Overhaul Overdraft Fees: Creating Pricing and Product Design Strategies with Big Data

Overhaul Overdraft Fees: Creatng Prcng and Product Desgn Strateges wth Bg Data Xao Lu, Alan Montgomery, Kannan Srnvasan September 30, 2014 Abstract In 2012, consumers pad an enormous $32 bllon overdraft fees. Consumer attrton and potental government regulatons to shut down the overdraft servce urge banks to come up wth fnancal nnovatons to overhaul the overdraft fees. However, no emprcal research has been done to explan consumers overdraft ncentves and evaluate alternatve prcng and product strateges. In ths paper, we buld a dynamc structural model wth consumer montorng cost and dssatsfacton. We fnd that on one hand, consumers heavly dscount the future and overdraw because of mpulsve spendng. On the other hand, a hgh montorng cost makes t hard for consumers to track ther fnances therefore they overdraw because of ratonal nattenton. In addton, consumers are dssatsfed by the overly hgh overdraft fee and close ther accounts. We apply the model to a bg dataset of more than 500,000 accounts for a span of 450 days. Our polcy smulatons show that alternatve prcng strateges may ncrease the bank s revenue. Sendng targeted and dynamc alerts to consumers can not only help consumers avod overdraft fees but mprove bank profts from hgher nterchange fees and less consumer attrton. To allevate the computatonal burden of solvng dynamc programmng problems on a large scale, we combne parallel computng technques wth a Bayesan Markov Chan Monte Carlo algorthm. The Bg Data allow us to detect the rare event of overdraft and reduce the samplng error wth mnmal computatonal costs. 1 Introducton An overdraft occurs when a consumer attempts to spend or wthdraw funds from her checkng accounts n an amount exceedng the account s avalable funds. In the US, banks allow consumers to overdraw ther accounts (subject to some restrctons at banks dscreton and charge an overdraft fee. Overdraft fees have become a major source of bank revenues snce banks started to offer free checkng accounts to attract consumers. In 2012, the total amount of overdraft fees n the US reached $32 bllon, accordng to Moebs Servces 1. Ths s equvalent to an average of $178 for each checkng account annually 2. Accordng to the Center for Responsble Lendng, US households spent more on overdraft fees than on fresh vegetables, postage and books n 2010. 3 We acknowledge support from the Dpankar and Sharmla Chakravart Fellowshp. All errors are our own. 1 http://www.moebs.com 2 Accordng to Evans, Ltan, and Schmalensee 2011, there are 180 mllon checkng accounts n the US. 3 http://www.blackenterprse.com/money/managng-credt-3-ways-overdraft-fees-wll-stll-haunt-you/ 1

The unfarly hgh overdraft fee has provoked a storm of consumer outrage and therefore caused many consumers to close the account. The US government has taken actons to regulate these overdraft fees through the Consumer Fnancal Protecton Agency 4 and may potentally shut down the overdraft servce 5. Wthout overhaulng the current overdraft fee, banks encounter the problem of losng valuable customers and possbly totally losng the revenue source from overdrafts. Fnancal nsttutons store massve amounts of nformaton about consumers. The advantages of technology and Bg Data enable banks to reverse the nformaton asymmetry (Kamenca, Mullanathan, and Thaler 2011 as they may be able to generate better forecasts about a consumer s fnancal state than consumers themselves can. In ths paper, we extract the valuable nformaton embedded n the Bg Data and harness t wth structural economc theores to explan consumers overdraft behavor. The large scale fnancal transacton panel data allows us to sort through consumers fnancal decson makng processes and dscover rch consumer heterogenety. As a consequence, we come up wth ndvdually customzed strateges that can ncrease both consumer welfare and bank revenue. In ths paper, we am to acheve two substantve goals. Frst, we leverage rch data about consumer spendng and balance checkng to understand the decson process for consumers to overdraw. We address the followng research questons. Are consumers fully attentve n montorng ther checkng account balances? How great s the montorng cost? Why do attentve consumers also overdraw? Are consumers dssatsfed because the overdraft fee? Second, we nvestgate prcng and new product desgn strateges that overhaul overdraft fees. Specfcally, we tackle these questons. Is the current overdraft fee structure optmal? How wll the bank revenue change under alternatve prcng strateges? More mportantly, what new revenue model can make the ncentves of the bank and consumers better algned? Can the bank beneft from helpng consumers make more nformed fnancal decsons, lke sendng alerts to consumers? If so, what s the optmal alert strategy? How can the bank leverage ts rch data about consumer fnancal behavors to reverse nformaton asymmetry and create targeted strateges? We estmate the dynamc structural model usng data from a large commercal bank n the US. The sample sze s over 500,000 accounts and the sample length s up to 450 days. We fnd that some consumers are nattentve n montorng ther fnances because of a substantally hgh montorng cost. In contrast, attentve consumers overdraw because they heavly dscount future utltes and are subject to mpulsve spendng. Consumers are dssatsfed to leave the bank after beng charged the unfarly hgh overdraft fees. In our counterfactual analyss, we show that a percentage fee or a quantty premum fee strategy can acheve hgher bank revenue compared to the current flat per-transacton fee strategy. Enabled by Bg Data, we also propose an optmal targeted alert strategy. The bank can beneft from sendng alerts to let consumers spend ther unused balances so that the bank can earn more nterchange fees. Helpng consumers make more nformed decsons wll also sgnfcantly reduce consumer attrton. The targeted dynamc alerts should be sent to consumers wth hgher montorng costs and both when they are underspendng and overspendng. Methodologcally, our paper makes two key contrbutons. Frst, we buld a dynamc structural model that ncorporates nattenton and dssatsfacton nto the lfe-tme consumpton model. Although we apply t to the overdraft context, the model framework can be generalzed to ana- 4 http://bankng-law.lawyers.com/consumer-bankng/consumers-and-congress-tackle-bg-bank-fees.html 5 http://fles.consumerfnance.gov/f/201306_cfpb_whtepaper_overdraft-practces.pdf 2

lyze other marketng problems regardng consumer dynamc budget allocaton, lke electrcty and cellphone usage. Second, we estmate the model on Bg Data wth the help of parallel computng technques. Structural models have the mert of producng polcy nvarate parameters that allow us to conduct counterfactual analyss. However, the nherent computatonal burden prevents t from beng wdely adopted by ndustres. Moreover, the data sze n a real settng s typcally much larger than what s used for research purposes. Companes, n our case a large bank, need to have methods that are easly scalable to generate targeted solutons for each consumer. Our proposed algorthm takes advantage of state-of-the-art parallel computng technques and estmaton methods that allevate computatonal burden and reduce the curse of dmensonalty. The rest of the paper s organzed as follows. In secton 2 we frst revew related lterature. Then we show summary statstcs n secton 3 whch motvate our model setup. Secton 4 descrbes our structural model and we provde detals of dentfcaton and estmaton procedures n secton 5. Then n sectons 6 and 7 we show estmaton results and counterfactual analyss. Secton 8 concludes and summarzes our lmtatons. 2 Related Lterature A varety of economc and psychologcal models can explan overdrafts, ncludng full-nformaton pure ratonal models and lmted attenton, as summarzed by Stango and Znman (2014. However, no emprcal paper has appled these theores to real consumer spendng data. Although Stango and Znman (2014 had a smlar dataset to ours, ther focus was on testng whether takng related surveys can reduce overdrafts. We develop a dynamc structural model that ncorporates theores of heavy dscountng, nattenton and dssatsfacton n a comprehensve framework. The model s flexble to address varous overdraft scenaros, thus t can be used by polcy makers and the bank to desgn targeted strateges to ncrease consumer welfare and bank revenue. Our model nherts from the tradtonal lfetme consumpton model but adds two novel features, nattenton and dssatsfacton. Frst of all, a large body of lterature n psychology and economcs has found that consumers pay lmted attenton to relevant nformaton. In the revew paper by Card, DellaVgna and Malmender (2011, they summarze fndngs ndcatng that consumers pay lmted attenton to 1 shppng costs, 2 tax (Chetty et. al. 2009 and 3 rankng (Pope 2009. Gabax and Labson (2006 fnd that consumers don t pay enough attenton to add-on prcng and Grubb (2014 shows consumers nattenton to ther cell-phone mnute balances. Many papers n the fnance and accountng doman have documented that nvestors and fnancal analysts are nattentve to varous fnancal nformaton (e.g., Hrshlefer and Teoh 2003, Peng and Xong 2006. We follow Stango and Znman (2014 to defne nattenton as ncomplete consderaton of account balances (realzed balance and avalable balance net of comng blls that would nform choces. We further explan nattenton wth a structural parameter, montorng cost, whch represents the tme and effort to know the exact amount of money n the checkng account. Wth ths parameter estmated, we are able to quantfy the economc value of sendng alerts to consumers and provde gudance for the bank to set ts prcng strategy. We also come up wth polcy smulatons about alerts because we thnk a drect remedy for consumers lmted attenton s to make nformaton more salent (Card, DellaVgna and Malmender 2011. Past lterature also fnds that remnders (Karlan et. al. 2010, mandatory dsclosure (Fshman and Hagerty 2003, and penal- 3

