Research Master Thesis

1

Research Master Thesis

Client value on the presence of credit

risk: A dynamic method to coordinate

customer life time value and credit

risk management decisions

J. Cardona Hernandez

August 16, 2013

Author: Johanna Cardona Hernandez Student number: s2097265

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______________________________________________________________

Abstract

Customer life time value (CLV) and delinquency behavior are two of the most important firm key performance indicators at the banking industry, and at any other business scenario in which customers are granted credits or loans. Coordination in decisions regarding these two important variables is essential to achieve overall firm performance. We develop a dynamic method that allows simultaneous coordination between CLV and credit risk management decisions while taking into account their time varying behavior, the observed and unobserved customer heterogeneity, and the panel data structure inherent to the CLV-credit risk context. The method, which is a combination of the extended Kalman filter for exponential family of distributions and a fixed effects model has not been illustrated and/or implemented before. Therefore, we show its validity form a theoretical and computational point of view, and how its results can be interpreted to make the desired decision coordination task using synthetic data.

Key words: customer life time value, credit risk management, extended Kalman filter.

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1. Introduction

3 The CLV perspective has been widely implemented within the financial and retailing industries given their ability to use sophisticated information systems to track and manage detailed customer behavior over time. Such detailed tracking offers great opportunities for companies to have an overall customer profitability evaluation that takes into account every angle of the customer relationship with the firm. Among firms such as banks or retailers offering the opportunity for its customers to buy on credit, the customer payment behavior is an interesting and important angle of the customer relationship. The profitability that each client generates for the firm directly depends on his/her ability to fully refund the granted loan or credit within an agreed deadline. Therefore, such ability it is an important element that should be considered when making profitability assessment, and subsequent marketing decisions, both at the customer and the firm level. In other words, the credit risk management process (credit granting decisions) should be a customer relationship angle considered when evaluating customer lifetime value. Moreover, coordination between CLV maximization and optimal credit risk decision making (and vice versa) can highly contribute to a firm’s overall profit. According to Finlay (2008b) making lending decisions based on the forecast of financial measures such as customer contribution to profit, is the true objective of commercial lenders. Such an objective allows companies to efficiently target profitable customers with special offers to improve retention, or to restrict credit limits on unprofitable customers to reduce losses.

Interestingly and to the best of our knowledge, the customer’s credit risk angle is not considered when evaluating customers’ value on both the academic and the business scenario. Moreover, credit risk and customer lifetime value decisions are typically made disjointly, and are not coordinated in terms of statistical approaches and available databases.

4 outcome period is in general weak. This sensitivity has made it difficult for researchers to adequately model the time-variant nature of profitability.

This dynamic behavior can be explained at the customer level by the different marketing actions used by the firm to retain the customer, and to ensure his/her good payment behavior, besides other customer specific aspects such as sociodemographic characteristics (income, education, etc.). In addition to these reasons for the lack of synchronization on the decision making process, we also believe that the time-varying behavior of the credit risk management related variables, such as payment behavior, also adds complexity to the task. In the same line of reasoning, a customer can be a good or a bad payer according to the firm actions, the customer socio demographic characteristics, the credit limits, and many other factors that also change over time.

The time-varying behavior of the CLV and credit risk indicators and their related variables, which are measured at the customer level, suggest the need of evaluating the CLV-credit risk decision making process within a panel data structure. Such structure allows taking into account the individual dynamic behavior to consequently incorporate it in an overall evaluation at the customer base level.

Apart from the previously mentioned obstacles, we also believe that CLV and credit risk actions are being taken disjointly mainly due to a lack of methods that actually allow managers to identify, study, and efficiently evaluate the synergies between the two variables. To the best of our knowledge there is not a single paper addressing such an important coordination task on the banking industry or on other environments where credit risk decisions have to be made.

5 The aim of this thesis is therefore to illustrate a method that allows managers to find new opportunities to coordinate different firm decisions so that they are simultaneously beneficial from a CLV and a credit risk point of view. Such a method should take into account three aspects: 1) the time varying behavior of the CLV and the credit risk related variables which comes from customer- and firm specific characteristics that also change over time, 2) the fact that those customer- and firm-specific characteristics make each customer different in terms of CLV and credit risk behavior, and therefore it is necessary to account for customer heterogeneity, and 3) the panel data structure inherent to the CLV-credit risk context.

In particular, we propose to use a generalized state space model using the extended Kalman filter which captures the time varying behavior of the context by updating the parameter estimates on time. Then, we account for customer heterogeneity and the panel data structure by combing the state space and filter with a fixed effects model. This method allows understanding how different marketing and credit risk management decisions (firm-specific characteristics) influence the time-variant behavior of both customer profitability (CLV) and customer payment behavior, through different customer-specific characteristics. It is noteworthy that to the best of our knowledge there are no studies using the extended Kalman filter for the case of two dependent variables, the CLV and the credit risk management variables that have different probability distributions: the normal and the binary distribution, respectively. Therefore it is also the aim of this thesis to show that the mentioned filter works well form a theoretical and computational point of view.

Summarizing, this thesis has both a theoretical and a managerial aim. We intent to show the theoretical and computational validity of the extended Kalman filter when having two different distributed dependent variables, and then how it can be used to coordinate CLV and credit risk management decisions, by combining it with a fixed effects model. The whole process will be illustrated using synthetic data.

6 coordinated decisions can be made. We end with a discussion of our findings and specify some directions for future research.

