; Denuit et al.,, 2007; Tan et al.,, 2015; Oh et al., 2020b, ).
The most common (and also one of the simplest) BMS is the $-1/+h$ system,
while other variations are also available, for example, in Tan et al., (2015).
However, many countries including Belgium, Korea, and Singapore adapt
transition rules with long memory. While such transition rules do not directly
possess Markov property, there are examples in the literature showing how
these can be dealt with (Lemaire,, 1995; Pitrebois et al., 2003b, ). Key
technique involves augmentation of BM levels, and the following section
generalizes these examples.

We remark that the optimization in (4) or (5) is not the only optimization
criterion that can be used. There are various criteria to define the
optimality of BM relativities, e.g. Denuit et al., (2007); Tan et al., (2015).
More importantly, BM relativities obtained from (4), (5), and Denuit et al.,
(2007); Tan et al., (2015) are known to create some systematic bias in the
prediction of premium. Such systematic bias is called the double-counting
problem, and we refer interested readers to Lemaire, (1995); Taylor, (1997);
Oh et al., 2020a  for the details of the double-counting problem and its
solution.

##  3 A modified BMS with augmented BM levels

In this section, the number of consecutive claim-free years for a policyholder
is taken into account in the transition rules. Assuming there are $z+1$ BM
levels, we let $L_{t}\in\\{0,1,\ldots,z\\}$ be the BM level at time
$t\in\\{0,1,\ldots\\}$ so that $L_{t}$ is the BM level applicable for the
$(t+1)$-th policy year, that is, from time $t$ to time $t+1$. Similar to
previous notation, for $t\in\\{1,2,\ldots\\}$ we use $N_{t}$ to denote the
number of claims in the $t$-th policy year.

###  3.1 Modified transition rules of $-1/+h/pen$

While the classical BMS usually adopts the so-called $-1/+h$ system for $h\leq
z$ as described in Section 2.2, we shall introduce an additional component,
namely “$pen$”, to this system. Specifically, under such a $-1/+h/pen$ system,
each reported claim increases the BM level by $h$ while $1+pen^{*}_{t}$
consecutive claim-free years will result in decrease of one BM level at time
$t+1$ when moving from the $(t+1)$-th policy year to the $(t+2)$-th. Here
$pen_{t}^{*}$ is defined as

$pen_{t}^{*}:=\min\\{pen,t\\},$ (6)

for $t\in\\{0,1,\ldots\\}$. This implies that the BM level at time $t=0$
starts with no penalty, and indeed a new policyholder can have his/her BM
level reduced by one every year as long as he/she maintains a no-claim record.
(This is also equivalent to letting $N_{0}=N_{-1}=\ldots=N_{-pen}=0$ and
saying that one requires $1+pen$ consecutive claim-free years to decrease BM
level by one.) Note that the classical BMS is retrieved from our extended
model by letting $pen=0$. We further assume that a new policyholder without
any driving history belongs to the BM level $\ell_{0}$ in the beginning. Then,
$L_{t}$ is mathematically represented as

$L_{t}:=\begin{cases}\min\\{L_{t-1}+hN_{t},z\\},&N_{t}>0,\\\
\max\\{L_{t-1}-1,0\\},&N_{t}=\ldots=N_{\max\\{t-pen^{*}_{t-1},1\\}}=0,\\\
L_{t-1},&\hbox{otherwise},\end{cases}$ (7)

for $t\in\\{1,2,\ldots\\}$, with $L_{0}=\ell_{0}$. It is also assumed that the
BM relativity ${\zeta}(\ell)$ is applied in the same manner as in the
classical $-1/+h$ system. That is, each BM level $\ell$ is bestowed with BM
relativity ${\zeta}(\ell)$. Compared to the classical $-1/+h$ system, an
additional $pen^{*}_{t}$ claim-free years are required to reduce BM level by
one, whereas the penalty of climbing $h$ BM levels per claim remains the same.
Such a system is introduced to motivate policyholders to drive carefully by
applying stricter rule of lowering the BM level (reward case) while keeping
the same rule of increasing the BM level (penalty case). This kind of the
transition rules are adopted in some countries such as Singapore and Korea.

While the Markov property in the transition probability appearing in certain
BMS is convenient for the analysis of BMS-related problems (e.g. Taylor,
(1997); Pitrebois et al., 2003b ; Denuit et al., (2007)), it is obvious that
with the current definition of $L_{t}$, the transition rules are not Markovian
as $L_{t}$ depends on not only $L_{t-1}$ but also the numbers of claims in the
past multiple periods. To resolve this issue, as explained in Lemaire, (1995)
and Pitrebois et al., 2003b , fictitious levels can be included to redefine
the state space of the BM levels. Subdividing some of the BM levels in order
to include the information of the number of claim-free years results in an
increase of the number of states compared to the classical case. More
precisely, for $\ell=h,\ldots,z$, the BM level $\ell$ is augmented into

$(\ell)_{0},(\ell)_{1},\ldots,(\ell)_{pen},$

where the subscript stands for the number of additional claim-free periods
(compared to the classical BMS) required to get rewarded, while
$\ell\in\\{0,\ldots,h-1\\}$ is just translated into $(\ell)_{0}$ without
augmentation.222No augmentation of the BM levels $0,\ldots,h-1$ is necessary
because these BM levels can only be reached as a result of reward from the
previous years. Hence, the original $(z+1)$ BM levels in the $-1/+h/pen$
system have been augmented to $[h+(z-h+1)\times(pen+1)]$ BM levels. Let us
denote the BM level at time $t$ under the augmented system as $L_{t}^{*}$.
Then, it is convenient to define the possible combinations of $(\ell)_{a}$ in
our proposed model (that is, the state space) as

$\mathcal{A}_{z,pen}:=\\{(\ell)_{0}\big{|}\ell=0,\ldots,h-1\\}\cup\\{(\ell)_{0},\ldots,(\ell)_{pen}\big{|}\ell=h,\ldots,z\\}.$
(8)

Furthermore, suppose that a policyholder who was at the BM level
$L_{t-1}^{*}=(\ell)_{a}\in\mathcal{A}_{z,pen}$ at time $t-1$ has reported at
time $t$ that $N_{t}$ accidents occurred during the $t$-th policy year. Then
the new BM level $L_{t}^{*}$ at time $t$ is determined as

$L_{t}^{*}:=\begin{cases}(\max\\{\ell-1,0\\})_{0},&\hbox{if}\quad
N_{t}=0\,\,\hbox{and}\,\,a=0,\\\ (\ell)_{a-1},&\hbox{if}\quad
N_{t}=0\,\,\hbox{and}\,\,a\neq 0,\\\
(\min\\{\ell+hN_{t},z\\})_{pen},&\hbox{if}\quad N_{t}>0,\\\ \end{cases}$ (9)

for $t\in\\{1,2,\ldots\\}$, where a new policyholder without driving history
starts with $L_{0}^{*}=(\ell_{0})_{0}$. The transitions of $L_{t}^{*}$ are now
Markovian, as $L_{t}^{*}$ depends on $L_{t-1}^{*}=(\ell)_{a}$ and the latest
claim number only. Also, we assume that the BM relativity in state
$(\ell)_{a}\in\mathcal{A}_{z,pen}$ under the augmented system defined via (9)
is the same as the one in $\ell\in\\{0,1,\ldots,z\\}$ for the original system
defined in (7) in the sense that the relativities depend on the BM level
$\ell$ but not the information $a$ which is artificially introduced to make
the transitions Markovian. In other words, under the proposed model one has
the relativities

$\zeta^{*}\left((\ell)_{a}\right):=\zeta(\ell),\qquad(\ell)_{a}\in\mathcal{A}_{z,pen}.$
(10)

We call the BMS with BM levels (8), transition rules (9) and relativities (10)
an augmented $-1/+h/pen$ system. The following example illustrates the
transition rules with a certain penalty period.

######  Example 1.

Let us consider the $-1/+2/2$ system with $z=20$. That is, the BM level is to
be increased by two per claim while having $1+pen^{*}_{t}$ consecutive claim-
free years is to be rewarded by a decrease of one BM level. We assume that a
new policyholder without driving history starts at BM level $L_{0}=10$ (so
$\ell=10$ at time 0). Given the BM level $L_{t}$ at time $t$ and the claim
frequency $N_{t+1}$ in the $(t+1)$-th year, the next year’s BM level $L_{t+1}$
according to (7) and alternatively $L^{\ast}_{t+1}$ according to (9) are
summarized in Table 1. (Note that, with $pen=2$, (6) implies that
$pen_{t}^{*}$ stays level at 2 once $t$ reaches 2.) The first few transitions
are explained as follows:

  1. i.

Transition from $t=0$ to $t=1$:
At time 0 one has $pen^{*}_{0}=0$ from (6), and therefore $1+pen^{*}_{0}=1$
claim-free year is required to reduce the BM level by one at time 1 (like the
classical BMS) and so we set $a=0$ at time 0. As it turns out that there is no
claim reported at $t=1$ (i.e. $N_{1}=0$), the BM level is reduced from
$L_{0}=10$ to $L_{1}=9$ (and thus $\ell=9$ at $t=1$). Now $pen^{*}_{1}=1$
according to (6), meaning that a total of $1+pen^{*}_{1}=2$ consecutive claim-
free years are needed to reduce one BM level next year at time 2. Since there
has already been one claim-free year, only one more claim-free year is needed
to earn such a reward (again like the classical BMS) and therefore one sets
$a=0$ at time 1.

  2. ii.

Transition from $t=1$ to $t=2$:
Since $N_{2}=0$ again, the condition of reward is satisfied and the BM level
moves down to $L_{2}=8$ (and $\ell=8$) at time $2$. With $pen^{*}_{2}=2$ at
time 2, it is known that $1+pen^{*}_{t}=3$ consecutive claim-free years are
needed to lower BM level by one. With two consecutive claim-free years already
in hand, we need only one more claim-free year to reduce BM level in the next
year and therefore $a=0$ at time 2.

  3. iii.

Transition from $t=2$ to $t=3$:
Since $N_{3}=2$ in the third year, the BM level moves up to 12 ($=8+2\times
2$) at time 3 (and so $\ell=12$). As $pen^{*}_{3}=2$ at time 3, one needs
$1+pen^{*}_{3}=3$ consecutive claim-free years to reduce BM level by one,
which means two claim-free years in additional to the one claim-free year
required in the classical BMS. Consequently, we have $a=2$ at $t=3$.

  4. iv.

Transition from $t=3$ to $t=4$:
Because $N_{4}=1$ in the forth year, the BM level increases to $L_{4}=14$
($=12+2\times 1$) at time 4. As there is a claim in this period, three
consecutive claim-free years are still needed to reduce BM level by one. So,
we have $a=2$ at $t=4$.

  5. v.

Transition from $t=4$ to $t=5$:
Although there is no claim in the fifth year as $N_{5}=0$, the condition of
having three consecutive claim-free years in order to enjoy a reward is not
satisfied, and therefore the BM level stays at the same level as last year so
that $L_{5}=14$. Then two more consecutive claim-free years are required to
lower the BM level by one. This is one additional year compared to the
classical $-1/+2$ system so $a=1$ at $t=5$.

