Document Sample

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. References Fisher, R. A. (1936). The use of multiple measurements in taxonomic problems, Annals of Eugenics, 7, 179-188. Freed and Glover (1981) A linear programming approach to the discriminate problem, Decision Science, 12, 68-74. Hand, D. J. (1981). Discrimination and Classification, John Wiley, Chichester, U.K. Mangasarian, O. L. (1965). Linear and nonlinear separation of patterns by linear programming, Operations Research, 13, 444-452. Narain, B. (1992). Survival analysis and the credit granting decision, In: Thomas, L. C., Crook, J. N. & Edelman, D. B. (Eds.), credit scoring and credit control, Oxford University Press, Oxford, 109-122. Stepanova, M. and Thomas, L. C. (2002). Survival analysis method for personal loan data, Operations Research, 50, 277-289. Thomas, L. C. (2000). A survey of credit and behavioural scoring: forecasting financial risk of lending to consumers, International Journal of Forecasting, 16, 149-172. Thomas, L. C., Banasik, J. and Crook, J. N. (1999). Not if but when loans default. Journal of Operational Research Society, 50, 1185-1190. Wiginton, J. C. (1980). A note on the comparison of logit and discriminate models of consumer credit behaviour, Journal of Financial and Quantitative Analysis, 15, 757-770. 12

DOCUMENT INFO

Shared By:

Categories:

Tags:
consumer acceptance, Y. Wu, food products, Systems Approach, Postharvest Handling, Genetic modification, Standard Deviation, genetically modified foods, GM food, processed products

Stats:

views: | 15 |

posted: | 3/10/2011 |

language: | English |

pages: | 12 |

OTHER DOCS BY hkksew3563rd

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.