Credit Card Pricing and Impact of Adverse Selection


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1 Credt Card Prcng and Impact of Adverse Selecton Bo Huang and Lyn C. Thomas Unversty of Southampton
2 Contents Background Aucton model of credt card solctaton  Errors n probablty of beng Good  Errors n the score Impact of Adverse Selecton on Rskbased Prcng Numercal Examples  Lnear Relatonshp Model  Logstc Model Conclusons
3 background Credt cards are probably the most convenent form of credt of all competng fnancal assets that nclude both payment and credt devces (Ayad, 1997). Varable rate loans have been legal snce the early 1980s. In the credt card market, however, a standard rate contnued to domnate untl the early 1990s. The development of the nternet and the telephone as new channels for loan applcatons has made the offer process more prvate to each ndvdual (Thomas 2008).
4 background However, Ausubel (1991) ponted out that competton n credt card prcng can result n adverse selecton. Adverse selecton occurs n a tradng stuaton where one sde processes nformaton (the nformed party) whch s relevant to hs tradng partner who s the unnformed sde. Adverse selecton has already been nvestgated n the nsurance ndustry and the secondhand cars market. In consumer lendng market, Thomas (2008) ponts out that adverse selecton s mportant n estmatng the nteracton between the qualty of the applcant and the chance of them takng the loan.
5 Aucton Model of Credt card Solctaton Ausubel (1999) suggested that the aucton model s a useful analogy to the credt card applcaton process. Dfferent ssuer wll have dfferent score cards, and the resultant scores determne not only whether the ssuer wll accept that customer but nowadays also the nterest rate that wll be charged Ths s what rsk based prcng seeks to do.
6 Aucton Model of Credt card Solctaton In the aucton, each lender, obtans nformaton on the applcant to obtan an applcaton score s for that applcant. One can translate the score to obtan whch s lender s probablty that the applcant s a Good and wll not default on the account n a prescrbed tme horzon. However these estmates of probablty of beng Good are lkely to have errors n them. ε s p
7 Aucton Model of Credt card Solctaton assume the applcant has a true probablty of beng good of p~, and so p = ~ p + ε. assume that the applcant wll choose the credt card offer wth the lowest nterest rate and that all the frms are usng the same rsk based prcng approach. So the applcant wll choose the frm whose probablty p = p of them beng Good s the largest. Ths s an example of the wnner s curse, n that the lender wll have a hgher assumed probablty of the applcant beng a good than s really the case.
8 Errors n probablty of beng Good We assume that lender, estmates va the applcaton score that the applcant s probablty of beng good s where the errors are ndependent random varable wth dstrbuton functon F ( ). We assume that ε p p = ~ p + are the probablty of beng Good and the error of the lender, whose credt card s chosen by the borrower ε { p }, ε max { ε } = max 1 N = 1 N
9 Errors n probablty of beng Good So the dstrbuton of the true probablty of beng good of the applcant accepted by the lender who perceves the applcant s probablty of beng good to be, s gven by, G t) = Pr ~ p t = Pr p ε t = Pr p t ε { } { } { } { max { ε } p t} = 1 Pr{ { ε } p t} ( = Pr 1 N max1 N = 1 F ( p t ) So the densty functon of p~ g ( t) = N f ( p Hence the expected value of s N t) F ( p t) N 1 + Exp( ~ p) = t g( t) dt = + N t f ( p t) F( p t) N 1 dt
10 Errors n probablty of beng Good Defnng u = p t, we get Exp ( ~ p ) = = + + p ( p u ) N f ( u ) F ( u ) N 1 N f ( u) F( u) du + N 1 u N du f ( u) F( u) N 1 du Hence snce and Exp + b N ( ~ p ) = p b N N [ F ( u ] N 1 N f ( u) F ( u) du = ) = + u N f ( u) F ( u) N 1 du The lnear relatonshp n ths equaton s smlar to that dscussed n Phllps (2005) and Thomas (2008) where Exp[ ~ p( r, p) ] = p dr, d 0 where t s called the lnear probablty adverse selecton functon. >
11 Errors n probablty of beng Good If one assumes the error n the probablty s a unform dstrbuton whch spreads more the hgher the rate charged,.e. dr,dr then the above calculaton gven [ ] b N = + u N f ( u) F( u) N 1 du= + dr dr 2 u dr u+ dr N ( ) 2dr N 1 ( N 1) du= dr ( N + 1) The argument s that f the lender ncreases the nterest rate, they are wllng to accept applcants wth a lower probablty of beng Good and ths wll lead to a wder range of errors
12 Errors n the score The probablty of the ndvdual applcant beng Good s not drectly observed by ssuers. Instead, the lenders collect data on prevous borrowers wth smlar characterstcs and translate ths nformaton nto a credt score s for that applcant. To model ths, suppose the customer wth characterstcs wll be gven credt score of s(x) whch relates to the probablty of the customer beng good. x
13 Errors n the score Thomas (2009) mply how use logstc regresson to buld a score card leads to a log odds score where the relatonshp between the credt score and the probablty of beng Good s gven by ( ) p( x) ( ) log s x p x = 1 ( ) e p( x) = 1 + e s x ( ) s x