tes (Haselhuhn et al. 2012 all serve the purpose of ncreasng salence and thus mtgatng the negatve consequences of nattenton. Second, as documented n prevous lterature, unfarly hgh prce may cause consumer dssatsfacton whch s one of the man causes of customer swtchng behavor (Keaveney 1995, Bolton 1998. We notce that consumers are more lkely to close the account after payng the overdraft fee and when the rato of the overdraft fee over the overdraft transacton amount s hgh. Ths s because gven the current bankng ndustry practce, a consumer pays a flat per-transacton fee regardless of the transacton amount. Therefore, the mpled nterest rate for an overdraft orgnated by a small transacton amount s much hgher than the socally accepted nterest rate (Matzler, Wurtele and Renzl 2006, leadng to prce dssatsfacton. We am to estmate ths nfnte horzon dynamc structural model on a large scale of data and obtan heterogeneous best response for each consumer to prepare targeted marketng strateges. After searchng among dfferent estmaton methods, ncludng the nested fxed pont algorthm (Rust 1987, the condtonal choce probablty estmaton (Arcdacono and Mller 2011 and the Bayesan estmaton method developed n Ima, Jan and Chng (2009 (IJC, we fnally choose the IJC method for the followng reasons. Frst of all, the herarchcal Bayes framework fts our goal of obtanng heterogeneous parameters. Second, n order to apply our model to a large scale of data, we need to estmate the model wth Bayesan MCMC so that we can mplement a parallel computng technque. Thrd, IJC s the state-of-the art Bayesan estmaton algorthm for nfnte horzon dynamc programmng models. It provdes two addtonal benefts n tacklng the computatonal challenges. One s that t allevates the computatonal burden by only evaluatng the value functon once n each MC teraton. Essentally, the algorthm solves the value functon and estmates the structural parameters smultaneously. So the computatonal burden of a dynamc problem s reduced by an order of magntude smlar to those computatonal costs of a statc model. The other s that the method reduces the curse of dmensonalty by allowng state space grd ponts to vary between estmaton teratons. On the other hand, as our sample sze s huge, tradtonal MCMC estmaton may take a prohbtvely, f not mpossbly, long tme, snce for N data ponts, most methods must perform O(N operatons to draw a sample. A natural way to reduce the computaton tme s to run the chan n parallel. Past methods of Parallel MCMC duplcate the data on multple machnes and cannot reduce the tme of burn-n. We nstead use a new technque developed by Neswanger, Wang and Xng (2014 to solve ths problem. The key dea of ths algorthm s that we can dstrbute data nto multple machnes and perform IJC estmaton n parallel. Once we obtan the posteror Markov Chans from each machne, we can algorthmcally combne these ndvdual chans to get the posteror chan of the whole sample. 3 Background and Model Free Evdence We obtaned data from a major commercal bank n the US. Durng our sample perod n 2012 and 2013, overdraft fees accounted for 47% of the revenue from depost account servce charges and 9.8% of the operatng revenue. The bank provdes a comprehensve overdraft soluton to consumers. (For general overdraft practces n the US, please refer to Stango and Znman (2014 for a good revew. Appendx A.1 tabulates current fee settngs n top US banks. In the standard overdraft servce, f the consumer 4

overdraws her account, the bank mght cover the transacton and charge $31 6 Overdraft Fee (OD or declne the transacton and charge a $31 Non-Suffcent-Fund Fee (NSF. Whether the transacton s accepted or declned s at the bank s dscreton. The OD/NSF fee s at a per-tem level. If a consumer performs several transactons when the account s already overdrawn, each transacton tem wll ncur a fee of 31 dollars. Wthn a day, a maxmum of four per-tem fees can be charged. If the account remans overdrawn for fve or more consecutve calendar days, a Contnuous Overdraft Fee of $6 wll be assessed up to a maxmum of $84. The bank also provdes an Overdraft Protecton Servce where the checkng account can lnk to another checkng account, a credt card or a lne of credt. In ths case, when the focal account s overdrawn, funds can be transferred to cover the negatve balance. The Overdraft Transfer Balance Fee s $9 for each transfer. As you can see, the fee structure for the bank s qute complcated. In the emprcal analyss below, we don t dstngush between dfferent types of overdraft fees and assume that money s fungble so that the consumer only cares about the total amount of overdraft fee rather than the underlyng prcng structure. The bank also provdes balance checkng servces through branch, automated teller machne (ATM, call center and onlne/moble bankng. Consumers can nqure about ther avalable balances and recent actvtes. There s also a notfcaton servce to consumers va emal or text message, named alerts. Consumers can set alerts when certan events take place, lke overdrafts, nsuffcent funds, transfers, deposts, etc. Unfortunately, our dataset only ncludes the balance checkng data but not the alert data. We ll dscuss ths lmtaton n secton 8. In 2009, the Federal Reserve Board made an amendment to Regulaton E (subsequently recodfed by the Consumer Fnancal Protecton Bureau (CFPB whch requres account holders to provde affrmatve consent (opt n for overdraft coverage of ATM and non-recurrng pont of sale (POS debt card transactons before banks can charge for payng such transactons 7. Ths Regulaton E amed to protect consumers aganst the heavy overdraft fees. The change became effectve for new accounts on July 1, 2010, and for exstng accounts on August 15, 2010. Our sample contans both opt-n and opt-out accounts. However, we don t know whch accounts have opted n unless we observe an ATM/POS ntated overdraft occason. We also dscuss ths data lmtaton n secton 8. 3.1 Summary Statstcs Our data can be dvded nto two categores, checkng account transactons and balance nqury actvtes. In our sample, there are between 500,000 and 1,000,000 8 accounts, among whch 15.8% had at least one overdraft ncdence durng the sample perod between June 2012 and Aug 2013. The proporton of accounts wth overdraft s lower than the 27% (across all banks and credt unons reported by the CFPB n 2012 9. In total, all the counts performed more than 200 mllon transactons, ncludng deposts, wthdrawals, transfers, and payments etc. For each transacton, we know the account number, transacton date, transacton amount, and transacton descrpton. The transac- 6 All dollar values n the paper have been rescaled by a number between.85 and 1.15 to help obfuscate the exact amounts wthout changng the substantve mplcatons. The bank also sets the frst tme overdraft fee for each consumer at $22. All the rest overdraft fees are set at $31. 7 http://www.occ.gov/news-ssuances/bulletns/2011/bulletn-2011-43.html 8 For the sake of prvacy, we can t dsclose the exact number. 9 http://fles.consumerfnance.gov/f/201306_cfpb_whtepaper_overdraft-practces.pdf 5

ton descrpton tells us the type of transacton (e.g., ATM wthdrawal or debt card purchase and locaton/assocated nsttuton of the transacton, lke merchant name or branch locaton. The descrpton helps us dentfy the cause of the overdraft, for nstance whether t s due to an electrcty bll or due to a grocery purchase. Table 1: Overdraft Frequency and Fee Dstrbuton Mean Std Medan Mn 99.85 Percentle OD Frequency 9.84 18.74 3 1 >100 OD Fee 245.46 523.04 77 10 >2730 As shown n Table 1, consumers who pad overdraft fees, on average, overdrew nearly 10 tmes and pad $245 durng the 15 month sample perod. Ths s consstent wth the fndng from the CFPB that the average overdraft- and NSF-related fees pad by all accounts that had one or more overdraft transactons n 2011 were $225 10. There s sgnfcant heterogenety n consumers overdraft frequency and the dstrbuton of overdraft frequency s qute skewed. The medan overdraft frequency s three and more than 25% of consumers overdrew only once. In contrast, the top 0.15% of heavy overdrafters overdrew more than 100 tmes. A smlar skewed pattern apples to the dstrbuton of overdraft fees. Whle the medan overdraft fee s $77, the top 0.15% of heavest overdrafters pad more than $2,730 n fees. Fgure 1: Overdraft Frequency and Fee Dstrbuton Now let s zoom n to take a look at the behavor of the majorty overdrafters that have overdrawn less than 40 tmes. The frst panel n Fgure 1 depcts the dstrbuton of overdraft frequency for those accounts. Notce that most consumers (> 50% only overdrew less than three tmes. The second panel shows the dstrbuton of the pad overdraft fee for accounts that have overdrawn less than $300. Consstent wth the fee structure where the standard per-tem overdraft fee s $22 or $31, we see spkes on these two numbers and ther multples. 10 http://fles.consumerfnance.gov/f/201306_cfpb_whtepaper_overdraft-practces.pdf 6

Table 2: Types of Transactons That Cause Overdraft Type Frequency Percentage Amount Debt Card Purchase 946,049 48.65% 29.50 ACH Transacton 267,854 13.77% 294.57 Check 227,128 11.68% 417.78 ATM Wthdrawal 68,328 3.51% 89.77 What types of transactons cause overdraft? We fnd that nearly 50% of overdrafts are caused by debt card purchases wth mean transacton amounts around $30. On the other hand, ACH (Automated Clearng House and Check transactons account for 13.77% and 11.68% of overdraft occasons. These transactons are generally for larger amounts, $294.57 and $417.78, respectvely. ATM wthdrawals lead to another 3.51% of the overdraft transactons wth an average amount of around $90. 3.2 Model Free Evdence Ths secton presents some patterns n the data that suggest the causes and effects of overdrafts. We show that heavy dscountng and nattenton may drve consumers overdraft behavors. And consumers are dssatsfed because of the overdraft fees. The model free evdence also hghlghts the varaton n the data that wll allow for the dentfcaton of the dscount factor, montorng cost and dssatsfacton senstvty. 3.2.1 Heavy Dscountng Frst of all, we argue that a consumer may overdraw because she prefers current consumpton much more than future consumpton,.e. she heavly dscounts future consumpton utlty. At the pont of sale, the consumer sharply dscounts the future cost of the overdraft fee to satsfy mmedate gratfcaton 11. If that s the case, then we should observe a steep downward slopng trend n the spendng pattern wthn a pay perod. That s, the consumer wll spend a lot rght after gettng a pay check and then reduce spendng durng the course of the month. But because of overspendng at the begnnng, the consumer s gong to run out of budget at the end of the pay perod and has to overdraw. We test ths hypothess wth the followng model specfcaton. We assume that the spendng for consumer at tme t Spendng t can be modeled as Spendng t = β LapsedTmeA fterincome t + µ + v t + ε t where LapsedTmeA fterincome t s the number of days after the consumer receved ncome (salary, µ s the ndvdual fxed effect and v t s the tme (day fxed effect. To control for the 11 We also consdered hyperbolc dscountng wth two dscount factors, a short term present bas parameter and a long term dscount factor. Wth more than three perods of data wthn a pay perod, hyperbolc dscount factors can be dentfed (Fang and Slverman 2009. However, our estmaton results show that the present bas parameter s not sgnfcantly dfferent from 1. Therefore we only keep one dscount factor n the current model. Estmaton results wth hyperbolc dscount factors are avalable upon requests. 7