$t$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
---|---|---|---|---|---|---|---|---|---|---
$N_{t+1}$ | 0 | 0 | 2 | 1 | 0 | 0 | 0 | 0 | 1 |
$L_{t}$ | 10 | 9 | 8 | 12 | 14 | 14 | 14 | 13 | 12 | 14
$pen_{t}^{*}$ | 0 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2
$L_{t}^{*}$ | $(10)_{0}$ | $(9)_{0}$ | $(8)_{0}$ | $(12)_{2}$ | $(14)_{2}$ | $(14)_{1}$ | $(14)_{0}$ | $(13)_{0}$ | $(12)_{0}$ | $(14)_{2}$
Table 1: Transition rules under a $-1/+2/2$ system

######  Example 2.

Suppose that we have the $-1/+2/1$ system with $z=7$. We consider Model 1
where a policyholder, say the $i$-th policyholder, has observed and unobserved
risk characteristics $\boldsymbol{X}_{i}=\boldsymbol{x}_{i}$ and
${\Theta}_{i}={\theta}_{i}$ respectively. Then, based on the augmented form of
BM levels defined in (8), one finds a total of $(2+(7-2+1)\times 2)=14$ BM
levels with the transition matrix $\boldsymbol{P}$ given by

$\blockarray{*{14}{c}l}&(0)_{0}(1)_{0}(2)_{0}(2)_{1}(3)_{0}(3)_{1}(4)_{0}(4)_{1}(5)_{0}(5)_{1}(6)_{0}(6)_{1}(7)_{0}(7)_{1}\\\
\block{c(*{13}{c}l)}(0)_{0}p_{0}00p_{1}000p_{2}000p_{3}01-p_{0}-p_{1}-p_{2}-p_{3}\\\
(1)_{0}p_{0}0000p_{1}000p_{2}0001-p_{0}-p_{1}-p_{2}\\\
(2)_{0}0p_{0}00000p_{1}000p_{2}01-p_{0}-p_{1}-p_{2}\\\
(2)_{1}00p_{0}0000p_{1}000p_{2}01-p_{0}-p_{1}-p_{2}\\\
(3)_{0}00p_{0}000000p_{1}0001-p_{0}-p_{1}\\\
(3)_{1}0000p_{0}0000p_{1}0001-p_{0}-p_{1}\\\
(4)_{0}0000p_{0}000000p_{1}01-p_{0}-p_{1}\\\
(4)_{1}000000p_{0}0000p_{1}01-p_{0}-p_{1}\\\
(5)_{0}000000p_{0}0000001-p_{0}\\\ (5)_{1}00000000p_{0}00001-p_{0}\\\
(6)_{0}00000000p_{0}00001-p_{0}\\\ (6)_{1}0000000000p_{0}001-p_{0}\\\
(7)_{0}0000000000p_{0}001-p_{0}\\\ (7)_{1}000000000000p_{0}1-p_{0}\\\ $

where $p_{n}$ in the matrix is defined by (see (23))

$p_{n}:=\mathbb{P}(N_{it}=n|\Theta_{i}=\theta_{i},\boldsymbol{X}_{i}=\boldsymbol{x}_{i}),\qquad
n=0,1,\ldots.$

For simplicity, the dependence of $p_{n}$ on $\theta_{i}$ and
$\boldsymbol{x}_{i}$ is suppressed. Similarly, if we assume Model 2 instead
and the $i$-th policyholder has observed and unobserved risk characteristics
given by
$(\boldsymbol{X}^{[1]}_{i},\boldsymbol{X}^{[2]}_{i})=(\boldsymbol{x}^{[1]}_{i},\boldsymbol{x}^{[2]}_{i})$
and $(\Theta^{[1]}_{i},\Theta^{[2]}_{i})=(\theta^{[1]}_{i},\theta^{[2]}_{i})$,
then $p_{n}$ in the matrix $\boldsymbol{P}$ is (see (25))

$p_{n}:=\mathbb{P}(N_{it}=n|\Theta^{[1]}_{i}=\theta^{[1]}_{i},\boldsymbol{X}^{[1]}_{i}=\boldsymbol{x}^{[1]}_{i}),\qquad
n=0,1,\ldots.$

###  3.2 Optimal relativities

In this section, under a $-1/+h/pen$ system, the optimal values of the
relativities $\zeta(\ell)$ for $\ell=0,1,\ldots,z$ are studied. Following the
classical BMS literature, we denote the stationary distribution of $L_{t}^{*}$
by $L^{*}$ for a randomly picked policyholder. Then, under the representation
in Model 1, one finds in an analogous manner to (3) that

$\mathbb{P}(L^{*}=(\ell)_{a})=\sum_{k=1}^{\mathcal{K}}w_{k}\int\pi_{(\ell)_{a}}^{*}(\lambda_{k}\theta,\psi)g(\theta){\rm
d}\theta,\qquad(\ell)_{a}\in\mathcal{A}_{z,pen},$ (11)

where $\pi_{(\ell)_{a}}^{*}(\lambda_{k}\theta,\psi)$ is the stationary
distribution for a policyholder to be in BM level $(\ell)_{a}$ given that
his/her a priori claim frequency is $\lambda_{k}$ and unobserved risk
characteristics are summarized in $\theta$. (Again, $\lambda_{k}$, $g(\cdot)$
and $\psi$ are replaced by $\lambda^{[1]}_{k}$, $g_{1}(\cdot)$ and
$\psi^{[1]}$ respectively if we consider Model 2 instead.) For example, the
14-dimensional row vector of stationary probabilities corresponding to Example
2, namely

$\boldsymbol{\pi}^{*}(\lambda_{k}\theta,\psi):=(\pi_{(0)_{0}}^{*}(\lambda_{k}\theta,\psi),\pi_{(1)_{0}}^{*}(\lambda_{k}\theta,\psi),\ldots,\pi_{(7)_{1}}^{*}(\lambda_{k}\theta,\psi)),$

can be obtained as the solution of

$\begin{cases}\boldsymbol{\pi}^{*}(\lambda_{k}\theta,\psi)=\boldsymbol{\pi}^{*}(\lambda_{k}\theta,\psi)\boldsymbol{P},\\\
\boldsymbol{\pi}^{*}(\lambda_{k}\theta,\psi)\boldsymbol{e}_{14}=1,\\\
\end{cases}$

where $\boldsymbol{P}$ is the transition matrix in Example 2 (calculated with
$\boldsymbol{x}_{k}$ and $\theta$ in place of $\boldsymbol{x}_{i}$ and
$\theta_{i}$ respectively), and $\boldsymbol{e}_{14}$ is a 14-dimensional
column vector of ones.

Under the augmented system, we consider the optimal relativities as the
solution of the optimization problem

$(\tilde{\zeta}(0),\ldots,\tilde{\zeta}(z)):=\operatorname*{arg\,min}_{({\zeta}(0),\ldots,{\zeta}(z))\in\mathbb{R}^{{z+1}}}\mathbb{E}[(\Lambda\Theta-\Lambda{\zeta}^{*}(L^{*}))^{2}]$
(12)

resembling (4) where we aim to predict the frequency under the frequency-only
Model 1. On the other hand, when we are interested in the prediction of the
aggregate claim, like (5) the optimal relativities are given by the solution
of the optimization problem

$(\tilde{\zeta}(0),\ldots,\tilde{\zeta}(z)):=\operatorname*{arg\,min}_{({\zeta}(0),\ldots,{\zeta}(z))\in\mathbb{R}^{{z+1}}}\mathbb{E}[(\Lambda^{[1]}\Lambda^{[2]}\Theta^{[1]}\Theta^{[2]}-\Lambda^{[1]}\Lambda^{[2]}{\zeta}^{*}(L^{*}))^{2}]$
(13)

under the frequency-severity Model 2. Recall that, by definition,
$\zeta^{*}:\mathcal{A}_{z,pen}\mapsto\mathbb{R}$ in (12) or (13) is completely
characterized by $\zeta(\ell)$ for $\ell=0,\ldots,z$ (see (10)).

######  Proposition 1.

Consider the $-1/+h/pen$ system under Model 1. The optimal relativities
defined as the solution of the optimization problem (12) are given by

$\tilde{\zeta}(\ell):=\frac{\mathbb{E}[\Lambda^{2}\Theta|L^{*}=(\ell)_{0}]}{\mathbb{E}[\Lambda^{2}|L^{*}=(\ell)_{0}]},\qquad\ell=0,\ldots,{h-1},$
(14)

and

$\tilde{\zeta}(\ell):=\frac{\sum_{a=0}^{pen}\mathbb{E}[\Lambda^{2}\Theta|L^{*}=(\ell)_{a}]\mathbb{P}(L^{*}=(\ell)_{a})}{\sum_{a=0}^{pen}\mathbb{E}[\Lambda^{2}|L^{*}=(\ell)_{a}]\mathbb{P}(L^{*}=(\ell)_{a})},\qquad\ell=h,\ldots,{z};a=0,\ldots,pen,$
(15)

where the numerator and denominator can be calculated based on

$\mathbb{E}[\Lambda^{2}\Theta|L^{*}=(\ell)_{a}]=\frac{\sum_{k=1}^{\mathcal{K}}w_{k}\lambda_{k}^{2}\int\theta\pi_{(\ell)_{a}}^{*}(\lambda_{k}\theta,\psi)g(\theta){\rm
d}\theta}{\mathbb{P}(L^{*}=(\ell)_{a})}$ (16)

and

$\mathbb{E}[\Lambda^{2}|L^{*}=(\ell)_{a}]=\frac{\sum_{k=1}^{\mathcal{K}}w_{k}\lambda_{k}^{2}\int\pi_{(\ell)_{a}}^{*}(\lambda_{k}\theta,\psi)g(\theta){\rm
d}\theta}{\mathbb{P}(L^{*}=(\ell)_{a})},$ (17)

respectively. Note that when $pen=0$ the optimal relativity
$\tilde{\zeta}(\ell)$ reduces to that in Tan et al., (2015) given in Lemma 1.

###### Proof.

Note that the objective function in (12) can be written as

$\displaystyle\mathbb{E}[(\Lambda\Theta-\Lambda{\zeta}^{*}(L^{*}))^{2}]=$
$\displaystyle\sum_{(\ell)_{a}\in\mathcal{A}_{z,pen}}\mathbb{E}[(\Lambda\Theta-\Lambda{\zeta}^{*}((\ell)_{a}))^{2}|L^{*}=(\ell)_{a}]\mathbb{P}(L^{*}=(\ell)_{a})$
$\displaystyle=$
$\displaystyle\sum_{(\ell)_{a}\in\mathcal{A}_{z,pen}}\mathbb{E}[(\Lambda\Theta-\Lambda{\zeta}(\ell))^{2}|L^{*}=(\ell)_{a}]\mathbb{P}(L^{*}=(\ell)_{a}),$
(18)

where the last line follows from (10). We observe from (8) that:

  * 1.

for each $\ell=0,\ldots,h-1$, the relativity $\zeta(\ell)$ is shared by only
one BM level $(\ell)_{0}$; and

  * 2.

for each $\ell=h,\ldots,z$, the same relativity $\zeta(\ell)$ is shared by the
augmented BM levels $(\ell)_{0},\ldots,(\ell)_{pen}$.

