Submitted to Expert Systems with Applications
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Selecting prospects for cross-selling financial products
using multivariate credibility
Fredrik Thuring, Jens Perch Nielsen, Montserrat Guill´en, Catalina Bolanc´e *
Cass Business School, City University, London, United Kingdom, Fredrik.Thuring.1@city.ac.uk (corresponding author)
Cass Business School, City University, London, United Kingdom, jens.nielsen.1@city.ac.uk
Insurance p olicies or credit instruments are financial products that involve a long-term relationship between
the customer and the company. For many companies a possible way to expand its business is to sell more
pro ducts to preferred customers in its portfolio. Data on the customers’ past behaviour is stored in the
company’s data base and these data can be used to assess whether or not more products should be offered to
a specific customer. In particular, data on past claiming history, for insurance products, or past information
on defaulting, for banking products, can be useful for determining how the client is expected to behave in
other financial products. This study implements a method for using historical information of each individual
customer, and the portfolio as a whole, to select a target group of customer to whom it would be interesting
to offer more pro ducts. This research can help to improve marketing to existing customers and to earn higher
profits for the company.
Key words : Cross-sale selections; Financial services industry; Multivariate credibility.
1. Introduction
Cross-selling means approaching the present customers of a company and encouraging them to
increase their engagement with the company by purchasing one or many additional products. It is
one of the main tools for managers to strengthen the customer relationship (Kamakura et al., 1991).
In the financial sector, customers have a long-term relationship with their service provider and
data on their characteristics, transactions, demographics and behaviour is stored in the company’s
data base, see Seng and Chen (2010) and Liao et al. (2011). This information can be used to select
preferred customers and cross-sell them products they do not yet possess.
* The authors acknowledge support by the Spanish Ministry of Science ECO2010-21787-C01-03.
1
Thuring, Nielsen, Guill´en, Bolanc´e: Selecting prospects for cross-selling
2 Article submitted to Expert Systems with Applications; manuscript no. 1
We present a method that describes how to model past behaviour in one or many financial
products in order to estimate a customer specific risk profile for a certain product not yet owned
by him or her, see e.g. Bae and Kim (2010) for other examples of modelling customer behaviour.
Thereafter, the risk profile estimate is used to select which customers, from the company’s portfolio,
to approach and attempt to make a cross-sale. Knowledge about past customer behaviour in one or
many financial products is known to explain the performance in other related products, of the same
customer (see e.g. Englund et al., 2009). However, there has been no attempt to implement this
as a cross-selling marketing instrument. Our objective is to show a case study of this method and
to explain how such a system can be implemented in practice. The general procedure is described
in Figure 1 where we see, from data analysis to customer selection, how a financial company can
select a target group of customers in order to cross-sell them a certain product. Initially, data
Figure 1 Workflow for cross-selling in the financial sector
on customers with several products are analysed and a model is specified. The model predicts an
individual score for each customer with respect to a financial product he/she does not own. In our
paper the score is called a risk profile because it predicts the customer behaviour, for a particular
Thuring, Nielsen, Guill´en, Bolanc´e: Selecting prospects for cross-selling
Article submitted to Expert Systems with Applications; manuscript no. 1 3
not owned product, given the individual information about the behaviour in other owned and
related products. The company can then select a target group for a marketing campaign based on
the predicted risk profiles, offer a specific product to this group and thereafter the success of the
cross-sale campaign can be analysed to refine the model, see also Malthouse (2010).