effect that consumers usually pay for ther blls (utltes, phone blls, credt card blls, etc after gettng the pay check, we exclude checks and ACH transactons whch are the common choces for bll payments from the daly spendngs and only keep debt card purchases, ATM wthdrawals and person-to-person transfers. We run ths OLS regresson for heavy overdrafters (whose overdraft frequency s n the top 20 percentle among all overdrafters, lght overdrafters (whose overdraft frequency s not n the top 20 percentle among all overdrafters and non-overdrafters (who ddn t overdraw durng the 15 months sample perod separately. The results are reported n column (1 (2 and (3 of Table 3. Table 3: Spendng Decreases wth Tme n a Pay Cycle (1 (2 (3 Heavy Overdrafters Lght Overdrafters Non- Overdrafters Lapsed Tme after Income (β 6.8374 0.00007815 0.00002195 (0.00006923 (0.00006540 (0.00002328 Fxed Effect Yes Yes Yes Number of Observatons 17, 810, 276 53, 845, 039 242, 598, 851 R 2 0.207 0.275 0.280 Note: *p<0.01;**p<0.001;***p<0.0001 We fnd that the coeffcent of LapsedTmeA fterincome t s negatve and sgnfcant for heavy overdrafters but not lght overdrafters or non-overdrafters. Ths suggests that heavy overdrafters have a steep downward slopng spendng pattern durng a pay perod whle lght overdrafters or non-overdrafters have a relatvely stable spendng stream. The heavy overdrafters are lkely to overdraw because they heavly dscount ther future consumptons. 3.2.2 Inattenton Next we explan the overdraft ncentves for the lght overdrafters wth nattenton. The dea s that consumers mght be nattentvely montorng ther checkng accounts so that they are uncertan about the exact balance amount. Sometmes the perceved balance can be hgher than the true balance and ths mght cause an overdraft. We frst present a representatve example of consumer nattenton. The example s based upon our data, but to protect the prvacy of the consumer and the merchants, amounts have been changed. However, the example remans representatve of the underlyng data. 8

Fgure 2: Overdraft due to Balance Percepton Error As shown n fgure 2, the consumer frst receved her salary on August 17th. After a seres of expenses she was left wth $21.16 on August 20th. As she had never checked her balance, she contnued spendng and overdrew her account for several small purchases, ncludng a $25 restaurant bll, a $17.12 beauty purchase, a $6.31 game and a $4.95 coffee purchase. These four transactons added up to $53.38 but caused her to pay four overdraft tem fees, a total of $124. We speculate that ths consumer was careless n montorng her account and overestmated her balance. Beyond ths example, we fnd more evdence of nattenton n the data. Intutvely, a drect support of nattenton s that the less frequent a consumer checks her balance, the more overdraft fee she pays. To test ths hypothess, we estmate the followng specfcaton: TotODPmt t =β 0 + β 1 BCFreq t + µ + v t + ε t where for consumer at tme t (month, TotODPmt t s the total overdraft payment, BCFreq t s the balance checkng frequency. We estmate ths model on lght overdrafters (whose overdraft frequency s not n the top 20 percentle and heavy overdrafters (whose overdraft frequency s n the top 20 percentle separately and report the result n the column (1 and (2 n Table 4. 9

Table 4: Frequent Balance Checkng Reduces Overdrafts for Lght Overdrafters (1 (2 (3 Lght Overdrafters Heavy Overdrafters All Overdrafters Balance Checkng Frequency (BCFreq, β 1 Overdraft Frequency (ODFreq, β 2 BCFreq ODFreq (β 3 0.5001 0.00001389 0.6823 (0.00000391 (0.00000894 (0.00000882 16.0294 (0.00002819 27.8136 (0.00000607 Number of Observatons 53, 845, 039 17, 810, 276 71, 655, 315 R 2 0.1417 0.1563 0.6742 Note: Fxed effects at ndvdual and day level; Robust standard errors, clustered at ndvdual level.*p<0.01;**p<0.001;***p<0.0001 The result suggests that more balance checkng decreases overdraft payment for lght overdrafters but not for heavy overdrafters. We further test ths effect by ncludng overdraft frequency (ODFreq t and an nteracton term of balance checkng frequency and overdraft frequency BCFreq t ODFreq t n the equaton below. The dea s that f the coeffcent for ths nteracton term s postve whle the coeffcent for balance checkng frequency (BCFreq t s negatve, then t mples that checkng balances more often only decreases the overdraft payment for consumers who overdraw nfrequently but not for those who do t wth hgh frequency. TotODPmt t =β 0 + β 1 BCFreq t + β 2 ODFreq t + β 3 BCFreq t ODFreq t + µ + v t + ε t The result n column (3 of Table 4 confrms our hypothess. Interestngly, we fnd that a consumer s balance percepton error accumulates overtme n the sense that the longer a consumer hasn t check balances, the more lkely that she s gong to overdraw and pay hgher amount of overdraft fees. Fgure 3 below exhbts the overdraft probablty across number of days snce a consumer checked balance last tme for lght overdrafters (whose overdraft frequency s not n the top 20 percentle. It suggests that the overdraft probablty ncreases moderately wth the number of days snce the last balance check. Fgure 3: Overdraft Lkelhood Increases wth Lapsed Tme Snce Last Balance Check 10

We confrm ths relatonshp wth the followng two specfcatons. We assume that overdraft ncdence I(OD t (where I(OD t = 1 denotes overdraft and I(OD t = 0 denotes no overdraft and overdraft fee payment amount ODFee t for consumer at tme t can be modeled as: I(OD t = Φ(ρ 0 + ρ 1 DaysSnceLastBalanceCheck t + ρ 2 BegnBal t + µ + v t ODFee t = ρ 0 + ρ 1 DaysSnceLastBalanceCheck t + ρ 2 BegnBal t + µ + v t + ε t where Φ s the cumulatve dstrbuton functon for standard normal dstrbuton. The term DaysSnceLastBalanceCheck t denotes the number of days consumer hasn t checked her balance untl tme t and BegnBal t s the begnnng balance at tme t. We control for the begnnng balance because t can be negatvely correlated wth the days snce last balance check due to the fact that consumers tend to check when the balance s low and a lower balance usually leads to an overdraft. Table 5: Reduced Form Evdence of Exstance of Montorng Cost I (OD ODFee Days Snce Last Balance Check (ρ 1 0.0415 0.0003 (0.00000027 (0.00000001 Begnnng Balance (ρ 2 0.7265 0.0439 (0.00000066 (0.00000038 Indvdual Fxed Effect Yes Yes Tme Fxed Effect Yes Yes Number of Observatons 53, 845, 039 53, 845, 039 R 2 0.5971 0.6448 Note: The estmaton sample only ncludes overdrafters. Margnal effects for the Probt model; Fxed effects at ndvdual and day level; robust standard errors, clustered at ndvdual level.*p<0.01;**p<0.001;***p<0.0001. Table 5 reports the estmaton results whch support our hypothess that the longer a consumer hasn t checked balance, the more lkely she overdraws and the hgher overdraft fee she pays. Snce checkng balances can effectvely help prevent overdrafts, why don t consumers do t often enough to avod overdraft fees? We argue that t s because montorng the account s costly n terms of tme, effort and mental resources. Therefore, a natural consequence s that f there s a means to save consumers tme, effort or mental resources, the consumer wll ndeed check balances more frequently. We fnd such support from the data about onlne bankng ownershp. Specfcally, for consumer we estmate the followng specfcaton: CheckBalFreq = β 0 + β 1 OnlneBankng +β 2 LowIncome + β 3 Age + ε where CheckBalFreq s the balance checkng frequency, OnlneBankng s onlne bankng ownershp (1 denotes the consumer has onlne bankng whle 0 denotes otherwse, LowIncome s whether the consumer belongs to the low ncome group (1 denotes yes and 0 denotes no and Age s age (n years. 11

Table 6: Reduced Form Evdence of Exstance of Montorng Cost Dependent varable Check Balance Frequency Onlne Bankng (β 1 58.4245 (0.5709 Low Income (β 2 3.3812 (0.4178 Age (β 3 0.6474 (0.0899 Number of Observatons 602,481 R 2 0.6448 *p<0.01;**p<0.001;***p<0.0001. Table 6 shows that after controllng for ncome and age, consumers wth onlne bankng accounts check the balance more frequently than those wthout, whch suggests that montorng costs exst and when they are reduced, consumers montor more frequently. 3.2.3 Dssatsfacton Table 7: Account Closure Frequency for Overdrafters vs Non-Overdrafters Total % Closed Heavy Overdrafters 23.36% Lght Overdrafters 10.56% Non-Overdrafters 7.87% We also fnd that overdrafters are more lkely to close ther accounts (Table 7. Among nonoverdrafters, 7.87% closed ther accounts durng the sample perod. Ths rato s much hgher for overdrafters. Specfcally, 23.36% of heavy overdrafters (whose overdraft frequency s n the top 20 percentle closed ther accounts, whle 10.56% of lght overdrafters (whose overdraft frequency s not n the top 20 percentle closed ther accounts. Table 8: Closure Reasons Overdraft Overdraft No Overdraft Forced Closure Voluntary Closure Voluntary Closure Heavy Overdrafters 86.34% 13.66% Lght Overdrafters 52.58% 47.42% Non-Overdrafters 100.00% From the descrpton feld n the data, we can dstngush the cause of account closure: forced closure by the bank because the consumer s unable or unwllng to pay back the negatve balances and the fee (charge-off or voluntary closure by the consumer. Among heavy overdrafters, 13.66% closed voluntarly and the rest (86.34% were forced to close by the bank (Table 8. In contrast, 47.42% of the lght overdrafters closed ther accounts voluntarly. We conjecture that the hgher 12

