A logistic regression approach to estimating customer profit loss due to lapses in insurance

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1 Insurance Markets and Companies: Analyses and Actuarial Computations, Volume, Issue, Montserrat Guillén Estany (Spain), Ana María Pérez-Marín (Spain), Manuela Alcañiz Zanón (Spain) A logistic regression approach to estimating customer proit loss due to lapses in insurance Abstract This article ocuses on business risk management in the insurance industry. It proposes a methodology or estimating the proit loss caused by each customer in the portolio due to policy cancellation. Using data rom a European insurance company, customer behavior over time is analyzed in order to estimate the probability o policy cancellation and the resulting potential proit loss due to cancellation. Customers may have up to two dierent lines o business contracts: motor insurance and other diverse insurance, such as, home contents, lie or accident insurance. The paper outlines the implications or understanding customer cancellation behavior as the core o business risk management. Keywords: policy cancellation, customer loyalty, proit loss, customer behavior. Introduction Under the ramework promulgated by the solvency regulation such as Solvency II in Europe, insurance companies ace an increasing number o challenges which require a change in the way business risk is currently measured and managed. Risk management under this perspective requires new approaches to all components o the insurance activity that may cause losses to the company. We will study policy cancellations and we will propose a method to measure potential proit loss that ollows rom customers that do not renew their insurance contracts. A real case study involving two lines o business will be described and we will outline qualitative implications. Insurance companies now operate in a more competitive environment than they used to in the past and customers easily switch rom one insurer to another. Cancellations and lapses have become one o the actors inluencing the level o risk an insurance company and its position in the market. The prolierating extension o the Internet played an instrumental role in reducing inormation-gathering costs or customers who wish to change insurers. Now, the central problem or insurance companies is not only to create and launch new products or the market, but additionally to achieve commercial success by retaining customers. As a consequence o this growing interest in increasing customer loyalty, the insurance business is no longer only product-oriented, but also customer-oriented. The luctuations in volumes and margins due to ongoing competition are a source o risk or the company which is called business risk (Nakada et al., 999 and Dhaene et al., 6). This type o risk, which is increasingly integral to a company s opera- Montserrat Guillén Estany, Ana María Pérez-Marín, Manuela Alcañiz Zanón,. We thank the Spanish Ministry o Science grants ECO-787- C- and ECO8-. Cancellations reer to policies that are not renewed. Equivalently, we also use the word lapse to indicate that an insurance contract is inished. tional risk, relects the reality o the market s impact on the stability o the company. Business risk management involves overcoming a number o diiculties, including the measurement o business risk itsel. Numerous actors inluence business risk, yet previous literature has not provided a way to quantiy and assess these actors. This could, in turn, clariy speciic actions to help protect companies against business risk. Despite the diiculties, business risk management provides beneits to companies. Companies increase overall results when they invest in retaining customers generating high proits, e.g., those who pay premiums or long periods without claims, rather than those with a bad claim history, contributing to reduce overall proits. Additionally, controlling business risk also contributes to the stability and solvency o a company by preventing losses caused by a potential decrease o its market share. Typically, insurance activity is managed separately or each line o business. The most aberrant implication o this classical managerial mistake is that dierent policies in the same company owned by the same customer are oten managed separately. Thus, events aecting one particular policy (claims, premium increases, etc.) are evaluated without consideration o concurrent events plausibly aecting other policies within the company. With such a onedimensional perspective, the possible relationship between these events and the customer s actions is ignored. As a result, neither the behavior o the customer nor the relationship o the company and the customer (which we call insurance relationship) are ully understood. In this article, we ocus on business risk management in the insurance industry and we account or the multidimensionality o the insurance relationship by taking all events aecting one policy into consideration when understanding what happens to We consider the same classiication o risks used by Nakada et al. (999) and Dhaene et al. (6).