In an insurance company the method presented in Section 3 can be used to detect customers
likely to report few insurance claims, with respect to a not yet owned insurance coverage, and
cross-sell them that specific coverage at p ossibly a discounted premium level. Insurance companies
normally have models for the expected (yearly) claim frequency, given certain characteristics of the
customer and the insured object, which have been estimated based on collateral data on historic
claims reported by past and present customers of the company, see Denuit et al. (2007) for details
on claims frequency models. When predicting the claim frequency of a specific customer, such
models do not usually take into consideration the individual claims experience of that customer, but
predict the claims frequency based on a risk categorization which is a function of the characteristics
with respect to the customer and the object. Since there can be customers with more or less risk
adverse (individual) behaviour, there are cases for which the claim frequency model over-estimates
or under-estimates the claim occurrence. If, for a certain customer, the claim frequency model
over-estimates the claim occurrence the customer is reporting ”fewer claims than expected”, on
the other hand if the claim frequency model under-estimates the claim occurrence the customer
reports ”more claims than expected”. By knowing about the individual behaviour (more or less
claims than expected) in one or many of a customer’s existing coverages, a similar behaviour can
be expected for another coverage, not yet owned by the customer. For instance, someone who has
a motor insurance policy coverage and who claimed less than expected is probably also going to be
claiming less than expected in other coverages such as house insurance. This phenomenon can be
explained by the attitude that individuals have towards risk (see Slovic et al., 2004 and Harrison et
al., 2007). People that are very much risk adverse drive carefully and also maintain their houses and
belongings in good conditions. As a result, there is a correlation between the number of claims that
they report to their insurance company in two different insurance coverages. On the other hand,
Thuring, Nielsen, Guill´en, Bolanc´e: Selecting prospects for cross-selling
4 Article submitted to Expert Systems with Applications; manuscript no. 1
some individuals have a completely different attitude towards risk, they are more aggressive when
driving and are therefore expected to be careless about their properties too. So, when cross-selling
house coverage to individuals who already have motor insurance with the company, it would be
wise to take into consideration the observed number of car claims (for the specific customer) in
comparison to the expected number of car claims. Note that the reverse is also true, the number
of past house insurance claims can help to predict future car insurance claims.
A similar argument can be made for the banking sector. Customers that have not defaulted
in the past on their loans and/or have a flawless credit card payment history, are the ones also
expected to be profitable for other credit instruments. As for insurance companies, banks and other
credit institutions have models and assessments for the likelihood of a customer not being able
to repay credit card loans or mortgages and the concept of ”fewer incident than expected” and
”more incidents than expected” is applicable here as well. In the proceeding, we will refer to the all
events leading to a customer induced loss for a financial company (insurance claims, loan defaults,
non-repayment of credit card loans, etc.) as incidents.
The rest of the paper is organized as follows. In Section 2 we present the background of cross-
selling and marketing of financial products. We show that selecting customers, based on behaviour
in other related products, is an issue that has not been explicitly discussed in existing works. Sec-
tion 2 also provides a short overview of credibility theory, which we use to estimate the individual
risk profile. In Section 3 we briefly show how the risk profile can be obtained, in the cross-selling
case, and Section 4 presents a real case study on customers from the database of a Swedish insur-
ance company. The results illustrate how the methods can be used in practice, they show that
implementation is straightforward and can lead to substantial profit improvement compared to a
strategy, for cross-selling, where customers are selected randomly. Finally, Section 5 concludes.
2. Background
We first review recent cross-sale studies and thereafter the concept of credibility theory, which is
the technique used for evaluating cross-sell prospects in this paper.
Thuring, Nielsen, Guill´en, Bolanc´e: Selecting prospects for cross-selling
Article submitted to Expert Systems with Applications; manuscript no. 1 5
2.1. Cross sale models
Understanding and using cross-selling techniques is crucially important for a company because as
the customers acquire more products from the same provider, the switching cost, associated with
leaving for a competitor, increases (Kamakura et al., 2003). Therefore, cross-selling is considered
a strong driver for lowering the customer churn, increasing the number of loyal customers and
obtaining higher customer lifetime value (Akura and Srinivasan, 2005). In addition to this, con-
sidering product features allows significant contributions for managers striving for valuable and
strong relationship with their current customer base (Larivi`ere and Van den Poel, 2004). Another
important, but not as obvious, benefit from cross-selling is that companies can learn more about
the customers’ preferences and buying behaviour (Kamakura et al., 2003) and cumulate various
types of data to their data warehouse e.g. demographic information (Ahn et al., 2011). Such infor-
mation can be used as explanatory variables to predict certain behaviours of the customers such
as customer retention and profitability outcomes (Larivi`ere and Van den Pol, 2005).
Other studies focus on modeling the probability of a successful cross-sale attempt. In an early
study by Kamakura et al. (1991) probabilistic predictions are made on whether or not a customer
would purchase a particular product/service based on their ownership of other products/services.
In Knott et al. (2002), different models are applied to predict which product a customer is expected
to buy next and the approach is further developed in Li et al. (2005), where also the appropriate
time to approach a specific customer is studied.