2. Research background

In this section we will discuss the most important elements and definitions regarding CLV and delinquency behavior, the type of methodologies used to model them, and their customer- and firm-related antecedents.

2.1 Customer life time value

The Customer Life Time Value (CLV) is a very useful metric to measure the long-term profitability of customers that allows identifying the value each client represents for the firm. It is generally defined as the present value of all future profits obtained from a customer over his or her life of relationship with a firm (Gupta et al. 2006). CLV is similar to the discounted cash flow approach used in finance, except from the fact that CLV is generally estimated at the individual level and incorporates the possibility of customers defecting to competitors in the future. CLV can be used to guide the firm's acquisition and retention activities to make a more efficient use of marketing resources, and can be aggregated over customers as a measure of firm or segment value (Blattberg et al. 2009). The CLV for a customer (omitting customer subscript) is as follows (Gupta, Lehmann and Stuart, 2004; Reinartz and Kumar, 2003):

∑ _{( )} ( ) -AC (1)

Where

pt = Revenues generated by the customer at time t, ct = direct cost of servicing the customer at time t, = discount rate or cost of capital for the firm,

rt = Retention probability: probability of customer repeat buying or being “alive” at time t, AC = acquisition cost,

7 The acquisition costs (AC) are included in equation (1) assuming that it is the lifetime value of an as-yet-to-be-acquired customer. In the case of existing customers AC is not included.

Based on equation (1) it is possible to conclude that the Customer Life Time Value has four main components: 1) the retention probability, 2) the generated revenues, 3) the incurred costs and 4) the discount rate. Each component can be computed or forecasted through different methodologies which will be explained on posterior sections. Once the components are predicted or computed, they are included in equation (1) in order to compute the Customer Life Time Value.

2.1.1 Models for the different CLV components

Retention

8 The always a share/migration/non-contractual setting considers that customers can reappear after some periods during which they did not make transactions, and traces their probability of ‘reactivating’ (Calciu, 2009). This setting is representative for catalogue buying, groceries stores, etc. Here not only surviving customers, but also customers allowed to reactivate after a given number of inactive periods, are included in the CLV computation. The basic migration model uses Markov Chains to forecast retention by defining ‘recency state j’ meaning that the customer’s last purchase from the company was j periods ago. The recency probabilities pj are then modeled as

the probability that the customer purchases in the current period, given that the customer last purchase was j periods ago (Blattberg, Kim and Neslin 2008). Other common method is the one proposed by Schmittlein et al. (1987): The Pareto/ Negative binomial distribution (NBD) model. It predicts the probability that a customer with a particular observed transaction history is still alive at time t, taking into account and measuring heterogeneity in observed behavior.

Revenues

Causal models such as multiple linear regression models are used to forecast future spending behavior. To do so, the log of spending (to avoid negative predictions) is used as dependent variable, and price, demographics and other behavioral variables can be used as independent variables (Blattberg, Kim and Neslin 2008). Other methodology useful to forecast future revenues from each customer is time series. Calculation of the trend in revenue per period per customer from the initial customer acquisition period to the end of the customer’s purchase series can be used to model the individual spending trend using a constant growth rate.

Costs

9

2.1.2 CLV Antecedents

According to Blattberg et al. (2009) the antecedents of CLV are defined as customer- or firm-related actions or behaviors that affect at least one of the CLV four main components. Among the customer-related antecedents, Reinartz and Kumar (2003) proposed different CLV antecedents. We discuss two of them: 1) customer spending, and 2) cross-buying.

The authors argue that those customers who buy more, buy more frequently, and buy more across different categories have a better fit with the vendor’s products and receive greater utility from them, making the relationship between the vendor and the customer more durable. In this way customer spending is positively related to retention, and in turn to CLV.

Based on the higher switching costs argument, the authors say that customers who cross-buy more, i.e. who consume from a variety of product lines, are less prone to terminate the relationship with the company. In other words, cross-buying is positively related with retention and consequently with CLV given all the effort that changing to another vendor would mean. Other authors such as Blattberg et al. (2009) highlight different papers where the relationship between customer satisfaction and relationship duration, and firm performance is studied (Bolton, 1998, Guo et al. 2004). They conclude that although the strength and magnitude of the effect of customer satisfaction on CLV (though retention) depends on the industry, it is an element that is worth studying and including as a CLV antecedent. Finally, Blattberg et al. (2009) also consider the Recency (time since the last purchase), Frequency (the number of previous purchases) and Monetary (the total amount spent) variables (RFM) as important elements when it comes to determine CLV. In their paper, the authors address the need for more research on the effect of the RFM variables on the CLV components. However, their view is that frequency and monetary are likely to have a positive effect and recency a negative effect (the longer a customer has been inactive the lower CLV is) on CLV.

10 aware of the latest promotions, the arrival of new products, or on the other hand, irritated by a huge amount of information they did not ask for. Therefore, and although Reinartz and Kumar (2003) suggest a positive relationship, we propose that mailings can have either a positive or negative effect on CLV via retention, depending on the type and amount of information being sent to the customers. Regarding other determinants, Anderson and Simester (2004) suggest that promotions increase short-term sales and decrease long-term sales due to stock piling effects. Therefore, it is important to efficiently decide the frequency and amount of promotions to be sent since they can affect purchasing behaviour and in turn CLV.

Within the banking scenario, the interest rates directly affect CLV since it becomes one of its main components: the discount rate. Moreover, it can also influence purchasing behaviour in the way that consumers find loans with lower interest rates more appealing. This can encourage them to acquire more products, increase their spending, and eventually increase the total revenues they generate for the firm and therefore, influence CLV.