2 Insurance Markets and Companies: Analyses and Actuarial Computations, Volume, Issue, other policies owned by the same customer. We posit that in order to successully manage business risk, an insurer needs to adopt the same integrated perspective o the insurance relationship as the one held by the customer. Thereore, the unit o analysis should be the individual policy holder (including all the policies he or she holds with the same company). However, as proposed by Guillén et al. (6) and Guillén et al. (8), a second level o analysis could also consider the household as a decisionmaking unit, as all adult members o the amily generally make decisions together about insurance policies that cover their common risks. Nevertheless, real data sets on policy cancellation do not normally include detailed inormation on households, so it is rather diicult to carry out in practice this second level o analysis. In this paper we analyze customer loyalty and the proit generated by a customer in dierent types o policies he may have in the same company. We develop a general procedure that disaggregates the proit in its dierent dimensions and inally we provide a valuation o the impact o cancellation on business risk in terms o the average expected yearly loss due to policy cancellation. An empirical investigation is also carried with real policy cancellation data. Our study is organized as ollows. Section briely summarizes a literature review on business risk management in the insurance industry. Section, describes three dimensions o the proit loss due to business risk: historical, prospective, and potential. Section presents the notation and ormulation or calculating proits. Section describes the data set used in the empirical application. Section 5 presents all the results obtained in our empirical investigation and inally, we discuss the managerial implications o our study in the last section.. Literature review Historically, there has not been much research on customer loyalty and business risk management in the actuarial ield, although this has changed in recent years. During the sixties, research interests ocused on investigating actors associated with the increasing demand or insurance, such as rising household incomes (Hammond et al., 967) and women entering the labor orce (Duker, 969). Later studies, based on the portolio theory, demonstrated that the demand or insurance products is determined simultaneously with the demand or other goods (Mayers and Smith, 98). On the other hand, Doherty (98) showed that the level o insurance increased as the number o insurable risks and their weight in the asset portolio also increases. Additionally, Babbel (985) claimed that underwriting lie insurance products was inversely related to changes in the real price index. Crosby and Stephens (987) were the irst to investigate the problem o customer loyalty in the insurance industry. They analyzed the eect o relationship marketing on lie insurance premiums and customer satisaction and retention. Their results suggested that non-lapsing customers reported a higher level o satisaction than lapsing ones (although the insureds were only ollowed or months). Interest in studying customer loyalty and satisaction grew during the eighties. Jackson (989) observed that very ew insurance companies calculated customer lietime value (CLV) at the time. At irst the author estimated CLV in the insurance sector, he emphasized its importance or determining strategies to increase the loyalty o high-valued customers. He proposed both a historical model to analyze the low o proits generated by customers and a predictive model to determine customer value in the long term. The calculation o the CLV was continued in the insurance sector throughout the late nineties and, even more intensively, in the most recent years. Berger and Nasr (998) believed that a systematic method o calculating CLV was missing and thereore they proposed a general mathematical model or calculating CLV in many dierent cases. The calculation was based on the discounted dierence between the revenues and costs, including all promotional costs. To illustrate their calculation, these authors used the case o an insurance product that required a contract or needed the contract to be renewed every year. However, their example was not based on actual empirical data. During the nineties, as the insurance market became more competitive, researchers began investigating the reasons why customers were switching companies. Schlesinger and Schulenburg (99), or instance, analyzed a sample o German automobile policy holders and ound two main reasons or changing insurers: () a avourable premium; and () a recommendation by a relative or riend. The authors also conducted parallel analyses or customers who changed insurers at some point in time and those who did not. Their indings suggested that those who changed insurers, were much more satisied with the claim handling process by the new insurance company. Several research articles rom the nineties also explored the relationship between the insurer s quality o service and customer satisaction. Wells and Staord (995) measured customer perception o service quality and compared it to the number o complaints registered by insurance companies. These authors observed that ewer complaints were signiicantly correlated with higher levels o perceived service quality. The results also suggested that customers who knew that they had the right to ile a complaint evaluated the service quality higher than those who did

3 Insurance Markets and Companies: Analyses and Actuarial Computations, Volume, Issue, not realize they had that right. Similarly, Staord et al. (998) ound that reliability (i.e., the ability to ulil the promised service in a timely manner) is the most important determinant o service quality and customer satisaction in the auto casualty claim process. In more recent research on loyalty strategies, Cooley () discussed a two-stage segmentation process to identiy our dierent groups o health insurance policy holders. Using such covariates as age, sex, type o coverage, and seniority, dierent loyalty strategies were applied to each group based on their particular needs and as a result, customer retention was observed to increase by approximately 7%. Following Mayers and Smith (98), Doherty (98), and Babbel (985), research interest in the demand or insurance products continued into the nineties as well. Showers and Shotick (99) investigated the eect o household characteristics and concluded that total household income and the number o