Even though many studies have been made on cross-selling as a method for increasing a com-
pany’s revenue, only few discuss potential heterogeneity in the profitability of the cross-sale
prospects. As pointed out in Larivi`ere and Van den Pol (2005), financial products are not the
typical grocery products such as milk, coffee or cookies, but products that are bought and owned
for a specific perio d in time. In addition to this, financial products are associated with uncertain
costs which are determined at some (uncertain) time after the product is sold. Therefore it is not
guaranteed that a successful cross-sale attempt, to a specific customer, will generate profit to the
company. Instead if the cross-sold product generates claims (for an insurance company) or a loan
Thuring, Nielsen, Guill´en, Bolanc´e: Selecting prospects for cross-selling
6 Article submitted to Expert Systems with Applications; manuscript no. 1
default (for a lending bank) the financial product actually generates a loss to the company, in most
cases far greater than the income at the point of sale (insurance premium or interest payment).
Englund et al. (2008) suggest that their multivariate credibility estimator could be used for evaluat-
ing cross-sale prospects by taking into account only information from the other insurance products
of these specific prospects. The resulting estimate of the risk profile can be used to identify the
expected profitable customers (having less than expected number of claims or loan defaults) and
hence increase the company’s total profit from cross-selling.
2.2. Credibility theory
In actuarial science, credibility theory is a technique widely used to price different insurance cov-
erage such as health, life and property insurance (Frees, 2003). In general, the idea is to weight
data, associated with an individual policyholder, with data from a collective of policyholders using
a credibility weight α,
individual estimate = α × individual data +
(
1 − α
)
× collective data .
A historical review of credibility theory starts with the papers by Mowbray (1914) and Whitney
(1918) in which the credibility weight is determined ad hoc, focusing on practical applications, and
not yet founded on concrete mathematical grounds. In B¨uhlmann (1967) (and in the more general
B¨uhlmann and Straub, 1970, where the B¨uhlmann-Straub credibility model is presented) this was
changed by viewing the determination of α as an optimisation problem where only the first and
second order moments of the data is needed for the optimal estimator (Norberg, 2004). The gener-
alisation of the credibility estimator to higher dimensions was introduced in Jewel (1973) and later
in a multivariate hierarchical framework by Venter (1985). In Jewel (1989) the specific problem of
multivariate predictions of first and second order are investigated, while a comprehensive reference
to (multivariate) credibility in general is B¨uhlmann and Gisler (2005). A specific interpretation
of the B¨uhlmann-Straub credibility model is found in Englund et al. (2008) and Englund et al.
(2009) where the dimensions, in the multidimensional credibility model, are interpreted as different
insurance coverages, between which the claim occurrence can be more or less correlated.
Thuring, Nielsen, Guill´en, Bolanc´e: Selecting prospects for cross-selling
Article submitted to Expert Systems with Applications; manuscript no. 1 7
3. Methodology
We use multivariate credibility theory to estimate a customer specific latent risk profile and there-
after evaluate if a specific additional product, of a specific customer, is expected to contribute
positively to the profit of the company, if that product is cross-sold to the customer. The profit
is measured as the customer specific deviation between the a priori expected number of incidents
(insurance claims, loan defaults, etc) and the corresponding observed number. In the next para-
graphs we present the methodology briefly and give reference to previous related work on the model
and estimation technique.
3.1. Estimation of the risk profile
We use the standard multivariate B¨uhlmann-Straub credibility model, see e.g. B¨uhlmann and
Gisler (2005, p. 178) and Englund et al. (2008). Individuals i = 1, . . . , I are customers to a financial
company and have been so during time periods j = 1, . . . , J
i
. During these time periods, every
customer has had l = 1, . . . , K different financial products. We alter between k, k
′
and l as index
for financial products in general. For each customer i in time period j and product l, we have
the a priori expected number of incidents λ
ijl
= e
ijl
g
l
(
Y
ijl
)
, which depends on the risk exposure
0 ≤ e
ijl
≤ 1, a regression function g
l
and of a set of explanatory variables Y
ijl
characterising the
customer and the insured object. This can be viewed as a categorisation of the customer and the
insured object into one of a large (but finite) number of risk categories. The function g
l
is common
for all customers i and time periods j and can be estimated, using a generalised linear model,
based on collateral data of the company. We assume that e
ijl
can take values between
[
0, 1
]
, where
e
ijl
= 0 means that the l-th product is not active (not owned) for customer i in time p eriod j and
correspondingly, e
ijl
= 1 means that the product l of customer i is active (owned) during the entire
time period j. We assume a Poisson distribution for the random variable N
ijl
, describing the actual
number of incidents for customer i in time p eriods j and product l. The observation of N
ijl
is n
ijl
.