Consequently, (3.2) becomes

$\displaystyle\mathbb{E}[(\Lambda\Theta-\Lambda{\zeta}^{*}(L^{*}))^{2}]=$
$\displaystyle~{}\sum_{\ell=0}^{h-1}\mathbb{E}[(\Lambda\Theta-\Lambda{\zeta}(\ell))^{2}|L^{*}=(\ell)_{0}]\mathbb{P}(L^{*}=(\ell)_{0})$
$\displaystyle+\sum_{\ell=h}^{z}\sum_{a=0}^{pen}\mathbb{E}[(\Lambda\Theta-\Lambda{\zeta}(\ell))^{2}|L^{*}=(\ell)_{a}]\mathbb{P}(L^{*}=(\ell)_{a}).$
(19)

For each fixed $\ell=0,\ldots,h-1$, differentiation of (3.2) with respect to
${\zeta}(\ell)$ for optimization yields

$\mathbb{E}[-2\Lambda(\Lambda\Theta-\Lambda\zeta(\ell))|L^{*}=(\ell)_{0}]\mathbb{P}(L^{*}=(\ell)_{0})=0,$

from which (14) follows. On the other hand, when $\ell=h,\ldots,{z}$, taking
derivative with respect to ${\zeta}(\ell)$ in (3.2) leads to

$\sum_{a=0}^{pen}\mathbb{E}[-2\Lambda(\Lambda\Theta-\Lambda\zeta(\ell))|L^{*}=(\ell)_{a}]\mathbb{P}(L^{*}=(\ell)_{a})=0,$

proving (15).

Here, the numerators of (14) and (15) can be calculated using

$\displaystyle\mathbb{E}[\Lambda^{2}\Theta|L^{*}=(\ell)_{a}]=$
$\displaystyle~{}\frac{1}{\mathbb{P}(L^{*}=(\ell)_{a})}\mathbb{E}[\Lambda^{2}\Theta
I(L^{*}=(\ell)_{a})]$ $\displaystyle=$
$\displaystyle~{}\frac{1}{\mathbb{P}(L^{*}=(\ell)_{a})}\sum_{k=1}^{\mathcal{K}}\int\lambda_{k}^{2}\theta\mathbb{P}(\Theta\in{\rm
d}\theta,\Lambda=\lambda_{k},L^{*}=(\ell)_{a})$ $\displaystyle=$
$\displaystyle~{}\frac{1}{\mathbb{P}(L^{*}=(\ell)_{a})}\sum_{k=1}^{\mathcal{K}}\int\lambda_{k}^{2}\theta\mathbb{P}(L^{*}=(\ell)_{a}|\Theta\in{\rm
d}\theta,\Lambda=\lambda_{k})\mathbb{P}(\Theta\in{\rm
d}\theta,\Lambda=\lambda_{k}).$ (20)

Since the a priori $\boldsymbol{X}$ and the a posteriori $\Theta$ are
independent and $\Lambda=\eta^{-1}(\boldsymbol{X}\boldsymbol{\beta})$, one has
that $\mathbb{P}(\Theta\in{\rm
d}\theta,\Lambda=\lambda_{k})=\mathbb{P}(\Lambda=\lambda_{k})\mathbb{P}(\Theta\in{\rm
d}\theta)=w_{k}g(\theta){\rm d}\theta$. Using this together with the fact that
$\mathbb{P}(L^{*}=(\ell)_{a}|\Theta\in{\rm
d}\theta,\Lambda=\lambda_{k})=\pi_{(\ell)_{a}}^{*}(\lambda_{k}\theta,\psi)$
confirms that (3.2) reduces to (16). The expectation (17) can be obtained in
almost identical manner and the details are omitted. ∎

######  Proposition 2.

Consider the $-1/+h/pen$ system under Model 2. The optimal relativities
defined as the solution of the optimization problem (13) are given by

$\tilde{\zeta}(\ell):=\frac{\mathbb{E}[(\Lambda^{[1]}\Lambda^{[2]})^{2}\Theta^{[1]}\Theta^{[2]}|L^{*}=(\ell)_{0}]}{\mathbb{E}[(\Lambda^{[1]}\Lambda^{[2]})^{2}|L^{*}=(\ell)_{0}]},\qquad\ell=0,\ldots,{h-1},$

and

$\tilde{\zeta}(\ell):=\frac{\sum_{a=0}^{pen}\mathbb{E}[(\Lambda^{[1]}\Lambda^{[2]})^{2}\Theta^{[1]}\Theta^{[2]}|L^{*}=(\ell)_{a}]\mathbb{P}(L^{*}=(\ell)_{a})}{\sum_{a=0}^{pen}\mathbb{E}[(\Lambda^{[1]}\Lambda^{[2]})^{2}|L^{*}=(\ell)_{a}]\mathbb{P}(L^{*}=(\ell)_{a})},\qquad\ell=h,\ldots,{z};a=0,\ldots,pen,$

where the numerator and denominator can be calculated based on

$\mathbb{E}[(\Lambda^{[1]}\Lambda^{[2]})^{2}\Theta^{[1]}\Theta^{[2]}|L^{*}=(\ell)_{a}]=\frac{\sum_{k=1}^{\mathcal{K}}w_{k}(\lambda_{k}^{[1]}\lambda_{k}^{[2]})^{2}\int\int\theta^{[1]}\theta^{[2]}\pi_{(\ell)_{a}}^{*}(\lambda_{k}^{[1]}\theta^{[1]},\psi^{[1]})h(\theta^{[1]},\theta^{[2]}){\rm
d}\theta^{[1]}{\rm d}\theta^{[2]}}{\mathbb{P}(L^{*}=(\ell)_{a})}$

and

$\mathbb{E}[(\Lambda^{[1]}\Lambda^{[2]})^{2}|L^{*}=(\ell)_{a}]=\frac{\sum_{k=1}^{\mathcal{K}}w_{k}(\lambda_{k}^{[1]}\lambda_{k}^{[2]})^{2}\int\pi_{(\ell)_{a}}^{*}(\lambda_{k}^{[1]}\theta^{[1]},\psi^{[1]})g_{1}(\theta^{[1]}){\rm
d}\theta^{[1]}}{\mathbb{P}(L^{*}=(\ell)_{a})},$

respectively. Note that when $pen=0$ the optimal relativity
$\tilde{\zeta}(\ell)$ reduces to that in Oh et al., 2020b  given in Lemma 2.

###### Proof.

The proof of Proposition 2 is similar to that of Proposition 1, and thus we
omit the details. ∎

###  3.3 Numerical Analysis

Here a simple example is provided to examine the effect of the period of
penalty on the optimal BM relativity $\tilde{\zeta}(\ell)$ and stationary
probability $\mathbb{P}(L=\ell)$. First, it is convenient to introduce the
relation

$\mathbb{P}(L=\ell)=\sum_{\\{a|(\ell)_{a}\in\mathcal{A}_{z,pen}\\}}\mathbb{P}(L^{*}=(\ell)_{a}),$
(21)

which follows from the definition of $L^{*}$. The performances of our modified
BM transition rules under a $-1/+1/pen$ system as well as a $-1/+2/pen$ system
for $pen=0,1,2,3$ are compared by computing the hypothetical mean square error
(HMSE). For Model 1 (the frequency random effect model) with the set of BM
relativities $\boldsymbol{\zeta}=\\{\zeta(\ell)\\}_{\ell=0}^{z}$, the HMSE is
expressed as

$\displaystyle{\rm HMSE}(\boldsymbol{\zeta})$
$\displaystyle:=\mathbb{E}[(\Lambda\Theta-\Lambda\zeta^{*}(L^{*}))^{2}]$
$\displaystyle=\sum_{k=1}^{\mathcal{K}}\sum_{(\ell)_{a}\in\mathcal{A}_{z,pen}}w_{k}\int(\lambda_{k}\theta-\lambda_{k}\zeta(\ell))^{2}\pi_{(\ell)_{a}}^{*}(\lambda_{k}\theta,\psi)g(\theta){\rm
d}\theta.$ (22)

If $\boldsymbol{\zeta}$ is replaced by the vector of optimal relativities
$\tilde{\boldsymbol{\zeta}}=\\{\tilde{\zeta}(\ell)\\}_{\ell=0}^{z}$ calculated
using Proposition 1, then ${\rm HMSE}(\tilde{\boldsymbol{\zeta}})$ is the
minimized value corresponding to the right-hand side of the optimization
problem (12).

Similarly, for Model 2 with the set of BM relativities
$\boldsymbol{\zeta}=\\{\zeta(\ell)\\}_{\ell=0}^{z}$, following the logic in Oh
et al., 2020b , the HMSE is expressed as

$\displaystyle{\rm HMSE}(\boldsymbol{\zeta})$
$\displaystyle:=\mathbb{E}[(\Lambda^{[1]}\Lambda^{[2]}\Theta^{[1]}\Theta^{[2]}-\Lambda^{[1]}\Lambda^{[2]}\zeta^{*}(L^{*}))^{2}]$
$\displaystyle=\sum_{k=1}^{\mathcal{K}}\sum_{(\ell)_{a}\in\mathcal{A}_{z,pen}}w_{k}\int\int\big{(}\lambda_{k}^{[1]}\lambda_{k}^{[2]}\theta^{[1]}\theta^{[2]}-\lambda_{k}^{[1]}\lambda_{k}^{[2]}\zeta(\ell)\big{)}^{2}\pi_{(\ell)_{a}}^{*}(\lambda_{k}\theta,\psi^{[1]})h(\theta^{[1]},\theta^{[2]}){\rm
d}\theta^{[1]}{\rm d}\theta^{[2]}.$

Again, ${\rm HMSE}(\tilde{\boldsymbol{\zeta}})$ is the minimum on the right-
hand side of the optimization problem (13) when
$\tilde{\boldsymbol{\zeta}}=\\{\tilde{\zeta}(\ell)\\}_{\ell=0}^{z}$ calculated
using Proposition 2.

######  Example 3.

Under the frequency model with random effect described as Model 1, it is
assumed that there is only one risk class, where the frequency $N$ follows a
Poisson distribution with a priori frequency $\lambda=\lambda_{0}$ (so that
the distribution function $F$ can be put in the form of (23)) and the random
effect $\Theta$ in (24) has a Lognormal distribution with mean 1 and
$\sigma^{2}=0.99$, that is,

$\displaystyle N|(\Lambda=\lambda,\Theta=\theta)$ $\displaystyle\sim{\rm
Poisson}(\lambda\theta)\qquad{\rm with}\quad\lambda=\lambda_{0},$
$\displaystyle\Theta$ $\displaystyle\sim{\rm
Lognormal}(-\sigma^{2}/2,\sigma^{2})\qquad{\rm with}\quad\sigma^{2}=0.99.$

With $z=9$ and $pen=0,1,2,3$, the stationary probabilities
$\mathbb{P}(L=\ell)$ and the optimal BM relativities $\tilde{\zeta}(\ell)$ for
all $\ell$’s and the resulting HMSE (under the optimal relativities) are
calculated under a $-1/+1/pen$ system and a $-1/+2/pen$ system. The results
when $\lambda_{0}=0.05$ are first summarized in Tables 2(a) and (b).

In obtaining the tables, we first construct the transition probability matrix
according to the transition rules of the BMS and the claim frequency
distribution with the assumed parameters (see Example 2) and obtain the
stationary probability $\pi_{(\ell)_{a}}^{*}$ as explained at the beginning of
Section 3.2. Then, $\mathbb{P}(L=\ell)$ can be calculated from (11) and (21).
Note that integration with respect to $\theta$ under the distributional
assumption on the density $g(\theta)$ of $\Theta$ is needed in (11). While
explicit evaluation of such one dimensional integral is not possible in
general, numerical integration can be easily implemented in most computing
software. In turn, the optimal relativities
$\tilde{\boldsymbol{\zeta}}=\\{\tilde{\zeta}(\ell)\\}_{\ell=0}^{z}$ can be
obtained from Proposition 1, where numerical integration is again performed
based on (16) and (17) (but the integral appearing in (17) is the same as that
in (11)). Calculation of HMSE can be similarly done via (3.3) once one has
calculated the optimal relativities $\tilde{\boldsymbol{\zeta}}$.