members contributing to this income are positively correlated with the demand or insurance products. They also observed that as amily size and age o the members increases, the marginal increment in insurance purchases decreases. Likewise, Ben-Arab et al. (996) constructed a model or the consumption o insurance products in which current individual consumption preerences depended on past consumption patterns. This study showed that the optimal level o purchased insurance products is higher when consumption habit ormation is included in the model. So, it is possible to explain why some individuals over-purchase insurance. Ryals and Knox (5) summarized the results o a number o studies on customer loyalty and concluded that a small increase in customer retention rom 85% to 9% results in net present value proits rising rom 5% to 95% amongst the business they examined. The authors also argued that CLV should be adjusted or the risk inherent in establishing a relationship with a customer, such as the volatility o uture income lows provided by a new customer. Thus, the authors proposed a measure called riskadjusted CLV, which combines the prediction o CLV with the uture risk o the relationship. This measure can also be interpreted as a measure o the economic value (EV) o the customer. Risk-adjusted CLV included: () the risk o having a claim (speciic to each customer, measured by the ratio between claims and premiums); and () the risk o establishing a relationship with the customer (measured by the probability o not retaining the customer). The authors concluded that the measurement o EV can be implemented as a beneicial management tool or addressing relationship marketing strategies. Verhoe and Donkers () and Donkers et al. (7) presented two o the most comprehensive studies on the calculation o CLV in the insurance industry. First, Verhoe and Donkers () predicted potential customer value in comparison to realized customer value. Later, Donkers et al. (7) compared the predictive perormance o various models used to calculate CLV in the insurance sector. They considered two types o approximations. The simplest ones include all policies a customer has with the same company, but only consider the total proit the company receives or them. These models are only based on the relationship. On the other hand, more complex models also take all policies into account, but disaggregate the contribution by each one and, thereore, these models are based on the product. The authors concluded that simple models provide good predictions o CLV that are only marginally improved by the more complex models. Recent papers, by Brockett et al. (8) and Guillén et al. (8), provided a irst approximation to business risk management in the insurance company. The authors analyzed the relationship between the customer and the insurer, the so-called insurance relationship, and based on that, their conclusions pointed to the need to consider dierent types o products simultaneously. In that context, the authors estimated the probability o policy cancellation and the customer lietime duration by using real policy data. Additionally, the authors also provided some guidance or targeting business risk management in the insurance industry. More recently, Lai et al. () ind empirical evidence o the moderating eects o inertia and switching costs on the satisaction-retention link in the auto liability insurance context. They show that the barriers made by switching costs and the behavioral lock-in eect produced by inertia create a pull-back eect, which prevents customers rom switching to another insurance provider even in the ace o dissatisaction with the quality o service by the existing provider. Only a ew empirical studies o customer behavior ocused on the particularities o the insurance sector exist and more research on business risk management is needed to control the impact o competitive markets on the stability o insurance irms.. Proit loss due to business risk It is necessary to distinguish the proit that a customer is going to generate and the lietime value o that customer rom a marketing perspective. Customer lietime value is the present value o the uture proit stream expected over a given time horizon o transacting with the customer (Kotler, 97). In this paper we only consider the proit that the customer is expected to generate or the company next year, in order to measure the expected impact o business risk on a

4 Insurance Markets and Companies: Analyses and Actuarial Computations, Volume, Issue, yearly basis. Proit here reers to the one that the company is getting rom each customer exclusively due to the insurance activity, essentially the balance between premiums and cost o claims. The analysis o the proit a customer generates is based on three dierent dimensions:. Historical proit, which is the proit accumulated during a period o time. We calculate the historical proit o a customer by incorporating available inormation on underwritten policies regarding claim histories and premiums during a certain period o time. We add the aggregated premiums and subtract the aggregated costs o the claims during that period or each policy the customer has with the company. Note that historical proit is based on the same principles as the concept o realized value used by Verhoe and Donkers (), i.e., a measure o what is the revenue that the customer has generated until now.. Prospective proit, when the customer does not cancel his or her policies. When we look at what is going to happen in the uture, prospective proit takes into account the revenue a company expects to receive rom the customer in the short term i the customer keeps the policies in orce. Contrary to historical proit which evaluates a customer s past behavior, prospective proit estimates uture behavior regarding the policies in orce. We consider how many policies the customer currently holds and measure the expected proit they would generate rom the inormation on those policies and the probability o uture renewal.. Potential proit, prospective added proit when the customer underwrites with the same company new policies o a dierent type than those he currently has. The proit that a customer is going to generate should also incorporate the probability o underwriting other types o products and the proits these products are expected to generate. Thereore, here we ocus on the cross-buying behavior o the customer. Note that potential proit mentioned above is not based on the same principles as customer potential value in Verhoe and Donkers (). When we look at what is going to