# Individual claims reserving using activation patterns

Marie Michaelides
Department of Mathematics
Université du Québec à Montréal
Montréal, QC H2X 3Y7
<EMAIL_ADDRESS>
&Mathieu Pigeon
Department of Mathematics
Université du Québec à Montréal
Montréal, QC H2X 3Y7
<EMAIL_ADDRESS>
&Hélène Cossette
École d’actuariat
Université Laval
Québec, QC G1V 0A6
<EMAIL_ADDRESS>
Corresponding author

###### Abstract

The occurrence of a claim often impacts not one but multiple insurance
coverages provided in the contract. To account for this multivariate feature,
we propose a new individual claims reserving model built around the activation
of the different coverages to predict the reserve amounts. Using the framework
of multinomial logistic regression, we model the activation of the different
insurance coverages for each claim and their development in the following
years, i.e. the activation of other coverages in the later years and all the
possible payments that might result from them. As such, the model allows us to
complete the individual development of the open claims in the portfolio. Using
a recent automobile dataset from a major Canadian insurance company, we
demonstrate that this approach generates accurate predictions of the total
reserves as well as of the reserves per insurance coverage. This allows the
insurer to get better insights in the dynamics of his claims reserves.

_Keywords_ Individual claims reserving $\cdot$ dependence modelling $\cdot$
insurance coverages

##  1 Introduction

Claims reserving is known to be one of the most crucial tasks performed by
actuaries in insurance companies all over the world. Insurers must accurately
predict the liabilities that will arise from open and future claims. This
allows them to answer the reporting standards that they are subject to and to
preserve a sufficient amount of capital with which they can set aside adequate
reserves to fulfill their obligations to the policyholders and avoid financial
ruin.

In the actuarial practice, claims data is commonly aggregated on an occurrence
year and development year basis in run-off triangles. These then help
actuaries to evaluate the reserves for the portfolio as a whole. One very
common method that uses such triangles is the Chain Ladder model introduced by
Mack (1993) and further discussed in Mack (1994), Mack (1999) or Mack and
Venter (2000).

Over the years, several authors have challenged the robustness of this model.
In particular, the recent increase in the quantity and availability of data
has contributed to questioning the use of such aggregate methods. The early
work of Buhlmann et al. (1980), Hachemeister (1980) and Norberg (1986)
attempted to benefit from these larger quantities of data. A few more years
were however necessary to obtain the necessary computing resources to move
from the classical run-off triangles to the so-called micro-level claims
reserving models. Antonio and Plat (2014) were the first to truly incorporate
information related to the policyholder or even the claim itself into their
model by building on the aforementioned work as well as on prior work
performed by Norberg (1993), Norberg (1999) and Haastrup and Arjas (1996).
They demonstrate that using the detailed information available to the insurer
at the claim level allows to obtain more accurate predictions for the
reserves.

Micro-level models now abound in the actuarial literature thanks to the
contributions of several authors. For example, Pigeon et al. (2013) propose a
discrete time model for the payments and then extend their work in Pigeon et
al. (2014) to include incurred losses as well. Other authors have also opened
the way to non-parametric approaches to claims reserving. Wüthrich (2018) was
the first to introduce the use of Breiman et al. (1984)’s Classification and
Regression Tree (CART) algorithm in a micro-level reserving model. Building on
this work, Lopez (2019) and Lopez et al. (2016) apply the CART algorithm with
censored data using, respectively, survival analysis and copulas to account
for the possible dependence between the development time of the claim and its
ultimate amount.

In addition to the constant increase in the quantity of data, the growing
diversification of the products offered by insurers contributes to further
complexify the work of actuaries. Evolving in a very competitive world and
taking advantage of the rise of new technologies, insurers must remain
constantly aware of changes and evolutions in their clients’ needs. To answer
them, they often diversify their offer, resulting in the multiplication of
coverages provided within a policy. Actuaries must thereby refine their models
to keep up with this diverseness in the portfolios.

In this paper, we propose a model that not only makes use of the large
quantity of data available to an insurer in a micro-level model but also takes
into account the different coverages that a policy offers and their dynamics
within the portfolio. We show that allowing dependence between coverages
increases prediction accuracy of both the total portfolio reserves and the
reserves per coverage. This in turn allows the insurer to gain further
insights into the dynamics of his portfolio. To the best of our knowledge,
such form of dependence between coverages has not yet been modelled in a
granular reserving framework. Zhou and Zhao (2010) and Lopez (2019) both used
copulas in such a context to model the dependence between the event times and
delay in the development of a claim or the development time and the final
amount of the claim. Pešta and Okhrin (2014) use time series and copulas to
take into account the dependence between payment amounts made at different
stages in the development of a claim. It is rather in the actuarial pricing
literature that we find examples of dependency modelling between insurance
coverages. Frees and Valdez (2008) and together with Frees et al. (2009) used
copulas to model the dependence between different claim types. They begin by
identifying the coverage(s) impacted by a claim, then those for which a
payment is made before predicting the associated severity. More recently, Côté
et al. (2022) extend their work by introducing a Bayesian model for
multivariate and multilevel claim amounts, therefore facilitating the
treatment of open claims which are of crucial importance in insurers’
datasets.

We propose a micro-level reserving model that builds upon the work of Frees
and Valdez (2008) and Frees et al. (2009) to include the dependence between
multiple insurance coverages. Section 2 presents the statistical model that is
based on activation patterns and illustrates how claims are handled at
different stages of their development. In Section 3, we apply the model to a
Canadian automobile insurance dataset with four different coverages. We begin
by presenting the data, the process of model fitting and finally the results
obtained. Section 4 draws the conclusion of our work.

##  2 An activation pattern model for claims reserving

This section introduces the model. We present the model components and the way
they evolve as the claim develops. We use an example to illustrate the model
and discuss the treatment of claims at various stages of their development.

Figure 1 shows the typical development process for a single claim. When a
claim occurs, the policyholder reports it to the insurance company. This can
take place either as soon as the claim occurs or with a certain reporting
delay. Once reported, the insurer records the claim and opens a new file in
his claims management system. Since insurance companies typically provide
their policyholders with multiple coverages under a single insurance policy,
the reporting of the claim activates at least one of these coverages. The
insurer can then start making payments towards the policyholder. During its
development, new information related to the claim can be brought to the
insurer that can result in the activation of one or more additional coverages
included in the insurance policy. Payments will then continue until settlement
of the claim.

Figure 1: Development process of a claim

We seek to model the way in which a claim can activate multiple coverages
either upon reporting or later, and the underlying dependence between them.
Note that in the remainder of this paper, we work in discrete time and group
the data per year. We could however have chosen any other time unit.

Although automobile claims often display shorter lifetimes than in other
branches of insurance, it is still fairly common to observe claims that remain
open for more than one year. We thereby structure our model based on the
development years. In Section 2.1, we present the model for a given
development year $j$ before moving on to the year $j+1$ in Section 2.2 and to
any additional development years in Section 2.3.

###  2.1 Development year $j$

Let $c=1,...,C$ denote a coverage provided by an insurer. Each policy in force
can incur a claim that will impact one or more of the $C$ insurance coverages
that the policyholder benefits from. For each claim $i$ with $i=1,...,n$ and a
given development year $j$ with $j=1,...,J$, we define the random vector
$\boldsymbol{A}_{i,j}$ of dimension $1\times C$. This vector indicates the
pattern in which claim $i$ activates the different coverages during year $j$.

With $C$ insurance coverages and assuming that at least one must be activated
when claim $i$ is reported, there are $V=2^{C}-1$ different activation pattern
vectors possible in year $j$. Let $\mathcal{V}$ be the ensemble of the $V$
possible activation pattern vectors in year $j$. We then define the
$v^{\text{th}}$ possible pattern, a realisation of the random vector
$\boldsymbol{A}_{i,j}$, as $\boldsymbol{a}_{i,j}^{v}\in\mathcal{V}$, with
$v=1,...,V$ such that:

$\displaystyle\boldsymbol{a}_{i,j}^{v}=\begin{bmatrix}a_{i,j,1}^{v}\hskip
5.69046pta_{i,j,2}^{v}\hskip 5.69046pt...\hskip
5.69046pta_{i,j,C}^{v}\end{bmatrix},$

where

$\displaystyle a_{i,j,c}^{v}$ $\displaystyle=\begin{cases}1,&\text{if coverage
$c$ is activated in the $v^{\text{th}}$ pattern for claim $i$ in year $j$}\\\
0,&\text{otherwise.}\end{cases}$

We use a multinomial logistic regression to model $\boldsymbol{A}_{i,j}$:

$\displaystyle\operatorname{P}[\boldsymbol{A}_{i,j}=\boldsymbol{a}_{i,j}^{v}|\boldsymbol{x}_{i},\boldsymbol{\beta}]=\frac{\exp{(\boldsymbol{x}_{i}^{\prime}\boldsymbol{\beta}_{j,v})}}{\sum_{k=1}^{V}\exp{(\boldsymbol{x}_{i}^{\prime}\boldsymbol{\beta}_{j,k}})},$
(1)

where $\boldsymbol{x}_{i}^{\prime}$ is a $1\times m$ vector of covariates for
claim $i$ that does not depend on the development year nor on the insurance
coverage and $\boldsymbol{\beta}_{j,v}$ is the $1\times m$ vector of
parameters that can vary with the different coverages and thus depends on the
$v^{th}$ pattern.