2.2 Credit risk management and scoring

Credit scoring is the set of predictive models and their underlying techniques that are useful for the assessment of the risk associated with granting credits. This risk is the possibility that counterparty in a financial contract or credit will not fulfill a contractual commitment to meet her/his obligation stated in the contract (Bielecki and Rutkowski, 2002). In other words, credit risk is the uncertainty about the client’s ability to fully refund the loan within the agreed maturity deadline. If the client does not fulfill the contractual agreement, we say that the client defaults or is delinquent, or that the default event occurs. Given its essential role on the credit granting decision making process, credit scoring models have become of primary importance in the financial environment (Mavri et al. 2008).

11 al. 2002). The relationship between historical information and future credit performance can be described by the following formula (Lahsasna et al. 2010):

yi=f(x1,x2,x3,….xm) (2) Where yi denotes if customer i is good or bad (non-delinquent or delinquent); the good/bad definition is based on three major components 1) the client’s number of days after the due date (days past due, DPD), 2) the amount past due, and 3) the time horizon in which these two components will be traced (Lewis, 1992). Each component is set to a specific value according to the type of financial product (mortgages have longer maturities than consumer loans), the company’s internal calculation, its marketing strategy and credit policy. Then, customers who have accounts fulfilling each of the three set values will be considered as a bad client. For example, a bad client can be defined as having more than 60 DPD in 12 months from the first due date with an amount past due higher than 3 euros.

The explanatory variables x1,x2,x3,….xm are customer and product-related features such age, income, past credit behavior or interest rates, and f is the function or the credit scoring model. According to Chuang and Huang (2011) credit scoring models can be classified into parametric statistical methods such as Linear Discriminant Analysis (Desai et al. 1996) and Logistic Regression (Laitinen, 1999), non-parametric statistical methods such as k nearest neighbor (Henley and Hand, 1996) and decision trees (West, 2000), and other computing approaches such as Artificial Neural Networks (Tsai and Wu, 2008).

2.2.1 Antecedents of delinquency

12 Other consumer variables such as income, region, credit experience, bankruptcy, assets, types of products purchased, and channels used to purchase also have a direct influence on delinquency.

When it comes to prevent delinquency, financial institutions can adopt different strategies to reduce consumer’s late payments and avoid financial losses. One option is to send via email or SMS, payment reminders to customers when their due date is soon. These alerts might prevent them from missing payments due to oversight. Another repayment incentive is related to the interest rates. Some institutions implement cash back strategies in which a reduction in the monthly interest rate is paid back at the conclusion of the loan, if customers make all their monthly payments on time.Other option is to reduce the interest rate of customer’s next loan if his/her current loan payments are all made on time. Other credit risk management decisions that can influence customer’s probability of default are the credit or loan-specific characteristics such as loan duration, number of installments in the case of credit cards, credit limits, etc.

3 Model specification

13 customer profitability from a customer life time value perspective and customer’s delinquency behavior have never been addressed using a state space model, nor the extended Kalman filter.

For ease of understanding, we first introduce our model specification in a normal state space format. Then, we specify the model within the generalized context. The normal state space model consists of two parts: the measurement equation and the state equation. The measurement equation is as follows: ( ) ( ) ( ) ( ) (3)

is the Customer Life Time Value (CLV) variable which is continuous and normally distributed. is a discrete and binary variable (Delinquency) which indicates if customer i is delinquent at

time period t (1 if the customer is delinquent, 0 otherwise). and are 1 x K1 and 1 x K2 matrices that contain the CLV and delinquency customer-specific antecedents, respectively. is

then defined as a 2 x K block matrix containing the and matrices ( customer-specific

characteristics and actions for subject i at time period t). ( ) is a K x 1 vector

denoting the state vector which is unknown. The error terms and are assumed to have

covariance different from zero, and normally and binomial distributions, respectively. The second equation in the normal state space model is the state or transition equation which describes the transitions of the state vector over time as follows: . However, since we

want to account for customer heterogeneity our state equation would be:

14 is a K x K matrix called the transition matrix; for our application F=I which specifies a multivariate regression and a logistic regression model with time-varying parameters.

( ) is a P-dimensional vector, including marketing and credit risk management decisions affecting each subject i at time period t. We include these variables to capture the observed consumer heterogeneity in their sensitivity to the variables. is therefore a K × P

matrix of time-fixed parameters ( ) that captures such observed customer

heterogeneity. The term ( ) is a K-dimensional column vector that captures the

unobserved heterogeneity in the state mean. We assume that the errors and are

uncorrelated. The hyperparameters , are unknown constants. Estimation of these

matrices will be addressed in the estimation procedure section.

Generalized linear dynamic model

Given the non-Gaussian nature of the delinquency variable ( ), and consequently, of the error

term , the previous state space model cannot be specified within the Gaussian context.