happen in the uture, the concept o customer potential value is measuring the proit or value delivered by a customer i this customer behaves ideally, i.e., the customer purchases all products or services he currently buys in the market at the ull prices at the ocal company (Verhoe and Donkers, ). Thereore, it is diicult or the insurer to measure potential value as he normally only knows which products the customer currently has in the ocal company. Nevertheless, Verhoe and Donkers () concluded that both socio-demographic and actual purchase inormation at the ocal company are useul predictors o purchase decisions determining potential value. Thereore, here we slightly depart rom the principles o potential value, but we still look at crossselling opportunities: we are interested in whether or not the customer will keep the policies he currently has (prospective proit) and i he is going to underwrite new types o policies (potential proit) in order to explore his potentiality o generating proit in other lines o business in the same company. Thereore, by adding prospective proit and potential proit we have some kind o approximation o customer potential value when it is measured in a yearly basis, as current inormation at the ocal company are useul predictors o purchase decisions determining potential customer value.. Notation Let K be the number o insurance product types, T be the total number o years considered in the historical analysis, and N t be the number o customers in year t, t =,,T. Let n itk be the number o policies o product type k, k =,, K the i th individual, i =,, N t has in year t. Furthermore, P itkl is the premium paid by the i th individual in year t, or the policy l, l =,, n itk o the product type k. Similarly, S itkl is the sum o the costs o claims compensated to the i th individual during year t, or the policy l o the product type k. Thereore, we can calculate the total amount o premiums or each product type k in a particular year t, that we denote by P tk : N t nitk i l P tk P itkl and the total costs o claims or each product type k in year t, S tk, by N t nitk. i l S tk S itkl In both cases, i n itk = then the term o the sum is equal to zero. The proit o each product type k in year t can be deined as a unction o the total premiums and claims. We will call this unction P tk, S ). ( tk This proit is requently measured by the dierence between the total premiums and costs o claims, which we express as D, D( P tk, S tk ) P tk S tk. For a longitudinal analysis o proits, we aggregate the results corresponding to dierent years as ollows: 5

5 Insurance Markets and Companies: Analyses and Actuarial Computations, Volume, Issue, D( k k k k T where P, S ) P S, P k P tk t and T S k S tk are the sums o t premiums and costs o claims accumulated during T years or product type k. The advantage o this latter expression is that it is additive, which means that D ( P, S k where k D ) ( P T t D ( P itk, Sitk ) tk, S P tk itk ) T S N t ( P, S D itk itk t i nitk n itk itk l P itkl ), l corresponds to the proit generated by the i th individual in year t, or the product type k. Obviously, i the customer has no policy o product type k in year n =, we would sum. t, itk.. Historical proit. The historical proit o the i th customer C is measured by C H D H K T i, Si ) D( Pitk, Sitk ) k t ( P. This measure, or a given customer, incorporates all inormation on dierent types o policies and years and provides the proit that the customer has generated during that period o time. It gives the same inormation as the realized value in Verhoe and Donkers ()... Prospective proit. The prospective proit o the customer is estimated to relect the possibility that a customer will renew the policies he or she currently has without underwriting new contracts in the next year. Let y itkl be a binary variable equal to when the i th individual has the l th policy, l =,, n itk o product type k, in year T, and we estimate the probability that this policy will be renewed next year, provided that it was already in orce the previous year, p itkl given by p itkl Pr( yi( T) kl yitkl ). Note that the estimation o this probability could incorporate other eatures characterizing the customer that could be used in a logistic regression model. I the customer renews the policy, then the company expects a proit. The company s proit can be calculated as the average o the proits obtained rom customers holding this particular type o product during the previous year, which we will call B DTk. That is, B DTk M M Tk Tk i D ( P itk, S itk ), S itkl where MTk is the number o customers holding the product type k in the year T. PRO The prospective proit C o the i th customer is obtained by summing the product o the probability o keeping policies and the expected proit rom them, or all types o products the customer holds. Thereore, C PRO K n itk k l p itkl B DTk... Potential proit. We estimate the potential proit by estimating the probability that a customer will underwrite next year new policies o a dierent type than those he currently has. We estimate the probability that n i(t+)k would be greater than zero, conditioned to n itk =, what we denote by p itk. That is, p itk itk Pr( ni ( T ) k n ), where p itk = i n itk >. Note that, again, the estimation o this probability could be done by using a logistic regression model. POT Thereore, we calculate the potential proit C by summing the product o the probability o a customer underwriting policies o a dierent type than his or her current policies and the average proit obtained or this type o product during the previous year B DTk, namely C POT K k p itk B DTk. Note that or prospective and potential proit calculation, we slightly depart rom the way proits were determined by Donkers et al. (7) as part o their calculation o customer value in the so-called relationship-level models. The authors assumed that the total proit each customer will generate next year will be the same as in the previous year, in case that the company retains that customer. As we are interested in analyzing the overall impact o business risk on the total portolio, instead, we assumed that the proit that will be generated by each policy the customer holds is the average o the proits obtained rom customers holding that particular type o policy the previous year. This procedure is similar to the way proit margins were determined in the so-called service level models by Donkers et al. (7)... The loss due to business risk. The expected loss due to business risk that the company will have next year L BR can be calculated as a unction o the PRO individual prospective and potential losses, C and POT C. We will consider two possible situations that can generate a loss next year: a partial and a total cancellation. A partial cancellation occurs when the 6