Knowing which coverages claim $i$ activates in year $j$ thanks to
$\boldsymbol{A}_{i,j}$, the insurer can move on to the next step of the claim
development process depicted in Figure 1 and determine which of the active
coverages will incur a payment within the year. We define
$\boldsymbol{P}_{i,j}$, a $1\times C$ vector in which each entry indicates
whether a payment has been made for the corresponding active coverage or not:

$\displaystyle P_{i,j,c}|(A_{i,j,c}=1)$
$\displaystyle=\begin{cases}1,&\text{if a payment has been made for claim
$i$}\\\ &\hskip 5.69046pt\text{and coverage $c$ in year $j$ given
$A_{i,j,c}=1$,}\\\ 0,&\text{otherwise.}\end{cases}$

For each insurance coverage $c$, we assume that

$\displaystyle
P_{i,j,c}|(A_{i,j,c}=1)\sim\text{Bernoulli}\big{(}\pi_{j,c}(\boldsymbol{x}_{i}^{\prime}\boldsymbol{\gamma}_{j,c})\big{)}.$
(2)

For development year $j$, the probability
$\pi_{j,c}(\boldsymbol{x}_{i}^{\prime}\boldsymbol{\gamma}_{j,c})$ is given by

$\displaystyle\pi_{j,c}(\boldsymbol{x}_{i}^{\prime}\boldsymbol{\gamma}_{j,c})=\frac{\exp{(\boldsymbol{x}_{i}^{\prime}\boldsymbol{\gamma}_{j,c})}}{1+\exp{(\boldsymbol{x}_{i}^{\prime}\boldsymbol{\gamma}_{j,c})}}.$

The vector of covariates $\boldsymbol{x}_{i}^{\prime}$ is the same as the one
used for the activation patterns in Equation 1 and $\boldsymbol{\gamma}_{j,c}$
is the $1\times m$ vector of coefficients varying with each coverage $c$ and
development year $j$.

Once we know which coverages are active in year $j$ and which of these have
incurred a payment, we can calculate the corresponding severity per coverage
for claim $i$. The actuarial literature contains many examples of the use of
Generalized Additive Models for the Location, Scale and Shape (GAMLSS) for
this purpose, thanks to the level of flexibility that these models provide
(see Stasinopoulos et al. (2017) for more details on the use of the GAMLSS).
We use them to predict the expected severity incurred by claim $i$:

$\displaystyle\operatorname{E}[Y_{i,j,c}|P_{i,j,c}=1,A_{i,j,c}=1]=g^{-1}(\boldsymbol{x}_{i}^{\prime}\boldsymbol{\alpha_{j,c}}+\alpha_{j,c}^{*}j).$
(3)

Here, $Y_{i,j,c}$ represents the severity associated to claim $i$ in year $j$
for insurance coverage $c$, $g(.)$ is the link function,
$\boldsymbol{\alpha}_{j,c}$ are the coefficients that depend on the insurance
coverages and $\alpha_{j,c}^{*}$ measures the effect of development year $j$
for coverage $c$.

With the activation patterns, payment vectors and corresponding severities, we
know precisely which coverages claim $i$ activated in year $j$ and the
severity associated to each of these coverages.

###  2.2 Development year $j+1$

Knowing $\boldsymbol{A}_{i,j}$, we are aware of the coverages that claim $i$
has activated at the start of year $j+1$. We assume that once an insurance
coverage is active, it remains active until the settlement of the claim. For
claim $i$ and coverage $c$ in development year $j$, we have

$\displaystyle A_{i,j+k,c}|(A_{i,j,c}=1)=1,\hskip 14.22636pt\text{for
all}\hskip 2.84544ptk\geq 1.$ (4)

As such, we now depict the set of possible patterns for
$\boldsymbol{A}_{i,j+1}$ as $\mathcal{V}^{*}\subset\mathcal{V}$. More
generally, we write

$\displaystyle\operatorname{P}[\boldsymbol{A}_{i,j+1}=\boldsymbol{a}_{i,j}^{v*}|\boldsymbol{A}_{i,j},\boldsymbol{x}_{i},\boldsymbol{\beta}]=\begin{cases}\frac{\operatorname{P}[\boldsymbol{A}_{i,j+1}=\boldsymbol{a}_{i,j}^{v*}|\boldsymbol{x^{\prime}}_{i},\boldsymbol{\beta}_{j+1}]}{\sum_{\boldsymbol{a}_{i,j}^{v*}\in\mathcal{V}^{*}}\operatorname{P}[\boldsymbol{A}_{i,j+1}=\boldsymbol{a}_{i,j}^{v*}|\boldsymbol{x^{\prime}}_{i},\boldsymbol{\beta}_{j+1}]},&\hskip
5.69046pt\text{if}\hskip
5.69046pt\boldsymbol{a}_{i,j}^{v*}\in\mathcal{V}^{*}\\\ 0,&\hskip
5.69046pt\text{otherwise,}\\\ \end{cases}$ (5)

where we further assume a Markovian property for the activation pattern
vectors and build them using only the information available in the previous
year. The vector of risk factors $\boldsymbol{x^{\prime}}_{i}$ remains
unchanged during the development of the claim and is thus the same as in year
$j$. The vector of parameters $\boldsymbol{\beta}_{j+1}$, that we can write
$\boldsymbol{\beta}_{j+1,v*}$ is still specific to the current development
year, i.e. $j+1$ and the new activation pattern observed.

Also assuming the Markovian property for the payment vectors, we then have:

$\displaystyle
P_{i,j+1,c}|(A_{i,j+1,c}=1),\boldsymbol{A}_{i,j}\sim\text{Bernoulli}\big{(}\pi_{j+1,c}(\boldsymbol{x}_{i}^{\prime}\boldsymbol{\gamma}_{j+1,c})\big{)},$
(6)

where

$\displaystyle\pi_{j+1,c}(\boldsymbol{x}_{i}^{\prime}\boldsymbol{\gamma}_{j+1,c})=\frac{\exp{(\boldsymbol{x}_{i}^{\prime}\boldsymbol{\gamma}_{j+1,c}})}{1+\exp{(\boldsymbol{x}_{i}^{\prime}\boldsymbol{\gamma}_{j+1,c})}},$

with the parameter vectors $\boldsymbol{\gamma}_{j+1,c}$ depending once again
on the specific insurance coverage $c$.

Knowing which coverages are active in year $j+1$ and which of them incurred a
payment, we can finally determine the severity of these payments. As in
development year $j$, we suggest the use of the GAMLSS to model the severity
per coverage using the resulting vectors of activation and payment patterns:

$\displaystyle\operatorname{E}[Y_{i,j+1,c}|P_{i,j+1,c}=1,A_{i,j+1,c}=1,\boldsymbol{A}_{i,j}]=g^{-1}(\boldsymbol{x}_{i}^{\prime}\boldsymbol{\alpha}_{j+1,c}+\alpha_{j+1,c}^{*}(j+1)).$
(7)

###  2.3 Development years $j+k$, for $k\geq 2$

If enough data is available and if the insurer deems it appropriate, he can
repeat the model described in Sections 2.1 and 2.2 in years $j+k$, for $k\geq
2$ until development year $j^{*}$. In year $j^{*}$, we assume that even if the
claim is still open, its development stabilizes, i.e. it does not activate any
additional coverage. Additional payments can however still incur and although
claims with such long settlement delays are rarer in automobile insurance, we
should not disregard them. A longest lifetime often indicates a larger
severity. If the claim is still open in development year $j^{*}$, we write its
remaining severity starting from that year as
$\boldsymbol{Y}_{i,j^{*}}|\boldsymbol{P}_{i,j+k},\boldsymbol{A}_{i,j+k},\boldsymbol{A}_{i,j+k-1},\boldsymbol{Y}_{i,j^{*}}\geq\boldsymbol{Y}_{i,j+k}$
where we note the term $\boldsymbol{Y}_{i,j^{*}}\geq\boldsymbol{Y}_{i,j+k}$ in
the condition. $\boldsymbol{Y}_{i,j^{*}}$ represents the severity of claim $i$
for the development years starting from year $j^{*}$ and encapsulates the
possible remaining amounts that might be paid at later dates for claims with
longer lifetimes.