Consider another random variable Θ
il
which represents hidden characteristics such as risk aver-
sion, attitude, etc. that are not captured by the explanatory variables. Θ
il
random variables are
Thuring, Nielsen, Guill´en, Bolanc´e: Selecting prospects for cross-selling
8 Article submitted to Expert Systems with Applications; manuscript no. 1
often called the random effects. Let the pairs
(
N
1jl
, Θ
1l
)
,
(
N
2jl
, Θ
2l
)
, . . . ,
(
N
Ijl
, Θ
Il
) be indepen-
dent. We assume E
[
N
ijl
]
= λ
ijl
θ
0l
where E
[
Θ
il
]
= θ
0l
and Cov
[
Θ
ik
, Θ
ik
′
]
= τ
2
kk
′
for k = 1, . . . , K and
k
′
= 1, . . . , K. Further we assume that the conditional expectation is E
[
N
ijl
| Θ
il
= θ
il
]
= λ
ijl
θ
il
. The
risk profile θ
il
describes the risk that is not captured by the model for the a priori expected number
of claims, of customer i and product l, and, as mentioned above, is sometimes called random effect.
We define F
ijl
as the deviation between the actual number of incidents N
ijl
and the a priori
expected number of incidents λ
ijl
,
F
ijl
=
N
ijl
λ
ijl
and F
i·l
=
N
i·l
λ
i·l
=
J
i
j=1
N
ijl
J
i
j=1
λ
ijl
.
Other definitions, of the deviation between the expected and observed risk, are possible see e.g.
Guill´en et al. (2011). We assume that Cov
[
F
ijk
, F
ijk
′
| Θ
ik
, Θ
ik
′
]
= 0, for k ̸= k
′
.
The homogeneous multivariate credibility estimator (1) is the best linear unbiased estimator of
θ
i
=
[
θ
i1
, . . . θ
iK
]
′
(see Englund et al., 2009 and B¨uhlmann and Gisler, 2005, p. 181).
θ
i
= θ
0
+ α
i
(
F
i·
− θ
0
)
(1)
with θ
0
=
[
θ
01
, . . . θ
0K
]
′
and F
i·
=
[
F
i·1
, . . . F
i·K
]
′
. The credibility weight α
i
= T Λ
i
(T Λ
i
+ S)
−1
where
T is a K by K matrix with elements τ
2
kk
′
, k = 1, . . . , K and k
′
= 1, . . . , K. The matrices Λ
i
and S
are diagonal matrices with, respectively, λ
i·l
, l = 1, . . . , K and σ
2
l
, l = 1, . . . , K in the diagonal. The
parameter σ
2
l
= E
[
σ
2
l
(Θ
il
)], where
σ
2
l
(Θ
il
) is the variance within an individual customer
i, for a
product l (for further details see B¨uhlmann and Gisler, 2005, p. 81). We also refer to B¨uhlmann
and Gisler (2005, pp. 185-186) for parameter estimation procedures of the matrices S and T and
the vector θ
0
.
Performing the matrix multiplication in
(1) and considering element
k of θ
i
we get
θ
ik
= θ
0k
+
K
k
′
=1
α
ikk
′
(
F
i·k
′
− θ
0k
′
)
Thuring, Nielsen, Guill´en, Bolanc´e: Selecting prospects for cross-selling
Article submitted to Expert Systems with Applications; manuscript no. 1 9
where α
ikk
′
is element kk
′
of the matrix α
i
. This can be rewritten as
θ
ik
= θ
0k
+ α
ikk
(
F
i·k
− θ
0k
) +
k
′
̸=k
α
ikk
′
(
F
i·k
′
− θ
0k
′
)
. (2)
We now assume that if product k is not active (not owned) by customer i, the risk exposure
e
ijk
= 0 for all j and consequently λ
ijk
= λ
i·k
= 0. It is p ossible to show that λ
i·k
= 0 implies that
α
ikk
= 0 and
(
2
)
becomes
θ
ik
= θ
0k
+
k
′
̸=k
α
ikk
′
(
F
i·k
′
− θ
0k
′
)
, (3)
where the α
ikk
′
is element kk
′
of α
i
when taken into consideration that λ
i·k
= 0 in Λ
i
.
Equation
(3) shows that even though a customer
i does not have an active product k, it is
possible to obtain his/her specific risk profile θ
ik
(with respect to product k) by using data of
F
i·k
′
=
N
i·k
′
λ
i·k
′
with respect to the other (owned) products k
′
∈ {1, . . . , k − 1, k + 1, . . . , K}. From a
company’s perspective, customers with a low risk profile are preferred and therefore the estimate
of θ
ik
can be used to assess which customers to cross-sell product k to.
