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Modeling Consumer Acceptance Probabilities

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Modeling Consumer Acceptance Probabilities Powered By Docstoc
					     MODELING CONSUMER ACCEPTANCE PROBABILITIES


                        Ki Mun Jung, L. C. Thomas, S.D.A Thomas
            School of Management, University of Southampton, Southampton, SO17 1BJ, UK


                                            ABSTRACT
      This paper investigates how to estimate the likelihood of a customer accepting a loan offer as a
function of the offer parameters and how to choose the optimal set of parameters for the offer to the
applicant in real time. There is no publicly available data set on whether customers accept the offer of a
financial product-the features of which are changing from offer to offer. Thus, we develop our own data
set using a Fantasy Student Current Account. In this paper, we suggest three approaches to determine the
probability that an applicant with characteristics will accept offer characteristics using the Fantasy
Student Current Account data. Firstly, a logistic regression model is applied to obtain the acceptance
probability. Secondly, a linear programming is adapted to obtain the acceptance probability model. To
build a model, we assume there is a dominant offer characteristic, where the probability of accepting the
offer increases (or decreases) monotonically as this characteristic’s value increases. Finally, an
accelerated life model is applied to obtain the probability of acceptance in the case where there is a
dominant offer characteristic.

Keywords: student bank account, acceptance probability, coarse classifying, logistic
regression model, linear programming, accelerated life model


1. Introduction

     Forecasting financial risk has over the last thirty years become one of the major
growth areas of statistics and probability modeling (Thomas, 2000). In retail or
consumer lending, the main approaches are credit scoring and behavioural scoring
which are based on statistical or operational research methods. The statistical methods
include discriminant analysis, logistic regression and survival analysis. Discriminant
analysis was proposed by Fisher (1936) as a discrimination and classification tools. It is
one of the first methods applied to building credit scoring models. Logistic regression is
a widely used statistical modeling method. Wiginton (1980) was one of the first to
describe the results of using logistic regression in credit scoring. The application of
survival analysis for building credit scoring models was introduced by Narain (1992)
and developed further by Thomas et al. (1999) and Stepanova and Thomas (2002). The
operational techniques include variants of linear programming. Mangasarian (1965)
was the first to recognize that linear programming could be used in classification
problems where there are two groups and there is a separating hyperplane, i.e., a linear
discriminate function, which can separate the groups exactly. Freed and Glover (1981)
and Hand (1981) recognized that linear programming could also be used to discriminate
when the two groups are not necessarily linear separable.




                                                    1
     In the same way as assessing default risk, statistical and operational research
method can be used to determine the probability that an applicant with certain
characteristics will accept a particular offer. In this paper, we investigate how to
estimate the likelihood of a customer accepting a loan offer as a function of the offer
parameters and how to choose the optimal set of parameters for the offer to the
applicant both in real time. This would need to be done without using many more
questions than are on the electronic application forms used for default estimation.

     New communication and marketing channels allow for offer features to be
determined during application process and the implication of customer relationship
management is that customers get products tailored to them. However, there is no
publicly available data set on whether customers accept the offer of a financial product-
the features of which are changing from offer to offer. Thus, we develop our own data
set using a Fantasy Student Current Account (FSCA). The data set of the application
and offer characteristics of 331 applicants was obtained from a website.

     This paper suggests three approaches to determine the probability that an applicant
with characteristics will accept offer characteristics using the FSCA data. Section 2
demonstrates the FSCA data, which is then used to build the model for consumer
acceptance probability. In section 3, the logistic regression (LR) model is applied to
obtain an acceptance probability estimate. To do so, the coarse classifying using the LR
model is considered. In Section 4, a linear programming approach is used to obtain the
acceptance probability model. To build a model, we assume there is a dominant offer
characteristic, where the probability of accepting the offer increases (or decreases)
monotonically as this characteristic’s value increases. The idea is that given the other
offer and applicant characteristics one can identify the value of this dominant
characteristic at which this applicant would accept the offer. Section 5 suggests
modeling acceptance probability by using accelerated life (AL) models. That is, the AL
model is applied to obtain the probability distribution of acceptance for a given value of
the dominant characteristic when all other offer and applicant characteristics are fixed.
This probability distribution of acceptance reflects changes in the individual’s desires,
economic circumstances and environment in which the offer is made, all of which can
fluctuate rapidly, as well as the applicant’s inability to make decisive judgments
between incrementally different offers. In applying the accelerated life model, one has
double censoring, since if a customer accepts an offer with a particular overdraft limit
one only knows the minimum acceptance value is below this value , while if he rejects
an offer one only knows the minimum acceptance value is above this offer value.