We schematize the model described in Sections 2.1, 2.2 and 2.3 in Figure 2. We
see how the different components interact and the dependence on the activation
pattern vector for year $j$, meaning $\boldsymbol{A}_{i,j}$. As shown in
Figure 2, we assume that a claim can activate a coverage at any time in a
given development year $j$ until the $30^{\text{th}}$ of December. The insurer
then records any payment made for the active coverages on the $31^{\text{st}}$
of December.

Figure 2: Summary of the activation patterns model

Example 1. To illustrate the model described in the previous sections, we
consider in Figure 3 a simple example. Assume that an insurer offers $C=2$
different insurance coverages. The set of initial activation patterns of
dimension $V=3$ is given by $\mathcal{V}=\\{(0\hskip 2.84544pt1)\hskip
5.69046pt(1\hskip 2.84544pt0)\hskip 5.69046pt(1\hskip 2.84544pt1)\\}$. We take
a closer look at the development of claim $i$ from development year $j=1$
onwards.

Figure 3: Illustration of the activation patterns model

  * •

Year $j=1$
Suppose that in development year $j=1$, we observe, for claim $i$, the
activation pattern $\boldsymbol{A}_{i,1}=\boldsymbol{v}_{i,1}^{1}=(0\hskip
2.84544pt1)$, i.e. claim $i$ activates the second coverage but not the first
one. Knowing $\boldsymbol{A}_{i,1}$, the possible payment pattern vectors are
$\boldsymbol{P}_{i,1}|\boldsymbol{A}_{i,1}=\\{(0\hskip 2.84544pt0)\hskip
5.69046pt(0\hskip 2.84544pt1)\\}$.

If we assume that the resulting pattern for the payments is $(0\hskip
2.84544pt1)$, then we have everything required to compute the severity for
claim $i$ in year 1. In this case, the severity for the first coverage, i.e.
$Y_{i,1,1}$, will be equal to 0 since claim $i$ did not activate it and hence,
no payment incurred. The severity for the second coverage, i.e. $Y_{i,1,2}$
can be estimated using simulations drawn from an appropriate distribution.
Let’s say we obtain $\boldsymbol{Y}_{i,j}$ = (0 ; 1,000), then the total
severity for claim $i$ in year 1 is equal to 1,000.

  * •

Year $j+1=2$
In the following year, the second coverage will remain active and the new set
of possible activation patterns $\mathcal{V}^{*}\subset\mathcal{V}$ is given
by $\mathcal{V}^{*}=\\{(0\hskip 2.84544pt1)\hskip 5.69046pt(1\hskip
2.84544pt1)\\}$.

Suppose that we obtain $\boldsymbol{A}_{i,2}|\boldsymbol{A}_{i,1}=(1\hskip
2.84544pt1)$. The possible payment patterns are then
$\boldsymbol{P}_{i,2}|\boldsymbol{A}_{i,2},\boldsymbol{A}_{i,1}\in\\{(0\hskip
2.84544pt0)\hskip 5.69046pt(0\hskip 2.84544pt1)\hskip 5.69046pt(1\hskip
2.84544pt0)\hskip 5.69046pt(1\hskip 2.84544pt1)\\}$ and we find, for example,
that $\boldsymbol{P}_{i,2}|\boldsymbol{A}_{i,2},\boldsymbol{A}_{i,1}=(1\hskip
2.84544pt1)$, i.e. the insurer recorded payments for both coverages.

We can finally compute the severity for both coverages for year 2 using the
appropriate models. We find for example that
$\boldsymbol{Y}_{i,j+1}|\boldsymbol{P}_{i,2},\boldsymbol{A}_{i,2},\boldsymbol{A}_{i,1}$
= (500 ; 100), which gives a total severity for year 2 of $600$.

The resulting total severity for claim $i$ is equal to $Y_{i}$ = 1,600.

###  2.4 IBNR, RBNS and RBNP claims

This section describes how the model handles claims at different stages of
their development. We first provide an illustration of two claims with varying
evaluation dates of the reserves and then describe the simulation routine that
we use to predict the reserves.

Example 2. We consider in Figures 4, 5 and 6 three different evaluation dates
of the reserves. In Figure 4, we evaluate the reserves in development year
$j=1$ at time $t_{1}$, before the 31st of December. At this time, claim 1
occurred and is reported since we observe its activation pattern
$\boldsymbol{A}_{1,1}$. However, the insurer has not recorded any payment yet,
making it a reported but not paid claim (RNBP). We have to estimate its
payment pattern $\boldsymbol{P}_{1,1}$ and corresponding severity for year $j$
(which corresponds to the first year of development for this specific claim),
$\boldsymbol{Y}_{1,1}$. Then we will also be able to simulate the necessary
components for year $j+1$, i.e. $\boldsymbol{A}_{1,2},\boldsymbol{P}_{1,2}$
and $\boldsymbol{Y}_{1,2}$. Claim 2 has not occurred yet and therefore, does
not appear in the analysis.

Figure 5 illustrates the case where we choose to evaluate the reserves at time
$t_{2}$, a few months after $t_{1}$ during the year $j=2$, for example on the
1st of January. This time, the insurer already recorded a payment for claim 1
but it is not settled yet. It is thus now a reported but not settled (RBNS)
claim. Knowing $\boldsymbol{A}_{1,1}$, we are able to predict
$\boldsymbol{A}_{1,2},\boldsymbol{P}_{1,2}$ and $\boldsymbol{Y}_{1,2}$. Claim
2 has occurred but has not been reported yet and is thus an incurred but not
reported (IBNR) claim. We will have to predict its first activation pattern
$\boldsymbol{A}_{2,1}$ and all subsequent components required for the
evaluation of the reserves.

Finally, we choose in Figure 6 $t_{3}$ as evaluation date. In this case, both
claims 1 and 2 have been reported although they are not settled yet and a
payment already took place for claim 1. Claim 1 is a RBNS claim while claim 2
is still a RBNP claim. For claim 1, we need to estimate the payment pattern
and severity for development year 2, i.e. $\boldsymbol{P}_{1,2}$ and
$\boldsymbol{Y}_{1,2}$. For claim 2 which is in its first year of
developement, we have observed $\boldsymbol{A}_{2,1}$ but will still need to
simulate
$\boldsymbol{P}_{2,1},\boldsymbol{Y}_{2,1},\boldsymbol{A}_{2,2},\boldsymbol{P}_{2,2}$
and $\boldsymbol{Y}_{2,2}$.

If the claims are still open after two development years, their final severity
will be given by, respectively,
$\boldsymbol{Y}_{1,2+k}|\boldsymbol{Y}_{1,2+k}\geq\boldsymbol{Y}_{1,2}$ and
$\boldsymbol{Y}_{2,2+k}|\boldsymbol{Y}_{2,2+k}\geq\boldsymbol{Y}_{2,2}$, as
discussed in Section 2.3.

Figure 4: Illustration of evaluation date $t_{1}$

Figure 5: Illustration of evaluation date $t_{2}$ Figure 6: Illustration of
evaluation date $t_{3}$

####  2.4.1 Simulation routine

To properly consider all open claims in a given dataset, we create a
simulation routine that we can tailor to the specific development stage of
different types of claims. We present the routine for the IBNR claims and the
RBNS claims, including those with longer development times for which we assume
that they will not activate any additional coverages.

##### IBNR claims

  1. 1.

For development year $j=1$:

    1. (a)

For each IBNR claim $i_{\text{IBNR}}$, with
$i_{\text{IBNR}}=1,...,n_{\text{IBNR}}$, simulate the first activation pattern
vectors $\boldsymbol{A}_{i_{\text{IBNR}},j}$ using the parameters estimated
for the multinomial logit model from in Section 2.1.

    2. (b)

For the inactive coverages, i.e. $A_{i_{\text{IBNR}},j,c}=0$, then the
corresponding $P_{i_{\text{IBNR}},j,c}$ is automatically generated as 0. For
the active coverages only, simulate the payment patterns using the Bernoulli
models with parameters $\hat{\pi}_{j,c}$.

    3. (c)

If the payment indicator $P_{i_{\text{IBNR}},j,c}=0$, either because the
corresponding coverage is not active or because we simulated it as such in the
previous step, then automatically set the corresponding severity for IBNR
claim $i_{\text{IBNR}}$ related to that coverage to 0, i.e.
$Y_{i_{\text{IBNR}},j,c}=0$. If $P_{i_{\text{IBNR}},j,c}=1$ from the previous
step, we simulate the severity from the distribution selected for the specific
coverage.

  2. 2.

