# yvonne

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```					  Using Adaptive Learning in
Credit Scoring to Estimate
Take-up Probability Distribution

Seow H.V. and Thomas L.C.
School of Management
University of Southampton
SO17 1BJ United Kingdom.
Contents
Literature Review
The Problem
The Model
Restless Bandit Problem
Empirical Results
Contributions
Further Research
ESI 2004 Ankara, Turkey   2
Literature Review
Credit scoring
Dates back to the 1950s – credit
decision
Application Scoring (Thomas et al.,
Satisfaction = better acceptance of new
product (Bolton, 1998)
Acceptance Scoring
ESI 2004 Ankara, Turkey   3
Literature Review
Customisation of Offers
‘Customising’ offers (Rodney et al.,
2002)
Based on usage (Banasik et al., 2001)
Acceptance Score
Benefits of Acceptance Scoring

ESI 2004 Ankara, Turkey    4
Literature Review
Meyer and Shi (1995), Banasik et al.
(2001), Stepanova and Thomas (2001)
The role of Bayesian Updating
Making Problem
Aid the model in selection of best offers
to maximise profits
ESI 2004 Ankara, Turkey       5
The Problem

Credit Scores minimise default rate
Growing demand for Acceptance
Scoring
Hence also a demand for a more
varied Score Card

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The Problem

programming with Bayesian
Updating to better estimate Take-
Up Probability distribution

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The Model
Example
Two financial products are offered: Offer
5% and Offer 10%
Think of them as interest rates charged
on loans or credit cards
The model is based on a Bernoulli trial,
using Bayes theorem with a priori belief
beta distribution parameterised by (r, n-r)

ESI 2004 Ankara, Turkey    8
Parameters of the Model
P5 = Profit gained if Offer 5% is accepted
P10 = Profit gained if Offer 10% is accepted
p5 = Probability (accept Offer 5%)
p10 = Probability (accept Offer 10% /
accepted Offer 5%)
m = size of population that has yet to be
offered any financial products

ESI 2004 Ankara, Turkey    9
Parameters of the Model
r5 = can be reinterpreted as the number
of people who already accepted Offer 5%
n5 = the number of people already offered
Offer 5%
r10 = can be reinterpreted as the number
of people who already accepted Offer
10%
n10 = the number of people already offered
Offer 10%
ESI 2004 Ankara, Turkey   10
Restrictions of The Model
There are some restrictions imposed for
this model:
(1) P  P
10   5

(2) P , P , r , n , r , n  0
5   10   5   5    10     10

(3) n5  r5
(4) n10  r10
(3) and (4) follow from a Beta Priori
ESI 2004 Ankara, Turkey   11
Restrictions of the Model
Unusual definition of p10 to ensure:
(1)Accepting 10% offer means 5% offer
would have been accepted
(2)Refusing 5 % offer means 10% offer
would have been refused
(3)Refusing 10% offer DOES NOT
mean that 5% offer would be refused
(4)Accepting 5% offer DOES NOT
mean that 10% offer is accepted
ESI 2004 Ankara, Turkey   12
The Objective of the Model
Maximises the expected profit
generated from acceptance of offers
‘Guide book’ to aid decision making
depending on the variables
Maximise profitable acceptance
v (r5, n5, r10, n 0,m) is optimal expected
profitabililty if m more customers to
come and current beliefs given by
(r5, n5, r10, n10 )
ESI 2004 Ankara, Turkey      13
The Equations of the Offers
 Offer 5%
v (r5, n5, r10, n 0,m) = max 
 Offer 10%
r5 r5                                       r 
Offer 5% = P5       vr5  1, n5  1, r10 , n10 , m  1  1  5 vr5 , n5  1, r10 , n10 , m  1
 n 
n5 n5                                          5 
r5 r10   r r
Offer 10% =     P10             5 10 v(r5  1, n5  1, r10  1, n10  1, m  1) 
n5 n10  n5 n10

r5    r10                                              r 
1     v(r5  1, n5  1, r10 , n10  1, m  1)  1  5 v(r5 , n5  1, r10 , n10 , m  1)
 n                                                n 
n5       10                                               5 

v(r5 , n5 , r10 , n10 ,0)  0

ESI 2004 Ankara, Turkey                                               14
Finite or Random Infinite
Population
These equations correspond to a finite
population
Optimal policy is hard to describe as it
will depend on m , the number of
customers that have yet to be offered
Easier (and more realistic) if population
has geometric distribution, parameter β

ESI 2004 Ankara, Turkey       15
Assume infinite number of customers
After each customer, there is a 1-β
probability of cancellation
Number of customer geometric with β
For p5 , is given by a Beta distribution
with parameters r5 and n5
For p10, is given by a Beta distribution
with parameters r10 and n10

ESI 2004 Ankara, Turkey     16
At any point, state is given by r5 , n5 , r10 , n10 
Insert element of discounting β , the
updated equation for Offer 5% and Offer
10%:
r5      r                                          r 
Offer 5% =     P5         { 5 vr5  1, n5  1, r10 , n10 , m  1  1  5 vr5 , n5  1, r10 , n10 , m  1}
 n 
n5      n5                                            5

r5 r10    r5 r10
Offer 10% = P10 n n   { n n v(r5  1, n5  1, r10  1, n10  1, m  1) 
5 10      5 10

r5    r10                                           r 
1  v(r5  1, n5  1, r10 , n10  1, m  1)  1  5 v(r5 , n5  1, r10 , n10 , m  1)}
 n                                             n 
n5      10                                             5