6 Insurance Markets and Companies: Analyses and Actuarial Computations, Volume, Issue, customer cancels some o his policies (but not all o them) and a total cancellation, when all policies are cancelled. We will assume that i a partial cancellation occurs, the loss generated by this customer is only the average prospective proit per policy. This is equivalent to assume that only one policy would be cancelled, and that policy would have generated a proit next year equal to the average prospective proit during the previous year. By doing so, we are assuming the most common and best possible situation when a partial cancellation occurs, as at least one policy would have been cancelled. On the other hand, i a total cancellation occurs, then we assume that the company looses all the prospective and potential proit. Based on these assumptions, the loss due to business risk that the company expects to have next year is: L BR i P( ni P( ni ( T K k ( T ) ) n it n it n it ) C ) n it, PRO POT C C PRO where n it n itk is the total number o policies the i th customer has in year t, t = T, T +.. Data Our sample included customers who had at least one policy with a European insurance company as o December, 5, o these types: automobile insurance (also called motor insurance), diverse nonautomobile (which includes house and contents, uneral and accident insurance), health, and agricultural insurance. We analyzed customer behavior over time rom December, 5 to March, 8, in order to estimate the probability o policy cancellation and determine the actors aecting customer loyalty. In Table we present sample characteristics at the beginning o the period o study. A majority o the customers were men (6.7% o the sample), while 8.6% were women (the rest were either irms or the gender was not identiied). The average age was 7. years (the standard deviation was.). Most customers were married (7.85%), although 5.6% were single and civil status was not declared by 5.5%. A majority o the sample (55.%) lived in rural villages/towns, but 7.% lived in urban areas. Approximately % o customers in the sample identiied their type o consumption as airmation (which means they had enough earnings to aord high-quality, permanent goods and still had suicient resources or leisure activities). The average seniority (meaning the average length o insurance relation- Unortunately, inormation on premiums and claims is only available or two lines o business: automobile and diverse. Thereore, only these ones are inally included in the calculations o proits. ship) in the company was 8.96 years (the standard deviation was 8.). In Table, we compare the composition o policies in orce or each customer on December, 5, with those in orce a year later. We observe that 8.% o customers had only one type o policy in orce at the end o 5 (mostly automobile or diverse policies), 5.69% had two types o policies, and less than % had three or more types o policies. More than 9% o customers with one type o policy at the end o 5 still had only that one type a year later. Most o the remaining 6% underwrote a new product. Among those with at least two types o policies, the percentage o customers who had the same types one year later ranges rom 7.% (those with automobile, diverse, and health policies) to 89.% (those with automobile and diverse policies). On the other hand, the percentage o those who cancelled their policies one year later ranges rom 9.9% (those with automobile, diverse, and agricultural policies) to 7.% (those with automobile, health and agricultural policies). Finally, the largest group o customers who underwrote a new policy was the one with health and agricultural policies and later added an automobile policy (7.7%). Table. Sample characteristics as o December, 5 Gender Marital status Address Type o consumption Frequency Percent Men 9.6 6,7% Women.77 8,6% Firm.79,99% Classiication not established.5 5,69% Total % Married 8.9 7,85% Divorced 5,7% Separated 559,7% Single.8 5,6% Widow 56,57% Classiication not established 8.9 5,5% Total % Rural (<. inhab.).7 55,% Urban (>. inhab.) ,% Unknown ,% Total % Luxurious ().,9% Socially improving () 8.777,% Airmation () 9.,% Emulation () 8.97,7% Subsistence (5) 7. 9,% Classiication not established.7 8,5% Total % Notes: () surplus resources allow or access to luxuries without aecting the uture possibility o consumption; () surplus resources allow or access to high status consumption; () earnings are suicient or buying high-quality, permanent goods with enough or leisure as well; () earnings are or goods which are immediately consumed; and (5) no regular earnings. 7

7 Insurance Markets and Companies: Analyses and Actuarial Computations, Volume, Issue, Table. Policies in orce on December, 5 vs. December, 6. Absolute requency and row percentage 5 ---G --S- --SG -D-- -D-G -DS- -DSG A--- A--G A-S- A-SG AD-- AD-G ADS- ADSG G --S- --SG -D-- -D-G -DS- -DSG A--- A--G A-S- A-SG AD-- AD-G ADS- ADSG Total row 8.8% 5.7% 5.6%.8% 87.%.%.7% 5.%.% 7 9.9%.5%.6% 5.% %.% %.9%.6% 5.%.%.% 76.9% 9.%.% 8.% % 7 5.6% 8 6.6% 58.%.%.6%.% 7.8% 7.% % 5.8% 8.7% 9.8%.% 7 7.9% 5.%.%.% 7.7% Total row % 5.% 9.%.% % 76 5.% 66.9% %.7%.% 6.8%.% 88.7% 8.%.7%.6% 8.%.%.% 7 7.8%.% 7.7%.% 6 7.7%.6%.% 8.6% 6.% 87.%.%.9% % 6.5% %.% 7.% 7.8%.%.% 88.8%.6%.6% 5.%.6% 6.6%.% 8 7.%.6%.%.6% 6 7.7% 6.68%.67% 6.% 8.56%.% 5.6% 9.% % 66.8%.56%.% 975.5% 6.58% 8.%.% Notes: The policies the customer has at any moment are represented by a string o our characters in the ollowing order A (automobile policy), D (diverse policy), S (health policy) and G (agricultural policy). The symbol - indicates that the customer does not have the corresponding policy. 8