2. Fantasy Student Current Account Data

    There is no publicly available data set on whether customers accept the offer of a
financial product-the features of which are changing from offer to offer. Thus, we
developed our own data set using a Fantasy Student Current Account (FSCA), which is
based around a website mimicking an on line application form for a student bank




                                            2
account. The website consists of three pages. The first is an application form for a
FSCA, which is similar to the bank account most students use in the UK for their
money transactions and their borrowing. The questions are created by looking at the
application forms of ten UK lenders including the four major UK retail banks. The
second page is an offer of such an account. There were six parameters of the account
that could be changed from offer to offer. In order to obtain to a reasonable spread of
offers combinations each applicant was randomly put into one of four offer categories.
In three of these categories everyone in that category received the same fixed offer
varying from £1,250 to £1,800 overdraft limit. In the fourth category which had the
largest probability the offer was given by one of 42 nodes of a decision tree arrived at
by splitting on the applicants’ characteristics. The decision tree was constructed
subjectively, using obvious associations and a desire to produce a wide spectrum of
offers. When the offer was made on the second page, the applicant had to submit
whether they accepted or rejected the offer. The final page asks applicants to rate the
importance of the offer characteristics in their decision and also how they feel about a
bank making different offers to different people.

             Table 1. Application and offer characteristics used in the model
         Characteristics                               Description
    Application characteristics
    Age                           Age of applicant
    Sex                           Sex of applicant
    Status                        Marital status of applicant
    Num_children                  Number of children
    Num_cards                     Number of credit card
    Wage                          Some income from wage
    Loan                          Some income from loan
    Contribution                  Some income from parental contribution
    Travel                        Interest in travel, True/False
    Music                         Interest in music, True/False
    Cars                          Interest in cars, True/False
    Cinema                        Interest in cinema, True/False
    Sports                        Interest in sports, True/False
    Clubbing                      Interest in clubbing, True/False
    Beer                          Interest in beer, True/False
    Country western               Interest in C&W music, True/False
    DIY                           Interest in DIY, True/False
    Gardening                     Interest in gardening , True/False
    Offer characteristics
    Overdraft                     Overdraft limit, 5 choices
    Creditcard                    Credit card included with account, 4 choices
    TM                            No fees on ordering foreign currency for travel, Yes/No
    Insurance                     Discounts on insurance, 4 choices
    Interest                      Interest paid when account in surplus, 4 choices
    Introductory                  Introductory free gift, 10 choice




                                            3
     The data set of the application and offer characteristics of 331 applicants was
obtained from the website. Table 1 shows the applicant and offer characteristic used in
analysis. There are 18 applicant characteristics and 6 offer characteristics. In Sections
3-5, we will deal with three approaches to determine the probability that an applicant
with characteristics will accept offer characteristics using the FSCA data.



3. Logistic regression based acceptance probability approach

3.1 Logistic regression model

     The logistic regression model is a widely used statistical modeling method in
which the probability of a dichotomous outcome is estimable. In general, the logistic
regression model has the form
                            log[ p (1 − p)] = β 0 + β 1 x1 + β 2 x 2 + L + β k x k
                                                                                                      (1)
                                            = xβ ,
where p is the probability of the outcome of interest, β 0 is the intercept term, β i is
the coefficient associated with the corresponding explanatory variable xi ,
 x = (1, x1 , x 2 , L, x k ) and β = ( β 0 , β 1 , L , β k )′ . So, in logistic regression, one estimates
the log of the probability odds by a linear combination of the characteristic variables.
Since p (1 − p ) takes values between 0 to ∞ , log[ p (1 − p )] takes values between − ∞
and + ∞ . Taking exponentials on both sides of (1) leads to the equation as follows.
                                                        exp(xβ)
                                                p=                   .
                                                      1 + exp(xβ)
     The LR model can be applied to obtain the acceptance probability for FSCA data.
To do so, let the applicant characteristics be x = ( x1 , L , x n ) and let o = (o1 ,L , om ) be
the offer characteristics. Then the basic logistic regression approach assume that the
probability that an applicant with characteristics x will accept offer o satisfies
                   log[ p (1 − p)] = β 0 + β 1 x1 + L + β n x n + β n +1o1 + L + β n + m om
                                                                                                      (2)
                                     = yβ ,
where y = (1, x1 , L, x n , o1 , L, om ) and β = ( β 0 , β 1 ,L , β n +m )′ . Next in sections 3.3-3.4,
the above model (2) is used to build the acceptance probability model for FSCA data.