4. Empirical study
In this section we describe the data set collected to test the cross-sale selection methodology and
our experiments with this data. We require a data set describing customers who own more than
one financial product.
We conduct the experiment by neglecting the data with respect to one of the products and
therefore imagine that this product is not owned by the customers. Instead the data for the other
products is used to investigate if we are able to identify customers with fewer (or more) than
expected number of incidents with respect to the discarded product.
4.1. Application data
The data sample is collected from the data base of a large Swedish insurance company writing
business in both personal and commercial lines, however our sample consists solely of personal lines
customers. The sample consist of a set of individuals who have been customers to the company
Thuring, Nielsen, Guill´en, Bolanc´e: Selecting prospects for cross-selling
10 Article submitted to Expert Systems with Applications; manuscript no. 1
between 1999 and 2004 and who, during this time period, have owned all of the K = 3 main
insurance coverages provided: motor, building and content insurance. The customers have not
owned the coverages for equally long time so the policy duration spans between J
i
= 3 and J
i
= 6
years.
We have collected data from I = 3395 customers and for each customer i we estimate the a priori
expected number of insurance claims
ˆ
λ
ijl
= e
ijl
ˆg
l
(
Y
ijl
)
(where ˆg
l
is estimated using a collateral
dataset from the same company) and collect the number of claims n
ijl
for each year j = 1, . . . , J
i
and for each of the three coverages l = 1 (motor), l = 2 (building) and l = 3 (content). The a priori
expected number of insurance claims
ˆ
λ
ijl
has been assessed with the claim frequency model ˆg
l
, in
force at the time, using the characteristics of each customer and insured object. We present the
mean and standard deviation of our data in Table 1, where it can be seen that the mean of the a
priori expected number of claims
ˆ
λ
ijl
is close to the mean of the observed number of claims n
ijl
with the exception for product l = 2 (building coverage). Note that the standard deviation of the
a priori expected number of claims is lower than the standard deviation of the observed number of
claims, which is the result of the random effects and justifies credibility estimation.
Table 1. Descriptive statistics for Swedish
insurer data from 1999 - 2004
Mean Std. dev.
Motor Expected 0.084 0.053
Observed 0.083 0.295
Building Expected 0.064 0.033
Observed 0.046 0.220
Content Expected 0.051 0.028
Observed 0.052 0.237
4.2. Experiment design and results
Our aim is to replicate the situation where the customers of a financial company have a set of
products but lacking one of the products offered by the company. We assume that the company is
interested in selecting customers with fewer than expected number of incidents. The company can
achieve this by estimating the risk profile θ
ik
for each customer i (with respect to the not owned
product k) and select those with low risk profile. With our data set we imagine not knowing about
Thuring, Nielsen, Guill´en, Bolanc´e: Selecting prospects for cross-selling
Article submitted to Expert Systems with Applications; manuscript no. 1 11
the data for one of the products k and thereafter estimate the risk profile θ
ik
with data from the
other products 1, . . . , k − 1, k + 1, . . . , K. Thereafter we order the data set by increasing
ˆ
θ
ik
and
partition it into a certain number M of subsets Φ
m
(of size ϕ
m
) with m = 1, . . . , M. The estimate
of the risk profile
ˆ
θ
ik
is
ˆ
θ
ik
=
ˆ
θ
0k
+
k
′
̸=k
ˆα
ikk
′
ˆ
F
i·k
′
−
ˆ
θ
0k
′
, where
ˆ
F
i·k
′
=
n
i·k
′
ˆ
λ
i·k
′
=
J
i
j=1
n
ijk
′
J
i
j=1
ˆ
λ
ijk
′
. (4)
The partitioning into subsets Φ
m
is needed for presenting the results in an understandable way,
we used different values of M and finally concluded that M = 5 is an appropriate number of subsets.