For development years $j>1$:

    1. (a)

Knowing the activation pattern $\boldsymbol{A}_{i_{\text{IBNR}},j-1}$ for the
previous development year, only keep the activation patterns that are now
still possible. Re-normalize the probabilities for the remaining activation
patterns and simulate them using the multinomial logit model to obtain the
vector $\boldsymbol{A}_{i_{\text{IBNR}},j}$.

    2. (b)

If $A_{i_{\text{IBNR}},j,c}=0$, $P_{i_{\text{IBNR}},j,c}=0$. For the active
coverages only, simulate the payment patterns using the Bernoulli models with
parameters $\hat{\pi}_{j,c}$.

    3. (c)

If the payment indicator $P_{i_{\text{IBNR}},j,c}=0$, generate
$Y_{i_{\text{IBNR}},j,c}=0$. If $P_{i_{\text{IBNR}},j,c}=1$ from the previous
step, simulate the severity from the distribution selected for the specific
coverage. The total severity simulated for the IBNR claims is given by
$\boldsymbol{Y}^{IBNR}=\sum_{i_{\text{IBNR}}=1}^{n_{\text{IBNR}}}\sum_{j=1}^{J}\sum_{c=1}^{C}Y_{i_{\text{IBNR}},j,c}$.

##### RBNS claims

At the time of evaluation, these claims are already in their $j^{\text{th}}$
development year. We already know the activation patterns, payment patterns
and severities for year $j-1$ and the simulation routine is:

  1. 1.

For development year $j>1$:

    1. (a)

Knowing $\boldsymbol{A}_{i_{RBNS},j-1}$, with $i_{RBNS}=1,...,n_{RBNS}$, re-
normalize the probabilities for the remaining possible activation patterns and
simulate them using the multinomial logit model from Section 2.2 to obtain the
vector $\boldsymbol{A}_{i_{RBNS},j}$.

    2. (b)

If $A_{i_{RBNS},j,c}=0$, $P_{i_{RBNS},j,c}=0$. For the active coverages only,
simulate the payment patterns using the Bernoulli models with parameters
$\hat{\pi}_{j,c}$.

    3. (c)

If $P_{i_{RBNS},j,c}=0$, $Y_{i_{RBNS},j,c}=0$. If $P_{i_{RBNS},j,c}=1$ from
the previous step, simulate the severity from the distribution selected for
the specific coverage. The total severity simulated for these claims is given
by
$\boldsymbol{Y}^{\text{RBNS}}=\sum_{i_{RBNS}=1}^{n_{RBNS}}\sum_{j>1}^{J}\sum_{c=1}^{C}Y_{i_{RBNS},j,c}$.

##### RBNS claims with longer development times

To remain thorough in our analysis, we must also consider the longer RBNS
claims that we previously mentioned in Section 2.3. After a certain
development year $j^{*}$, we assume that the activation pattern for claim
$i_{RBNS+}$, with $i_{RBNS+}=1,...,n_{RBNS+}$,
$\boldsymbol{A}_{i_{RBNS+},j^{*}}$ remains unchanged. We therefore simulate
the additional severity that this claim can incur using the same severity
distribution as the one we selected for year $j+k<j^{*}$, where $j+k$ denotes
the last development year in which the activation patterns might have changed.
Note that $Y_{i_{RBNS+},j*,c}$ must be at least equal to
$Y_{i_{RBNS+},j+k,c}$. In what follows, we detail our method for coverage $c$.
For claim $i_{RBNS+}$, we work with the distribution of the random variable
$Z_{i_{RBNS+},c}=\max(Y_{i_{RBNS+},j+k,c},Y_{i_{RBNS+},j^{*},c})$. Its
cumulative distribution function is given by

$\displaystyle F_{Z_{i_{RBNS+},c}}(z)$
$\displaystyle=\operatorname{P}[Z_{i_{RBNS+},c}\leq
z]=\operatorname{P}[\max(Y_{i_{RBNS+},j+k,c},Y_{i_{RBNS+},j^{*},c})\leq z].$

We can rewrite this as

$\displaystyle F_{Z_{i_{RBNS+},c}}(z)=\begin{cases}0,&z\leq
Y_{i_{RBNS+},j^{*},c}\\\
F_{\boldsymbol{Y}_{j+k,c}}(z),&z>Y_{i_{RBNS+},j^{*},c}.\end{cases}$

The inverse cumulative distribution function of $Z$ is then given by

$\displaystyle
F_{Z_{i_{RBNS+},c}}^{-1}(u)=\begin{cases}Y_{i_{RBNS+},j^{*},c},&0\leq u\leq
F_{\boldsymbol{Y}_{j+k,c}}^{-1}(Y_{i_{RBNS+},j^{*},c})\\\
F_{\boldsymbol{Y}_{j+k,c}}^{-1}(u),&u>F_{\boldsymbol{Y}_{j+k,c}}^{-1}(Y_{i_{RBNS+},j^{*},c}).\end{cases}$
(8)

##  3 Numerical Application

We dedicate this section to the application of the model described in Section
2 to an automobile dataset from a Canadian insurance company. We first provide
an overview and some exploratory analysis of the dataset before detailing the
results obtained from the estimation of the different parameters of our model.
We finally present the results obtained by simulations for the final reserves
and compare them with those obtained using three other models that we describe
in greater details below.

###  3.1 Data exploration

Our dataset contains 656,153 claims which have occurred between the
$1^{\text{st}}$ of January 2015 and the $30^{\text{th}}$ of June 2021. Each of
these claims comes with information related either to the insured, the car
driven or the claim itself, and impacts at least one of four possible
insurance coverages provided by the insurer.

####  3.1.1 Risk factors

Table 1 presents the risk factors that we will use in our analysis to build
the different models. We provide further insights in Figure 7. Note that the
risk factors GENDER and YOB contained a significant proportion of missing
values. We used the R package mice to fill them in. This package uses Fully
Conditional Specification (FCS) where binary data (GENDER) is imputed via
logistic regression while unordered categorical data (YOB) is imputed with
polytomous logistic regression. More details can be found in van Buuren and
Groothuis-Oudshoorn (2011). Note that this data filling procedure did not
cause any significant changes in the results that we will discuss later on.

Table 1: Description of the risk factors provided for each claim in the
dataset Risk factors
---
GENDER |  Gender of the insured.
YOB |  Year of birth: decade during which the insured was born.
VU |  Use of the vehicle made by the insured.
AM |  Annual distance driven by the insured (in km).
PROV |  Place of occurrence of the claim: one of the Canadian provinces or the USA.
FR |  Fault rating: evaluation of the insured’s level of responsability in the accident.

Our dataset includes around $60\%$ of male against $40\%$ of female insureds,
more than half of them being born between 1960 and 1989. Around half of the
insureds use their vehicle for commuting purposes while an additional $37\%$
use it for pleasure. The remaining $13\%$ use their vehicle for commercial or
business reasons, or did not disclose that information to the insurer. $76\%$
of the insureds drive between 10,000 and 20,000 kilometers per year according
to the annual mileage risk factor which is the only continuous one in our
dataset. A bit more than $50\%$ of the claims occur in Ontario, $20\%$ in
Alberta and the remaining $30\%$ take place in the other Canadian provinces or
in the United States of America. Finally, the insurance company assesses the
insured’s level of responsibility in an accident and issues a fault rating. In
$37\%$ of the claims, the insured was not considered to be at fault. The
rating could not be applied for $32\%$ of the claims and in $27\%$ of the
cases, the insured was deemed at fault. In very few cases could the company
determine a partial level of responsibility or conclude that no fault was
committed by neither of the parties involved in the claim.

Figure 7: Risk factors.

####  3.1.2 Insurance coverages

As previously mentioned, each of the 656,153 claims in our dataset must impact
at least one of four coverages offered to the policyholders to be activated in
the systems of the insurer. We provide a brief definition of these four
coverages in Table 2.

Table 2: Description of the four insurance coverages provided within the
policy Insurance coverages
---
Accident Benefits |  Compensation for loss of revenue, funeral expenses, medical expenses, death,…
Bodily Injury |  Compensation for medical expenses or loss of revenue of a third party.
Vehicle Damage |  Compensation for the damages incurred to the insured or another party’s vehicle.
Loss of Use |  Compensation of costs in case of the temporary replacement of a vehicle or any other alternative transportation means used during vehicle repairs.

To better understand the dynamics of our portfolio, Table 3 shows the
importance of each coverage in terms of the proportion of claims and
proportion of the total cost. Unsurprisingly, the Vehicle Damage coverage is
the one that most often comes into play with $96.39\%$ of the 656,153 claims
activating it. A bit more than half of the claims impact the Loss of Use
coverage. Only $5.70\%$ of the claims in the portfolio activate the Bodily
Injury coverage. Similarly and as expected, less than $10\%$ of the total
claims impact the Accident Benefits coverage.