2.2 Coarse classifying using the logistic regression model

     Firstly, we consider coarse classifying the characteristics to obtain the acceptance
probability using the LR model. In general, the coarse classifying of characteristics
procedure splits the values of a continuous characteristic into bands and the values of a
discrete characteristic with many values are grouped together. It helps ensure that the
credit scoring systems are robust, i.e., predictive rather than descriptive of data because




                                                   4
it allows the prediction to be non-monotonic in the characteristic value. The traditional
approaches of finding the suitable splits involve looking at the accept-reject ratio for
different attributes ( or percentile groups for a continuous characteristic) values of the
characteristic and then grouping the values with similar odds.
The following method can be used to find the best split for the continuous or discrete
characteristic.

Step 1. (Continuous characteristic) Split the characteristic into n binary variables with
        approximately equal number of observations in each variable.
        (Discrete characteristic) A binary variable is created for each attribute of the
        characteristic.
Step 2. Apply LR model with these binary variables.
Step 3. Chart parameter estimates.
Step 4. Choose the splits based on similarity of parameter estimates.

Example 1 demonstrates the coarse classifying of the discrete characteristic using the
LR model.

Example 1 To illustrate coarse classifying using the LR model, the introductory gift
characteristic is considered. There are 10 different attributes of the characteristic, so
that the LR model is fitted to 10 binary variables. That is,
           log[ p (1 − p)] = β 0 + β 1 ( Introductory1) + L + β 10 ( Introductory10)
Figure 1 shows the histogram of the parameter estimates. Then there binary variables
are created as Table 2. These three binary variables are included in the LR model
considering the next section.
                               1.2

                                 1
                               0.8
          Parameter estimate




                               0.6
                               0.4
                               0.2
                                 0
                               -0.2
                               -0.4
                               -0.6
                                      1   2   3   4       5   6   7   8   9   10

          Figure 1. LR parameter estimates for introductory gift characteristic




                                                      5
           Table 2. Coarse classifying of the introductory gift characteristic
                                                         No. of observations
           Binary variable    Type of introductory
                                                         Accept        Reject
           Introductory_1       1, 8, 9                    42            55
           Introductory_2       2, 6, 7                   127            56
           Introductory_3       3, 4, 5, 10                27            26

For all characteristics excepting the binary variables, the coarse classifying using the
LR model can be applied. Thus, we can create the binary variable for all characteristics
to use the LR model.

3.3 Acceptance probability using the logistic regression model

     This subsection applies the LR model to estimate the probability that an applicant
with characteristics x will accept offer o . The model is built on a training sample of
265 cases and tested on a holdout of 66 cases . The indicator variables in the LR model
are created by the coarse classifying using the LR model as suggested in previous
subsection. Thus, using (2) and the results of the coarse classifying, LR model with
only indicator variables is built as follows.
                   log[ p (1 − p)] = β 0 + β 1 (Var1 ) + β 2 (Var2 ) + L + β k (Vark ) ,
where Var1 , Var2 , L, Vark are the indicator variables, β 0 , β 1 ,L , β k are the parameters
to be estimated. In this preliminary investigation, all variables are included in the model.
The variables which have the major impact on the score are overdraft, interest, and
insurance characteristic. Table 3 shows the results on a training, holdout and whole
sample.