8 5. Empirical application We estimate the probability o policy cancellation within three time periods, starting rom December, 5: short term (8 days), medium term ( year), and long term ( years and months, the whole time period being analyzed). We use logistic regression to predict the probability o observing a cancellation in each time period, based on a set o explanatory covariates. The variable descriptions and regression results are presented in the Appendix (Tables A to A). We use these models to estimate the probability o cancellation or dierent types o customers. In Table, we ix the customer s age, seniority (longevity in the company), and premium at the sample mean. Then, we calculate the probability o cancellation depending on gender, marital status, and type o policy (assuming the customer only has one) or each time period. For example, the probability o cancellation or a married man with a health policy is.% in the short term, 8.8% in the medium term, and 6.9% in the long term. Furthermore, the probability o cancellation or a single woman with an automobile policy ranges rom.6% (short term) to 6.9% (long term). In Table, we show the average historical proit generated by customers and also the estimation o prospective and potential proits or next year depending on gender, marital status and types o policies in orce. As previously mentioned, data Insurance Markets and Companies: Analyses and Actuarial Computations, Volume, Issue, on premiums and claims are only available or automobile and diverse policies, thereore, in these predictions we only consider customers having one o these two types o policies. The historical proit in Table is simply the average proit (average o the dierence between premiums and costs o claims) generated by these customers during 6 and 7. Married men with a diverse policy have the highest historical proit at 86,88 euros. In general, the historical proit o customers with policies in the diverse line o business is higher than o those with automobile policies, except or single women. Women have lower historical values than men, and single customers have higher historical values than those who are married. To calculate the prospective proit, we use a logistic regression model (see Appendix, Tables A5 to A8) to predict the probability o renewal or each customer and then multiply that probability by the average proit o the corresponding type o policy or the previous year. We ind that, on average, customers in the diverse line o business have slightly higher prospective proits than those in the automobile line o business. On the other hand, there are only very small dierences among customers in terms o gender or marital status. Actually, the estimation o prospective proit or customers having a diverse policy is the same or dierent gender or marital status because these two variables have no signiicant parameters in the corresponding logistic regression model. Table. Probability o cancellation by customer characteristics, time period, and policy Men Women Short term Medium term Long term Medium term Short term Long term Policy Marital status Auto Health Diverse Agro Married.7%.%.96%.% Single.77% 9.5%.6%.8% Married 6.6% 8.8% 9.8% 7.55% Single.%.6% 6.%.8% Married.6% 6.9% 9.77% 5.95% Single 9.9% 8.%.%.% Married.6%.8%.9%.% Single.6% 8.6%.9%.97% Married 5.% 5.9% 7.9% 6.7% Single.8%.% 5.8%.86% Married.9%.5% 5.5%.8% Single 6.9%.%.% 7.89% Age is 7. years; longevity in the company is 8.96 years; and premium is 8.7 euros. The average proit or the automobile line o business is.67 euros i the customer has only one policy and 5. euros i he has two or more policies. Average proit or the diverse line o business is 85. euros or only one policy and 5.78 euros or two or more policies. 9

9 Insurance Markets and Companies: Analyses and Actuarial Computations, Volume, Issue, Table. Historical, prospective and potential average proits (premiums minus total claims) or 6-7 in euros by type o policy Men Women Marital status Married Single Married Single Proit Policy Auto Diverse Historical Prospective Potential Historical Prospective Potential 5.6. Historical Prospective Potential Historical Prospective Potential.5.5 To estimate potential proit, we speciy a logistic regression model (see Appendix, Tables A9 and A) to predict the probability o underwriting policies o a dierent type than those the customer currently has and multiply that probability by the average proit o the corresponding type o policy during the previous year. We show in Table that customers with one policy covering diverse risks have higher average potential values than those with one policy o automobile. This is in part because it is more likely that a customer having a diverse policy would underwrite an automobile policy than the other way around. We also observe that men generally have a higher potential proit than women, and married customers have a higher proit than single customers. The expected proit loss or next year due to business risk can be estimated as described in section. Here we split the contribution o partial cancellations and total cancellation on the expected total loss. The expected loss due to partial cancellations or the total sample o customers is.87 million euros. Here, the probability o cancellation has been estimated using logistic regression model in Table A in the Appendix, which corresponds to cancellations in the medium term (one year time horizon). On the other hand, in order to measure the expected impact o total cancellations on the loss due to business risk we need to know the probability o a total cancellation, which we assumed the one observed in the sample, which is.76%. We multiply this probability by the total customer proit (prospective proit plus expected proit) and ind an additional loss o. million euros. The average proit is 99.7 or automobile and 66.6 or diverse policies (in the calculation o the average, all customers having these particular type o policies have been considered, independently on how many o them). The inal sample size was. ater delating some observations containing missing values. Thereore, the total expected loss this sample portolio will incur due to cancellations is =.9 million euros. Namely,.9 million euros / =. per customer. However, this estimation is only an expected value and does not represent a risk measure, which should be calculated with conidence levels o 95% or obtaining the quantile value o loss at this level. The most straightorward way to optimize the way unds aimed at increasing retention are used, is to invest them in customers generating the highest proits, while trying to avoid those with the lowest contribution to proits. In Table 5, we identiy our groups o customers according to their loyalty and proit. Proits here are calculated by adding prospective and expected proits, that is, revenue the company anticipates in the next year. Historical proit is not included, as it is a measure o the customer s past value, but its calculation is essencial as we saw that uture proit estimation is based on the realized or historical proit observed in the past. We divide customers by high and low proit (proit that is greater or smaller than the median o approximately ) and by high and low loyalty (probability o cancellation that is smaller or greater than the median o 6.%). We classiy the total number o customers in our sample accordingly in order to suggest dierent retention strategies. Finally, we ind that the company in our example should concentrate its retention eorts on the customer generating the highest proit (the right column o Table 5). Table 5. Loyalty vs. proit Loyalty Low loyalty Prob. 6.% Low proit Proit < Age: 9.55 Longevity: % men 89.5% married 7.75% single Value High proit Proit Age: 7.5 Longevity:.9 8.5% men 8.5% married.86% single 5