voluntary closures may be due to customer dssatsfacton wth the bank, wth evdence shown below. Fgure 4: Days to Closure After Last Overdraft Frst, we fnd that overdrafters who closed voluntarly were very lkely to close soon after the overdraft. In Fgure 4 we plot the hstogram of number of days t took the account to close after ts last overdraft occason. It shows that more than 60% of accounts closed wthn 30 days after the overdraft occason. Fgure 5: Percentage of Accounts Closed Increases wth Fee/Transacton Amount Rato Second, lght overdrafters are also more lkely to close ther accounts when the rato of overdraft fee over the transacton amount that caused the overdraft fee s hgher. In other words, the more unfar the overdraft fee (hgher rato of overdraft fee over the transacton amount that caused the overdraft fee, the more lkely t s that she wll close the account. We show ths pattern n the left panel of Fgure 5. However, ths effect doesn t seem to be present for heavy overdrafters (rght panel of Fgure 5. The model free evdence ndcate that consumer heavy dscountng and nattenton can help explan consumers overdraft behavors as consumers mght be dssatsfed after beng charged the overdraft fees. Below we ll buld a structural model that ncorporates consumer heavy dscountng, nattenton and dssatsfacton. 4 Model We model a consumer s daly decson about non-preauthorzed spendng n her checkng account. Alternatvely we could descrbe ths non-preauthorzed spendng as mmedate or dscretonary; not dscretonary n the sense that economsts tradtonally use the term, but n the sense that mmedate spendng lkely could have been delayed. To focus on ratonalzng the consumer s overdraft 13

behavor, we make the followng assumptons. Frst, we abstract away from the complexty assocated wth our data and assume that the consumer s ncome and preauthorzed spendngs are exogenously gven. We refer to preauthorzed spendng to mean those expenses for whch the spendng decson was made pror to payment. For example, a telephone bll or a mortgage due are usually arranged before the date that the actual payment occurs. We assume that decsons for preauthorzed spendng are hard to change on a daly bass after they are authorzed and more lkely to be related to consumpton that has medum or long-run consequences. In contrast, nonpreauthorzed spendng nvolves a consumer s frequent day-to-day decsons and the consumer can adjust the spendng amount flexbly. We make ths dstncton because non-preauthorzed spendng s at the consumer s dscreton and thus affects the overdraft outcome drectly. To ease explanaton, we use comng blls to represent preauthorzed spendng for the rest of the paper. Second, we allow the consumer to be nattentve to montorng her account balance and comng blls. But she can decde whether to check her balance. When a consumer hasn t checked the balance, she comes up wth an estmate of the avalable balance and forms an expectaton about comng blls. If she makes a wrong estmate or expectaton, she faces the rsk of overdrawng her account. Last, as consumpton s not observed n the data, we make a bold assumpton that spendng s equvalent to consumpton n terms of generatng utlty. That s, the more a consumer spends, the more she consumes, the hgher utlty she obtans. In what follows, we use consumpton and spendng nterchangeably. We ll descrbe the model n the next four parts: (1 tmng, (2 basc model (3 nattenton and balance checkng and (4 dssatsfacton and account closng. 4.1 Tmng The tmng of the model s as follows (Fgure 6. On each day: 1. The consumer receves ncome, f there s any. 2. Her blls arrve f there s any. 3. Balance checkng stage (CB: She decdes whether to check her balance. If she checks, she ncurs a cost and knows today s begnnng balance and the bll amount. If not, she recalls an estmate of the balance and bll amount. 4. Spendng stage (SP: She makes the dscretonary spendng decson (Choose C to maxmze total dscounted utlty V (or expected total dscounted utlty EV f she ddn t check balancefor today and spends the money. 5. Overdraft fee s charged f the endng balance s below zero. 6. Account closng stage (AC: She decdes whether to close the account (after payng the overdraft fee f there s any. If she closes the account, she receves an outsde opton. If she doesn t chose the account, she goes to 7. 7. Balance updates and the next day comes. 14

Fgure 6: Model Tmng 4.2 Basc Model We assume the consumer s per-perod consumpton utlty at tme t s a constant relatve rsk averse utlty (Arrow 1963: u C (C t = C1 θ t t (1 1 θ t where θ t s the relatve rsk averse coeffcent whch represents the consumer s preference about consumpton. The hgher θ t, the hgher utlty the consumer can derve from a margnal unt of consumpton. θ t = exp(θ + ε t ε t N ( 0,ς 2 As consumers preference for consumpton mght change over tme and the relatve rsk averse coeffcent s always postve, we allow θ t to follow a log-normal dstrbuton. Essentally, θ t s the exponental of the sum of a tme-nvarant mean θ and a random shock ε t. The shocks capture unexpected needs for consumpton and follow a normal dstrbuton wth mean 0 and varance ς 2 (Yao et. al. 2012. Notce that the consumpton plan C t depends on the consumer s budget constrant, whch further depends on her current balance B t, ncome Y t and future blls Ψ t. For example, when the comng bll s for a small amount, the consumpton can be hgher than when the bll s for a large amount. 4.3 Inattenton and Balance Checkng In practce, the consumer may not be fully attentve to her fnancal well-beng. Because montorng her account balance takes tme and effort, she may not check her balance frequently. As a 15

consequence, nstead of knowng the exact (avalable balance B t 12, she recalls a perceved balance B t. Followng Mehta, Rajv and Srnvasan (2003, we allow the perceved balance B t to be the sum of the true balance B t and a percepton error η t ω t. The frst component of the percepton error η t s a random draw from the standard normal dstrbuton 13 and the second component s the standard devaton of the percepton error, ω t. So B t follows a normal dstrbuton B t N ( B t + η t ω t,ωt 2 The varance of the percepton error ωt 2 measures the extent of uncertanty. Based on the evdence from secton 3.2.2, we allow ths extent of uncertanty to accumulate through tme whch mples that the longer the consumer goes wthout checkng her balance, the more naccurate her perceved balance s. That s, ω 2 t = ργ t (2 where Γ t denotes the lapsed tme snce the consumer last checked her balance, and ρ denotes the senstvty to lapsed tme as shown n the equaton (2 above 14. Notce that the expected utlty s decreasng n the varance of the percepton error ω 2 t. Ths s true because the larger the varance of the percepton error, the less accurate the consumer s estmate of her true balance, and the more lkely she s gong to mstakenly overdraw, whch lowers her utlty. We further assume that the consumer s sophstcated nattentve 15 n the sense that she s aware of her own nattenton (Grubb 2014. Sophstcated nattentve consumers are ratonal n that they choose to be nattentve due to the hgh cost of montorng her balances from day-to-day. We also model the consumer s balance checkng behavor. We denote the balance checkng choce as Q t {1,0} where 1 means check and 0 otherwse. If a consumer checks her balance, she ncurs a montorng cost but knows exactly what her balance s. So the percepton error s reduced to zero and she can make her optmal spendng decson wth all nformaton. In mathematcs form, her consumpton utlty functon changes to u t = C1 θ t t Q t ξ + χ tqt (3 1 θ t where ξ s her balance checkng cost and χ Qt s the dosyncratc shock that affects her balance checkng cost. The shock χ tqt can come from random events lke a consumer checks balance because she s also performng other types of transactons (lke onlne bll payments or she s on vacaton wthout access to any bank channels so t s hard for her to check balances. The equaton 12 Avalable balance means the ntal balance plus ncome mnus blls. For the ease of exposton, we omt the word "avalable" and only use "balance". 13 The mean balance percepton error η cannot be separately dentfed from the varance parameters ρ because the dentfcaton sources both come from consumers overdraft fee payment. Specfcally, the hgh overdraft payment for a consumer can be ether explaned by a postve balance percepton error or large percepton error varance caused by large ρ. So we fx η at zero,.e. the percepton error s assumed to be unbased. 14 We consdered other specfcatons for the relatonshp between percepton error varance and lapsed tme snce last balance check. Results reman qualtatvely unchanged 15 Consumers can also be navely nattentve, but we don t allow t here. See dscusson n Grubb 2014. 16

1 mples that f the consumer checks her balance, then her utlty decreases by a monetary equvalence of [(1 θ t ξ ] 1 θt. We assume that χ tqt are d and follow a type I extreme value dstrbuton. If she doesn t check, she recalls her balance B t wth the percepton error η t. So her perceved balance s B t Q t B t + (1 Q t N ( B t + η t ω t,ωt 2 She forms an expected utlty based on her knowledge about the dstrbuton of her percepton error. The optmal spendng wll maxmze her expected utlty after ntegratng out the balance percepton error, whch s ˆ ˆ ( u t = (C t ; B t df (η t df B t B t η t u t 4.4 Dssatsfacton and Account Closng We assume that the consumer also has the opton of closng the account (e.g., an outsde opton. If she chooses to close the account, she mght swtch to other competng banks or become unbanked. Wth support from secton 3.1, we make an assumpton that consumers are senstve to the rato of the overdraft fee to the overdraft transacton amount and we useξ t to denote ths rato as a state varable. We assume that the hgher the rato, the more lkely t s that the consumer wll be dssatsfed to close the account because the forward-lookng consumer antcpates that she s gong to accumulate more dssatsfacton (as well as lost consumpton utlty due to overdrafts n the future so that t s not benefcal for her to keep the account open any more. Furthermore, we assume that consumers keep updatng her belef of the rato and only remembers the hghest rato that has ever ncurred. That s f we use t to denote the per-perod rato then and t = OD t B t C t E [Ξ t+1 Ξ t ] = max(ξ t, t. Ths assumpton reflects a consumer s learnng behavor over tme n the sense that after experencng many overdrafts, a consumer realzes how costly (or dssatsfed t could be for her to keep the account open. When she learns that the rato can be hgh enough so that t s not benefcal for her to keep the account open any more, she ll choose to close the account. Once she chooses to close the account, she receves an outsde opton wth a value normalzed to 0 for dentfcaton purposes 16. More specfcally, let W denote the choce to close the account, where W = 1 s closng the account and W = 0 s keepng the account open. Then the per-perod utlty functon for the consumer becomes { u t ϒ t I[B t C t < 0] + ϖ t0 f W t = 0 U t = ϖ t1 f W t = 1 16 Although the outsde opton s normalzed to zero for all consumers, the mplct assumpton s that we allow for heterogeneous utlty of the outsde opton. The heterogenety s reflected by the other structural parameters, ncludng the dssatsfacton senstvty. 17