Therefore, it is necessary to use a generalized linear dynamic approach to define it. Since both and follow distributions that can be classified into the exponential family of distributions,

it is possible to assume that the error term also belongs to that family. West, Harrison, and

Migon (1985) proposed the generalized linear dynamic model in order to take into account this specific type of distributions when estimating a state space model. Assuming ( )

normally distributed, it would be possible to reformulate (3) as follows (Fahrmeir and Tutz, 1994):

( ) (6)

Using this notation it is easy to illustrate the model for the exponential family of distributions. The conditional density ( ) for the exponential family of distributions has conditional

mean ( ) ( ) . This conditional mean together with

the exponential family assumption (the distribution of and belong to the exponential

family of distributions) replaces the measurement equation in linear normal models. Since in this case is a 2-dimensional vector containing two different dependent variables, each with

15 (( ) ( )) ( ) ( ( ) ( )) (7)

Given that is normally distributed, is the link function for the Gaussian case: ( ) ( ) with variance ( ) . Since is a binomial variable and we

would like to estimate a logistic regression, is the logit function ( ) ( ). It

is noteworthy that a probit model would also be suitable to model the binary variable ‘delinquency’. Moreover, such a model would avoid the non-normality issue by using the cumulative normal distribution as link function. However, we choose to use the logistic regression because historically, it has been used and preferred in the credit risk area due to its easier interpretation. In this way, the use of the logistic model make the results of this thesis have more practical value and more likely to be applied in real business scenarios.

Consequently, we assume that has a distribution with zero mean and variance given by the

logistic variance ( ) ( )( ( )). The covariance between and is

defined as ( ) √ ( ) ( ) √ ( )( ( )). Finally, is

specified as:

( ( )

( ) ( )( ( )))

( ) (8)

Once , g(.) and have been defined, the measurement equation for the generalized linear

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3.1 Estimation procedure

Kalman filter

When it comes to estimating and its correspondent covariance matrix, different estimation

procedures are available. The most widely used option is the Kalman filter algorithm (Kalman, 1960) which tries to find optimal values of given its covariance matrix which is estimated by means of maximum likelihood (Fahrmeir and Tutz (1994) pp. 263-264). However, and given the non-Gaussian nature of the delinquency variable and consequently of the error term , the

standard Kalman filter recursions are not valid in our case. A most suitable approach would be the extended Kalman filter which allows the estimation of the state parameter in the case of

exponential family of distributions. This filter bases its estimates on the posterior mode instead of the posterior mean. The posterior mode or the maximum a posteriori estimate is obtained by maximizing ( ) with respect to . The recursive formulas of the extended Kalman filter

and smoother for the parameter updates can be found in Appendix A.1. Moreover, Fahrmeir and Tutz (1994) pp. 296-297 also present the generalized Kalman filter for panel data, which performs an additional loop across individuals to incorporate the longitudinal structure of the data.

At first sight the latter methodology seems to be perfectly suitable for the panel data structure and exponential family distributions we are dealing with. However, the two previously mentioned models have an important assumption that contradicts our desired model specification from equation (4). They assume that , i.e. that all i=1,..,n share an underlying structure, and

therefore it does not account for heterogeneity across individuals. This means that it would be wrong to let depend on the firm specific decisions , which play an essential role in our

intended method. In this way it is possible to conclude that the extended Kalman filter and the generalized Kalman filter for panel data do not go in line with the aim of this thesis.

Kalman filter and fixed effects model

17 it is possible to use the extended Kalman filter and therefore account for the non-normal distribution of the error (See appendix A1 for the formulas of the extended Kalman filter). Once the filter has been run separately for each of the n customers and estimations for each of the ( ) parameters are estimated; we incorporate the observed and

unobserved customer heterogeneity and the panel data structure by fitting a fixed effects model as follows:

̂ ̅̅̅̅̅ ̂ ( ) ( )

Where ̅̅̅̅̅ is the average across individuals of the estimations of ̂ at time period t.

We choose to estimate a fixed-effects model rather than a random-effects model. This is because we assume that each customer is different due the application of different firm actions that then influence their customer-specific variables (and consequently, their CLV and delinquency behavior), and to some other omitted variables. In other words, we assume the variables to

be correlated to the . It is noteworthy that the parameters do not change across

customers because we want to coordinate decisions at the customer base level and not at the customer level. Those parameters still account for customer heterogeneity because their estimation is based on differences among customers related to firm actions i.e. the variables.

The fixed effects model assumes that the variables considerably change both across time and

individuals, and therefore provides valid estimations for the parameters . Then, these

parameters will be used to coordinate the marketing and credit risk decisions. We include the ̅̅̅̅̅̅̅to take into account the base line from which all the ̂ vary according to each

customer’s values and other possibly omitted variables ( ).

Kalman smoother

Assuming that extra information is available, i.e. that at time t there are future observations t+1, t+2,…,T it is possible to use smoothing procedures to reduce the noise on the filtered estimates of

. More specifically, clearer estimates are provided by using the difference between the

18 backwards recursions to get smoothed estimates of the state vector and covariance matrices. This smoother is suitable for both normal and exponential family distributed errors.

EM-type algorithm

The iterative updating principle of the Kalman filter requires values for the unknown variance matrix , and the so called priors a0 and Q0 to generate the estimates for . Given that those

matrices play a role in the estimation of the state parameters, it is important to ensure that the selected values help the model to generate better state estimations. Such selection will be done using optimization procedures based on the log-likelihood function of the model and the repeated application of the modified Kalman filter in an Expectation-Maximization (EM) type of algorithm. More specifically, the method consists on identifying the values of

a

₀

,

Q

₀

,

Q

that maximize the log-likelihood function of the generalized linear dynamic model. These values are then used in the extended Kalman filter to obtain estimations of the state parameters. This procedure is done within an EM-type of algorithm context (iteratively) until convergence is reached, i.e. until we find the values of

a

₀

,

Q

₀

,

Q

that better accommodate the data, and the state parameter’s estimation. The (penalized) log-likelihood function that needs to be maximized both for the Kalman filter estimations and the hyperparameters estimation for the i-th individual is:

(

) ∑

(

)

(

)

(

)

₍

₎

∑

(

)

(

) (10)

Where

and are the initial values for the Kalman filter, and

(

)

and

(

)

are the normal and logistic log-likelihood contributions of individual i,

respectively, with

( ) and

( ).