We first look at the $-1/+1/pen$ system in Table 2(a). For each fixed value of
$pen=0,1,2,3$, most of the population is in the lowest BM level 0 (as
$\mathbb{P}(L=0)$ is at least $82\%$), which can be explained by the
relatively low mean claim frequency of $\lambda_{0}=0.05$. By inspecting
$\mathbb{P}(L=\ell)$ for $pen=0,1,2,3$, we observe that an increase in $pen$
tends to move some of the population toward higher BM levels, leading to a
diversification of the stationary BM levels. This is because a higher $pen$
means that more claim-free years are needed to reduce the BM level by one,
rendering it more difficult for drivers at a high BM level to move downward.
As mentioned in Proposition 1, the optimal relativities under $pen=0$ are the
same as those in Tan et al., (2015). As $pen$ increases, it is important to
observe that the optimal relativity $\tilde{\zeta}(\ell)$ for each fixed BM
level $\ell$ decreases. Although a lower BM relativity means that a driver
occupying a given BM level pays less expensive premium in a $-1/+1/pen$ system
with a higher $pen$, the insurer’s premium income is in turn compensated by an
increased portion of drivers occupying higher BM levels as $pen$ increases. A
higher $pen$ also results in an improvement of the HMSE in this example.
Similar patterns can be found in $-1/+2/pen$ system as shown in Table 2(b).
However, for fixed $pen=1,2,3$, the optimal relativity $\tilde{\zeta}(\ell)$
in Table 2(b) is not necessarily increasing in $\ell$ especially for low BM
levels. For example, it is observed that $\tilde{\zeta}(2)$ is slightly
smaller than $\tilde{\zeta}(1)$. With $h=2$, a possible explanation for
$\tilde{\zeta}(2)<\tilde{\zeta}(1)$ is that certain policyholders occupying BM
level 2 could be those who were in the lowest BM level 0 (and had one claim)
in the previous year, and these are still good drivers. On the other hand,
those in BM level 1 could not have come directly from BM level 0. It is
interesting to note that, although $\tilde{\zeta}(\ell)$ is not always
increasing in $\ell$ in Table 2(b), reporting $n\geq 1$ claims moves a
policyholder upward by $2n$ BM levels and this always leads to an increase in
$\tilde{\zeta}(\ell)$. Nevertheless, potential problems arise when a
policyholder moves down from BM level 2 to BM level 1 after a number of claim-
free years and then suffers an increase in the optimal relativity (and hence
premium). For future work, constrained optimization may be considered by
requiring $\tilde{\zeta}(\ell)$ to be increasing in $\ell$, but it is unlikely
that this will lead to optimal relativities that admit expressions as explicit
as those in Propositions 1 and 2.

When $\lambda_{0}=1$, we consider BM levels up to $z=14$ because the
assumption of a higher mean claim frequency will result in a larger proportion
of policyholders occupying higher BM levels. The $-1/+1/pen$ system in Table
2(c) shows some similar features: as $pen$ increases, $\tilde{\zeta}(\ell)$
decreases and the distribution (at equilibrium) of policyholders shifts from
lower BM levels to higher ones. However, since a significant fraction of the
population is already in the highest BM level 14 when $pen=0$, increasing
$pen$ anti-diversifies the stationary distribution of BM levels. We observe
that HMSE is increased with higher $pen$.

We also notice one interesting phenomenon when comparing the cases where the
claim frequencies are of different magnitude. In Tables 2(a) and (b) for which
$\lambda_{0}=0.05$, the optimal relativity $\tilde{\zeta}(0)$ for BM level 0
is still at least 69% while that for the highest BM level can be as high as
17. An intuitive explanation is that, given the low claim frequency
$\lambda_{0}=0.05$, it is not unusual that drivers do not file claims. As a
result, claim-free drivers (those occupying the lowest BM level) are not
necessary much better drivers than others and therefore they are not rewarded
with much premium discount. On the other hand, drivers occupying the highest
BM level must have reported some claims in the past years, and they are more
likely to be worse drivers with more risky unobserved risk characteristics
(that is, higher $\Theta$) and thus more severely penalized. Moving to Table
2(c) for which $\lambda_{0}=1$, it becomes much more common for drivers to
report claims and occupy higher BM levels, so they are not penalized too much
with the highest BM relativity $\tilde{\zeta}(14)$ around 2. But it is those
at lower BM levels who have claim-free years that should be regarded as
significantly better drivers, and they are given great discount with the BM
relativity $\tilde{\zeta}(0)$ no larger than 24.6%.

From this example, we conclude that while the extension of the classical
$-1/+h$ system to the $-1/+h/pen$ system provides room for improvement of the
prediction power via lowering the HMSE, such an improvement is not always
guaranteed.

######  Example 4.

Under the collective risk model with random effect described as Model 2, we
assume one risk class only, where the distribution function $F_{1}$ in (25)
for the frequency part follows a Poisson distribution with
$\lambda^{[1]}=\lambda_{0}^{[1]}$ and the distribution function $F_{2}$ in
(27) for the severity part follows a Gamma distribution with
$\lambda^{[2]}=\lambda_{0}^{[2]}$ and $\psi^{[2]}=\psi_{0}^{[2]}$. The
marginal distributions of the random effects $(\Theta^{[1]},\Theta^{[2]})$ are
specified as

$\begin{cases}\Theta^{[1]}~{}\sim&{\rm
Lognormal}(-\sigma_{1}^{2}/2,\sigma_{1}^{2})\qquad{\rm
with}\quad\sigma_{1}^{2}=0.99,\\\ \Theta^{[2]}~{}\sim&{\rm
Lognormal}(-\sigma_{2}^{2}/2,\sigma_{2}^{2})\qquad{\rm
with}\quad\sigma_{2}^{2}=0.29,\end{cases}$

with distribution functions $G_{1}$ and $G_{2}$ respectively, and the
dependence is described by a Gaussian copula $C$ with correlation coefficient
$\rho=-0.45$ (so that the joint distribution function is given by
$H=C(G_{1},G_{2})$ in (28)). With $pen=0,1,2,3$, we have calculated the values
of the stationary probabilities $\mathbb{P}(L=\ell)$ and the optimal BM
relativities $\tilde{\zeta}(\ell)$ for all $\ell$’s together with the HMSE
under a $-1/+1/pen$ system for the cases $z=9$ and $\lambda_{0}^{[1]}=0.05$
(Table 3(a)) as well as $z=14$ and $\lambda_{0}^{[1]}=1$ (Table 3(c)) and
under a $-1/+2/pen$ system for the case $z=9$ and $\lambda_{0}^{[1]}=0.05$
(Table 3(b)). In constructing Table 3, the parameters
$\lambda_{0}^{[2]}=\exp(8.00)$, and $\psi_{0}^{[2]}=1/0.670$ are kept fixed.
Note that the values of HMSE in Table 3 are much higher than those in Table 2
concerning Example 3, and this is simply because of the fact that the HMSE in
the present example measures the error concerning the aggregate claim (and the
individual severity has large magnitude as $\lambda_{0}^{[2]}=\exp(8.00)$)
while the HMSE in Example 3 is about the error in the claim number. Moreover,
some of the optimal relativities in Table 3 are quite different from those in
Table 2, showing the importance of taking into account the dependence between
claim frequency and claim severity for modeling purposes. Nonetheless, similar
patterns to those in Table 2 are observed in Table 3. Note also that the
stationary probabilities in Table 3 are identical to their counterparts in
Table 2 because the probabilities only depend on the distribution of the claim
frequency.

Table 2: (Example 3) Distribution of $L$ and optimal relativities under the
frequency-only Model 1

(a) $-1/+1/pen$ systems with various $pen$ for $\lambda_{0}=0.05$ |
---|---
| $pen$ | 0 |  | 1 |  | 2 |  | 3
Level $\ell$ |  | $\tilde{\zeta}(\ell)$ | $\mathbb{P}(L=\ell)$ |  | $\tilde{\zeta}(\ell)$ | $\mathbb{P}(L=\ell)$ |  | $\tilde{\zeta}(\ell)$ | $\mathbb{P}(L=\ell)$ |  | $\tilde{\zeta}(\ell)$ | $\mathbb{P}(L=\ell)$
9 |  | 17.096 | 0.001 |  | 12.635 | 0.002 |  | 10.288 | 0.004 |  | 8.801 | 0.008
8 |  | 13.947 | 0.000 |  | 9.626 | 0.001 |  | 7.468 | 0.002 |  | 6.151 | 0.002
7 |  | 11.978 | 0.000 |  | 8.413 | 0.001 |  | 6.597 | 0.002 |  | 5.477 | 0.002
6 |  | 10.384 | 0.000 |  | 7.331 | 0.001 |  | 5.786 | 0.002 |  | 4.832 | 0.003
5 |  | 9.408 | 0.001 |  | 6.282 | 0.001 |  | 4.985 | 0.003 |  | 4.187 | 0.004
4 |  | 7.304 | 0.001 |  | 5.197 | 0.002 |  | 4.159 | 0.004 |  | 3.520 | 0.006
3 |  | 5.552 | 0.002 |  | 4.035 | 0.005 |  | 3.284 | 0.008 |  | 2.815 | 0.012
2 |  | 3.676 | 0.007 |  | 2.816 | 0.015 |  | 2.367 | 0.023 |  | 2.076 | 0.030
1 |  | 2.003 | 0.044 |  | 1.676 | 0.073 |  | 1.483 | 0.095 |  | 1.348 | 0.111
0 |  | 0.887 | 0.944 |  | 0.826 | 0.898 |  | 0.779 | 0.858 |  | 0.740 | 0.821
HMSE |  | 0.00287 |  | 0.00249 |  | 0.00227 |  | 0.00213

(b) $-1/+2/pen$ systems with various $pen$ for $\lambda_{0}=0.05$ |
---|---
| $pen$ | 0 |  | 1 |  | 2 |  | 3
Level $\ell$ |  | $\tilde{\zeta}(\ell)$ | $\mathbb{P}(L=\ell)$ |  | $\tilde{\zeta}(\ell)$ | $\mathbb{P}(L=\ell)$ |  | $\tilde{\zeta}(\ell)$ | $\mathbb{P}(L=\ell)$ |  | $\tilde{\zeta}(\ell)$ | $\mathbb{P}(L=\ell)$
9 |  | 10.288 | 0.002 |  | 8.105 | 0.007 |  | 6.858 | 0.013 |  | 6.030 | 0.020
8 |  | 8.106 | 0.002 |  | 5.793 | 0.004 |  | 4.567 | 0.005 |  | 3.807 | 0.007
7 |  | 6.755 | 0.002 |  | 5.101 | 0.004 |  | 4.209 | 0.005 |  | 3.633 | 0.006
6 |  | 5.410 | 0.003 |  | 4.006 | 0.005 |  | 3.263 | 0.008 |  | 2.796 | 0.011
5 |  | 4.638 | 0.003 |  | 3.674 | 0.005 |  | 3.135 | 0.007 |  | 2.772 | 0.008
4 |  | 3.349 | 0.007 |  | 2.599 | 0.014 |  | 2.205 | 0.020 |  | 1.950 | 0.027
3 |  | 2.973 | 0.009 |  | 2.497 | 0.012 |  | 2.206 | 0.013 |  | 2.000 | 0.014
2 |  | 1.779 | 0.041 |  | 1.508 | 0.066 |  | 1.351 | 0.086 |  | 1.241 | 0.101
1 |  | 1.677 | 0.038 |  | 1.512 | 0.034 |  | 1.394 | 0.030 |  | 1.303 | 0.028
0 |  | 0.811 | 0.892 |  | 0.762 | 0.850 |  | 0.723 | 0.812 |  | 0.691 | 0.778
HMSE |  | 0.00260 |  | 0.00244 |  | 0.00235 |  | 0.00229