whereu t s defned n equaton 3. We use ϒ to model the dssatsfacton senstvty,.e., the mpact of chargng an overdraft fee on a consumer s decson to close the account. ϖ 0 and ϖ 1 are the dosyncratc shocks that determne a consumer s account closng decson. Sources of the shocks may nclude (1 the consumer moved address; (2 competng bank entered the market, and so on. We assume these shocks follow a type I extreme value dstrbuton. 4.5 State Varables and the Transton Process We have explaned the followng state varables n the model: (begnnng balance B t, ncome Y t, comng bll ψ t, lapsed tme snce last balance check Γ t, overdraft fee OD t, rato of overdraft fee to the overdraft transacton amount Ξ t, preference shock ε t, balance checkng cost shock χ t and account closure utlty shock ϖ t. The other state varable to be ntroduced later, DL t, s nvolved n the transton process. For (avalable balance B t, the transton process satsfes the consumer s budget constrant, whch s B t+1 = B t C t OD t I (B t C t < 0 +Y t+1 ψ t+1 where OD t s the overdraft fee. As we model the consumer s spendng decson at the daly level rather than transacton level, we aggregate all overdraft fees pad and assume the consumer knows the per-tem fee structure stated n secton 3. Ths assumpton s realstc n our settng because we have already dstngushed between nattentve and attentve consumers. The argument that a consumer mght not be fully aware of the per-tem fee s ndrectly captured by the balance percepton error n the sense that the uncertan overdraft fee s equvalent to the uncertan balance because they both tghten the consumer s budget constrant. As for the attentve consumer who overdraws because of heavy dscountng, she should be fully aware of the potental cost of overdraft. So n both cases we argue that the assumpton of a known total overdraft fee s reasonable. The state varable OD t s assumed to be d over tme and to follow a dscrete dstrbuton wth support vector and probablty vector {X, p}. The support vector contans multples of the per-tem overdraft fee. Consstent wth our data, we assume an ncome dstrbuton as follows Y t = Y I (DL t = PC where Y s the stable perodc (monthly/weekly/bweekly ncome, DL t s the number of days left untl the next payday and PC s the length of the pay cycle. The transton process of DL s determnstc DL t+1 = DL t 1 + PC I (DL t = 1 where t decreases by one for each perod ahead and goes back to the full length when one pay cycle ends. The comng blls are assumed to be d draws from a compound Posson dstrbuton wth arrval rate φ and jump sze dstrbuton G, Ψ t CP(φ,G. Ths dstrbuton can capture the pattern of blls arrvng randomly accordng to a Posson process and bll szes are sums of fxed components (each separate bll. The tme snce last checkng the balance also evolves determnstcally based on the balance checkng behavor. Formally, we have 18

Γ t+1 = 1 + Γ t (1 Q t whch means that f the consumer checks her balance n the current perod, then the lapsed tme goes back to 1 but f she doesn t check, the lapsed tme accumulates by one more perod. The rato of the overdraft fee to the overdraft transacton amount evolves by keepng the maxmum amount over tme. E [Ξ t+1 Ξ t ] = max(ξ t, t The shocks ε t, χ t and ϖ t are all assumed to be d over tme. { } In summary, the whole state space for consumer s S t = B t,ψ t,y t,dl t,od t,γ t,ξ t,ε t, χ t,ϖ t. In our dataset, { we observe } Ŝ t = {B t,ψ t,y t,dl t,od t,γ t,ξ t } and our unobservable state varables are S t = B t,η t,ε t, χ t,ϖ t. S t = Ŝ t S t {B t,ψ t }. Notce here that consumers also have unobserved states B t and ψ t due to nattenton, whch means that the consumer doesn t know the true balance (B t or the bll amount (ψ t f she doesn t check her balance but only the perceved balance ( B t and expected bll (Ψ t. 4.6 The Dynamc Optmzaton Problem and Intertemporal Tradeoff The consumer chooses an nfnte sequence of decson rules {C t,q t,w t } t=1 n order to maxmze the expected total dscounted utlty: { where U t (C t,q t,w t ;S t =. max {C t,q t,w t } t=0 E {St } t=1 [ˆ Let V (S t denote the value functon: B t V (S t = ˆ η t U 0 (C 0,Q 0,W 0 ;S 0 + t=1 β t U t (C t,q t,w t ;S t S 0 } { } C 1 θ t ( t Q t ξ + χ tqt df (η t df B t ϒ OD t I[B t C t < 0] + ϖ t0 ](1 W t +W t ϖ t1 1 θ t B t C t max {C τ,q τ,w τ } τ=t E {{Sτ } τ=t+1} { U t (S t + τ=t+1 β τ t U τ (S τ S t } accordng to Bellman (1957, ths nfnte perod dynamc optmzaton problem can be solved through the Bellman Equaton V (S t = max C,Q,W E S t+1 {U (C,Q,W;S t + βv (S t+1 S t } (5 In the nfnte horzon dynamc programmng problem, the polcy functon doesn t depend on tme. So we can elmnate the tme subscrpt. Then we have the followng choce specfc value functon: (4 19

v ( C,Q,W; B,Ψ,Y,DL,OD,Γ,Ξ,ε, χ,ϖ u C (C ξ + χ 1 ϒ OD I[B C<0] B C + ϖ ( 0 ] +βe S+1 [V B +1,Ψ +1,Y +1,DL +1,OD +1,1,Ξ +1 ε +1, χ +1,ϖ +1 f Q = 1&W = 0 ( = B t η t [u C (C + χ 0 ]df (η t df B t ϒ OD I[B C<0] B C + ϖ 0 ( ] +βe S+1 [V B +1,Ψ +1,Y +1,DL +1,OD +1,Γ + 1,Ξ +1,ε +1, χ +1,ϖ +1 f Q = 0&W = 0 ϖ 1 f W = 1 (6 where subscrpt+1 denotes the next tme perod. So the optmal polcy s gven by the followng soluton {C,Q,W } = argmaxv ( C,Q,W; B,Ψ,Y,DL,OD,Γ,Ξ,ε, χ,ϖ One thng that s worth notcng s that there s a dstncton between ths dynamc programmng problem and tradtonal ones. Because of the percepton error, the consumer observes B t = B t + η t ω t but doesn t know B t or η t. She only knows the dstrbuton N(B t + η t ω t,ω 2 t. The consumer makes a decson C ( B t based on the perceved balance B t. But as researchers, we don t know the realzed percepton error η t. We observe the true balance B t and the consumer s spendng C ( B t. So we can only assume C ( B t maxmzes the expected ex-ante value functon. Later we look for parameters such that the lkelhood for C ( B t maxmzes the expected ex-ante value functon attans maxmum. Followng Rust (1987, we obtan the ex-ante value functon whch ntegrates out the cost shocks, preference shocks, account closng shocks and unobserved mean balance error. ˆ EV (B,ψ,Y,DL,OD,Γ,Ξ = ϖ ˆ ˆ χ ε ˆ η v ( C,Q,W ; B,Ψ,Y,DL,OD,Γ,Ξ,ε, χ,ϖ dηdεdχdϖ Consumers ntertemporal trade-offs are assocated wth the three dynamc decsons. Frst of all, gven the budget constrant, a consumer wll evaluate the utlty of spendng (or consumng today versus tomorrow. The hgher amount she spends today, the lower amount she can spend tomorrow. So spendng s essentally a dynamc decson and the optmal choce for the consumer s to smooth out consumpton over the tme. Second, when decdng when to check balance, the consumer wll compare the montorng cost wth the expected gan from avodng the overdraft fee. She ll only check when the expected overdraft fee s hgher than her montorng cost. As the consumer s balance percepton error mght accumulate wth tme, the consumer s overdraft probablty also ncreases wth the lapse tme snce the last balance check. As a result, the consumer wll wat untl the overdraft probablty reaches the certan threshold (when the expected overdraft fee equals the montorng cost to check the balance. Fnally, the decson to close the account s an optmal stoppng problem. The consumer wll compare the total dscounted utlty of keepng the account wth the utlty from the outsde opton to decde when to close the account. When expectng too much overdraft fees as well as the accompaned dssatsfacton, the consumer wll fnd t more attractve to take the outsde opton and close the account. 20