Summarizing, we will estimate a generalized linear dynamic model for the case of the exponential family of distributions (extended Kalman filter). We will estimate the hyperparameters ( ) using an EM-type of algorithm. Those estimates will then be used to generate filtered

parameter estimates. Afterwards, the filtered vector is smoothed using the fixed interval

19 where we will estimate the parameters . These parameters will allow us to coordinate the CLV

and credit risk management decisions. In this way, we have a method that accounts for the time-variant behavior of the involved variables, the customer heterogeneity, the panel data structure and the exponential family distribution of the error term. To the best of our knowledge there are not standard packages available to estimate this specific model; therefore we developed a code using the R language (The R Project for Statistical Computing).

4 Simulation

Given the lack of real data to test the previous methodology we will generate synthetic data using the model specification from equations (4) and (9). This type of data will not allow us to actually make coordinated decisions for CLV and delinquency. However, it will be useful to show the adequacy of the extended Kalman filter from a computational and a theoretical point of view, and to illustrate how and which type of decisions can be made with the method.

The data simulation procedure is executed for 1000customers, using a time span of four years (T=4), in which we believe CLV and delinquency are very likely to expose their time-variant behavior. We simulate K=6 customer-specific variables and P=4 marketing and credit risk management decisions.

The state equation

The first step in the data generation procedure is to obtain the states parameters that will then be used to generate the dependent variables. Using some previously set and random initial values for , a normally distributed random error, and the state equation from equation (9), we

simulate the state parameters ( ). Table 1 presents the summary statistics of

20 Table 1 Summary statistics of the simulated state parameters

The measurement equation

For the measurement equation it is necessary to simulate two dependent variables: CLV and delinquency, and several customer-related variables that affect each of those two key performance indicators. As mentioned in section 2.1.2 we identified several antecedents for CLV from a customer point of view. These variables are contained in matrix and are customer

spending, customer cross-buying, customer satisfaction and the RFM variables (recency, frequency, and monetary). On section 2.2.2 we also highlighted different customer variables that influence delinquency: past credit history, income, credit experience, bankruptcy, assets, and types of products. These variables are included in . We simulate the determinants of CLV and

delinquency generating random numbers using the normal distribution. Although we identify several antecedents, we only include six simulated explanatory variables on the measurement equation (K1=3 explanatory variables for CLV, and K2=3 variables to explain delinquency, K=6). The reason is that the aim of this thesis is to prove that the developed method works computationally and theoretically, and not to give specific information on the effect of determinants on the dependent variables. Summary statistics of the K=6 simulated explanatory variables can be found in Appendix A.2.

To simulate the dependent variables following the model specification, we use the generated states and the simulated . The CLV variable ( ) is the result of the multiplication of

21 with probability of success equal to ( )/ ( ( ) )), where success

means that the customer was delinquent during a specific time period. Summary statistics of the CLV and delinquency variables can be found in Appendix A.3.

The fixed-effects model

The fixed effects model includes the firm decision variables that are supposed to affect both CLV and delinquency through their antecedents, or more specifically through the state parameter estimations. As mentioned in sections 2.1.2 and 2.2.2, the firm decisions that affect these key performance indicators are payment reminders, interest rates reductions, loan duration, number of installments in the case of credit cards, credit limits, mailing, promotions, and interest rates. In a real data demonstration of the method, these and other variables should be included in the matrix in the fixed effects model. For this thesis purposes we will include four synthetic

variables that will act as firm actions (P=4), and will be generated through the normal distribution. Summary statistics of the simulated P variables can be found in Appendix A.4.

5 Results

In this section, we discuss the estimation results of the extended Kalman filter and the fixed effects models. We assess the extended Kalman filter predictive ability, and present the interpretation of the fixed effects parameter estimates within a decision making scenario.

The extended Kalman Filter

The first step is to generate the state parameter estimates for each customer. This is done running the extended Kalman filter and smoother using the mentioned number of explanatory variables (K=6). The initial values used for the estimation were ( ) ( ) . The

transition matrix is defined as , which specifies a regression model with time-varying parameters. In average, each customer’s filter convergence, via the EM-type algorithm, was achieved within 2 iterations.

22 T time periods (taking into account all the individuals in the simulated customer base) (Table 2). It is possible to say that the estimations of the six parameters along the four time periods are in average reasonable and do not present any strange behavior that could highlight a malfunctioning of the extended Kalman filter. This conclusion is based on the values of the first and third quartile of the distribution of all parameters at each time period which are, alike the distribution of the true parameters, between -1.5 and 1.7. It is although noteworthy that the minimum and maximum values for all parameters are somehow big, meaning that for some customers the model is generating unusual values. It is also possible to see that in general the three first parameter estimates and (parameters that correspond to ) have more

extreme values (maximum and minimum) than the ones corresponding to the delinquency variable ( and ). These unusual values however, are only present in approximately 2% of

the customer base given the values of the 1-st and 99-Th percentile of the distributions. Regarding the average standard deviation of each parameter, the values also seem to be reasonable compared to the true values from Table 1.