The proportions are slightly different when we look at the repartition of the
total portfolio cost over the four coverages. In particular, Loss of Use is
the coverage that weights the less in terms of cost, representing less than
$4\%$ of the total cost of the portfolio. Despite being activated by less than
$6\%$ of all claims, the Bodily Injury coverage still represents around $13\%$
of the total portfolio cost.

Table 3: Weight of each coverage in the portfolio Coverage |  $\%$ of claims |  $\%$ of the total cost
---|---|---
Accident Benefits | 9.42 | 12.82
Bodily Injury | 5.70 | 13.13
Vehicle Damage | 96.39 | 70.44
Loss of Use | 51.89 | 3.61

Table 4 shows some descriptive statistics for each of the four coverages.
Bodily Injury claims are those for which the average payments are the biggest,
followed by the Accident Benefits coverage. The Loss of Use coverage presents
by far the lowest payments as was already hinted by Table 3. We also notice
that the payments for the Accident Benefits, Bodily Injury and Vehicle Damage
coverages present large values in the higher quantiles, indicating that their
corresponding distributions are probably heavy-tailed.

Table 4: Descriptive statistics for the four insurance coverages Coverage | Mean | Std. dev. | Quantiles | Max.
---|---|---|---|---
0.5 | 0.75 | 0.95 | 0.99
Accident Benefits | 12,386 | 53,561 | 3,215 | 6,909 | 47,757 | 127,896 | 2,435,334
Bodily Injury | 23,271 | 76,027 | 4,000 | 15,150 | 98,449 | 322,612 | 2,039,570
Vehicle Damage | 5,040 | 8,121 | 2,605 | 5,830 | 17,984 | 40,611 | 149,399
Loss of Use | 545 | 620 | 419 | 714 | 1,000 | 2,336 | 52,777

####  3.1.3 Activation and payment delays

In Table 5, we take a look at the average activation delays per coverage, i.e.
the average delays between the reporting of a claim and the activation of the
coverages in the insurer’s systems.

The Accident Benefits, Vehicle Damage and Loss of Use coverages are typically
activated without delay for the vast majority of the claims. For the Bodily
Injury coverage however, around $10\%$ of the claims only activate it with a
delay of at least one year after the reporting date.

Example 3. To illustrate this, consider the case where a policyholder is at
fault in an accident with a third party. The policyholder reports the claim to
the insurer who, considering the reparations needed to the vehicle, records it
as a Vehicle Damage claim. The following year, the third party realises that
he has been injured in the accident and demands a compensation. The insurer
thereby needs to record the claim as a Bodily Injury as well as Vehicle Damage
claim. This illustrates how in some cases, some coverages become active at a
later time during the development of the claim rather than directly upon its
reporting to the insurance company.

Appendix Appendix A: Observed activation patterns further illustrates the
dynamics between the different coverages by presenting the activation patterns
observed in the dataset for the first development year of the claims.

Table 5: Percentage of claims with different activation delays for the four coverages | Activation delays
---|---
Coverage | No delay | 1 year |  $\geq$2 years
Accident Benefits | 96.65 | 3.18 | 0.11
Bodily Injury | 89.89 | 7.70 | 2.11
Vehicle Damage | 98.20 | 1.73 | 0.07
Loss of Use | 97.80 | 2.17 | 0.03

Once a claim activates a coverage, we are interested in knowing when the first
payment will take place. The payment delays refer to the time elapsed between
the reporting date of a claim and the date at which the insurer records a
first payment for the coverage(s) that the claim activated. Thereby, we do not
assume that the payments directly follow the activation of the coverages. This
is clearly shown in our data and illustrated in Figure 8. The Vehicle Damage
coverage is the one for which payments happen the fastest with a payment
taking place in the same year as the reporting year for $77\%$ of the claims
that activate that coverage. Payments come directly in the first development
year for $66\%$ of the claims that activate the Loss of Use coverage but only
for $50\%$ of the claims that activate the Accident Benefits coverage and
$27\%$ of the claims for the Bodily Injury coverage. For the rest of the
claims, almost all remaining payments take place in the second year following
the reporting date for the Vehicle Damage and Loss of Use coverages, such that
when we reach the third development year, these coverages practically never
incur additional payments. For the other two coverages however, the claimant
might have to wait a bit longer to receive his payments. When we reach the
fourth development year, only $63\%$ of the claims have received their first
payment for the Accident Benefits coverage against only $46\%$ for the Bodily
Injury coverage. It is only starting from the fifth development year that
almost all claims have received their first payment regardless of the coverage
impacted.

Example 3 (continued). Consider again our earlier example. Even though claim
activated the Bodily Injury coverage in the second year after reporting, the
third party that was injured might still wait a little to receive his payment
while the insurance company consults with medical or legal experts to
determine the final amount that the claimant is owed.

Figure 8: Payment delays per coverage

###  3.2 Estimation

We apply our model to the data introduced in Section 3.1. We present the
results of the model fitting performed for the three components of the model,
namely the activation patterns, payment patterns and severities. Since with
work with automobile claims and given the limited number of development years
available in our dataset, we work with the activation patterns and payment
vectors for development years $j=1$ and $j=2$. For the claims that are already
in at least their third development year at the evaluation of the reserves, we
apply the methodology laid out in Section 2.3.

####  3.2.1 Activation patterns

Using the same notation as in Section 2, we have $C=4$ insurance coverages,
leading to $V=15$ possible activation patterns. We fit the multinomial logit
model for the activation patterns using maximum likelihood estimation with the
likelihood function given by

$\displaystyle\mathcal{L}(\boldsymbol{A}_{i,j}|\boldsymbol{x}_{i}\boldsymbol{\beta})=\prod_{i=1}^{n}\frac{\exp{(\boldsymbol{x}_{i}^{\prime}\boldsymbol{\beta}_{j,v})}}{\sum_{k=1}^{V}\exp{(\boldsymbol{x}_{i}^{\prime}\boldsymbol{\beta}_{j,k}})}.$
(9)

In Appendix Appendix B: Multinomial logit model, we provide the full list of
parameters estimated by maximum likelihood for each of the $V=15$ possible
activation patterns.

We observe that the impact of some risk factors becomes very important for the
$5^{\text{th}}$ pattern, i.e. the one in which a claim simultaneously
activates the Bodily Injury and Loss of Use coverages. In particular, the
probability of observing that specific activation pattern increases quite
substantially if the claim occurred in the USA. However, if the claim occurred
in Quebec or in Saskatchewan, the probability of activating both the Bodily
Injury and the Loss of Use coverages greatly decreases. This does not come as
a surprise due to the legislation regarding automobile insurance in these two
states. In Quebec, the Bodily Injury coverage does not exist: if an individual
is injured in a car accident, he is covered by the public automobile insurance
plan provided by the Société de l’assurance automobile du Quebec (SAAQ) rather
than by a private insurance company. In Saskatchewan, the insured must choose
between a “no fault" and a “tort" auto injury insurance coverage offered by
the Saskatchewan driver’s licensing and vehicle registration. Almost all
residents choose the “tort" coverage under which they are insured regardless
of whether they are at fault or not. As such, the Bodily Injury coverage very
rarely appears for claims that occur in that province. Consequently, we see
that the parameter estimates for Quebec and Saskatchewan are always quite
negative for all the activation patterns that include the Bodily Injury
coverage.

Moreover, when the fault rating is either not available or not applicable or
when the degree of fault of the insured is evaluated at $25\%$, the
probability of the claim activating both the Bodily Injury and Loss of Use
coverages decreases. Once again, since the provincial legislation often
impacts the way Bodily Injury claims are handled, we know that for many of
those claims, insurers will not record a fault rating or deem it inapplicable
to the specific situation. We can thus expect for example that when the fault
rating is not applicable for a claim, this claim has occurred in Quebec where
the Bodily Injury coverage does not apply and hence, the probability of
activating it decreases.

Finally, the year of birth ia also a big driver for this particular pattern
with the probability of observing it increasing for the younger insured
individuals born in the 1990s and after 2000. It is commonly observed in
automobile insurance data that younger drivers tend to cause more frequent and
serious accidents. As a consequence, we can expect to observe more accidents
with injuries or even casualties and loss of use of a vehicle in the younger
population. Appendix Appendix B: Multinomial logit model illustrates this
well, with the insureds born after 2000 being the only cohort that always has
an increasing impact on the probability of observing each activation pattern,
even though they only represent $1.52\%$ of all the insureds.