4.7 Heterogenety In our data, consumers exhbt dfferent responses to ther state condtons. For example, some consumers have never checked ther balances and frequently overdraw whle other consumers frequently check ther balances and rarely overdraw. We hypothesze that t s due to ther heterogeneous dscount factors and montorng costs. Therefore, our model needs to account for unobserved heterogenety. We follow a herarchcal Bayesan framework (Ross, McCulloch and Allenby 2005 and ncorporate heterogenety by assumng that all parameters: β (dscount factor, ς (standard devaton of rsk averse coeffcent,ξ (montorng cost, ρ (senstvty of error varance to lapsed tme snce last checkng balance and ϒ (dssatsfacton senstvty have a random coeffcent specfcaton. For each of these parameters, ϑ {β,ς,λ,ξ,ρ }, the pror dstrbuton s defned as ϑ N ( µ ϑ,σ 2 ϑ. The hyper-pror dstrbuton s assumed to be dffuse. 4.8 Numercal Example Here we use a numercal example to show that nattenton can explan the observed overdraft occasons n the data. More mportantly, we dsplay an nterestng case n whch an unbased percepton can make the consumer spend less than the desred level. In ths example, there are two perods, t {1, 2}. The consumer chooses the optmal consumpton to maxmze the expected total dscounted utlty. In order to obtan an analytcal soluton for the optmal spendng, we assume a CARA utlty u C (C t = θ 1 exp( θc t and the comng bll followng a normal dstrbuton Ψ 2 N ( ψ 2,ζ2 2. The ntal balance s B1 and the consumer receves ncome Y 1 and Y 2. As perod 2 s the termnaton perod, the consumer wll spend whatever s left from perod 1,.e., C 2 = B 1 +Y 1 ψ 1 C 1 OD (B 1 +Y ψ 1 C 1 ψ 2 +Y 2 ψ 2. So the only decson s how much to spend for perod 1: C 1. Let θ = 0.07, B 1 = 3.8, Y 1 = 3, Y 2 = 3, ψ 2 = 1,ζ 2 = 3.9,β = 0.99, OD = 3.58 (The values seem small compared to spendng n realty because we apply log to all monetary values. 4.8.1 Effect of Overdraft Fgure 7: Optmal Spendng wth Neutral vs Negatve Shock In ths example n Fgure 7, when there s no bll to pay n the frst perod (ψ 1 = 0 n the left panel, the total budget for the consumer s 6.8 and she would lke to spend 4.2 to attan the maxmum utlty. However, when she has to pay for a bll of 6 (rght panel, she s left wth only 0.8. Her optmal choce s to spend 0.8 and just clear the budget because the dsutlty of overdraft (utlty functon wth overdraft s the black lne labeled as OD s too hgh. Ths example shows that snce 21

the overdraft fee s equvalent to an extremely hgh nterest rate short-term loan, the consumer wouldn t want to overdraw her account. 4.8.2 Effect of Inattenton Overdraft Fgure 8: Inattenton Leads to Overdraft Balance Error B 1 > B 1 In a dfferent scenaro (Fgure 8, f the consumer overestmates her balance to be 7 (her true balance s 3.8,.e., she has a postve percepton error regardng her true balance, then she would spend 2.8 whch s the optmal amount based on ths mspercepton. Ths percepton error leads her to an overdraft. 4.8.3 Effect of Inattenton Error Constrants Spendng Fgure 9: Inattenton Leads to Underspendng Fnally, we dscover an nterestng case where nattenton may cause the consumer to spend less than her optmal spendng level. Ths happens because the consumer knows that she s nattentve,.e., she mght overestmate her effectve balance to run nto overdraft. In order to prevent ths, the consumer tends to constran her spendng. As shown n Fgure 9, though the optmal spendng s 0.8 as n the prevous example (secton 4.8.1, the nattentve consumer chooses to spend 0.5 to prevent overdraft. Ths example suggests a new revenue source for the bank. If the bank provdes automatc alerts to consumers to nform them of ther exact balances, the consumers won t have to 22

take precautons to avod overdrafts. As a consequence, consumers wll spend more and the bank can beneft from the ncreased nterchange fees. 5 Identfcaton and Estmaton We now dscuss the dentfcaton of the parameters and the estmaton procedure. 5.1 Identfcaton The unknown structural parameters n the model nclude {θ,β,ς,ξ,ρ,ϒ} where θ s the logarthm of the mean rsk averse coeffcent, β s the dscount factor, ς s the standard devaton of the rsk averse coeffcent, ξ s the montorng cost, ρ s the senstvty of balance error varance to the lapsed tme snce last balance checkng, and ϒ s the dssatsfacton senstvty. Next we provde an nformal ratonale for dentfcaton of each parameter. Frst of all, as we know from Rust (1987, the dscount factor β cannot be separately dentfed from the statc utlty parameter, whch n our case, the rsk averson coeffcent θ. The reason s that lowerng θ tends to ncrease consumpton/spendng, an effect whch can also be acheved by lowerng β. As we are more nterested n the consumers tme preference rather than rsk preference, we fx the rsk averse coeffcent θ, whch allows me to dentfy the dscount factor 17. Ths practce s also used n Gopalakrshnan, Iyengar, Meyer 2014. As to the rsk averse coeffcent, we choose θ = 0.74, followng the latest lterature by Andersen et al. (2008 where they jontly elct rsk and tme preferences 18. After fxng θ, β can be well dentfed by the sequences of consumpton (spendng wthn a pay perod. A large dscount factor (close to 1 mples a stable consumpton stream whle a small dscount factor mples a downward slopng consumpton stream. Because a dscount factor s constraned above by 1, we do a transformaton to set 1 β = 1+exp(λ and estmate λ nstead. Second, the standard devaton of rsk averse coeffcent ς s dentfed by the varaton of consumptons on the same day of the pay perod but across dfferent pay perods. Moreover, accordng to the ntertemporal tradeoff, the longer the consumer goes wthout checkng her balance, the more lkely she wll be to overdraw due to the balance error. The observed data pattern of more overdraft fees pad longer after a balance checkng nqury can help pn down the structural parameters ρ. Intutvely, the montorng cost ξ s dentfed by the expected overdraft payment amount. Recall that the tradeoff regardng balance checkng s that a consumer only checks balance when ξ s smaller than the expected overdraft payment amount. In the data we observe the balance checkng frequency. Combnng ths wth the calculated ρ we can compute the expected overdraft 17 We also tred to fx the dscount factor (at 0.9998 and estmate the rsk averse coeffcents. Other structural parameter estmates are not sgnfcantly unaffected under ths specfcaton. Our results confrm that the rsk averse coeffcent and the dscount factor are mathematcally substtutes (Andersen et al. 2008. Estmaton results wth fxed dscount factor are avalable upon requests. 18 We also tred other values for the relatve rsk averse coeffcent θ, the estmated dscount factor β values change wth dfferent θ s, but other structural parameter values reman the same. The polcy smulaton results are also robust wth dfferent values of θ s. 23

probablty and further the expected overdraft payment amount, whch s the dentfed ξ. Gven ρ, a consumer wth few balance checkng nqures must have a hgher balance checkng cost ξ. Lastly, the dssatsfacton senstvty parameter ϒ can be dentfed by the data pattern that consumers account closure probablty vares wth the rato of overdraft fee over the overdraft transacton amount, as shown n secton 3.1. Note that asde from these structural parameters, there s another set of parameters that govern the transton process. These parameters can be dentfed pror to structural estmaton from the observed state varables n our data. The set ncludes {φ,g,x, p}. In sum, the structural parameters to be estmated nclude {λ,ς,ξ,ρ,ϒ }. 5.2 Lkelhood The full lkelhood functon s {{Ct } } L(,Q t,w t ; Sˆ T I t t=1 =1 ( { = L { } } C t,q t,w t; Sˆ T I { t L( f { } } Sˆ T I } I t=1 t Ŝ t 1 L( =1 t=1 {Ŝ0 =1 =1 where Ŝ t = {B t,ψ t,y t,dl t,od t,γ t,ξ t }. As the lkelhood for the optmal choces and that for the state transton process are addtvely separable when we apply log to the lkelhood functon, we can frst estmate the state transton process from the data, then maxmze the lkelhood for the optmal choces. The lkelhood functon for the optmal choce s } I L( {{Ct,Q t,w t ;Ŝ t } T t=1 = = L T =1 t=1 L T =1 t=1 =1 L ( C t ;Ŝ t L ( Qt ;Ŝ t L ( Wt ;Ŝ t f (ε t C t Pr (χ t Q t,c t Pr (ϖ t W t,q t,c t where f (ε t C t s estmated from the normal kernel densty estmator to be explaned n secton 5.3.1, Pr (χ t C t,q t and Pr (ϖ t C t,q t,w t follow the standard logt model gven the choce specfc value functon n equaton 6. In specfc, Pr ( ˆ ˆ ˆ { ( } exp v Ct,Q t = 1,W t ;Ŝ t Q t = 1;Ŝ t = η t Qt exp { v ( } C t,q t,w t ;Ŝ t Pr ( W t = 1;Ŝ t = ˆ 5.3 Estmaton: Ima, Jan and Chng (2009 5.3.1 Modfed IJC ϖ t χ t ˆ ε t ε t ˆ { ( } exp v Ct,Q t,w t = 1;Ŝ t η t Wt exp { v ( } C t,q t,w t ;Ŝ t We use the Bayesan estmaton method developed by Ima, Jan and Chng (2009 to estmate the dynamc choce problem wth heterogeneous parameters. As our model nvolves a contnuous 24

choce varable, spendng, we adjust the IJC algorthm 19 to obtan the choce probablty through kernel densty estmaton. We now show the detals of the estmaton procedure. The whole parameter space s dvded nto two sets (Ω = {Ω 1,Ω 2 }, where the frst one contans hyper-parameters n the dstrbuton of the heterogeneous parameters (Ω 1 = { } µ λ, µ ς, µ ξ, µ ρ, µ ϒ,σ λ,σ ς,σ ξ,σ ρ,σ ϒ, and the second set contans heterogeneous parameters (Ω 2 = {λ,ς,ξ,ρ,ϒ } I =1. We allow all heterogeneous parameters (represented by ϑ to follow a normal dstrbuton wth parameters mean µ ϑ and standard devaton σ ϑ. Let the observed choces be O d = { O d } I =1 = { C d,qd,w d } where C d { Ct d, t}, Q d { Q d t, t} and W d { Wt d, t}. Each MCMC teraton manly conssts of two blocks. ( Draw Ω r 1, that s, draw µr ϑ f µ ϑ (ϑ σϑ r 1,Ωr 1 2 and σϑ r f σ ϑ (σ ϑ µ ϑ r,ωr 1 2 (ϑ {λ,ς,ξ,ρ,ϒ}, the parameters that capture the dstrbuton of ϑ for the populaton where f µϑ and f σϑ are the condtonal posteror dstrbutons. ( Draw Ω r 2, that s, draw ndvdual parameters ϑ ( f ϑ O d 1,Ωr by the Metropols- Hastngs (M-H algorthm. More detals of the estmaton algorthm s presented n Appendx A.2. 5.3.2 Parallel Computng: Neswanger, Wang and Xng (2014 We adopt the parallel computng algorthm by Neswanger, Wang and Xng (2014 to estmate our model wth data from more than 500,000 consumers. The logc behnd ths algorthm s that the full lkelhood functon s a multplcatve of the ndvdual lkelhood. p ( ϑ x N p(ϑ p ( x N ϑ = p(ϑ N =1 p(x ϑ So we can partton the data onto multple machnes, and then perform MCMC samplng on each usng only the subset of data on that machne (n parallel, wthout any communcaton. Fnally, we can combne the subposteror samples to algorthmcally construct samples from the full-data posteror. In detals, the procedure s: (1 Partton data x N nto M subsets {x n 1,...,x n M}. (2 For m = 1,...,M (n parallel: (a Sample from the subposteror p m, where p m (ϑ x n m p(ϑ M 1 p(x n m ϑ (3 Combne the subposteror samples to produce samples from an estmate of the subposteror densty product p 1...p M, whch s proportonal to the full-data posteror,.e. p 1...p M (ϑ p ( ϑ x N. Gven T samples {ϑ t } t=1 T from a subposteror p m, we can wrte the kernel densty estmator as pˆ m (ϑ, 19 The IJC method s desgned for dynamc dscrete choce problems. Zhou (2012 also appled t to a contnuous choce problem. 25