(c) $-1/+1/pen$ systems with various $pen$ for $\lambda_{0}=1$ |
---|---
| $pen$ | 0 |  | 1 |  | 2 |  | 3
Level $\ell$ |  | $\tilde{\zeta}(\ell)$ | $\mathbb{P}(L=\ell)$ |  | $\tilde{\zeta}(\ell)$ | $\mathbb{P}(L=\ell)$ |  | $\tilde{\zeta}(\ell)$ | $\mathbb{P}(L=\ell)$ |  | $\tilde{\zeta}(\ell)$ | $\mathbb{P}(L=\ell)$
14 |  | 2.092 | 0.311 |  | 1.713 | 0.463 |  | 1.528 | 0.562 |  | 1.416 | 0.634
13 |  | 1.147 | 0.089 |  | 0.791 | 0.060 |  | 0.618 | 0.045 |  | 0.512 | 0.035
12 |  | 0.914 | 0.044 |  | 0.677 | 0.036 |  | 0.546 | 0.030 |  | 0.462 | 0.025
11 |  | 0.794 | 0.027 |  | 0.602 | 0.025 |  | 0.494 | 0.022 |  | 0.422 | 0.020
10 |  | 0.718 | 0.020 |  | 0.548 | 0.019 |  | 0.453 | 0.018 |  | 0.390 | 0.016
9 |  | 0.662 | 0.016 |  | 0.505 | 0.016 |  | 0.419 | 0.015 |  | 0.362 | 0.014
8 |  | 0.618 | 0.014 |  | 0.470 | 0.015 |  | 0.390 | 0.014 |  | 0.337 | 0.013
7 |  | 0.579 | 0.014 |  | 0.439 | 0.014 |  | 0.363 | 0.014 |  | 0.314 | 0.013
6 |  | 0.543 | 0.014 |  | 0.410 | 0.015 |  | 0.339 | 0.014 |  | 0.293 | 0.013
5 |  | 0.508 | 0.016 |  | 0.382 | 0.016 |  | 0.315 | 0.016 |  | 0.273 | 0.014
4 |  | 0.470 | 0.020 |  | 0.353 | 0.020 |  | 0.291 | 0.018 |  | 0.252 | 0.017
3 |  | 0.429 | 0.027 |  | 0.322 | 0.026 |  | 0.266 | 0.024 |  | 0.230 | 0.021
2 |  | 0.380 | 0.042 |  | 0.286 | 0.038 |  | 0.238 | 0.033 |  | 0.207 | 0.029
1 |  | 0.321 | 0.078 |  | 0.246 | 0.065 |  | 0.206 | 0.053 |  | 0.180 | 0.044
0 |  | 0.246 | 0.269 |  | 0.197 | 0.172 |  | 0.169 | 0.122 |  | 0.150 | 0.091
HMSE |  | 1.08320 |  | 1.23048 |  | 1.32241 |  | 1.38605

Table 3: (Example 4) Distribution of $L$ and optimal relativities under the
frequency-severity Model 2 with dependence

(a) $-1/+1/pen$ systems with various $pen$ for $\lambda_{0}^{[1]}=0.05$ |
---|---
| $pen$ | 0 |  | 1 |  | 2 |  | 3
Level $\ell$ |  | $\tilde{\zeta}(\ell)$ | $\mathbb{P}(L=\ell)$ |  | $\tilde{\zeta}(\ell)$ | $\mathbb{P}(L=\ell)$ |  | $\tilde{\zeta}(\ell)$ | $\mathbb{P}(L=\ell)$ |  | $\tilde{\zeta}(\ell)$ | $\mathbb{P}(L=\ell)$
9 |  | 7.217 | 0.001 |  | 5.749 | 0.002 |  | 4.917 | 0.004 |  | 4.366 | 0.008
8 |  | 6.246 | 0.000 |  | 4.715 | 0.001 |  | 3.889 | 0.002 |  | 3.357 | 0.002
7 |  | 5.573 | 0.000 |  | 4.259 | 0.001 |  | 3.540 | 0.002 |  | 3.073 | 0.002
6 |  | 5.000 | 0.000 |  | 3.834 | 0.001 |  | 3.202 | 0.002 |  | 2.792 | 0.003
5 |  | 4.695 | 0.001 |  | 3.404 | 0.001 |  | 2.854 | 0.003 |  | 2.500 | 0.004
4 |  | 3.805 | 0.001 |  | 2.936 | 0.002 |  | 2.479 | 0.004 |  | 2.184 | 0.006
3 |  | 3.065 | 0.002 |  | 2.407 | 0.005 |  | 2.060 | 0.008 |  | 1.834 | 0.012
2 |  | 2.210 | 0.007 |  | 1.812 | 0.015 |  | 1.591 | 0.023 |  | 1.443 | 0.030
1 |  | 1.369 | 0.044 |  | 1.203 | 0.073 |  | 1.101 | 0.095 |  | 1.026 | 0.111
0 |  | 0.727 | 0.944 |  | 0.692 | 0.898 |  | 0.665 | 0.858 |  | 0.641 | 0.821
HMSE |  | 14179.63 |  | 13189.89 |  | 12525.65 |  | 12053.38

(b) $-1/+2/pen$ systems with various $pen$ for $\lambda_{0}^{[1]}=0.05$ |
---|---
| $pen$ | 0 |  | 1 |  | 2 |  | 3
Level $\ell$ |  | $\tilde{\zeta}(\ell)$ | $\mathbb{P}(L=\ell)$ |  | $\tilde{\zeta}(\ell)$ | $\mathbb{P}(L=\ell)$ |  | $\tilde{\zeta}(\ell)$ | $\mathbb{P}(L=\ell)$ |  | $\tilde{\zeta}(\ell)$ | $\mathbb{P}(L=\ell)$
9 |  | 4.876 | 0.002 |  | 4.069 | 0.007 |  | 3.582 | 0.013 |  | 3.247 | 0.020
8 |  | 4.093 | 0.002 |  | 3.170 | 0.004 |  | 2.647 | 0.005 |  | 2.307 | 0.007
7 |  | 3.568 | 0.002 |  | 2.882 | 0.004 |  | 2.491 | 0.005 |  | 2.229 | 0.006
6 |  | 3.005 | 0.003 |  | 2.389 | 0.005 |  | 2.045 | 0.008 |  | 1.819 | 0.011
5 |  | 2.672 | 0.003 |  | 2.239 | 0.005 |  | 1.985 | 0.007 |  | 1.809 | 0.008
4 |  | 2.067 | 0.007 |  | 1.706 | 0.014 |  | 1.509 | 0.020 |  | 1.376 | 0.027
3 |  | 1.887 | 0.009 |  | 1.656 | 0.012 |  | 1.510 | 0.013 |  | 1.403 | 0.014
2 |  | 1.259 | 0.041 |  | 1.115 | 0.066 |  | 1.029 | 0.086 |  | 0.966 | 0.101
1 |  | 1.205 | 0.038 |  | 1.117 | 0.034 |  | 1.053 | 0.030 |  | 1.002 | 0.028
0 |  | 0.684 | 0.892 |  | 0.655 | 0.850 |  | 0.631 | 0.812 |  | 0.611 | 0.778
HMSE |  | 13230.45 |  | 12675.38 |  | 12303.48 |  | 12042.78

(c) $-1/+1/pen$ systems with various $pen$ for $\lambda_{0}^{[1]}=1$ |
---|---
| $pen$ | 0 |  | 1 |  | 2 |  | 3
Level $\ell$ |  | $\tilde{\zeta}(\ell)$ | $\mathbb{P}(L=\ell)$ |  | $\tilde{\zeta}(\ell)$ | $\mathbb{P}(L=\ell)$ |  | $\tilde{\zeta}(\ell)$ | $\mathbb{P}(L=\ell)$ |  | $\tilde{\zeta}(\ell)$ | $\mathbb{P}(L=\ell)$
14 |  | 1.439 | 0.311 |  | 1.230 | 0.463 |  | 1.123 | 0.562 |  | 1.055 | 0.634
13 |  | 0.939 | 0.089 |  | 0.711 | 0.060 |  | 0.590 | 0.045 |  | 0.513 | 0.035
12 |  | 0.796 | 0.044 |  | 0.634 | 0.036 |  | 0.539 | 0.030 |  | 0.475 | 0.025
11 |  | 0.717 | 0.027 |  | 0.581 | 0.025 |  | 0.500 | 0.022 |  | 0.444 | 0.020
10 |  | 0.666 | 0.020 |  | 0.542 | 0.019 |  | 0.468 | 0.018 |  | 0.418 | 0.016
9 |  | 0.627 | 0.016 |  | 0.510 | 0.016 |  | 0.442 | 0.015 |  | 0.395 | 0.014
8 |  | 0.595 | 0.014 |  | 0.483 | 0.015 |  | 0.419 | 0.014 |  | 0.375 | 0.013
7 |  | 0.567 | 0.014 |  | 0.459 | 0.014 |  | 0.397 | 0.014 |  | 0.356 | 0.013
6 |  | 0.540 | 0.014 |  | 0.436 | 0.015 |  | 0.377 | 0.014 |  | 0.337 | 0.013
5 |  | 0.513 | 0.016 |  | 0.413 | 0.016 |  | 0.357 | 0.016 |  | 0.319 | 0.014
4 |  | 0.483 | 0.020 |  | 0.388 | 0.020 |  | 0.335 | 0.018 |  | 0.300 | 0.017
3 |  | 0.450 | 0.027 |  | 0.361 | 0.026 |  | 0.313 | 0.024 |  | 0.280 | 0.021
2 |  | 0.409 | 0.042 |  | 0.330 | 0.038 |  | 0.287 | 0.033 |  | 0.258 | 0.029
1 |  | 0.359 | 0.078 |  | 0.293 | 0.065 |  | 0.256 | 0.053 |  | 0.232 | 0.044
0 |  | 0.291 | 0.269 |  | 0.246 | 0.172 |  | 0.219 | 0.122 |  | 0.201 | 0.091
HMSE |  | 4649638 |  | 5052378 |  | 5328804 |  | 5532048

##  4 Data analysis

###  4.1 Summary of data estimation results

In this subsection, we summarize the data estimation results using real data.
In order to examine the effect of dependence on ratemaking, we utilize a data
set concerning collision coverage for new and old vehicles from the Wisconsin
Local Government Property Insurance Fund (LGPIF) (Frees et al.,, 2016), where
detailed information on the project can be found at the LGPIF project website.
Such collision coverage provides cover for impact of vehicle with an object,
impact of vehicle with an attached vehicle, and overturn of a vehicle. The
observations include policyholders who have either new collision coverage or
old collision coverage or both. In our data analysis, longitudinal data over
the policy years from 2006 to 2010 with 497 governmental entities are used.
There are two categorical variables: the entity type with six levels, and the
coverage with three levels as shown in Table 4 (which is the same as Table C.6
in Oh et al., 2020b ).