In this way, Φ
1
contains 20% of the customers associated with the lowest
ˆ
θ
ik
, Φ
2
contains the next
20%, etc.. The number ϕ
m
= 679, for m = 1, . . . 5. Since the data sample is ordered by increasing
ˆ
θ
ik
before the partitioning into subsets Φ
m
, we expect to capture customers with fewest incidents,
compared to the a priori expected number, in subset Φ
1
and the customers with the most incidents,
in comparison to the a priori expected number, in subset Φ
5
. This can be validated by analysing
the observed number of claims n
i·k
in comparison to the a priori expected number λ
i·k
for the
customers in the different subsets Φ
m
, with respect to the previously imagined not owned product
k. For each subset Φ
m
, we are interested in the deviation ∆ of the observed number of claims in
comparison to the a priori expected number expressed as a percentage as follows,
∆ (Φ
m
)
= 100
i∈Φ
m
n
i·k
i∈Φ
m
ˆ
λ
i·k
− 1
, m ∈ {1, 2, 3, 4, 5}. (5)
Figure 2 describes our experiment with the data, for the situation where we are interested in
identifying subsets Φ
m
for product 2, using data from products 1 and 3. We use the notation
ˆ
θ
i,213
meaning that the risk profile θ
i2
is estimated using data from products 1 and 3.
It is not uncommon that some customers of a financial company only have one of the many
products offered by the company. The presented methodology works in this specific case as well by
setting e
ijk
= 0 for the all products k which the customers does not own. I.e. for our data sample,
we can also estimate the risk profile θ
ik
by using information from only one of the two remaining
Thuring, Nielsen, Guill´en, Bolanc´e: Selecting prospects for cross-selling
12 Article submitted to Expert Systems with Applications; manuscript no. 1
Figure 2 The design of the experiment for the particular case of creating subsets Φ
1
to Φ
5
based on the
estimated risk profile for product 2 using information from products 1 and 3.
products in the data set. For instance, for the estimate of the risk profile of product k = 1, θ
i1
, we
use the notation
ˆ
θ
i,12
if only data from product 2 is used in the estimation, and correspondingly
for the other products.
The evaluation criteria
(5), applied to investigate the deviation between of the observed number
of claims n
i·k
and the a priori estimated expected number
ˆ
λ
i·k
, in the 5 subsets Φ
m
, is presented
in Figures 3 to 5. In Figure 3, we imagine that product k = 1 (car coverage) is not owned by
the customers and we use data from either product k
′
= 2 (building coverage) or product k
′
= 3
(content coverage) or data from both building and content coverage to estimate the risk profile
θ
i1
, with respect to product 1. Thereafter, for each of the three different estimators, we order the
data set by increasing value of the risk profile estimate and partition the data into the subsets
Thuring, Nielsen, Guill´en, Bolanc´e: Selecting prospects for cross-selling
Article submitted to Expert Systems with Applications; manuscript no. 1 13
Φ
m
with m = 1, . . . , 5 for calculation of ∆
(Φ
m
), see equation (5).
As seen in Figure 3, the credibility estimator
ˆ
θ
i,12
, which uses data from product 2, does only
slightly differentiate the customers with respect to claiming (n
i·1
) in comparison to the a priori
expected claiming (λ
i·1
) (left sub-figure of Figure 3). However, when ordering the data with respect
to
ˆ
θ
i,13
, which uses information from product 3, subset Φ
1
contains customers with on average 6%
lower claims frequency than expected and subset Φ
5
contains customers with 22% more claims
than exp ected, see center sub-figure of Figure 3. When using data from both product 2 and 3 the
result is improved slightly and Φ
1
contains customers with 8% less claims than expected and Φ
5
contains customers with 26% more claims than expected.
1 2 3 4 5
Ordering by increasing θ
^
i,12
m
∆(Φ
m
) (%)
−20 −10 0 10 20 30
−1 −3 −5 2 2
1 2 3 4 5
Ordering by increasing θ
^
i,13
m
∆(Φ
m
) (%)
−20 −10 0 10 20 30
−6 −11 −6 −8 22
1 2 3 4 5
Ordering by increasing θ
^
i,123
m
∆(Φ
m
) (%)
−20 −10 0 10 20 30
−8 −5 −6 −13 26
Figure 3 Average deviation between observed number of claims and a priori expected number with respect to
product 1 (car coverage). The subsets Φ
m
are created using only information from building coverage (left
sub-figure), using only information from content coverage (center sub-figure) or using information from both
building and content coverages (right sub-figure)
In Figure 4 we imagine that product 2 (building coverage) is not owned by the customers. We see
Thuring, Nielsen, Guill´en, Bolanc´e: Selecting prospects for cross-selling
14 Article submitted to Expert Systems with Applications; manuscript no. 1
that almost all subsets Φ
m
contain customers with fewer claims than expected because (according
to Table 1) the average value of
ˆ
λ
i·2
is far greater than the average value of n
i·2
since almost all
customers have reported fewer claims than a priori expected. Still, the credibility estimators
ˆ
θ
i,23
(center sub-figure) and
ˆ
θ
i,213
(right sub-figure) is able to differentiate between subsets containing
customers with less than expected claiming and more than expected claiming.