10 Loyalty High loyalty Prob. < 6.% Conclusions Table 5 (cont.). Loyalty vs. proit Low proit Proit < Age: 6.5 Longevity: % men 5.% married 9.% single Insurance Markets and Companies: Analyses and Actuarial Computations, Volume, Issue, Value High proit Proit Age: 5. Longevity:.7 57.% men 85.% married.6% single An analysis o loyalty and the proit generated by the customer is the oundation or managing business risk. In this article, we propose a method or determining expected losses due to policy cancellation by estimating the proit generated by each customer and predicting their probability o cancellation. The analysis o the proit the customer generates is done in three dimensions: historical, prospective and potential proit. The irst o them considers the proit generated in the past and the other two are the ones that can be generated in the next period as a result o keeping the policies in orce (prospective proit) or underwriting policies o a dierent type (potential proit). Based on them, the expected proit loss due to business risk is ormulated. Reerences An empirical application is carried out by analyzing a sample o customers. Dierent products are considered in our analysis simultaneously. We observe that actors such as gender or civil status aects the probability o cancellation, and also that health policies are more likely to be cancelled than the other types o policies being considered here. We additionally calculate the historical, prospective and potential proits or customers in our sample holding diverse or automobile policies. Based on these calculations, the total expected loss the company will incur due to business risk is calculated or our sample. This is not a risk measure but only the expected value o the loss due to business risk. Nevertheless, it is a valuable inormation or managers, as it is the limit o unds that should be invested in customer loyalty. Segmentation strategies can be applied in order to decide how to invest these unds in customer retention. The research carried out here can be extended in order to deine suitable measures that could better represent the exposure o the company to business risk, in the context o classical risk measures.. Babbel, D. (985). The price elasticity o demand or whole lie insurance, The Journal o Finance, (), pp Ben-Arab, M., Brys, E. & Schlesinger, H. (996). Habit ormation and the demand or insurance, Journal o Risk and Insurance, 6 (), pp Berger, P.D. & Nasr, N. (998). Customer lietime value: marketing models and applications, Journal o Interactive Marketing,, pp Brockett, P.L., Golden, L., Guillén, M., Nielsen, J.P., Parner, J. & Pérez-Marín, A.M. (8). Survival analysis o household insurance policies: How Much Time Do You Have to Stop Total Customer Deection? Journal o Risk and Insurance, 75 (), pp Cooley, S. (). Loyalty strategy development using applied member-cohort segmentation, Journal o Consumer Marketing, 9 (7), pp Crosby, L.A. & Stephens, N. (987). Eects o relationship marketing on satisaction, retention, and prices in the lie insurance industry, Journal o Marketing Research, (), pp Dhaene, J., Vanduel, S., Goovaerts, M.J., Kaas, R., Tang, Q. & Vyncke, D. (6). Risk measures and comonotonicity: a review, Stochastic Models,, pp Doherty, N.A. (98). Portolio eicient insurance buying strategies, The Journal o Risk and Insurance, 5 (), pp Donkers, B., Verhoe, P.C. & Jong, M.G. (7). Modeling CLV: a test o competing models in the insurance industry, Quantitative Marketing and Economics, 5 (), pp Duker, J.M. (969). Expenditures or lie insurance among working-wie amilies, The Journal o Risk and Insurance, 6 (5), pp Guillén, M., Nielsen, J.P. & Pérez-Marín, A.M. (6). La duración de distintos contratos de seguros en los hogares, Un enoque integrado, Gerencia de Riesgos y Seguros, 96, pp. -.. Guillén, M., Nielsen, J.P. & Pérez-Marín, A.M. (8). The need to monitor customer loyalty and business risk in the European insurance industry, Geneva Papers on Risk and Insurance Issues and Practice,, pp Hammond, J.D., Houston, D.B. & Melander, E.R. (967). Determinants o household lie insurance premium expenditures: an empirical investigation, The Journal o Risk and Insurance, (), pp Jackson, D. (989). Determining a customer s lietime value, part three, Direct Marketing, 5 (), pp Kotler, P. (97). Marketing during periods o shortage, Journal o Marketing, 8 (), pp Lai, Li-Hua, Liu, Chun Ting, Lin, Jinn Tyan (). The moderating eects o switching costs and inertia on the customer satisaction-retention link: auto liability insurance service in Taiwan, Insurance Markets and Companies: Analyses and Actuarial Computations, (), pp Mayers, D. & Smith, C.W.Jr. (98). The interdependence o individual portolio decisions and the demand or insurance, The Journal o Political Economy, 9 (), pp. -. 5

11 Insurance Markets and Companies: Analyses and Actuarial Computations, Volume, Issue, 8. Nakada, P., Shah, H., Koyluoglu, H.U. & Collignon, O. (999). P&C RAROC: a catalyst or improved capital management in the property and casualty insurance industry, The Journal o Risk Finance, (), pp Ryals, L.J. & Knox, S. (5). Measuring risk-adjusted customer lietime value and its impact on relationship marketing strategies and shareholder value, European Journal o Marketing, 9 (5/6), pp Schlesinger, H. & Schulenburg, J.M. (99). Customer inormation and decisions to switch insurers, Journal o Risk and Insurance, 6 (), pp Showers, V.E. and Shotick, J.A. (99). The eects o household characteristics on demand or insurance: a tobit analysis, Journal o Risk and Insurance, 6, pp Staord, M.R., Staord, T.F. & Wells, B.P. (998). Determinants o service quality and satisaction in the auto casualty claims process, Journal o Services Marketing, (6), pp Verhoe, P.C. and B. Donkers (). Predicting customer potential value: an application to the insurance industry, Decision Support Systems,, pp Wells, B.P. & Staord, M.R. (995). Service quality in the insurance industry. Customer perception versus regulatory perceptions, Journal o Insurance Regulation, (), pp Appendix