            Table 3. Classification results using the logistic regression model
                      Training data             Holdout data               Whole data
                  Actual numbers LR         Actual numbers LR          Actual numbers LR
Y-predicted Y          155          121           39           29           195          150
Y-predicted N            0           34            0           10             0           44
N-predicted N          110           57           27           25           137           72
N-predicted Y            0           53            0           12             0           65

4. Overdraft exact cut-off approach

The second approach to developing a model to determine which offer to make assumes
that there is a dominant offer characteristic, where the probability of accepting the offer
increases (or decreases) monotonically as this characteristic’s value increases. The idea
is that given the other offer and applicant characteristics one can identify the value of
this dominant characteristic at which this applicant would accept the offer. One could
use this value in profit calculations to see whether it is profitable to make such an offer.




                                              6
In the credit card context, this dominant characteristic could be the APR charged for
borrowers or credit limit for transactors, but for student bank accounts, where the
overdraft is interest free, this characteristic would be the overdraft limit.

So again assume applicant characteristics x= (x1,….,xn ) ,offer characteristics
o = (o 2 ...o n ) with O = o1 being the dominant offer characteristic and the interaction
characteristics i . We are interested in determining the accept/reject level of O, O* , as a
linear function of x, o and i. Hence we assume.

O* = c 0 + c 1 .x + c 2 .o + c 3 i = c.y where y = (x, o, i )

Taking a sample of previous applicants, if applicant i (with characteristics yi ) accepted
an offer of oi then oi ≥ c.yi while if applicant j (with characteristics y j) rejected on
offer of oj then oj ≤ c.yj , where we assume the probability of acceptance increases as
O increases. Hence we can use linear programming to determine the coefficients c as
follows.

Let the sample of previous customers be n(a)+n( r ) where i=1,...n(a) accepted the offer
and j=n(a)+1,…..,n(a)+n( r) rejected the offer. Let applicant i have applicant/offer
                         i
characteristics y i = ( y1 ,... y ip ) and be made an offer o i . Then to find the coefficients c that
give the best overdraft accept/reject level we want to solve the following linear
programme.
Minimise     e1 + ....e n ( a ) + n ( r )
Subject to

o i + e i ≥ c i y ii + c 2 y 2 + ... c p y ip
                             i
                                                    i=1.....n(a)
                  j           j             j
 o i − e j ≤ ci yi + c2 y 2 + ...c p y p
                                      j=n(a)+1,...n(a) + n(r)
                                 ei ≥ 0 i=1,...n(a) +n(r)
Applying this to the Fantasy account data set , using the variables and their
characteristics identified in section three, gave the following results. All the variables
are used and the coarse classifying developed in the previous section is used to identify
23 binary variables to put into the linear programme.
The results were as follows.

            Table 4: Classification results using linear programming model
               Training data             Holdout data           Whole data
               Actual numbers LP Actual numbers LP Actual numbers                              LP
Y predicted Y 155                  137 39                   30 194                             167
Y predicted N 0                    18 0                     9   0                              27
N predicted N 110                  88 27                    22 137                             110
N predicted Y 0                    22 0                     5   0                              27




                                                    7
        One could imagine that linear programming does particularly well here because
of the number of variables compared with the size of the training sample. As the sample
sizes increases it is likely the strength of this approach may diminish a little.

The variables which had significant coefficients were as follows

                 Table 5: Coefficients in Linear programming approach

Attribute   Age18-   Age    0       1        2        3+        0%         1+%        Intro offer-   Other
            26       27+   Credit   Credit   Credit   Credit    interest   interest   CD player or   intro
                           card     Card     Card     Card                            £40            offers
Score       0        250   525      0        275      1000      0          275        275            0

This says that older student wanted an extra £250 on their overdraft limit before they
were likely to accept the offer. Those with 3 or more credit cards wanted £1000 more
on their overdraft limit compared with those with 1 credit card, though those with 0
credit cards also wanted a £525 higher overdraft limit. Some of the other aspects of the
attribute values suggested banks might want to reconsider their marketing. Offering a
CD player or £40 voucher meant students would want a higher overdraft limit by £275
compared with the other offers while offering interest when in credit also increased the
cut-off level of when the account was likely to be accepted by £275.