1 2 3 4 5
Ordering by increasing θ
^
i,21
m
∆(Φ
m
) (%)
−80 −60 −40 −20 0 20
−41 −28 −23 −21 −25
1 2 3 4 5
Ordering by increasing θ
^
i,23
m
∆(Φ
m
) (%)
−80 −60 −40 −20 0 20
−43 −40 −35 −36 6
1 2 3 4 5
Ordering by increasing θ
^
i,213
m
∆(Φ
m
) (%)
−80 −60 −40 −20 0 20
−45 −36 −29 −36 3
Figure 4 Average deviation between observed number of claims and a priori expected number with respect to
product 2 (building coverage). The subsets Φ
m
are created using only information from car coverage (left
sub-figure), using only information from content coverage (center sub-figure) or using information from both car
and content coverages (right sub-figure)
In Figure 5, we imagine that product 3 (content coverage) is not owned by the customers. We
see that all credibility estimators (
ˆ
θ
i,31
,
ˆ
θ
i,32
,
ˆ
θ
i,312
) are identifying the customers in subset Φ
5
as
having much more claims than exp ected. Especially the estimator
ˆ
θ
i,312
(right sub-figure) is able to
identify, in the subset Φ
5
, customers who have on average 64% more claims than a priori expected
while also identifying the customers in the subset Φ
1
with on average 10% less claims than a priori
Thuring, Nielsen, Guill´en, Bolanc´e: Selecting prospects for cross-selling
Article submitted to Expert Systems with Applications; manuscript no. 1 15
expected.
1 2 3 4 5
Ordering by increasing θ
^
i,31
m
∆(Φ
m
) (%)
−40 −20 0 20 40 60
−3 −13 −14 12 31
1 2 3 4 5
Ordering by increasing θ
^
i,32
m
∆(Φ
m
) (%)
−40 −20 0 20 40 60
2 −7 −28 −15 48
1 2 3 4 5
Ordering by increasing θ
^
i,312
m
∆(Φ
m
) (%)
−40 −20 0 20 40 60
−10 −27 −19 −3 64
Figure 5 Average deviation between observed number of claims and a priori expected number with respect to
product 3 (content coverage). The subsets Φ
m
are created using only information from car coverage (left
sub-figure), using only information from building coverage (center sub-figure) or using information from both car
and building coverages (right sub-figure)
In Figures 3 to 5 it would be expected and preferred that the deviation of the observed number
of claims in comparison to the a priori expected number, ∆
(
Φ
m
)
, would be lowest for m = 1.
However, this is not the case for many of the estimators and especially for cross-selling product
k = 1 (car) in Figure 3 the lowest ∆
(
Φ
m
)
is recorded for m = 3, m = 2 and m = 4 for the credibility
estimators
ˆ
θ
i,12
,
ˆ
θ
i,13
and
ˆ
θ
i,123
, respectively. A similar note can be made with regards to Figure
5. We draw the conclusion that for the collected data sample it is more efficient to identify a
small group of customers to avoid to cross-sale to (Φ
5
) than a small group of customers to target
(Φ
1
). Consequently, we find that by avoiding the 20% of the customers associated with the highest
risk profile estimates
ˆ
θ
ik
(Φ
5
) and targeting the remaining 80% the company would increase its
Thuring, Nielsen, Guill´en, Bolanc´e: Selecting prospects for cross-selling
16 Article submitted to Expert Systems with Applications; manuscript no. 1
profit significantly. In Table 2 we compare ∆
(Φ
5
) to ∆ (
∪
4
m=1
Φ
m
) = ∆ (Φ
1
∪ Φ
2
∪ Φ
3
∪ Φ
4
) where
Φ
m
∪ Φ
m+1
denotes the union of Φ
m
and Φ
m+1
.
Table 2. Percentage deviation between observed and expected number of claims. Note
that a positive value indicates that the subset of customers has reported more claims
a priori expected.