Table 4: (Data analysis) Observable policy characteristics used as covariates Categorical | Description |  | Proportions
---|---|---|---
variables |
Entity type | Type of local government entity |  |  |
|  Miscellaneous |  | 5.03$\%$
|  City |  | 9.66$\%$
|  County |  | 11.47$\%$
|  School |  | 36.42$\%$
|  Town |  | 16.90$\%$
|  Village |  | 20.52$\%$
Coverage | Collision coverage amount for old and new vehicles |  |  |
|  Coverage $\in(0,0.14]=1$ |  | 33.40$\%$
|  Coverage $\in(0.14,0.74]=2$ |  | 33.20$\%$
|  Coverage $\in(0.74,\infty)=3$ |  | 33.40$\%$

Specifically, for Model 1 in the data analysis, as in (23) we consider

$N_{it}|(\Theta_{i}=\theta_{i},\boldsymbol{X}_{i}=\boldsymbol{x}_{i})\sim{\rm
Poisson}(\lambda_{i}\theta_{i})\quad\hbox{with}\quad\lambda_{i}:=\exp(\boldsymbol{x}_{i}\boldsymbol{\beta})$

where $\boldsymbol{x}_{i}$ is a row vector containing the $i$-th
policyholder’s information of the categorical variables in Table 4 for the
frequency, and the column vector $\boldsymbol{\beta}$ contains the
corresponding parameters. For the random effect $\Theta$, we assume a
Lognormal distribution with mean 1 and variance of $\exp(\sigma^{2})-1$, that
is,

$\Theta\sim{\rm Lognormal}(-\sigma^{2}/2,\sigma^{2}).$

Summary statistics of the posterior samples for the parameters in Model 1
using Bayesian approach are presented in Table 5. The table includes the
posterior median (Est), the posterior standard deviation (Std.dev), and the
95% highest posterior density Bayesian credible interval (95$\%$ CI). Note
that a $*$ sign indicates that the parameters are significant at 0.05 level.
In estimating the parameters in Table 5, we have run 30,000 MCMC iterations
saving every 5th sample after burn-in of 15,000 iterations by using JAGS.
Standard MCMC diagnostics gave no indication of lack of convergence. For Model
2, in line with (25) and (26), we assume

$N_{it}|(\Theta_{i}^{[1]}=\theta_{i}^{[1]},\boldsymbol{X}_{i}^{[1]}=\boldsymbol{x}_{i}^{[1]})\sim{\rm
Poisson}(\lambda_{i}^{[1]}\theta_{i}^{[1]})$

for the frequency, and

$Y_{itj}|(\Theta_{i}^{[2]}=\theta_{i}^{[2]},\boldsymbol{X}_{i}^{[2]}=\boldsymbol{x}_{i}^{[2]})\sim{\rm
Gamma}(\lambda_{i}^{[2]}\theta_{i}^{[2]},1/\psi^{[2]})$

for the individual claim severity, where $\lambda_{i}^{[2]}\theta_{i}^{[2]}$
is the mean and $1/\psi^{[2]}$ is the shape parameter of the Gamma
distribution. With log link, we assume

$\lambda_{i}^{[1]}=\exp(\boldsymbol{x}_{i}^{[1]}\boldsymbol{\beta}^{[1]})\quad\hbox{and}\quad\lambda_{i}^{[2]}=\exp(\boldsymbol{x}_{i}^{[2]}\boldsymbol{\beta}^{[2]}),$

where we take $\boldsymbol{x}_{i}^{[1]}=\boldsymbol{x}_{i}^{[2]}$, and
$\boldsymbol{\beta}^{[1]}$ and $\boldsymbol{\beta}^{[2]}$ are the
corresponding parameters.

For the bivariate random effect $(\Theta^{[1]},\Theta^{[2]})$, we assume a
Gaussian copula $C$ in (28) with correlation coefficient $\rho$, and the
marginal distributions are assumed to follow Lognormal distributions with
different parameters specified as

$\begin{cases}\Theta^{[1]}~{}\sim&{\rm
Lognormal}(-\sigma_{1}^{2}/2,\sigma_{1}^{2}),\\\ \Theta^{[2]}~{}\sim&{\rm
Lognormal}(-\sigma_{2}^{2}/2,\sigma_{2}^{2}).\end{cases}$

The estimation results from Oh et al., 2020b  are summarized in Table 7, and
we refer interested readers to Oh et al., 2020b  for the details of the
estimation procedure and further results. In particular, the dependence
parameter of the Gaussian copula is estimated to be $\rho=-0.447$, suggesting
a significant negative dependence between the random effects $\Theta^{[1]}$
and $\Theta^{[2]}$ of the frequency and the severity respectively.

###  4.2 Analysis of optimal relativities in modified BMS

Table 5: (Data analysis) Estimation results under the frequency-only Model 1 parameter | Est | Std.dev | 95$\%$ CI |
---|---|---|---|---
|  |  | lower | upper |
Fixed effect |  |
(Intercept) | -2.798 | 0.312 | -3.433 | -2.225 | *
Type=City | 0.601 | 0.337 | -0.044 | 1.280 |
Type=County | 1.923 | 0.329 | 1.257 | 2.543 | *
Type=School | 0.438 | 0.303 | -0.172 | 1.023 |
Type=Town | -1.343 | 0.380 | -2.087 | -0.600 | *
Type=Village | -0.005 | 0.320 | -0.630 | 0.618 |
Coverage=2 | 1.254 | 0.214 | 0.820 | 1.651 | *
Coverage=3 | 2.156 | 0.231 | 1.688 | 2.587 | *
Random effect |  |
$\sigma^{2}$ | 0.993 | 0.14 | 0.733 | 1.276 | *

Using the posterior median as estimates of the parameters in Models 1 and 2
when $z=14$, we calculate the optimal relativity and stationary probability
for each BM level as well as the values of HMSE under the modified BM
transition rules for a $-1/+1/pen$ system and a $-1/+2/pen$ system when
$pen=0,1,2,3$. The results are summarized in Tables 6 and 8. We remark that,
unlike in Examples 3 and 4 where the stationary probabilities
$\mathbb{P}(L=\ell)$ are identical across Tables 2 and 3, the stationary
probabilities in Tables 6 and 8 are close but not identical. The reason is
that the frequency parameters for Model 2 in Table 7 are estimated jointly
with the severity parameters and therefore the results are slightly different
from those for Model 1 given in Table 5 where only frequency parameters are
estimated. Note also that the parameters are estimated using MCMC sampling
which has also possibly contributed to some differences.

Comparing across Tables 6 and 8, one can observe the impact of frequency-
severity dependence on the optimal relativities. In particular, under negative
dependence between frequency and severity, which is implied from the negative
dependence between the two random effects with $\rho=-0.447$, the optimal
relativities tend to decrease for higher BM levels but increase for lower BM
levels. Consequently, the optimal relativities in Table 8 are less spread out
than those in Table 6. It is noted that all optimal BM relativities in Table 8
are less than 1, which may be counter-intuitive at a first glance. However, it
is noted that, roughly speaking, $\tilde{\zeta}(\ell)$ may be regarded as an
estimate of $\Theta^{[1]}\Theta^{[2]}$ when $L^{*}=(\ell)_{a}$ for some $a$
(see (10) and the optimization (13) in Model 2). Since $\Theta^{[1]}$ and
$\Theta^{[2]}$ are negatively dependent (with both having mean 1), when one of
them is large the other is more likely to be small, making the product
unlikely to be larger than 1.

Both Tables 6 and 8 confirm that as $pen$ increases, each optimal relativity
$\tilde{\zeta}(\ell)$ (for fixed $\ell$) decreases and the proportion of
policyholders in higher BM levels increases. Explanations similar to those in
Examples 3 and 4 are applicable. Certain degree of diversification effects on
the BM levels as $pen$ increases resembling Tables 2(a) and (b) in Example 3
and Tables 3(a) and (b) in Example 4 can be observed from Tables 6 and 8.
Meanwhile, the HMSE values show that a higher $pen$ leads to a decrease in
prediction power in this example. The higher HMSE in Table 8 compared to Table
6 is again attributed to the fact that Model 2 is concerned with the aggregate
claim while Model 1 is about claim number, and interested readers are referred
to Tables A4 and A5 in Oh et al., 2020b  for the magnitude of the average
claim severity in Model 2.

Lastly, we can compare the stationary distribution $L$ of the BM level and the
optimal relativities $\tilde{\zeta}(\ell)$ under two different BM systems: (i)
a $-1/+1/2$ system with a small increase of BM level per claim ($h=1$) but a
period of penalty ($pen=2$); and (ii) a $-1/+2/0$ system with a larger
increase of BM level per reported claim ($h=2$) but without period of penalty
($pen=0$). From Tables 6(a) and (b) concerning Model 1, the presence of a
period of penalty leads to a smaller proportion of policyholders in the lowest
level 0 (with $\mathbb{P}(L=0)=0.505$ in the former model and
$\mathbb{P}(L=0)=0.542$ in the latter) but a larger proportion in the highest
level 14 (with $\mathbb{P}(L=14)=0.237$ in the former model and
$\mathbb{P}(L=14)=0.173$ in the latter). For a given BM level $\ell$, the
optimal relativity $\tilde{\zeta}(\ell)$ in the former model is always lower
than the latter, suggesting that a BMS with a period of penalty instead of a
higher increase of the BM level per claim can look more attractive to the
market if the relativities are available to potential customers. Such effect
is even more pronounced if one assumes $pen=3$ instead of $pen=2$. Hence, this
numerical example clearly illustrates that different outcomes can be
anticipated in terms of the distribution of the BM level as well as the
optimal relativities depending on how more rigid the BMS is designed. Instead
of (or in addition to) imposing a higher increase of the BM level per claim,
our model makes it harder for a policyholder to transit to lower BM levels
once he/she has reported a claim by requiring consecutive and multiple claim-
free years to enjoy a bonus. We also remark that the same conclusion can be
drawn from Tables 8(a) and (b) by comparing the $-1/+1/2$ (or $-1/+1/3$)
system and the $-1/+2/0$ system under Model 2.