####  3.2.2 Payment patterns

As described in Section 2, the payment patterns help actuaries determine
whether or not a payment took place for an active coverage $c$ of claim $i$ in
development year $j$ i.e. in the case where $A_{i,j,c}=1$. We fit in total
eight Bernoulli regressions: one for each of the four coverages in years $j=1$
and $j=2+$ and use again maximum likelihood optimization to obtain the
parameter estimates for each of the models. Table 6 displays the results of
these estimations. When comparing them with the payment delays observed in
Figure 8, we see that the models provide overall a good fit. The Vehicle
Damage coverage is the one for which payments come the fastest with around
$80\%$ of the claims receiving a payment in their reporting year, i.e. in
development year $j=1$. The Loss of Use coverage comes in second place with an
estimated $70\%$ of claims receiving a payment in the first year against
around $65\%$ of claims in the observed data. The estimated percentage
$\pi_{j,\text{AB}}$ for the Accident Benefits is very similar to the observed
value with a payment happening in $j=1$ for almost half of the claims.
Finally, while we observe that a bit less than $30\%$ of the claims receive a
payment in the first year for the Bodily Injury coverage, the estimation
obtained from our model is around $35\%$.

Table 6: Fitted average probabilities of observing a payment Probability | $j=1$ | $j=2+$
---|---|---
$\pi_{j,\text{AB}}$ | 0.5155 | 0.3578
$\pi_{j,\text{BI}}$ | 0.3484 | 0.2812
$\pi_{j,\text{VD}}$ | 0.8203 | 0.1138
$\pi_{j,\text{LoU}}$ | 0.7070 | 0.0720

####  3.2.3 Payment severities

From the observations in Table 4, we note the importance of considering long-
tailed distributions to model the severity of payments, as is usually done in
the actuarial literature. Among others, Frees et al. (2009) opt for the
Generalized Beta of the second kind distribution to accommodate the long-tail
nature of claims. Figure 9 shows the histogram of the payments made for the
four coverages. Note that for the Accident Benefits and Bodily Injury
coverages, we cut out most of the tail of the histograms in the right part of
the graph to ease the readability. These graphs further illustrate the need on
the one hand to consider long-tailed distributions for the coverages and on
the other hand the benefits that we can earn from selecting different
distributions for the coverages rather than a single distribution common to
all. We are thereby able to tailor the model to the specific distribution
profiles of the different types of claims.

For each coverage, we consider five distributions commonly used. We fit the
following severity models for both years $j=1$ and $j=2+$: Log-Normal, Gamma,
Pareto, Generalized Beta of the second kind and Weibull distributions.
Appendix Appendix C: Choice of the severity models presents the Akaike
Information Criterion (AIC) and Bayesian Information Criterion (BIC) for all
the models for both time periods considered and all four insurance coverages.
For the first development year, the Generalized Beta of the second kind seems
to be the most appropriate distributional choice for Accident Benefits,
Vehicle Damage and Loss of Use while the Pareto distribution is more suited to
model the payments incurred in the Bodily Injury coverage. For the other
development years, the Generalized Beta of the second kind is again the
distribution that presents the lowest AIC for Accident Benefits, Vehicle
Damage and Loss of Use while we prefer the Weibull distribution to model
Bodily Injury claims. This illustrates the advantages of separating the
severity modelling of the different coverages. From the model fitting, we see
that the distribution profile of the Bodily Injury coverage is different than
that of the others and varies with the development periods.

Figure 9: Histograms of the observed payments per coverage

###  3.3 Predictive distributions and model comparisons

With the data presented in Section 3.1 and the models fitted in Section 3.2,
we can now focus in this section on presenting the results obtained after
performing the evaluation of the reserves for our portfolio of automobile
insurance claims, choosing the $1^{\text{st}}$ of January 2019 as our
evaluation date. We present the predictive distributions for the reserve of
the whole portfolio and the four coverages separately. We compare these
results to the true reserve and to that obtained from fitting some more
classical and commonly used reserving models.

For all the models that we will consider in this section, we perform 5,000
simulations using the simulation routine described in Section 2.4.1. As shown
in Appendix Appendix D: Results stability, this is a sufficiently large number
to bring stability to our results. We first present the estimations from the
activation patterns model before comparing them to other reserving models.

Figure 10 displays the predictive distribution for the total reserves in our
portfolio, i.e. for both the IBNR and RBNS claims on the left and the total
reserves only for the RBNS claims on the right. In both cases and in all
graphs that will follow, the red line marks the true reserve, the blue dotted
line shows the average value of the predictions and the continuous blue line
depicts the $95^{\text{th}}$ quantile of the distribution. For all claims, the
true reserve amounts to 542.55M CAD, whereas our model produces an average
predicted amount of 561.62M CAD. For the RBNS claims only, the true reserve is
equal to 524.91M CAD and the simulations give 536.86M CAD. Note that the true
reserve amounts that we provide here are minimum amounts since at the
evaluation date, a bit more than $3\%$ of the claims are still open in the
portfolio. To handle them, we chose to consider the date of the last observed
payment as the settlement date. Thereby, we must be aware that the true
severity of these claims will in all likelihood be greater than the one
observed.

Figure 10: Simulated reserves as of the 1st of January 2019. The red line,
dashed blue line and continuous blue lines depict, respectively, the true
reserve amount, the average and the $95^{\text{th}}$ quantile of the
simulations

With the IBNR claims representing only $3.25\%$ of the total final cost in our
evaluation dataset, we choose to focus on the results of the RBNS claims only.
Figure 11 presents the predictive distributions obtained for the reserves of
the four insurance coverages. As in Figure 10, the red lines show the true
reserve amounts while the dotted and continuous blue lines represent,
respectively, the mean and the $95^{\text{th}}$ quantile of the distributions.
In Table 7, we provide further details on these results. Since financial
legislation around the world typically require insurers to set aside an amount
based on a high quantile of the predictive distribution of their reserves, we
provide the $95\%$ Value-at-Risk (VaR) in addition to the mean for each of the
four coverages and for the portfolio as a whole.

Figure 11: Simulated reserves per insurance coverage (RBNS). The red lines, dashed blue lines and continuous blue lines depict, respectively, the true reserve amounts, the average and the $95^{\text{th}}$ quantiles of the simulations Table 7: Simulated RBNS reserves (in M CAD). Coverage | True reserves | Activation Patterns
---|---|---
Mean | $\text{VaR}_{0.95}$
Accident Benefits |  $>$158.90 | 162.15 | 196.64
Bodily Injury |  $>$288.17 | 300.02 | 312.58
Vehicle Damage |  $>$73.63 | 70.80 | 72.95
Loss of Use |  $>$4.20 | 3.91 | 4.01
Total |  $>$524.91 | 536.86 | 574.58

For the full portfolio, the activation patterns model overestimates the true
reserves, providing rather conservative estimates. This is also the case for
the Accident Benefits and Bodily Injury coverages. For the Vehicle Damage and
Loss of Use coverages that represent together around $15\%$ of the total RBNS
reserve, the predictions slightly underestimate the true amounts.

In Sections 3.3.1, 3.3.2 and 3.3.3, we pursue the analysis of our results by
comparing them to those obtained using, respectively, an aggregate model, a
replica of the activation patterns model but without taking dependence into
account and the individual reserving model proposed by Antonio and Plat
(2014).

####  3.3.1 Activation patterns model vs Aggregate models

Table 8 compares the reserve estimates obtained with the proposed model based
on the activation patterns to those of the classical Overdispersed Poisson
(ODP) Chain Ladder and to the true reserves, both for the portfolio as a whole
and for the four insurance coverages taken separately.

On average, the aggregate model underestimates the true reserves in each case.
We need to take the $95\%$ VaR to obtain an estimate of the whole RBNS reserve
that is above the minimum observed value of 524.91M CAD. Considering that we
expect this amount to be higher due to the open claims, our model gives a more
conservative estimate of the total RBNS reserve with an average simulated
amount of 536,86M CAD. Most importantly, the aggregate model does not perform
well at the coverage level. In each case and particularly for the Accident
Benefits and Bodily Injury coverages, the predictions underestimate the true
reserve amounts. Using such a model, the insurer is thus blind to the dynamics
of his portfolio and can only get an idea of what his total reserve amounts
to.

In addition, since our data spans from 2015 to mid-2021, we only have a
limited number of development years and thereby rather small claim triangles
to perform the analysis with aggregate models. To obtain more appropriate
estimates in this specific case, the insurer should consider making use of
tail factors.

Table 8: Comparison between the activation patterns model and the Overdispersed Poisson Chain Ladder (in M CAD) Coverage | True reserve | Activation patterns | ODP
---|---|---|---
Mean | $\text{VaR}_{0.95}$ | Mean | $\text{VaR}_{0.95}$
Accident Benefits |  $>$158.90 | 162.15 | 196.64 | 71.01 | 81.66
Bodily Injury |  $>$288.17 | 300.02 | 312.58 | 177.64 | 204.63
Vehicle Damage |  $>$73.63 | 70.80 | 72.95 | 40.87 | 48.43
Loss of Use |  $>$4.20 | 3.91 | 4.01 | 3.02 | 3.88
Total |  $>$524.91 | 536.86 | 574.58 | 493.14 | 537.16

####  3.3.2 Activation patterns model vs independence model

To strengthen our argument in favour of a model that takes into account the
possible dependence between the insurance coverages, we perform a series of
independence tests on our data. First, we test for independence between each
possible pair of insurance coverages using the chi-squared goodness of fit
test. Each of these two-way interaction tests clearly rejects the hypothesis
of independence. We also perform likelihood ratio tests where we compare
models with no interaction, two, three and four-way interactions. Once again,
the tests clearly show the presence of dependence between the coverages and
reject the simpler models in favour of the more complex ones in each scenario.