5. Overdraft cut off distribution

    This section develops a model to determine the probability of accepting the offer
using an accelerated life model which is one of the survival analysis models. Survival
analysis is the area of statistics that deals with analysis of lifetime data. Especially,
proportional hazard model and accelerated life model were used to building credit
scoring models (Narian 1992, Thomas et al. 1999, Stepanova and Thomas 2002). To
apply the accelerated life model for FSCA data, we again assume that there is an
important offer characteristic and the probability of accepting increases or decreases
monotonically as this dominant offer characteristic’s value increases. The idea is that
given the other offer and applicant characteristics one can identify the value of this
dominant characteristic at which this applicant would accept the offer.

5.1 Accelerated life model for overdraft cut off distribution

     In this subsection, the accelerated life model is used to estimate the probability of
the customers rejecting the offer. Let T be the dominant offer characteristic. If one has
applicant characteristics x and offer characteristic (t ,o) where t is the value of
monotone characteristic T , then we are interested in the probability of an applicant
with x accepting offer (t ,o) . Thus, if T is the lowest value of t at which offer is
accepted, then




                                             8
                 Prob{individual with characteristic x accepts offer (t ,o) }
                  = Prob{T ≤ t | y = (x, o)}
                  = F (t | y = (x, o))
where y = ( x1 , L , xl , o1 ,L , om ) . An argument for assuming this is a probability
distribution of acceptance rather than an exact cut-off point as in the previous section is
that changes in the individual’s desires, economic circumstances and the environment
in which the offer is made, all of which can fluctuate rapidly, could mean the same
person making different decisions to the same offer at different times. It might also
reflect an applicant’s inability to make decisive judgments between incrementally
different offers. So, the probability of an individual with characteristic x rejecting offer
(t , o) is given by
                              S (t | y = (x, o)) = 1 − F ((t | y = (x, o)) .
This is known in survival analysis as the survival function. The accelerated life model
can be applied to estimate the reject probability for FSCA Data. where Table 1 shows
the applicant and offer characteristic used in analysis.

     In the credit card context, the dominant characteristic could be the ARP charged,
but for student bank accounts, where the overdraft is interest free, this characteristic
would be the overdraft limit. Thus, we consider the overdraft as an important offer
characteristic. Let T be the level of overdraft at which an applicant with characteristics
x accepts an offer where other features are o . Define y = (x, o) . Notice that all the
data will be either right or left censored. If applicant i with characteristics x i accepts
offer (t , o i ) then all we can say is T ≤ t . It applicant j with characteristics x j rejects
offer (t , o j ) then all we can say is T ≥ t . Thus, we can not observe uncensored data. In
accelerated life model, one define the survival (reject) function by
                                          S (t | y ) = S 0 (e yβ t ) ,                              (3)
where y = (1, x1 ,L , xl , o1 ,L , om ) , β = (1, β 0 , β 1 ,L, β l + m )′ and S 0 (⋅) is a baseline
survival function. One common accelerated life model in survival analysis is to take
 S 0 (⋅) to be the Weibull distribution. Thus, we can have the baseline survival function
as follows.
                                                      {      }
                                          S 0 (t ) = exp − (λt ) k ,                                (4)
where λ and k are the scale and shape parameters of the Weibull distribution . From
(3) and (4), the survival function in accelerated life model can be expressed as follows.
                     {                                                                       }
         S (t | y ) = exp − (λ exp(b0 + b1 x1 + b2 x 2 + L + bl xl + bl +1ol + L + bl + m om )t )
                                                                                                  k

                                                                                                    (5)
                     {                  }
                    = exp − (λ exp(yβ)t ) .
                                         k


       By applying the accelerated life model of doubly censored data, we can obtain the
likelihood function as follows.




                                                  9
                                      n(a)                n( a )+ n ( r )
          L(θ ) = ∏ (1 − S (t i | y i ))                     ∏              S (t j | y j )
                                       i =1                j = n ( a ) +1


                                              [   {                               }] ∏                     {                   }
                                      n( a)                                        n( a )+ n ( r )
                                 = ∏ 1 − exp − (λ exp(y i β)t i )                                    exp − (λ exp(y j β)t j ) .
                                                                              k                                            k

                                      i =1                                          j = n ( a ) +1

One can then find the maximum likelihood estimates of λ , k and β in Weibull based
accelerated life by regression techniques.