Table 6: (Data analysis) Distribution of $L$ and optimal relativities under
frequency-only Model 1

(a) $-1/+1/pen$ system with various $pen$ |
---|---
| $pen$ | 0 |  | 1 |  | 2 |  | 3
Level $\ell$ |  | $\tilde{\zeta}(\ell)$ | $\mathbb{P}(L=\ell)$ |  | $\tilde{\zeta}(\ell)$ | $\mathbb{P}(L=\ell)$ |  | $\tilde{\zeta}(\ell)$ | $\mathbb{P}(L=\ell)$ |  | $\tilde{\zeta}(\ell)$ | $\mathbb{P}(L=\ell)$
14 |  | 1.295 | 0.143 |  | 1.167 | 0.197 |  | 1.114 | 0.237 |  | 1.086 | 0.268
13 |  | 0.484 | 0.031 |  | 0.362 | 0.022 |  | 0.312 | 0.018 |  | 0.285 | 0.016
12 |  | 0.385 | 0.015 |  | 0.321 | 0.014 |  | 0.290 | 0.013 |  | 0.271 | 0.012
11 |  | 0.342 | 0.009 |  | 0.298 | 0.010 |  | 0.276 | 0.010 |  | 0.262 | 0.010
10 |  | 0.319 | 0.007 |  | 0.284 | 0.008 |  | 0.266 | 0.008 |  | 0.255 | 0.009
9 |  | 0.304 | 0.006 |  | 0.274 | 0.007 |  | 0.259 | 0.008 |  | 0.249 | 0.008
8 |  | 0.293 | 0.005 |  | 0.266 | 0.007 |  | 0.253 | 0.007 |  | 0.244 | 0.008
7 |  | 0.285 | 0.005 |  | 0.260 | 0.007 |  | 0.248 | 0.008 |  | 0.241 | 0.009
6 |  | 0.277 | 0.006 |  | 0.255 | 0.007 |  | 0.244 | 0.009 |  | 0.238 | 0.010
5 |  | 0.271 | 0.007 |  | 0.251 | 0.009 |  | 0.241 | 0.010 |  | 0.235 | 0.012
4 |  | 0.265 | 0.009 |  | 0.247 | 0.012 |  | 0.237 | 0.014 |  | 0.233 | 0.015
3 |  | 0.259 | 0.014 |  | 0.242 | 0.018 |  | 0.234 | 0.021 |  | 0.229 | 0.023
2 |  | 0.252 | 0.026 |  | 0.237 | 0.034 |  | 0.229 | 0.038 |  | 0.225 | 0.041
1 |  | 0.243 | 0.072 |  | 0.230 | 0.087 |  | 0.223 | 0.094 |  | 0.218 | 0.098
0 |  | 0.228 | 0.645 |  | 0.217 | 0.562 |  | 0.210 | 0.505 |  | 0.205 | 0.462
HMSE |  | 2.42280 |  | 2.54650 |  | 2.60334 |  | 2.63588

(b) $-1/+2/pen$ system with various $pen$ |
---|---
| $pen$ | 0 |  | 1 |  | 2 |  | 3
Level $\ell$ |  | $\tilde{\zeta}(\ell)$ | $\mathbb{P}(L=\ell)$ |  | $\tilde{\zeta}(\ell)$ | $\mathbb{P}(L=\ell)$ |  | $\tilde{\zeta}(\ell)$ | $\mathbb{P}(L=\ell)$ |  | $\tilde{\zeta}(\ell)$ | $\mathbb{P}(L=\ell)$
14 |  | 1.240 | 0.173 |  | 1.130 | 0.236 |  | 1.088 | 0.280 |  | 1.065 | 0.315
13 |  | 0.452 | 0.047 |  | 0.346 | 0.031 |  | 0.301 | 0.024 |  | 0.276 | 0.020
12 |  | 0.354 | 0.025 |  | 0.305 | 0.021 |  | 0.278 | 0.018 |  | 0.262 | 0.017
11 |  | 0.312 | 0.017 |  | 0.282 | 0.015 |  | 0.265 | 0.015 |  | 0.254 | 0.014
10 |  | 0.288 | 0.013 |  | 0.267 | 0.013 |  | 0.255 | 0.013 |  | 0.246 | 0.013
9 |  | 0.273 | 0.011 |  | 0.258 | 0.011 |  | 0.248 | 0.012 |  | 0.242 | 0.011
8 |  | 0.263 | 0.010 |  | 0.250 | 0.011 |  | 0.242 | 0.013 |  | 0.236 | 0.013
7 |  | 0.256 | 0.010 |  | 0.245 | 0.011 |  | 0.239 | 0.011 |  | 0.234 | 0.011
6 |  | 0.250 | 0.011 |  | 0.240 | 0.014 |  | 0.233 | 0.016 |  | 0.228 | 0.017
5 |  | 0.245 | 0.012 |  | 0.237 | 0.013 |  | 0.232 | 0.013 |  | 0.228 | 0.012
4 |  | 0.240 | 0.018 |  | 0.231 | 0.023 |  | 0.225 | 0.027 |  | 0.221 | 0.030
3 |  | 0.237 | 0.019 |  | 0.230 | 0.019 |  | 0.225 | 0.017 |  | 0.221 | 0.016
2 |  | 0.230 | 0.050 |  | 0.221 | 0.064 |  | 0.216 | 0.072 |  | 0.211 | 0.077
1 |  | 0.228 | 0.043 |  | 0.221 | 0.033 |  | 0.217 | 0.026 |  | 0.213 | 0.022
0 |  | 0.213 | 0.542 |  | 0.206 | 0.486 |  | 0.201 | 0.444 |  | 0.196 | 0.410
HMSE |  | 2.48426 |  | 2.58912 |  | 2.63540 |  | 2.66122

Table 7: (Data analysis) Estimation results under the frequency-severity Model 2 with dependence |  |  | 95$\%$ CI |  |
---|---|---|---|---|---
parameter | Est | Std.dev | lower | upper |  |
Frequency part |  |  |  |
Intercept | -2.767 | 0.318 | -3.417 | -2.153 | * |
City | 0.597 | 0.337 | -0.051 | 1.272 |  |
County | 1.907 | 0.335 | 1.271 | 2.587 | * |
School | 0.411 | 0.304 | -0.181 | 1.014 |  |
Town | -1.351 | 0.384 | -2.103 | -0.584 | * |
Village | -0.012 | 0.323 | -0.626 | 0.654 |  |
Coverage2 | 1.247 | 0.212 | 0.829 | 1.667 | * |
Coverage3 | 2.139 | 0.230 | 1.713 | 2.615 | * |
Severity part |  |  |  |
Intercept | 8.829 | 0.375 | 8.103 | 9.588 | * |
City | -0.036 | 0.353 | -0.737 | 0.637 |  |
County | 0.341 | 0.338 | -0.336 | 0.980 |  |
School | -0.173 | 0.328 | -0.805 | 0.484 |  |
Town | 0.497 | 0.440 | -0.356 | 1.349 |  |
Village | 0.316 | 0.346 | -0.357 | 0.994 |  |
Coverage2 | 0.180 | 0.244 | -0.308 | 0.646 |  |
Coverage3 | -0.027 | 0.261 | -0.533 | 0.493 |  |
$1/\psi^{[2]}$ | 0.670 | 0.041 | 0.592 | 0.752 | * |
Copula part |  |  |  |
$\sigma^{2}_{1}$ | 0.992 | 0.142 | 0.746 | 1.292 | * |
$\sigma^{2}_{2}$ | 0.293 | 0.067 | 0.176 | 0.433 | * |
$\rho$ | -0.447 | 0.130 | -0.690 | -0.190 | * |
Table 8: (Data analysis) Distribution of $L$ and optimal relativities under
the frequency-severity Model 2 with dependence

(a) $-1/+1/pen$ system with various $pen$ |
---|---
| $pen$ | 0 |  | 1 |  | 2 |  | 3
Level $\ell$ |  | $\tilde{\zeta}(\ell)$ | $\mathbb{P}(L=\ell)$ |  | $\tilde{\zeta}(\ell)$ | $\mathbb{P}(L=\ell)$ |  | $\tilde{\zeta}(\ell)$ | $\mathbb{P}(L=\ell)$ |  | $\tilde{\zeta}(\ell)$ | $\mathbb{P}(L=\ell)$
14 |  | 0.968 | 0.144 |  | 0.888 | 0.199 |  | 0.855 | 0.238 |  | 0.837 | 0.270
13 |  | 0.459 | 0.031 |  | 0.356 | 0.022 |  | 0.309 | 0.018 |  | 0.282 | 0.016
12 |  | 0.376 | 0.015 |  | 0.318 | 0.014 |  | 0.287 | 0.013 |  | 0.268 | 0.012
11 |  | 0.338 | 0.010 |  | 0.295 | 0.010 |  | 0.272 | 0.010 |  | 0.259 | 0.010
10 |  | 0.316 | 0.007 |  | 0.280 | 0.008 |  | 0.262 | 0.008 |  | 0.252 | 0.009
9 |  | 0.301 | 0.006 |  | 0.270 | 0.007 |  | 0.255 | 0.008 |  | 0.247 | 0.008
8 |  | 0.290 | 0.005 |  | 0.262 | 0.007 |  | 0.250 | 0.008 |  | 0.243 | 0.008
7 |  | 0.281 | 0.005 |  | 0.256 | 0.007 |  | 0.245 | 0.008 |  | 0.240 | 0.009
6 |  | 0.274 | 0.006 |  | 0.251 | 0.007 |  | 0.242 | 0.009 |  | 0.239 | 0.010
5 |  | 0.267 | 0.007 |  | 0.247 | 0.009 |  | 0.240 | 0.011 |  | 0.237 | 0.012
4 |  | 0.261 | 0.009 |  | 0.244 | 0.012 |  | 0.238 | 0.014 |  | 0.237 | 0.015
3 |  | 0.256 | 0.014 |  | 0.241 | 0.018 |  | 0.237 | 0.021 |  | 0.237 | 0.023
2 |  | 0.250 | 0.026 |  | 0.239 | 0.034 |  | 0.237 | 0.038 |  | 0.237 | 0.041
1 |  | 0.245 | 0.072 |  | 0.238 | 0.088 |  | 0.238 | 0.095 |  | 0.238 | 0.098
0 |  | 0.240 | 0.643 |  | 0.237 | 0.560 |  | 0.236 | 0.503 |  | 0.236 | 0.460
HMSE |  | 91069385 |  | 95366837 |  | 97457899 |  | 98678459

(b) $-1/+2/pen$ system with various $pen$ |
---|---
| $pen$ | 0 |  | 1 |  | 2 |  | 3
Level $\ell$ |  | $\tilde{\zeta}(\ell)$ | $\mathbb{P}(L=\ell)$ |  | $\tilde{\zeta}(\ell)$ | $\mathbb{P}(L=\ell)$ |  | $\tilde{\zeta}(\ell)$ | $\mathbb{P}(L=\ell)$ |  | $\tilde{\zeta}(\ell)$ | $\mathbb{P}(L=\ell)$
14 |  | 0.933 | 0.174 |  | 0.865 | 0.238 |  | 0.838 | 0.282 |  | 0.824 | 0.316
13 |  | 0.431 | 0.047 |  | 0.341 | 0.031 |  | 0.298 | 0.024 |  | 0.275 | 0.021
12 |  | 0.348 | 0.025 |  | 0.302 | 0.021 |  | 0.277 | 0.018 |  | 0.262 | 0.017
11 |  | 0.309 | 0.017 |  | 0.280 | 0.016 |  | 0.263 | 0.015 |  | 0.254 | 0.014
10 |  | 0.286 | 0.013 |  | 0.265 | 0.013 |  | 0.254 | 0.013 |  | 0.247 | 0.013
9 |  | 0.271 | 0.011 |  | 0.256 | 0.011 |  | 0.248 | 0.012 |  | 0.244 | 0.011
8 |  | 0.260 | 0.010 |  | 0.249 | 0.011 |  | 0.244 | 0.013 |  | 0.241 | 0.014
7 |  | 0.253 | 0.010 |  | 0.245 | 0.011 |  | 0.241 | 0.011 |  | 0.240 | 0.011
6 |  | 0.248 | 0.011 |  | 0.241 | 0.014 |  | 0.239 | 0.016 |  | 0.239 | 0.017
5 |  | 0.244 | 0.012 |  | 0.240 | 0.013 |  | 0.239 | 0.013 |  | 0.239 | 0.013
4 |  | 0.241 | 0.018 |  | 0.238 | 0.023 |  | 0.238 | 0.027 |  | 0.239 | 0.030
3 |  | 0.240 | 0.019 |  | 0.238 | 0.019 |  | 0.238 | 0.017 |  | 0.239 | 0.016
2 |  | 0.238 | 0.050 |  | 0.238 | 0.064 |  | 0.239 | 0.072 |  | 0.239 | 0.077
1 |  | 0.238 | 0.043 |  | 0.238 | 0.033 |  | 0.239 | 0.026 |  | 0.239 | 0.022
0 |  | 0.237 | 0.540 |  | 0.236 | 0.483 |  | 0.236 | 0.441 |  | 0.235 | 0.407
HMSE |  | 93352237 |  | 96970265 |  | 98659783 |  | 99617247