To assess the impact of modelling this dependence on the reserve estimates, we
build an independence model. We reproduce the activation patterns model but
rather than modelling the activation patterns $\boldsymbol{A_{i,j}}$ and
$\boldsymbol{A_{i,j+1}}$ with the multinomial logit model described in Section
2, we model the activation of the coverages using four separate and
independent Bernoulli regressions for development periods $j=1$ and $j=2+$. We
present the results obtained with this model in Table 9 and compare them to
the true reserves and the estimates from our activation patterns model.

With an average estimated amount and average $95\%$ VaR of, respectively,
495.32M CAD and 526.93M CAD, the independence model clearly underestimates the
true reserves. The predictions are also less accurate for each of the four
coverages and the activation patterns model outperforms the independence model
in each case. Leaving the dependence between the coverages out of the
modelling process can thereby lead to underestimated reserves which can put
the insurer at risk of financial sanctions.

Table 9: Comparison between the activation patterns model and the independence model (in M CAD) Coverage | True reserve | Activation patterns | Independence
---|---|---|---
Mean | $\text{VaR}_{0.95}$ | Mean | $\text{VaR}_{0.95}$
Accident Benefits |  $>$158.90 | 162.15 | 196.64 | 143.41 | 173.50
Bodily Injury |  $>$288.17 | 300.02 | 312.58 | 278.18 | 290.31
Vehicle Damage |  $>$73.63 | 70.80 | 72.95 | 69.94 | 72.07
Loss of Use |  $>$4.20 | 3.41 | 4.01 | 3.78 | 3.89
Total |  $>$524.91 | 536.86 | 574.58 | 495.32 | 526.93

####  3.3.3 Activation patterns model vs another individual model

Next to the aggregate and independence models, we compare the model based on
the activation patterns to the individual reserving model first introduced by
K. Antonio and R. Plat in 2014. Working in continuous time, the authors use a
hierarchical process to estimate the reserves for the RBNS claims. The
interested reader can find more details on their model in Antonio and Plat
(2014).

We focus here solely on the Accident Benefits coverage. Figure 12 shows the
predictive distributions for the RBNS reserves of this coverage obtained with
the activation patterns model (in blue) and Antonio and Plat (2014)’s
individual model (in grey). As before, the red line shows the true amount of
the reserves equal to 158,90M CAD while the dotted and continuous lines
represent, respectively, the average reserve estimates and $95\%$ VaR obtained
with each model. For the activation patterns model and Antonio and Plat
(2014)’s model, the average predictions are, respectively, 162,15M CAD and
143,46M CAD while the $95\%$ VaR amount to 196,64M CAD and 168,30M CAD. The
predictive distribution of the activation patterns model is thereby closer on
average to the true reserve. However, Antonio and Plat (2014)’s model gets a
closer estimate in terms of the VaR whereas in this case, our model
overestimates the reserve.

Figure 12: Predictive distributions of the RBNS reserves obtained with the
activation patterns model and the model of Antonio and Plat Antonio and Plat
(2014). The red line shows the true reserve amount and the blue and grey
dotted and continuous lines depict the average predictions and $95\%$ VaR
obtained with, respectively, Antonio and Plat Antonio and Plat (2014)’s model
and the model based on the activation patterns

####  3.3.4 Results summary

Table 10 summarizes the results discussed in Sections 3.3.1 and 3.3.2. Of the
four different models, the independence model is the one that underestimates
the total reserve the most in terms of the Value-at-Risk.

The aggregate model presents the lowest average estimates but its predictive
distribution is more heavy-tailed than that of the independence model and the
$95\%$ VaR of 537,16M CAD seems to be a more appropriate estimate. However,
this model performs very poorly when we take the coverages separately and
consistently underestimates the true reserves. It is thereby not fitted to
gain a better understanding of the dynamics of the portfolio. In addition and
as mentioned in Section 3.3.1, the limited amount of development years
available forces the insurer to use tail factors, making the model less
reliable or apt for predictions.

The model based on the activation patterns provides a greater level of
accuracy in the estimation of the total reserves both on average and in the
higher quantiles of the predictive distributions. Even though it is the model
that overestimates the true reserve the most, especially in terms of the
VaRwhere we could prefer the total estimate of the aggregate model, it
provides more accurate predictive distributions at the level of the different
insurance coverages.

Table 10: Model comparison - summary (in M CAD) Coverage | Observed | Simul. | Act. pat. | ODP | Ind. | [1]
---|---|---|---|---|---|---
Accident Benefits | $>$158.90 | Mean | 162.15 | 71.01 | 143.41 | 143.46
$\text{VaR}_{0.95}$ | 196.64 | 81.66 | 173.50 | 169.30
Bodily Injury | $>$288.17 | Mean | 300.02 | 177.64 | 278.18 | -
$\text{VaR}_{0.95}$ | 312.58 | 204.63 | 290.31 | -
Vehicle Damage | $>$73.63 | Mean | 70.80 | 40.87 | 69.94 | -
$\text{VaR}_{0.95}$ | 72.95 | 48.43 | 72.07 | -
Loss of Use | $>$4.20 | Mean | 3.91 | 3.02 | 3.78 | -
$\text{VaR}_{0.95}$ | 4.01 | 3.88 | 3.89 | -
Total | $>$524.91 | Mean | 536.86 | 493.14 | 495.32 | -
$\text{VaR}_{0.95}$ | 574.58 | 537.16 | 526.93 | -

##  4 Conclusion

In this paper, we introduce a model based on activation patterns for the
insurance coverages to estimate the claims reserves on an individual basis
while taking into account the dependence between the different coverages
provided by an insurer. More specifically, we analyze the way in which a
single claim can simultaneously activate multiple coverages. Based on the
predictions of the so-called activation patterns of the coverages in year $j$,
we then predict whether an active coverage will incur a payment and its
corresponding amount. Using a Canadian automobile dataset, we apply the model
and compare the results obtained to those of three additional models: the
classical Overdispersed Poisson Chain Ladder, a replica of the activation
patterns models that does not take the dependence into account and Antonio and
Plat (2014)’s individual reserving model. We observe that when taking the
dependence between the coverages into account, we reach on average more
accurate estimates of the reserves than the other models. The predictive
distribution that we obtain is however more heavy-tailed than the others,
leading to larger Value-at-Risk. We can however consider these as more
conservative compared to other models for which the estimates are sometimes
too close to the observed amount. The main contribution of our model resides
in its capacity to predict the specific reserve estimates for each insurance
coverages more accurately, thereby allowing the insurer to better understand
the dynamics of his portfolio. This is particularly interesting considering
some regulatory frameworks such as Solvency II that requires a detailed
analysis of the reserves on a line of business basis and that consistently
advocates for the use of more prudent and conservative modelling techniques.

## Appendix A: Observed activation patterns

Table 11 presents the activation patterns $\boldsymbol{A}_{i,1}$ observed in
the dataset for the first development year of the claims. The value $1$ stands
for the activation of the coverage. $85.67\%$ of all claims activate the
Vehicle Damage coverage either alone or simultaneously with the Loss of Use
coverage. The Accident Benefits coverage is most often activated together with
the Vehicle Damage coverage or on its own. We also observe that around $1\%$
of all claims simultaneously activate all four coverages upon their reporting.

Table 11: Frequency of the activation patterns observed in the first development year Accident Benefits | Bodily Injury | Vehicle Damage | Loss of Use |  $\%$ of claims
---|---|---|---|---
0 | 0 | 1 | 0 | 42.95
0 | 0 | 1 | 1 | 42.72
1 | 0 | 1 | 0 | 4.98
0 | 1 | 1 | 1 | 1.99
1 | 0 | 0 | 0 | 1.44
0 | 1 | 1 | 0 | 1.24
1 | 0 | 1 | 0 | 1.07
0 | 0 | 0 | 1 | 1.04
1 | 1 | 1 | 1 | 1.01
0 | 1 | 0 | 0 | 0.61
1 | 1 | 1 | 0 | 0.43
1 | 1 | 0 | 0 | 0.38
1 | 0 | 0 | 1 | 0.10
0 | 1 | 0 | 1 | 0.04
1 | 1 | 0 | 1 | 0.01

## Appendix B: Multinomial logit model

Table 12: Parameter estimates for the multinomial logit model.