pˆ m (ϑ = 1 T = 1 T = 1 T T t=1 T t=1 T t=1 1 h d K( ϑ ϑ t h ( 2πh 2 d 2 I d 1 2 exp { 1 2h 2 (ϑ ϑ t I 1 d (ϑ ϑ t N ( ϑ ϑ t,h 2 I d where we have used a Gaussan kernel wth bandwdth parameter h. After we have obtaned the kernel densty estmator pˆ m (ϑ for M subposterors, we defne our nonparametrc densty product estmator for the full posteror as } p 1 p m (ϑ = pˆ 1 pˆ m (ϑ = 1 T M T t 1 =1 T t 1 =1 T t 1 =1 T t M =1 T t M =1 T t M =1 M m=1 N (ϑ ϑ t, h2 w t N N ( ϑ ϑ m t m,h 2 I d M d M I N ( ϑt m m ϑ t,h 2 I d m=1 (ϑ ϑ t, h2 M I d Ths estmate s the probablty densty functon (pdf of a mxture of TM Gaussans wth unnormalzed mxture weghts w t Here, we use t = {t 1,...,t M } to denote the set of ndces for the M samples { ϑt 1 1,...,ϑt M } M (each from one machne assocated wth a gven mxture component, and let w t = N (ϑ ϑ t, h2 M I d ϑ t = 1 M M ϑt m m m=1 (4 Gven the herarchcal Bayes framework, after obtanng the posteror dstrbuton of the populaton parameter ϑ, use M-H algorthm once more to obtan the ndvdual parameters (detals n Appendx A.2 Step 4 The samplng algorthm s presented n Appendx A.3. 26

6 Results 6.1 Model Comparson Table 9: Model Comparson A: No Forward Lookng B: No Inattenton C: No Heterogenety D: Proposed Log-Margnal Densty -2943.28-3636.59-2764.56-1758.33 Ht Rate: Overdraft 0.499 0.351 0.504 0.870 Ht Rate: Check Balance 0.405 0.226 0.632 0.841 Ht Rate: Close Account 0.660 0.727 0.696 0.758 We compare our model aganst the other four benchmark models n order to nvestgate the contrbuton of each element of the structural model. Models A to C are our proposed model wthout forward-lookng, nattenton and unobserved heterogenety respectvely and model D s our proposed model. Table 9 shows the log-margnal densty (Kass and Raftery 1995 and the ht rate for overdraft, check balance and close account ncdences (we only consder when these events happen because no event takes place the majorty of the tme. All four measures show that our proposed model sgnfcantly outperforms the benchmark models. Notably nattenton contrbutes the most to model ft whch s consstent wth our conjecture n secton 3. 6.2 Value of Parallel IJC Table 10: Estmaton Tme Comparson Sze\Method (seconds Parallel IJC IJC CCP FIML 1,000 518 1579 526 5,010 10,000 3,199 12,560 4,679 54,280 100,000 4,059 14,0813 55,226 640,360 >500,000 5,308 788,294 399,337 3,372,660 (1.5 hr (9 days (5 days (39 days 27

Table 11: Monte Carlo Results when N=100,000 Var True Value Parallel IJC IJC CCP FIML µ β 0.9 Mean 0.878 0.883 0.851 0.892 Std 0.041 0.039 0.036 0.025 µ ς 1.5 Mean 1.505 1.502 1.508 1.501 Std 0.131 0.124 0.199 0.103 µ ξ 0.5 Mean 0.482 0.507 0.515 0.502 Std 0.056 0.039 0.071 0.044 µ ρ 1 Mean 1.006 1.003 1.015 1.002 Std 0.027 0.022 0.026 0.019 µ ϒ 5 Mean 5.032 5.011 4.943 4.987 Std 0.023 0.010 0.124 0.008 σ β 0.1 Mean 0.113 0.095 0.084 0.104 Std 0.016 0.014 0.015 0.010 σ ς 0.3 Mean 0.332 0.318 0.277 0.309 Std 0.024 0.015 0.029 0.021 σ ξ 0.1 Mean 0.112 0.091 0.080 0.090 Std 0.055 0.029 0.025 0.025 σ ρ 0.1 Mean 0.107 0.107 0.085 0.105 Std 0.008 0.006 0.010 0.006 σ ϒ 0.1 Mean 0.092 0.109 0.111 0.100 Std 0.014 0.013 0.021 0.009 We report the computatonal performance of dfferent estmaton methods n Table 10. All the experments are done on a server wth an Intel Xeon CPU, 144 cores and 64 GB RAM. The frst column s the performance of our proposed method, IJC wth parallel computng. We compare t wth the orgnal IJC method, the Condtonal Choce Probablty (CCP method by Arcdacono and Mller (2011 20 and the Full Informaton Maxmum Lkelhood (FIML method by Rust (1987 (or Nested Fxed Pont Algorthm 21. As the sample sze ncreases, the comparatve advantage of our proposed method s more notable. To run the model on the full dataset wth more than 500,000 accounts takes roughly 1.5 hours compared to 9 days wth the orgnal IJC method. The reason for the decrease n computng tme s that our method takes advantage of multple machnes that run n parallel. We further run a smulaton study to see f the varous methods are able to accurately estmate all parameters. Table 11 shows that dfferent methods produce qute smlar estmates and all mean parameter estmates are wthn two standard errors of the true values. The Parallel IJC method s slghtly less accurate than the orgnal IJC method. The parallel IJC s almost 600 tmes faster than FIML. Ths happens because the full soluton method solves the dynamc programmng problem at each canddate value for the parameter estmates, whereas ths IJC estmator only evaluates the value functon once n each teraton. 20 We use the fnte mxture model to capture unobserved heterogenety and apply the EM algorthm to solve for the unobserved heterogenety. More detals of the estmaton results can be obtaned upon requests. 21 We use the random coeffcent model to capture unobserved heterogenety. More detals of the estmaton results can be obtaned upon requests. 28

6.3 Parameter Estmates Table 12: Structural Model Estmaton Results Var Interpretaton Mean (µ ϑ Standard devaton (σ ϑ β Dscount factor 0.9997 0.362 (0.00005 (0.058 ς Standard devaton of relatve rsk averson 0.257 0.028 (0.014 (0.003 ξ Montorng cost 0.708 0.255 (0.084 (0.041 ρ Inattenton Dynamcs lapsed tme 7.865 0.648 (0.334 (0.097 ϒ Dssatsfacton Senstvty 5.479 1.276 (1.329 (0.109 Table 12 presents the results of the structural model. We fnd that the hgher the age, the more rsk averse the consumer s. The montorng cost s estmated to be 0.708. Usng the rsk averse coeffcent, we can evaluate the montorng cost n monetary terms. It turns out to be $2.03. We also obtaned the cost measure for each ndvdual consumer. The varance of the balance percepton error ncreases wth the lapsed tme snce the last tme to check balance and wth the mean balance level. Notably the varance of the balance percepton error s qute large. If we take the average number of days to check the balance from the data, whch s 9, then the standard devaton s 7.865 9 = 70.79. Ths suggests a very wdely spread dstrbuton of the balance percepton error. The estmated dssatsfacton senstvty parameter confrms our hypothess that consumers can be strongly affected by the bank fee and close the account as a consequence of dssatsfacton. If we consder an average overdraft transacton amount at $33, then the relatve magntude of the effect of dssatsfacton s comparable to $171. Ths suggests that unless the bank would lke to offer a $171 compensaton to the consumer, the dssatsfed consumer wll close the current account and swtch. Moreover, consstent wth the evdence n Fgure 5, the dssatsfacton senstvty s stronger for lght overdrafters (whose average s 5.911 than for heavy overdrafters (whose average s 3.387. And keepng the average overdraft transacton amount as fxed, a 1% ncrease n the overdraft fee can ncrease the closng probablty by 0.12%. 7 Counterfactuals 7.1 Prcng The structurally estmated model allows us to examne the effect of changng the prcng structure on consumers spendng pattern and more mportantly, ther overdraft behavor. We test three alternatve prcng schemes: a reduced per-tem flat fee, a percentage fee, and a quantty premum. 29