Table A. Explanatory covariates Variable Description Longevity Number o whole years passed since the customer underwrote the irst policy in the company until December, 5. Age Age o the customer as o December, 5. Mars Gender Premium Marital status, speciied by our binary covariates: Mars_M married, Mars_W widow, Mars_D Divorced, and Mars_S Separated. The reerence group is composed o single individuals. Those or whom the marital status is unknown have been eliminated rom the analysis, including irms. Gender o the customer, indicated by a binary covariate which is equal to when the customer is male. Premium paid by the customer. I a cancellation has occurred, we took the premium paid or the policy being cancelled as representative o the premium level o that customer. I there has not been any cancellation, we took the premium o the irst policy that has been modiied during the period considered in the analysis, usually the irst one that was underwritten or renewed during the period. Auto Automobile policy. Binary covariate equal to i the customer has an automobile policy in orce as o December, 5. Health Health policy. Binary covariate equal to i the customer has a health policy in orce as o December, 5. Diverse Diverse policy. Binary covariate equal to i the customer has a diverse policy in orce as o December, 5. Agro Agricultural policy. Binary covariate equal to i the customer has an agricultural policy in orce as o December, 5. Table A. Logistic regression model estimation or the probability o cancellation in the short term (*) Constant <. Longevity <. Age <. Mars_M <. Mars _W <. Mars _D Mars _S Gender Premium Auto <. Health <. Diverse <. Agro <. Notes: (*) Ater deleting observations with incomplete inormation, we estimated the probability o cancellation in the short term or a sample o 8798 customers. This includes 5 cancellations (.%) during the irst 8 days o 6. Results support the overall signiicance o the model. The likelihood ratio is 8.6, with degrees o reedom, and p-value less than.. Table A. Logistic regression model estimation or the probability o cancellation in the medium term (*) Constant <. Longevity <. Age <. Mars_M <. Tari premium without taxes, gross yearly amount. 5

12 Insurance Markets and Companies: Analyses and Actuarial Computations, Volume, Issue, Table A (cont.). Logistic regression model estimation or the probability o cancellation in the medium term (*) Mars _D Mars _S Gender <. Premium Auto <. Health <. Diverse <. Agro <. Notes: (*) Our sample o 8798 customers includes 7 cancellations (8.9%) during 6. Results support the overall signiicance o the model. The likelihood ratio is 885.6, with degrees o reedom, and p-value less than.. Table A. Logistic regression model estimation or the probability o cancellationin the long term (*) Constant <. Longevity <. Age <. Mars_M <. Mars _W <. Mars _D Mars _S Gender <. Premium <. Auto <. Health <. Diverse <. Agro <. Notes: (*) Our sample o 8798 customers includes 87 cancellations (7.%) rom December, 5 to March, 8. Results support the overall signiicance o the model. The likelihood ratio is 8.6, with degrees o reedom, and p-value less than.. Table A5. Estimation o the logistic regression model or the probability o renewing an automobile policy i that policy was in orce in the previous year (*) Constant <. Longevity Age Mars_M Mars_D Mars_S <. Gender Diverse Notes: (*) The sample used consists o observations corresponding to customers with one automobile policy in orce at the end o 6. Most o them, 7 customers, renewed their policies (99.%) the next year. Results support the overall signiicance o the model. The likelihood ratio is 8.57, with 7 degrees o reedom, and p-value less than.. Table A6. Estimation o the logistic regression model or the probability o renewing two or more automobile policies i that policies were in orce in the previous year (*) Constant <. Longevity <. Mars_M Notes: (*) The sample used consists o 67 observations corresponding to customers with two or more automobile policies in orce at the end o 6. Most o them, 879 customers, renewed their policies (8.6%) the next year. Results support the overall signiicance o the model. The likelihood ratio is 5.7, with degrees o reedom, and p-value less than.. 5

13 Insurance Markets and Companies: Analyses and Actuarial Computations, Volume, Issue, Table A7. Estimation o the logistic regression model or the probability o renewing a diverse policy i that policy was in orce in the previous year (*) Constant <. Longevity Age <. Notes: (*) The sample used consists o 858 observations corresponding to customers with one diverse policy in orce at the end o 6. Most o them, 799 customers, renewed their policies (9.%) the next year. Results support the overall signiicance o the model. The likelihood ratio is 6.9, with degrees o reedom, and p-value less than.. Table A8. Estimation o the logistic regression model or the probability o renewing two or more diverse policies i that policies were in orce in the previous year (*) Constant <. Longevity Age Gender Notes: (*) The sample used consists o 5 observations corresponding to customers with two or more automobile policies in orce at the end o 6. Most o them, 6 customers, renewed their policies (88.6%) the next year. Results support the overall signiicance o the model. The likelihood ratio is.8, with degrees o reedom, and p-value less than.. Table A9. Estimation o the logistic regression model or the probability o underwriting an automobile policy i the customer did not have any policy o this type the previous year (*) Constant <. Longevity <. Age <. Mars_M <. Mars_W Mars_S <. Gender <. Notes: (*) The sample used consists o 5 observations corresponding to customers who do not have any automobile policy in orce at the end o 6. Only 8 o them underwrote an automobile policy the next year (7.87%). Results support the overall signiicance o the model. The likelihood ratio is 78.5, with 6 degrees o reedom, and p-value less than.. Table A. Estimation o the logistic regression model or the probability o underwriting a diverse policy i the customer did not have any policy o this type the previous year (*) Constant <. Age Mars_M Mars_W Mars_D Gender Notes: (*) The sample used consists o 859 observations corresponding to customers who do not have any diverse policy in orce at the end o 6. Only 9 o them underwrote an automobile policy the next year (.%). Results support the overall signiicance o the model. The likelihood ratio is 6.5, with 5 degrees o reedom, and p-value less than.. 5

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