It is important to highlight that when running our computational code for the extended Kalman filter, we noted that the (2,2) element of the matrix that needs to be inverted in the Kalman gain

formula:

[

1, '

]

' it it it t i it it

X

V

X

D

D







(See appendix A.1 for details on the matrix D and the Kalman gain formulation), becomes very small when becomes large. This is because this

(2,2) element is computed using the (2,2) element of the D matrix which is equal to D[2,2]

( )

_{( ) . We believe that such small values, which become large when the}

mentioned matrix is inverted, are one of the causes for which 2% of the estimations are very

large. We consider this issue as a minor limitation of our method, since it only affects a small fraction of the customer base. Moreover, its solution is out of the scope of this thesis and should be studied further to ensure a 100% validity of the Kalman gain.

23 useful. This is due to 1) the fact that the measure the relationship between the two dependent

variables and the customer-specific variables, which are already known and assumed to be antecedents of CLV and delinquency, and 2) the fact that in our method the are only used as

input in the fixed effects model so that the parameters can be estimated, and the CLV and

credit risk management decisions can be coordinated.

Once the individual parameter estimations ̂ are obtained and validated, it is possible to

generate the fitted ̂ values to assess the extended Kalman filter predictive performance. For the

case of the continuous variable such assessment is done using the mean absolute

percentage error (MAPE). It is a relative measure of the deviations between the true (synthetic) and the fitted ̂ for each individual over time:

∑ | ̂

|

( )

When having a perfect fit, MAPE is zero. Regarding its upper level there is no restriction or threshold to define an acceptable fit. We do not use other predictive performance measures such as the mean squared prediction error (MSE) or the root mean squared prediction error (RMSE), since they are not relative, and therefore are more suitable for scenarios in which it is necessary to compare different models.

Figure 2 shows the distribution of the MAPE values for all customers. It is possible to observe that the extended Kalman filter has poor predictive performance for some individuals, since their MAPE values are considerably high. Such result is a direct consequence of the identified large estimation of the parameter estimates illustrated in table 2. Figure 3 shows the true and fitted CLV values, and their correspondent MAPE, for some of those customers over time. This figure suggests that in spite of very big deviations, the fitted ̂ for those customers with high MAPE,

somehow follow the true pattern of the values over time. This is somehow counterintuitive,

24 Observation of the MAPE formula from equation (11) and its components, for the customers illustrated in Figure 3, shed some light on such a counterintuitive result. Extreme MAPE values occur when the denominator, which is equal to the true CLV value, is very small and/or near to zero. This small denominator renders the whole computation to very large numbers that then have to be summed over time. Moreover, the formula gives even larger numbers when the numerator becomes also larger. In other words, the smaller the true CLV value and the larger the difference between the true and fitted CLV values, the larger the MAPE value would be. This is the case for the customers’ values in Figure 3. All of them have at least one true CLV value near to zero, and its correspondent difference between the true and the fitted CLV value is considerably large, and actually the largest among all differences over time. Consequently, it is possible to conclude that the MAPE only measures distance between fitted and true values, without taking into account the pattern over time. Therefore, extreme MAPE values can correspond to fitted CLV values that do follow the true values pattern, as it can be seen in Figure 3.

From a managerial point of view, the previous discussion implies that although these customers’ CLV level is not being accurately estimated at every time period, some of their CLV behavior and changes over time (pattern) are actually being captured by the extended Kalman filter. Capturing such a time-varying behavior is very valuable when it comes to coordinate and manage firm decisions that influence the changes on the profitability of customers.

As already said, the MAPE does not have a standard threshold to decide whether the model has good or bad predictive performance. Therefore, we set a threshold of 100% for the MAPE values. This is, those models generating fitted CLV values that are, as maximum, twice as big as the true value , will be considered to have acceptable predictive performance. Following this threshold

25 Table 2 Summary statistics for the estimated state parameters

26 Figure 2 True vs. Fitted CLV for customers with high MAPE values

Table 3 MAPE Summary statistics

Following with the assessment of the predictive performance of the extended Kalman filter, we now focus on the binary variable Delinquency. We opt to execute such assessment using the

27 not included in the training sample. Table 4 shows the three measures along the four time periods using the whole customer base (training sample). All of them suggest that the extended Kalman filter is doing a god job identifying and separating delinquent customers from non-delinquent ones. This is, the top decile lift values for the four time periods are higher than 1 and very near from two, and the total and delinquent hit rates are very near from 100%. Since the three measures are considerably high for the four time periods, it is possible to say that the extended Kalman filter is actually capturing the dynamics of the delinquent behavior of customers over time. It is noteworthy however, that the hit rates slightly decrease over time.

Table 4 Predictive performance measures for the delinquency variable

Summarizing, it is valid to assume that the extended Kalman filter is working well. In comparison to the distribution of the true simulated parameters, the estimated distribution shows similar measures of central tendency. There are some estimated state parameters that present extreme values, this however occurs for less than 2% of the customer base. This issue is due to data specific aspects that influence the extended Kalman filter formulation. Moreover, we concluded that the model presents good predictive ability given that 60% of the customers have fitted CLV with MAPE values lower than 100%, and the delinquency variable is accurately predicted for more than 80% of the customer base.

The fixed effects model

After having estimated valid state parameters, it is possible to assume that they represent the true relationship between the two dependent variables and the customer-specific antecedents .