14 Errors n the score Suppose the errors that the lenders made are drectly n the score they gve the applcant, whch translates nto errors n the probablty of the applcant beng Good. Let s~ be the true score of the applcant and ths corresponds to ~ p, the true probablty of the applcant beng Good. Assume lender has a scorecard whch gves that applcant a score, where ~ s =. s ε s Agan we assume the applcant wll choose the lender who gves the hghest score snce under rsk based prcng ths wll lead to the offer whch the lowest rate requred on the credt card. Assume that there are N potental lenders and the scorng errors made by each lender are ndependent and have a common dstrbuton wth dstrbuton functon F( ). ε
15 Errors n the score We can follow the calculatons of the prevous error type to get the followng results. snce p% t t G( t) = Pr{ p% t} = Pr log log = Pr s% log 1 p% 1 t 1 t t p t = Pr s ε log = Pr log log ε 1 t 1 p 1 t { } [ ] = 1 We have G ( t ) { } [ ] Pr ε y = Pr max ε, ε, ε..., ε y = 1 Pr{max ε, ε, ε..., ε < y} N N N = 1 Pr{ ε < y} = 1 F( y) N p = 1 F log log 1 p 1 t t N
16 Errors n the score Then, the probablty densty functon s N p t p t g( t) = f log log log log F t ( 1 t ) 1 p 1 t 1 p 1 t So, for the lender who has taken the applcant assumng hs score s, then the expected true credt score of that applcant wll be gven by 1 ( ) ( ) ( ) ( ) ( ) N % = ε = = N E s E s E s yf y F y dy s b In terms of probabltes of beng Good ths becomes p% p p N 1 p E(log ) = E(log ) E( ε ) = E log yf ( y) F( y) dy = E log b 1 p% 1 p 1 p 1 p N 1 N
17 Errors n the score In the case when the score errors are unformly dstrbuted from to dr, where agan s the nterest rate charged one gets 1 (log p % ) (log p ) ( ) log p E E E dr N = ε = 1 p% 1 p 1 p N + 1 dr whch s strongly related to the lnear log odds selecton functon suggested n (Thomas 2008) and (Phllps 2005).
18 Impact of Adverse Selecton on Rskbased Prcng Rskbased prcng means that the nterest rate charged on a loan to a potental borrower depends on the lender s vew of the borrower s default rsk. In partcular we are nterested n the fact that these wnner s curse selecton errors are a form of adverse selecton snce they ncrease as the nterest rate charged ncreases. We use a rate model whch looks at the proftablty of decdng whether to lendng one unt to an applcant ( see Thomas 2008).
19 Impact of Adverse Selecton on Rskbased Prcng In the prevous secton we descrbed how the wnner s curse can lead to a lnear or logstc relatonshp between these two probabltes. In ths case, the lender s assumng that the probablty of the lender beng Good s that probablty p that would E ( ~ p ) ~ p 1 ~ p ) p 1 p = p er, and E(log = log er In general we denote ths relatonshp as ~ p ( r, p )
20 Impact of Adverse Selecton on Rskbased Prcng We assume that the rsk free rate at whch the lender can borrower the money s r F and the loss gven default (the percentage of defaulted loan fnally lost) s. r If the lender charges a rate to an applcant whose probablty of beng Good s p, then the take probablty, (the chance the applcant wll accept such a loan) s q( r, p. ) l D
21 Impact of Adverse Selecton on Rskbased Prcng If the lender beleves the borrower has a probablty p of beng Good, then the lender beleves the expected proft f a rate r(p) s charged to be EP ( r, p ) = q( r, p )[( r ( p ) r ) p ( l + r ) ( 1 p )] In order to fnd the optmal nterest rate, we dfferentate ths equaton wth respect to r and set the dervate to zero, to fnd when the proft s optmsed. Ths gves a rsk based nterest rate of r ( p) = r + ( l + r ) F D F F 1 p p q r D ( r, p) q( r, p) F
22 Impact of Adverse Selecton on Rskbased Prcng The realty through s that the lender s estmate of the probablty of the borrower beng Good s p, where the true probablty s ~ p. The optmal proft the lender would possbly obtan from such a borrower f the lender had the correct vew of the borrower s probablty of beng Good would be EP opt [ r ( p) ~, p] = q r ( ~ p) (, ~ p) [ ( r ( ~ p) r ) ~ p ( l + r ) ( 1 ~ p) ] ~ However, the lender s estmate of the borrower s probablty of beng Good s, and so what the lender expects the proft to be s EP [ r ( p ), p ] = q( r ( p ), p ) [ ( r ( p ) r ) p ( l + r ) ( p )] exp F D F 1 even though the borrower s true probablty s F D p~ F 1) 2)
23 Impact of Adverse Selecton on Rskbased Prcng In fact, the borrower wll not lve up to ths expectaton and the true expected proft the lender wll get s EP true [ r ( p ) ~ p] q( r ( p ) ~, =, p) [ ( r ( p ) r ) ~ p ( l + r ) ( 1 ~ p) ] F D F 3)
24 Numercal Examples Consder two examples, the frst s where the take functon and the adverse selecton functon are lnear n form and the second s where both are logstc n form. Take functons of those two forms were dscussed n Phllp (2005). In both cases we assume the rsk free rate s 5%, r F =0. 05and the loss gven default l D =0.5.