ESI 2004 Ankara, Turkey                                               17
Assume at state r5 , n5 , r10 , n10  ,the customer
is required to pay a fee of  1  n v(r , n  1, r


r


5
5   5       10 , n10 )
                  5

Revised optimality equation:

vr5  1, n5  1, r , n10 , m  1]
r5      P    r5 ~
Offer 5% =    (     )[  5

n5          n5
10

r5     P r      r ~
Offer 10% =   (
n5
 )[ 10 10  10 v(r5  1, n5  1, r10  1, n10  1, m  1) 
 n10 n10

    r       ~
1  10       v(r5  1, n5  1, r10 , n10  1, m  1)]
            
   n10      

ESI 2004 Ankara, Turkey                                  18
define
r5          r 1                 r  s 1
1        , 2  5      , ...,  s  5         
n5         n5  1               n5  s  1

where s = number of people offered the offers

Let   Ws r5 , n5 , r10 , n10 , m     = max
 Offer 5%

 Offer 10%
Ws 1 r5  s  1, n5  s  1, r10 , n10 , m  1]
P5    r5
Offer 5% =          s[        
     n5
Offer 10% =              P r       r
 s [ 10 10  10 Ws 1 (r5  s  1, n5  s  1, r10  1, n10  1, m  1) 
 n10 n10
    r 
1  10 Ws 1 (r5  s  1, n5  s  1, r10 , n10  1, m  1)]
   n10 
       

ESI 2004 Ankara, Turkey                                 19
Note that     Ws r5 , n5 , r10 , n10 , m   depends on s, r10
and n10

Redefine optimality and value equations
as:                       P
        [  W (r , n )]         5
s 1 10
W r , n  = max 
s                   10

s   10   10      r P r                r 
 s [ 10 10  10 W s 1 (r10  1, n10  1)  1  10 W s 1 (r10 , n10  1)]
     n10                                     n 
             n10                                 10 

This is a Dynamic Program with varying
discount rate where βs increases

ESI 2004 Ankara, Turkey                                     20
Let’s define
P                       
s   5
 Ws 1 r10 , n10   Ws5 r10 , n10                         and
                       

r P                                        r                       
 s  10 10  10 Ws 1 r10  1, n10  1  1  10 Ws 1 r10 , n10  1  Ws10 r10 , n10 
r
 n 
 n10 
         n10                                10                     


For W , we have
                                            P5
 5[         W (r10 , n10 )]
W r10 , n10  = max 
                                            
                                                               
  5 [ r10 P  r10 W (r10  1, n10  1)  1  r10 W (r10 , n10  1)]
10
 n10                                          n10 
               n10                                 

ESI 2004 Ankara, Turkey                                      21
The action of accepting Offer 5%
continuously will result in the value
P5
function of W r10 , n10  =  1   
5


So as s approaches , Offer 5%
becomes fixed
Can be said W r , n  increases with r10
10
    10   10

So if at (r10*, n10) Offer 10% is chosen, it
will be the same for all r10 > r10*.

ESI 2004 Ankara, Turkey         22
Restless Bandit Problems
Problem with two arms
Choose one arm (active), other is passive
Classical Restless Bandit Problems
(Gittins, 1979)
The static arms are able to evolve
statistically, hence do not remain static
Restless Bandit Problem by Whittle
(1988)

ESI 2004 Ankara, Turkey   23
Comparisons of Model and
Restless Bandit Problem
Probability of population acceptance
unknown
Do not know which arm is best
However, there is a reward – observation
on each offer extended provides insights
Two arms (offers) compete to be chosen
over a time epoch, t, over infinite time
horizon
ESI 2004 Ankara, Turkey   24
Every time an arm is chosen, a
discounted reward is earned
Objective is to maximise profit expected
profit
Through definition of discount, the arms
are found to be independent
Hence, this is a Multi-armed Restless
Bandit Problem

ESI 2004 Ankara, Turkey   25
Empirical Results
Value of n5 = 20.          Value of P5 = 30.000.
Value of n10 = 15.         Value of P10 = 60.000.                 Value of m = 15.
r5          n5       r10              n10            Profit (£)       Decision

10          20       0                10             225.00           Offer 5%

10          20       1                10             225.00           Offer 5%

10          20       2                10             225.00           Offer 5%

10          20       3                10             225.00           Offer 5%

10          20       4                10             225.00           Offer 5%

10          20       5                10             235.62           Offer 10%

10          20       6                10             271.42           Offer 10%

10          20       7                10             315.09           Offer 10%

10          20       8                10             360.00           Offer 10%

10          20       9                10             405.00           Offer 10%

10          20       10               10             450.00           Offer 10%

ESI 2004 Ankara, Turkey                                26
r5         n5    r10             n10             Profit (£)   Decision

2          6     0               9               150.00       Offer 5%

2          6     1               9               150.00       Offer 5%

2          6     2               9               150.00       Offer 5%

2          6     3               9               150.00       Offer 5%

2          6     4               9               150.08       Offer 5%

2          6     5               9               169.17       Offer 10%

2          6     6               9               200.19       Offer 10%

2          6     7               9               233.34       Offer 10%

2          6     8               9               266.67       Offer 10%

2          6     9               9               300.00       Offer 10%

learning
Aids selection of offers
ESI 2004 Ankara, Turkey                            27
Contributions
Bayesian Updating as part of the
Behaviour of customers
‘Guide Book’ to aid in selection of offers
Acceptance Scoring

ESI 2004 Ankara, Turkey   28
Further Research Efforts
Defining the switch equation is ongoing
Proving that the model is a Restless
Bandit Problem
Development of a dynamic
programming-based approach to aid
question selections to achieve high
take-up of offer and optimal profit

ESI 2004 Ankara, Turkey     29
Thank you

Questions???

```
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