Therefore, such estimations can be used to estimate the parameters that capture observed

28 We proceed to estimate the six fixed effects models for each of the six state parameters following equation (10), and their correspondent validity tests and

measures (R-squared, F-statistics, residual standard error, and the magnitude and significance of the estimated parameters). Those measures are used to assess a model as valid or appropriate, only on the basis of its fit with the data used as input (if the theoretical specification of the model is known to be valid and appropriate for the type of data). Given that our aim is not to show the validity of the fixed effects model from a theoretical or computational point of view, the variables

were randomly generated with no specific correlation with the dependent variables CLV and

delinquency. Therefore, we conclude that validity or predictive performance assessment of the fixed effects models is not necessary, informative, or within the aim of this thesis1. The fixed effects models, which are already known as theoretically correct and are included in almost all statistical softwares, are only a tool to generate the estimates of the that will be used to

illustrate the CLV-delinquency decision coordination. In this way, we limit ourselves to conclude that the fixed effects models for each of the state parameters are valid from a theoretical point of view, given the arguments presented in section 3.1. We assume that under a real life scenario in which the variables are assumed to have an underlying correlation structure with both

the and the variables, the validity test measures will indicate the adequacy of the model.

In spite of the previous discussion, and for illustrative purposes only, we present the results of the six effects models in appendix A.5. Summarizing, the estimated parameters and their

standard deviations for all the models, do not present any extreme or unusual values: all of them are between -1 and 3. Moreover, all the models present acceptable goodness of fit given their high R-squared values, significant F-statistics, and residual standard errors between one and six. The high R-squared values however are probably also due the large number of dummies that are included in the model to account for the customer-fixed effects (the larger the number of variables, the larger the R-squared).

1

It is noteworthy that the validity and predictive performance assessment of the extended Kalman filter was done because the variables and were simulated in such a way that they depend on the variables through the true parameters.

Therefore, their estimations could be compared to their true values, and could be used to compute fitted values that could also be

29 Interpretation of the parameter estimates

So far we showed that the extended Kalman filter is giving valid estimations of the , and

therefore they can be used as dependent variable in the fixed effects model to estimate the parameters . We also concluded that those fixed effects models are valid, and therefore our

method to coordinate CLV and credit risk management decisions has been validated from a theoretical and computational point of view. We now focus on the interpretation of the estimates of the fixed effects parameters .

The consumer-specific level shifters present a considerable amount of mean variance for each

of the six models (Appendix A.5), indicating the appropriateness of having controlled for unobserved differences among customers. The parameters represent the mean effect (across

both time and consumers) of the variable on the correspondent dependent variable (CLV or delinquency). Therefore their interpretation is not very informative for our firm decision coordination purposes.

Regarding the interpretation of , it is useful to think that if one were to use both the Kalman

filter and the fixed effects model estimations to obtain fitted values for the dependent variables, one would have to replace , from equation (3), with equation (10). Such a replacement leads to

the conclusion that the parameters are actually expressing the effect of both and on

the dependent variables CLV and delinquency, i.e. they are interaction parameters. Therefore, their interpretation is presented as such. For ease of understanding we present the estimations of the parameters in an interaction fashion (Table 6) between the variables and , so it is

clear which relationship each represents. Table 6 does not show the significance of the

30 Table 5 Parameter estimates of the parameters

If is significant and positive, it is possible to conclude that an increase in , has a

higher increasing effect on ( ) for customers with high levels of , than for those with

low levels of . Now if is significant and negative, one can come to the conclusion that

when large, individuals with higher levels is of would tend to have lower values of

than those customers with lower levels of . Such interpretation becomes very useful when it

comes to decide how a specific firm action ( ) affect CLV, delinquency and their antecedents.

Let’s think of a hypothetical case in which is one of the mentioned CLV antecedents, let’s say

customer spending (letting k=1, we have ), is the payment

reminders variable and their interaction parameter estimate is (Table 6:

number in bold at the CLV side) is significant and negative. In this case, the conclusion is that customers with high spending and who receive a large number of payment reminders have lower CLV than those who receive less reminders. This can be a very important finding that would mean that customers who have high spending on the company, and therefore generate big revenues, get irritated and pushed by those reminders and in turn can reduce their willingness to stay with the company, reducing their total profitability (CLV). Now, to efficiently make any possible decision about the number of reminders that should be sent to high spenders, it is necessary to first see the same effect on delinquency. This could be done by including customer spending as an antecedent of delinquency on the measurement equation. Let’s now assume that is

customer spending (delinquency antecedent), i.e. is the delinquency variable, is the payment reminders variable, and (Table 6: number in bold

31 who have high spending levels, and receive many payment reminders have lower probability of being delinquent than those who do not receive that many reminders. In this case we would have concluded that sending to many reminders can simultaneously decrease both the customer’s delinquency probability, and his/her CLV. From a managerial point of view, this would be a controversial finding which would demand agreements between the company’s credit risk and marketing departments. Under a customer oriented scenario in which the goal is to maximize profit, the path to follow is to find the number of reminders that maximizes the CLV under a previously set maximum credit risk limit. In this way, customers with high spending levels are guaranteed to keep generating revenues for the firm without irritating them, and ensuring an acceptable credit risk level that will efficiently benefit both the customer and the firm. The same interpretation path can be followed for all the parameters, so that their interaction effect

between and can be used as a tool to understand the synergies between the two dependent variables, and coordinated decisions can be made. It is noteworthy that the optimization part of the decision making process just described is out of the scope of this thesis. However, it is important to highlight that our method allow managers to identify if specific firm decisions affect contradictorily, or in a suboptimal way, two of the most important key performance indicators: CLV and delinquency. Therefore, it brings new opportunities to optimize decision making processes and consequently maximize profits, while making a balance between marketing and credit risk goals.