25 Lnear Relatonshp Model For the lnear, take probablty or response rate functon defne ( r, p) mn{ max[ 0,1 b( r r ) + c ( 1 p),1] } Then the optmal nterest rate s for q = L 0 p 1 r ( p) = r + ( l + r ) F D F 1 p 1 b p ( r r ) + c ( 1 p) We choose the value wth r =0.04, =2.5, and If we assume the relatonshp between and p s gven by a unform error, then we get a lnear relatonshp between p~ and p.so that, from prevous secton, we have ~ n 1 p = p dr( ), d > 0 n + 1 L L b b c = 2 p~
26 Lnear Relatonshp Model ~ and f we assume N = 500, d = 0.15 then p = p r The results of applyng these relatonshps n 1), 2) and 3) lead to the results n Table 1: p~ EPtrue EP opt p EPexp Table 1: Results of a lnear probablty adverse selecton functon.
27 Lnear Relatonshp Model
28 Logstc Model The logstc rskbased response functon or take rate s q e = 1 + a br cp ( r, p) a br cp The optmal nterest rate n ths case s r ( p) = r + ( l + r ) F D F Usng the cost structure of the prevous case wth rskfree rate beng 0.05 and the loss gven default l D beng 0.5, we assume the parameters for the logstc response rate functon are a = 54, b = 32,and c = 50. e 1 p p a 1+ e b br cp
29 Logstc Model p~ Assume the relatonshp between and p s lnear n the log odds correspondng to the stuaton n secton 2 where the error s n the score. We assume that t s gven by ~ p p n 1 log ~ = log dr( ), d > 0 1 p 1 p n+ 1 ~ p = (1 p p ) e dr N N p We take the values N = 500 and d = 4 so that error between the p log odds s ~ p = 3. (1 p) e 98r + p
30 Logstc Model The results of applyng these relatonshps n 1), 2) and 3) lead to the results n Table 2 p~ EPtrue EP opt p EPexp Table 2: Results of a lnear log odds probablty adverse selecton.
31 Logstc Model
32 Conclusons We show how modellng the way a borrower selects a loan as an aucton, means that wnner s curse leads to adverse selecton. We show that the relatonshps between the actual default rsk of a borrower and the lender s perceved vew of ths rsk are very smple ones, whatever the dstrbuton of the errors the lender makes. By buldng a smple model of the proft a lender makes from a loan, we are able to examne the effect of these adverse selecton errors.
33 Conclusons One way out of ths dlemma s for the lender to allow for the fact he wll msrepresent the rsk of the borrower, when calculatng the optmal rate to charge. The dffcultly wth ths s that the populaton who take the loan depend crtcally on the varable rates beng offered, and one of the strengths of varable prcng s that one can vary the rates to respond to charges n the market.
34 References Ausubel, L. (1991). The Falure of Cometton n the Credt Card Market, Amercan Economc Revew, 81(1), Ausubel, L. (1999). Adverse Selecton n the Credt Card Market, Workng Paper, Unversty of Maryland. Phllps, L.R. (2005), Prcng and revenue optmzaton, Stanford Unversty press. Stegltz, J. and Wess, A. (1981) Credt ratonng n markets wth mperfect nformaton, Amercan Economc Revew Thomas, L. (2009), Consumer credt models: prcng, proft, and portfolos. Oxford Unversty Press.
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