Risk factors | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
Intercept | 3.45 | 3.44 | -0.26 | -12.78 | 1.31 | 1.93 | 0.30 | -1.04 | 1.14 | 1.67 | 0.28 | -1.68 | 0.56 | 1.69
PROV | New Brunswick | 0.21 | -0.16 | 1.11 | 0.43 | 0.05 | -0.49 | -0.93 | 0.29 | -0.18 | 0.35 | 0.48 | 1.07 | 0.40 | 0.35
Newfoundland and Labrador | 0.55 | 0.02 | 0.58 | 0.63 | 0.09 | 0.27 | -0.06 | -0.16 | -0.09 | 0.26 | 0.79 | 0.83 | 0.61 | 0.65
Nova Scotia | 0.36 | -0.05 | 1.09 | 0.18 | -0.08 | -0.72 | -1.08 | 0.23 | -0.73 | 0.30 | 0.48 | 0.01 | -0.01 | 0.29
Ontario | -0.14 | -0.12 | 0.93 | 0.25 | -0.35 | -0.32 | -0.50 | -0.15 | -1.26 | -0.34 | 0.15 | 0.41 | -0.65 | -0.31
Other | 0.57 | -0.14 | 0.61 | -0.62 | -0.06 | -0.08 | 0.25 | 0.10 | 0.34 | -0.28 | 0.76 | 1.87 | 0.71 | 0.29
Prince Edward Island | 0.93 | -0.13 | 0.93 | 0.98 | 0.17 | -0.75 | -0.57 | 0.31 | 0.01 | 0.31 | 0.58 | -3.21 | 1.11 | 0.61
Quebec | 1.64 | -0.11 | -2.10 | -19.74 | -2.77 | -3.31 | -2.34 | -2.10 | -1.97 | -2.01 | -1.66 | -0.22 | -2.46 | -2.34
Saskatchewan | 1.53 | -0.40 | -1.21 | -23.02 | -1.04 | -2.84 | -1.05 | -0.78 | -0.41 | -1.27 | -2.73 | -390.59 | -0.60 | -2.37
USA | -5.52 | -1.50 | -3.83 | 11.02 | -5.16 | -2.24 | -7.22 | 0.32 | -5.19 | -0.99 | -3.53 | 0.10 | -3.45 | -0.86
YOB | 2000+ | 0.71 | 0.70 | 3.71 | 14.79 | 3.87 | 3.80 | 3.90 | 2.92 | 3.61 | 3.48 | 3.94 | 3.49 | 3.84 | 3.62
1930s | -0.34 | -0.21 | -1.13 | 9.95 | -1.09 | -0.84 | -0.96 | -1.10 | -0.72 | -1.0 | -1.47 | -1.95 | -1.07 | -1.04
1940s | -0.41 | -0.26 | -0.72 | 10.46 | -0.49 | -0.56 | -1.39 | -1.82 | -1.16 | -1.08 | -1.90 | -2.24 | -1.68 | -1.64
1950s | -0.55 | -0.27 | -0.82 | 10.36 | -0.56 | -0.73 | -0.84 | -1.57 | -0.69 | -0.82 | -1.54 | -1.81 | -1.36 | -1.29
1960s | -0.55 | -0.29 | -0.58 | 10.44 | -0.33 | -0.44 | -0.81 | -1.31 | -0.82 | -0.88 | -1.79 | -1.36 | -1.57 | -1.60
1970s | -0.60 | -0.26 | -0.54 | 10.37 | -0.30 | -0.37 | -0.75 | -1.32 | -0.74 | -0.81 | -1.88 | -1.43 | -1.52 | -1.51
1980s | -0.73 | -0.33 | -0.48 | 10.18 | -0.29 | -0.40 | -0.69 | -1.14 | -0.78 | -0.71 | -1.62 | -1.18 | -1.27 | -1.26
1990s | -0.44 | -0.30 | -0.30 | 11.04 | -0.08 | -0.12 | 0.15 | -1.20 | -0.07 | -0.48 | -0.50 | -0.50 | -0.39 | -0.63
GENDER | Male | 0.24 | 0.08 | -0.22 | -0.44 | -0.23 | -0.28 | -0.26 | -0.89 | -0.35 | -0.89 | -0.28 | -0.69 | -0.45 | -0.92
AM | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
VU | Commercial | 0.68 | -0.11 | 0.83 | 0.25 | 0.71 | -0.19 | 0.28 | -0.299 | 0.89 | -0.19 | 0.71 | -0.55 | 0.84 | -0.27
Commute | -0.13 | -0.00 | -0.08 | -0.20 | -0.05 | -0.14 | 0.03 | -0.11 | 0.15 | 0.09 | 0.07 | -0.34 | 0.17 | -0.02
Not Available | 2.37 | 0.13 | 2.73 | -0.07 | 2.59 | 0.28 | 2.25 | -1.88 | 1.44 | -1.13 | 0.11 | -3.26 | 1.28 | -1.50
Pleasure | 0.15 | -0.03 | -0.00 | -0.15 | 0.05 | -0.21 | 0.25 | -0.22 | 0.57 | 0.04 | 0.34 | -0.46 | 0.57 | -0.06
FR | Insured not at fault | -0.94 | -0.06 | -2.53 | -2.32 | -3.38 | -2.70 | 0.55 | 0.84 | -0.23 | 0.72 | -1.28 | -1.39 | -1.82 | -1.35
Insured Partly at fault $25\%$ | 1.10 | 1.37 | 2.96 | -12.13 | 1.62 | 2.13 | 0.89 | -9.41 | 2.23 | 2.10 | 3.51 | -2.41 | 3.43 | 2.98
Insured Partly at fault $50\%$ | 0.34 | 0.56 | -0.01 | -0.42 | -0.12 | -0.01 | -0.92 | -0.38 | 0.18 | 0.71 | -0.07 | 0.16 | 0.17 | 0.50
Insured Partly at fault $75\%$ | -0.55 | -0.27 | 0.61 | -8.55 | 0.88 | 0.34 | 0.93 | -6.36 | 0.61 | 0.67 | 2.64 | -3.71 | 0.80 | 1.49
No fault | 4.30 | 3.23 | -1.19 | -5.06 | -1.25 | 0.05 | 6.37 | 1.61 | 2.30 | 4.04 | 1.29 | -2.13 | -0.83 | 0.87
Not applicable | 1.46 | -0.07 | -1.88 | -28.36 | -3.83 | -4.53 | 1.05 | -1.28 | -1.12 | -1.36 | -0.77 | -2.98 | -2.35 | -3.16
Not Available | 5.81 | 1.34 | 3.74 | -12.24 | 4.27 | 1.26 | 7.88 | 0.74 | 4.41 | 0.99 | 4.16 | -0.43 | 3.32 | 1.06

## Appendix C: Choice of the severity models

Table 13: Choice of the distributions for the severity of the payments Coverage | Model | $j=1$ | $j=2+$
---|---|---|---
AIC | BIC | AIC | BIC
Accident Benefits | Log-Normal | 266,408 | 266,651 | 181,577 | 181,827
Gamma | 264,411 | 264,654 | 183,670 | 183,921
Pareto | 262,903 | 263,146 | 181,281 | 181,532
Generalized Beta II | 261,110 | 261,368 | 181,222 | 181,487
Weibull | 264,101 | 264,344 | 182,545 | 182,796
Bodily Injury | Log-Normal | 100,523 | 100,735 | 106,985 | 107,214
Gamma | 99,675 | 99,887 | 107,096 | 107,325
Pareto | 98,971 | 99,183 | 106,691 | 106,919
Generalized Beta II | 99,966 | 100,185 | 106,884 | 107,119
Weibull | 99,187 | 99,400 | 106,673 | 106,902
Vehicle Damage | Log-Normal | 5,375,907 | 5,376,246 | 620,136 | 620,431
Gamma | 5,385,180 | 5,385,519 | 619,456 | 619,751
Pareto | 5,342,022 | 5,342,361 | 617,247 | 617,542
Generalized Beta II | 5,330,989 | 5,331,349 | 617,245 | 617,557
Weibull | 5,371,787 | 5,372,126 | 618,531 | 618,827
Loss of Use | Log-Normal | 1,874,185 | 1,874,498 | 193,705 | 193,967
Gamma | 1,868,867 | 1,869,180 | 192,929 | 193,191
Pareto | 1,894,927 | 1,895,241 | 194,376 | 194,638
Generalized Beta II | 1,861,585 | 1,861,918 | 192,212 | 192,489
Weibull | 1,877,508 | 1,877,821 | 193,459 | 193,721

## Appendix D: Results stability

Figure 13: Results of the RBNS reserves ($95\%$ VaR) based on the number of
simulations performed

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