5.2 Coarse classifying using the accelerated life model (AL)

     In this subsection, the accelerated life model approach is used for the coarse
classifying. This overcomes the problem that in the traditional approach we are not
worrying about the actual offer made even though there is a strong interaction between
the offer level and the accept-reject decision. The coarse classifying method using the
AL model is similar with method given in Subsection 3.2.
Step 1. (Continuous characteristic) Split the characteristic into n binary variables with
        approximately equal number of observations in each variable.
        (Discrete characteristic) A binary variable is created for each attribute of the
        characteristic.
Step 2. Apply AL model with these binary variables.
Step 3. Chart parameter estimates.
Step 4. Choose the splits based on similarity of parameter estimates.

Example 2 To illustrate coarse classifying using the AL model, the introductory gift
characteristic is considered. This characteristic is also considered in Example 1. Figure
2 shows the histogram of the parameter estimates and the resulting coarse classifying is
in Table 6. It can be seen that the histogram of the AL and RL model parameter
estimates are different and may suggest different groupings.
                               0.4
                               0.3
                               0.2
          Parameter estimate




                               0.1
                                 0

                               -0.1
                               -0.2

                               -0.3
                               -0.4
                               -0.5
                                              1   2   3           4            5           6           7       8    9     10

         Figure 2. AL parameter estimates for introductory gift characteristic




                                                                            10
           Table 6. Coarse classifying of the introductory gift characteristic
                                                         No. of observations
           Binary variable    Type of introductory
                                                         Accept        Reject
           Introductory_1       1,10                       36            45
           Introductory_2       1, 6                      107            47
           Introductory_3       2, 3, 4, 5, 7, 8           53            45


5.3 Modeling of acceptance probability using accelerated life model

    The Accelerated life model is fitted so as to estimate the probability that an
applicant rejects the offer. The model is built on a training sample and tested on a
holdout. The variables used in the model are those created by coarse classifying using
the AL model as suggested in previous subsection. Using (5) and the results of the
coarse classifying, the accelerated life model is fitted as follows.
                               {                                        }
                  S (t | y ) = exp − (λ exp( β 0 + β 1Var1 + L + β qVarq )t ) ,
                                                                             k


where Var1 ,Var2 ,L, Varq are the indicator variables, β 0 , β 1 ,L , β q are the parameters
to be estimated and T is the accepted overdraft by an applicant with y .
     To compare AL model with the LR and the LP models, all the variables are
included in the AL model. Table 7 shows the results on a training, holdout and whole
sample.

                 Table 7. Classification results using AR and LR model
                     Training data             Holdout data            Whole data
                  Actual                    Actual                 Actual
                 Numbers LR AR numbers LR AR numbers LR AR
Y-predicted Y       155      121 124         39         29 28       194     150 152
Y-predicted N          0     34 31             0        10 11          0      44 42
N-predicted Y       110      57 50           27         15 12       137       72 62
N-predicted N          0     53 60             0        12 15          0      65 75

6. Conclusions

        This paper introduces three techniques which can be used to build models of the
probabilities that a particular consumer will accept different variants of a generic
borrowing product like a credit card or account with an overdraft facility. It derives a
data set based on students acceptance or rejection of a Fantasy Student Account offer.
Thus one must include many of the caveats when one uses data which is essentially
obtained from gaming experiments rather than from real experience. However the paper
does show that it is possible to build acceptance probability models using such data and
makes a preliminary investigation of three different approaches. We believe that two of
these – linear programming and accelerated life models – have not been tried before in
this context. All three approaches are technically feasible and can result in real time




                                            11
decisions about which variant of the product to offer the current applicant in order to
maximize profit, though two of them do require the notion of a dominant offer
characteristic. For many products it does seem reasonable to assume that one
characteristic has the necessary monotone properties to apply such procedures.
We believe these probability acceptance models will become increasingly important as
the consumer lending market matures and it becomes a buyers rather than a sellers
market. It also satisfies the customer relationship marketing credo of tailoring the
product to the customer.


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