##  5 Conclusion

In automobile third-party liability insurance, BMS has been broadly used as a
posteriori ratemaking mechanism which helps insurers rate a policyholder’s
risk more accurately and thus premium can be calculated more fairly to reflect
his/her risk. BMS is also designed to stimulate drivers to practise safer
driving by adjusting the premium on policy renewal based on the claim history
in the previous year. However, a typical BMS immediately offering a reward to
policyholders without claim in the previous year may tend to move
policyholders easily towards lower BM levels. Hence, to resolve the unbalanced
issue due to a concentration of policyholders in the lowest BM level, to
prevent a quick recovery (in terms of paying lower premium) for those drivers
who have a good claim history only for a single period, and to better
distinguish between drivers who are consistently good and those who are only
temporarily good, in this paper we introduce a “period of penalty” to count
the number of consecutive claim-free years required to lower the BM level.
Although other more rigid BMS transition rules such as a $-1/+h$ system with
$h=2$ or $h=3$ to severely penalize claims in the previous year may put less
pressure on the premium income of the insurer, these may weaken the product’s
competitiveness in the market and the insurers may lose better customers.
Through numerical illustrations with a data set, we have demonstrated that,
compared to a $-1/+2$ system, the introduction of a penalty period to a
$-1/+1$ system can result in (i) lower values of optimal BM relativities which
can potentially improve marketability of the product; and (ii) better
separation of risks because the BM level 0 is less concentrated as drivers who
are not consistently good are moved to higher BM levels. With the
policyholders aware of the requirement of consecutive claim-free years to
enjoy lower BM relativities at lower BM levels, they can be well motivated to
drive more safely.

Moreover, in the afore-mentioned modified BMS with a penalty period, we take
into account the randomness of (i) frequency only or (ii) both frequency and
severity when modeling the unobserved risk characteristics and to construct
the optimal relativities associated with the BM levels. These relativities are
then used to determine the premium actually charged to the policyholders
staying in the corresponding BM levels. We hope that our extended BMS allowing
for dependency between the frequency and the severity of claims via dependency
of the unobserved risk characteristics provides some insights for improving
the classical BMS, in particular, when the insurers have the freedom to alter
the features of the BMS to reflect claim experience more accurately based on
the claim history of policyholders in multiple years with product
competitiveness in mind.

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##  Appendix A Generalized linear models (GLMs)

The exponential dispersion family (EDF) in McCullagh and Nelder, (1989) can be
considered for modeling the random components of the frequency and the
severity of insurance claims in the GLMs. The EDF with mean $\mu$ and
dispersion $\psi$, whose distribution function is denoted by
$F(\cdot;\mu,\psi)$, has the probability density/mass function (in $y$)

$p(y|\vartheta,\psi)=\exp[(y\vartheta-b(\vartheta))/\psi+c(y,\psi)],$

where $\vartheta$ is the canonical parameter, and $b(\cdot)$ and $c(\cdot)$
are predetermined functions. The mean of the distribution can be expressed as
$\mu=b^{\prime}(\vartheta)$ and the variance is
$b^{\prime\prime}(\vartheta)\psi\equiv V(\mu)$, where the inverse of
$b^{\prime}(\cdot)$ is known as the canonical link function and $V(\cdot)$ is
called the variance function.

##  Appendix B Model 1

In the frequency random effect model:

  1. i.

For the $i$-th policyholder, the conditional distribution of the number of
claims $N_{it}$ in the $t$-th policy year given the observed risk
characteristics $\boldsymbol{X}_{i}=\boldsymbol{x}_{i}$ and unobserved risk
characteristics $\Theta_{i}=\theta_{i}$ is specified as

$N_{it}|(\Theta_{i}=\theta_{i},\boldsymbol{X}_{i}=\boldsymbol{x}_{i})\buildrel{\rm
i.i.d.}\over{\sim}{F}(\cdot;\lambda_{i}\theta_{i},\psi),$ (23)

where the distribution function $F$ has mean parameter $\lambda_{i}\theta_{i}$
with $\lambda_{i}=\eta^{-1}(\boldsymbol{x}_{i}\boldsymbol{\beta})$ and some
parameter $\psi$. The “i.i.d.” in (23) means that, conditional on
$\Theta_{i}=\theta_{i}$ and $\boldsymbol{X}_{i}=\boldsymbol{x}_{i}$, the claim
numbers $N_{it}$’s for different $t$’s are independent and identically
distributed. When $F$ is in the class of EDF in A, $\psi$ is the dispersion
parameter.

  2. ii.

The random variable $\Theta_{i}$ for the $i$-th policyholder’s unobserved risk
characteristics concerning the claim frequency is assumed to be independent of
the observed risk characteristics $\boldsymbol{X}_{i}$. The $\Theta_{i}$’s are
i.i.d. having distribution function $G$ and density $g=G^{\prime}$, and we
write

$\Theta_{i}\buildrel{\rm i.i.d.}\over{\sim}G.$ (24)

A generic variable with distribution $G$ is denoted by $\Theta$, and we assume
$\mathbb{E}[\Theta]=1$ for convenience.

##  Appendix C Model 2

To account for various forms of dependence such as frequency-frequency,
severity-severity and frequency-severity, we consider a copula-based random
effect model as in Oh et al., 2020b :

  1. i.

The frequency component $N_{it}$ is specified using a count regression model
conditional on the observed risk characteristics $\boldsymbol{X}_{i}^{[1]}$
and the unobserved risk $\Theta_{i}^{[1]}$ such that

$N_{it}|(\Theta_{i}^{[1]}=\theta_{i}^{[1]},\boldsymbol{X}_{i}^{[1]}=\boldsymbol{x}_{i}^{[1]})\buildrel{\rm
i.i.d.}\over{\sim}{F}_{1}(\cdot;\lambda_{i}^{[1]}\theta_{i}^{[1]},\psi^{[1]}),$
(25)

where the distribution function $F_{1}$ has mean parameter
$\lambda_{i}^{[1]}\theta_{i}^{[1]}$ with
$\lambda_{i}^{[1]}=\eta_{1}^{-1}(\boldsymbol{x}_{i}^{[1]}\boldsymbol{\beta}^{[1]})$
and some parameter $\psi^{[1]}$. The “i.i.d.” in (25) means that $N_{it}$’s
for different $t$’s are (conditionally) independent and identically
distributed. If $F_{1}$ belongs to EDF, then $\psi^{[1]}$ corresponds to the
dispersion parameter. Here we implicitly assume that the conditional
distribution in (25) does not change even if we are given further information
on the risk characteristics $\boldsymbol{X}_{i}^{[2]}$ and $\Theta_{i}^{[2]}$
pertaining to the severity.

  2. ii.

The distribution of the $i$-th policyholder’s $j$-th claim severity $Y_{itj}$
in the $t$-th year conditional on the observed risk characteristics
$\boldsymbol{X}_{i}^{[2]}$ and the unobserved risk characteristics
$\Theta_{i}^{[2]}$ is specified as

$Y_{itj}|(\Theta_{i}^{[2]}=\theta_{i}^{[2]},\boldsymbol{X}_{i}^{[2]}=\boldsymbol{x}_{i}^{[2]})\buildrel{\rm
i.i.d.}\over{\sim}{F}_{2}(\cdot\,;\,\lambda_{i}^{[2]}\theta_{i}^{[2]},\psi^{[2]}),$
(26)

where the distribution function $F_{2}$ is in EDF with mean parameter
$\lambda_{i}^{[2]}\theta_{i}^{[2]}$ (where
$\lambda_{i}^{[2]}=\eta_{2}^{-1}(\boldsymbol{x}_{i}^{[2]}\boldsymbol{\beta}^{[2]})$)
and dispersion parameter $\psi^{[2]}$. The “i.i.d.” in (26) means that
$Y_{itj}$’s for different $t$’s and different $j$’s are (conditionally)
independent and identically distributed. It is understood that the conditional
distribution in (26) remains the same even if we are further given the risk
characteristics $\boldsymbol{X}_{i}^{[1]}$ and $\Theta_{i}^{[1]}$ pertaining
to the frequency. By the property of EDF, the distribution of the average
claim amount $M_{it}$ (see (1)) conditional on the observed risk
characteristics $\boldsymbol{X}_{i}^{[2]}$, the unobserved risk
characteristics $\Theta_{i}^{[2]}$, and the frequency $N_{it}$ is specified
via

$M_{it}|(N_{it}=n_{it},\Theta_{i}^{[2]}=\theta_{i}^{[2]},\boldsymbol{X}_{i}^{[2]}=\boldsymbol{x}_{i}^{[2]})\buildrel{\rm
i.i.d.}\over{\sim}{F}_{2}(\cdot\,;\,\lambda_{i}^{[2]}\theta_{i}^{[2]},\psi^{[2]}/n_{it}),\qquad
n_{it}>0,$ (27)

with updated dispersion parameter $\psi^{[2]}/n_{it}$, and

$\mathbb{P}(M_{it}=0|N_{it}=n_{it},\Theta_{i}^{[2]}=\theta_{i}^{[2]},\boldsymbol{X}_{i}^{[2]}=\boldsymbol{x}_{i}^{[2]})=1,\qquad
n_{it}=0.$

The “i.i.d.” in (27) means that $M_{it}$’s for different $t$’s are
(conditionally) independent and identically distributed.

  3. iii.

The bivariate random vector of unobserved risk characteristics
$(\Theta_{i}^{[1]},\Theta_{i}^{[2]})$ of the $i$-th policyholder, which is
assumed to be independent of the observed risk characteristics
$(\boldsymbol{X}_{i}^{[1]},\boldsymbol{X}_{i}^{[2]})$, has joint distribution
function $H$ specified via

$(\Theta_{i}^{[1]},\Theta_{i}^{[2]})\buildrel{\rm
i.i.d.}\over{\sim}H=C(G_{1},G_{2}),$ (28)

where $G_{1}$ and $G_{2}$ denote the marginal distribution functions of
$\Theta_{i}^{[1]}$ and $\Theta_{i}^{[2]}$ respectively, and $C$ is a bivariate
copula. The “i.i.d.” in (28) means that
$(\Theta_{i}^{[1]},\Theta_{i}^{[2]})$’s for different $i$’s are independent
and identically distributed. A generic pair of
$(\Theta_{i}^{[1]},\Theta_{i}^{[2]})$ will be denoted by
$(\Theta^{[1]},\Theta^{[2]})$. We shall use $g_{1}$, $g_{2}$, and $h$ to
denote the density versions of $G_{1}$, $G_{2}$, and $H$, respectively. For
convenience, it is further assumed that

$\mathbb{E}[\Theta^{[1]}]=1\quad\hbox{and}\quad\mathbb{E}[\Theta^{[2]}]=1.$