Car Building Content
Order ∆
(
∪
4
m=1
Φ
m
)
∆
(
Φ
5
)
Order ∆
(
∪
4
m=1
Φ
m
)
∆
(
Φ
5
)
Order ∆
(
∪
4
m=1
Φ
m
)
∆
(
Φ
5
)
Random 0% -4% Random -29% -30% Random 4% 3%
ˆ
θ
i,12
-2% 2%
ˆ
θ
i,21
-30% -25%
ˆ
θ
i31
-4% 31%
ˆ
θ
i,13
-8% 22%
ˆ
θ
i,23
-40% 6%
ˆ
θ
i32
-9% 48%
ˆ
θ
i,123
-8% 26%
ˆ
θ
i,213
-38% 3%
ˆ
θ
i312
-13% 64%
Table 2 shows that by selecting the 80% (∪
4
m=1
Φ
m
) most favorable customers, with respect to
the estimate of the risk profile θ
ik
, the company is able to avoid customers associated with up to
64% more claims than a priori expected (content coverage, product 3). In the table we have also
included results produced when the data sample has been randomly ordered and partitioned into
80% of the data and 20% of the data. The random order does not differentiate between subsets of
customers with respect to percentage deviation between observed and expected number of claims.
We see a similar pattern for product 1 (car) where the 80% most favorable customers are associated
with 8% less claims than a priori expected while the remaining 20% are associated with 26%
more claims than a priori expected. The performance of the credibility estimators in Product 2
(building) is difficult to interpret because almost all customers are associated with lower observed
claim occurrence than a priori expected. However, even for this particular situation a subset Φ
5
can
be identified consisting of customers with on average 6% more claims than a priori expected. Note
that this is not received for the credibility estimator which uses all available information (
ˆ
θ
i,213
)
but the estimator which only uses data from the content product k
′
= 3,
ˆ
θ
i,23
.
5. Discussion
This study investigates identification of customers to whom additional pro ducts should be offered,
by estimating a customers specific risk profile with the use of behavioural data from other products
of the specific customers. We use a standard multivariate credibility model applied to a portfolio
of customers, of a financial company, owning several financial products from the company. The
Thuring, Nielsen, Guill´en, Bolanc´e: Selecting prospects for cross-selling
Article submitted to Expert Systems with Applications; manuscript no. 1 17
model allows us to take into consideration the possible (positive) correlation in customer behaviour
between different financial products and estimate the customer specific risk profiles, for a specific
product not owned by the customer, without having observed any customer specific information
with respect to that particular product. Instead, data on customer behaviour, with respect to the
other (owned) products, is the only necessity for estimating the risk profile.
The methodology uses only two observables: the a priori expected number of incidents and
the observed number of incidents. We assume that the financial company has a model for the
expected number of incidents or is able to assess a value specific for each customer or category
of customers. When estimating such models it is unusual to incorporate information about the
number of incidents related to a specific customer. Instead the company finds patterns which can
be used to categorise the customers, with respect to the expected occurrence of incidents, based
on customers’ characteristics. It is not uncommon that customers are associated with more or less
number of incidents, than suggested by the categorisation, based on their attitude towards risk.
In our methodology we use that the attitude towards risk seems to be similar across different
financial products. I.e. if a customer is associated with more or less number of incidents, than a
priori expected in some products, it is likely that this pattern will also emerge in other related
products.
With the presented credibility estimators we are able to assign, to each customer, a specific
estimate of his/her risk profile based on data which the company has available. We use the estimate,
of each customer’s risk profile, to identify subsets from the data containing customers associated
with more or less incidents than a priori expected. In this way the company receives knowledge
about which customer to target for cross-selling and which to avoid.
In our empirical study we analyse our methodology on real data from a large Swedish insurance
company, consisting of personal lines customer with three different insurance coverages. We find
that there are subsets of the data sample with large heterogeneity with respect to claiming in
comparison to expected claiming. Furthermore, we find that these subsets are identifiable by using
Thuring, Nielsen, Guill´en, Bolanc´e: Selecting prospects for cross-selling
18 Article submitted to Expert Systems with Applications; manuscript no. 1
an appropriate credibility estimator of the risk profiles. The appropriateness of a specific credibility
estimator is dependent of the considered product, but in most cases an estimator which uses
all available information is preferable. We find that it is easier to identify the 20% of the data
containing customers to avoid than the 20% of the data containing customers to target. In fact,
by targeting all customers but the worst 20%, the company could expect a subset of customer
associated with less claims than a priori expected indifferent of which product is considered. The
remaining 20% of the data sample consist of customers with up to 64% more claims than a priori
expected.
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