6 Discussion

32 the lack of real data, we test our method using simulated longitudinal data that follows the extended Kalman filter specification.

The results show that the extended Kalman filter with two different distributed dependent variables is able to produce valid state parameter estimates for the majority of the customer base. Therefore, the generated computational code is useful to estimate this type of models for the CLV and credit risk management scenario, and for any other context in which the synergies between normal and binary variables are to be studied. The distribution of the estimated state parameters taking into account 98% of the individuals is very similar to the distribution of the simulated states. For the remaining 2% of the population, the model generated extreme values in comparison to the true ones. Those unusual values are believed to be due to data specific situations in which the Kalman gain matrix becomes large. Solution of such an issue is out of the scope of this thesis. Regarding predictive performance, the extended Kalman gain is also doing an acceptable job generating reasonable fitted values. 60% of the CLV fitted values are within the threshold of acceptable predictive error (based on the MAPE), and between 80% and 98% of the delinquent customers were predicted as such by the model. We believe that one of the reasons for which the remaining percentage of the population is obtaining fitted values that are not that close to their true values (large MAPE values, or misclassification), is the previously mentioned issue with the Kalman gain matrix. Further research should investigate the reasons for this extreme state parameter estimates. We concluded that extreme values in the parameter estimates and fitted values far from their true value, are present only in a small portion of the whole customer base, and therefore they are not limiting for our purposes.

Next, we estimated the fixed effects model for each of the state parameters. Validation of their results seems useless given the simulated nature of the variables and the aim of this thesis. In

33 In this way we provide an innovative method that has not been used before to coordinate two types of decisions that are commonly made disjointly: marketing and credit risk management actions. Such a method accounts for the fact that these actions have different probability distributions, the fact that each customer is different and therefore customer heterogeneity should be modeled, and the panel data structure natural of time varying variables such as CLV and delinquency.

7 Implications

Theoretical implications

Fahrmeir and Tutz (1994) present the extended Kalman filter for the case of exponential family of distributions. Although the method is to be valid for multiple response variables, they only show validated results for one dependent variable. Therefore, the main theoretical implication of this thesis is that the method works well for more than one variable, even in the case when they have different probability distributions. This is an opportunity to understand complex time-varying relationships between variables that could not be studied before, given the lack of methods that could simultaneously predict different probability distributions. This holds not only for the credit risk or marketing scenario, but for every context in which this type of studies are necessary. Moreover, the extended Kalman filter in combination with the fixed effects models, provide a valid method to not only understand the synergies between different distributed variables, but also to account for observed and unobserved subject-heterogeneity that can improve the state parameters estimations. This is very valuable especially in cases where the aim, unlike us, is not to understand the interaction effect between the and on the response variables, but to

accurately predict the dependent variables on time. Managerial implications

34 The presented method allows understanding the simultaneous effect of any firm action in the customer’s CLV and delinquency behavior. Therefore, it is not only a powerful tool for optimal decision making processes that take into account two of the most important angles of the customer profile, but also it represents the opportunity to bring together two departments that are usually rivals within a company’s structure. It has always been though that the credit risk and the marketing departments have different goals, and therefore joint decisions are seen as suboptimal or nor realistic. Our method provides a tool to reconcile those differences and create a customer oriented vision where all departments work towards the same goal: the profitability of the firm.

8 Limitations and further research

Although the extended Kalman filter is generating valid state parameter estimates for the majority of the customers included in the simulated customer base, it is still possible to find some customers for whom the results are not that desirable. Further studies will have to deeply investigate the reasons for the extreme values in the parameter estimates. In specific, we believe that it is worth assessing the true effect of the previously mentioned (2,2) element of the D matrix (the matrix containing the differentiation of the link function with respect to the linear predictor of each of the dependent variables) on the Kalman gain computation, when is a binary variable.

One way of doing so is reproducing our method using different distributions from the exponential family to conclude if the extreme values are due to the link function of the binary variable, or if they are due to some other issues. Identifying and solving those issues should be the ultimate goal to ensure a perfectly working extended Kalman filter for all types of data, distributions, and number of dependent variables.

35 Another limitation of this thesis is the fact that we do not use real data to test the method, and therefore actual decision coordination is not possible. Further studies should include real data so that the method can be better illustrated and used as a tool to coordinate CLV and credit risk management decisions. Moreover, to make the results and decisions drawn from the method generalized, one could use real data from different industries, countries, and financial institutions.

36

9 Appendix

A1. Extended Kalman Filter (Fahrmeir and Tutz (1994) pp. 277-278) Initialization:

a

_0/0



a

₀

V

_0/0



Q

₀ For t=1,…,T: Prediction step:

a

t/t1



F

t

a

t/t1 V_{t t} F_tV_t F_t'Q 1 1 / Correction step: Initial values: a0,t at/t1 V0,t Vt/1 For i=1,….,n: ait ai1,t Kit(yit 



it)

V

it

I

K

it

D

it

X

it

V

i 1,t '

)

(



_



Kalman gain: 1, ' 1 ' ' , 1

[

]

  







_{i t it it it it i t it it it} it

V

X

D

D

X

V

X

D

K

with

a

it=

a

t and

37 A.2 Summary statistics of the K explanatory variables (measurement equation)

39 A.5 Fixed effects model estimations for each state parameter

40

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