Table 13: Overdraft Fee under Alternatve Prcng Prcng Current Reduced Flat Percentage Quantty Premum $31 $29.27 15.8% 8.5% *I (OD 10 + $31 *I (OD > 10 Overdraft Revenue $18,654,510 $19,262,647 $19,982,711 $20,297,972 Overdraft Freq 544,997 590,093 610,288 631,325 % Revenue +3.26% +7.12% +8.81% % Freq +2.77% +11.98% +15.84% % Check Balance -3.58% +2.83% +3.31% % Close Account -1.01% -1.35% -1.94% Notce here that the underlyng assumpton for all these smulatons s fungblty,.e., consumers reacton only depends on the fee amount rather than the fee structure. If two dfferent fee structures result n the same fee amount, then the consumer should respond n the same fashon. In the frst scenaro, we keep the per-tem flat fee scheme but reduce t to $29.27 per tem. Because of law of demand, there s a negatve relatonshp between the per-tem overdraft fee and overdraft frequency. So we further pursue an optmzaton task where we try to solve the optmal per-tem fee. As we aggregate data to the daly level, we calculate the average transacton amount for each tem, whch s $44, and use t to derve the total overdraft fee. For example, f a consumer overspent $170, then the consumer had to pay four overdraft tem fees. The optmzaton s a nested algorthm where n the outer loop we search for the per-tem overdraft fee, and n the nner loop we solve the consumer s best response, ncludng optmal spendng, balance checkng and account closng gven the fee sze. We found that the optmal per-tem overdraft fee s $29.27 under whch the bank s revenue wll ncrease by 3.26%. Ths suggests that the current overdraft fee s too hgh because the bank fals to take nto account consumer s negatve reacton to the overdraft fee, whch results n huge loss n the consumers lfetme value (I calculate the lfetme value of a consumer n a conservatve way by multplyng the accounts spendngs by the nterchange rate. In the second scenaro, the per-tem flat fee s changed to a percentage fee of 15.8% (optmzed n a smlar way as descrbed n the frst scenaro. Ths s lower than the 17% calculated from the rato of the total fee pad over the total transacton amount that caused the fees n the data. Agan ths suggests that the bank mght be chargng a too hgh fee currently. Intutvely, the percentage structure should encourage consumers to overdraw on transactons of a small amount but deter them from overdrawng on transactons of a large amount. As there are more transactons of a small amount than transactons of a large amount, the total fees generated soars by 7.12%. Therefore, the percentage overdraft fee nvtes more consumers to use the overdraft servce. It s ths market expanson effect that ncreases the bank s overdraft revenue. In the last scenaro, a quantty premum structure s employed, where when a consumer overdraws no more than 10 tmes, she pays a 8.5% percentage fee and f she overdraws more than 10 tmes, she pays a flat fee at $31. Ths quantty premum can ncrease the bank s revenue by 8.81%, because the quantty premum uses the second degree prce dscrmnaton to segment two types of overdrafters. The bank wll earn more overdraft fee from the heavy overdrafters who are wllng to pay for the flat fee whle retanng the lfetme value for the lght overdrafters who prefer the percentage fee (due to the hgh dssatsfacton senstvty. 30

7.2 Alerts Beneft Consumers And the Bank Although the changed prcng strateges can help the bank mprove revenue, the bank s stll explotng consumer nattenton and may exacerbate consumer attrton. In ths counterfactual, we propose a new product desgn strategy (specfc desgn to be ntroduced n secton 7.3 to help consumers prevent overdrafts: sendng automatc alerts to nform consumers about ther balances. As alerts elmnate consumers balance percepton error, the total amount of overdraft fee pad by consumers decreases by 49.53% (Table 14. Ths s n comparson to the overdraft revenue under the optmal Quantty Premum prcng strategy n Table 13. Table 14: Effect of Alerts on Bank s Revenue Amount Percentage Change Overdraft revenue $10,243,529-49.53% Interchange revenue from ncreased spendngs $1,997,488 +9.84% Lfetme value from retaned consumers $8,430,424 +41.53% Total $20,671,441 +1.84% Although alerts beneft consumers by helpng them avod the hgh overdraft fees, the bank mght not have ncentves to send out alerts as ts objectve s to earn more revenue. However, we fnd that alerts can beneft the bank too for two reasons. Frst of all, as shown n secton 4.8.3, due to nattenton consumers are constranng spendngs to prevent overdrafts. Wth alerts, consumers precautonary motve s releved so that they wll ncrease spendngs. As a result, the bank can gan more nterchange fees. We calculate ths gan of more nterchange fee from the ncreased amount of spendng by multplyng the ncreased spendng wth an average nterchange fee rate of 0.8% 22. We fnd that sendng alerts to consumers can offset 9.84% of the loss n overdraft fees because of the gan n the nterchange fees. Moreover, wthout beng dssatsfed by the overdraft fee, consumers are less lkely to close ther accounts. We fnd that alerts reduce the number of closed accounts from 16.37% to 8.25% whch ncreases the bank s revenue by gettng the lfetme value from these retaned consumers. As shown n Table 14, the ncreased lfetme value from retaned consumers and the ncrease n nterchange fee from ncreased spendngs not only offset the loss n overdraft revenue but ncrease t by 1.84%. 7.3 Optmal Alert Strategy Fnally, we explan how we desgn the optmal alert that can help the bank ncrease ts revenue n secton 7.2. We show the effect of the proposed alert wth an example n Fgure 10. Consder a consumer who receves a weekly wage of $2000. Ths consumer s dscount factor s 0.8. 23 She sets a threshold alert at $300 orgnally thus wll only receve the alert when the account balance s below $300. But our proposed alert wll be trggered both when the consumer s overspendng and underspendng. As shown n the fgure, as long as the consumer s spendng falls out of the range between the overspendng and underspendng lnes, an alert wll be receved. So when the consumer s balance s below $700 on day 2, she wll receve an alert although the threshold s 22 http://www.federalreserve.gov/paymentsystems/reg-average-nterchange-fee.htm 23 For the ease of exposton, we choose a relatvely small dscount factor. 31

not reached yet. The optmal alert s earler than the threshold alert to gve the consumer more tme to adjust her spendng rather than to wat untl the last moment when she can hardly make any mmedate change. On the other hand, f the consumer s balance s below $300 on day 5, the threshold alert wll be trggered whle the consumer s stll n a safe zone. Recevng the threshold alert doesn t help consumers because her percepton error accumulates too fast to make day 6 and 7 danger days prone to overdrafts agan. Therefore, the dynamc alert can correct the defects of the threshold alerts of beng ether too late or too early. Fgure 10: Dynamc Optmal Alert Notfes Overspendng and Underspendng Another mbedded feature of the dynamc alert s that t accounts for consumers dsutlty to receve too many alerts. In realty, consumers dslke frequent alerts that spam ther malboxes. We ncorporate ths alert-averse effect nto an optmzaton task where we choose the optmal tmng to send the alerts gven the estmated structural parameters. The objectve functon s as follows max {At } N =1t=1 β t 1 [ U t ( C t,w t ;Ŝ t κ ] Ŝ t = A t S t + (1 A t S t where A t s a bnary choce of whether to send an alert to the consumer at tme t. The second equaton means that f the alert s sent, the consumer knows the exact balance and comng blls, denoted as the true state varable S t ; f not, the consumer only knows the dstrbuton of the perceved balance and comng blls, denoted as S t. The consumers dsutlty of recevng the alert s summarzed by a tme nvarate the parameter κ. We solve the optmzaton problem n a nested algorthm where n the outer loop we test for all combnatons of alert opportuntes, and n the nner loop we solve the consumer s best response, ncludng optmal spendng and account closng gven the alert profle. (We assume that consumers don t have to make the balance checkng decson because of the automatc alerts. We frst test the optmal alert strategy assumng that all consumers have the same structural parameters (we use the posteror mean of the hyper-dstrbuton parameters. We set ths dsutlty as the nverse of the estmated montorng cost (µ ξ because the consumer who ncurs a hgh montorng cost mght not know how to use onlne bankng or call centers so automatc message 32

alerts are favored. As Table 15 reports, ths alert servce ncreases total consumer utlty by 1.11% when the threshold rule of $300 s appled and 2.85% when the dynamc rule s appled. We further allow all structural parameters to be heterogeneous across consumers and solve the optmal alert tmng specfc to each ndvdual. We fnd that targeted alerts can ncrease consumer utltes sx tmes more than the unform threshold alert (6.65%. Table 15: Utlty Impact of Dfferent Types of Alerts Alert Type Alert Tmng Utlty Gan Unform Threshold 1.11% Dynamc 2.85% Targeted Threshold 4.39% Dynamc 6.65% 8 Contrbutons and Lmtatons The $32 bllon dollar annual overdraft fee has caused consumer attrton and may nduce potentally tghter regulaton. However there s lttle quanttatve research on consumers fnancal decson makng processes that explans ther overdraft behavors. The lack of well-calbrated models prevent fnancal nsttutons from desgnng prcng strateges and mprovng fnancal products. Wth the ad of Bg Data assocated wth consumers spendng patterns and fnancal management actvtes, banks can use adverse targetng (Kamenca, Mullanathan, and Thaler 2011 to help consumers know themselves better and make better fnancal decsons. In ths paper we buld a dynamc structural model of consumer daly spendng that ncorporates nattenton to ratonalze consumers overdraft behavor. We quantfy the dscount factor, montorng cost and dssatsfacton senstvty for each consumer and use these to desgn new strateges. Frst we compare the current prcng scheme wth several alternatve prcng strateges. We fnd that a percentage fee structure can ncrease the bank s revenue through market expanson and the quantty premum structure can ncrease the bank s revenue because of second degree prce dscrmnaton. More mportantly, we propose an alert strategy to make the ncentve of the bank and the ncentve of the consumers better algned. The optmal alert can be sent to the rght consumer at the rght tme to prevent overdrafts. Ths customzed dynamc alert product can be sx tmes more effectve than a unform threshold alert. Not only does ths alert benefts consumers, t can also beneft the bank through ncreased nterchange fees and lower consumer attrton. We calbrated our model at an ndvdual level on a sample of more than 500,000 accounts. Ths Bg Data provde great value for our analyss. Frst of all, an overdraft s stll a relatvely rare event compared to numerous other transactons. Wthout a large amount of data, we cannot detect these rare but detrmental events, let alone ther dverse causes. Second, as summarzed by Enav and Levn (2014, Bg Data contan rch mcro-level varaton that can used to dentfy novel behavor and develop predctve models that are harder wth smaller samples, fewer varables, and more aggregaton. We leverage the varaton n consumer daly spendng and balance checkng behavors to evaluate the effect of heterogeneous polcy nstruments. These evaluatons can be useful for bank managers to desgn new products and polcy makers to create new regulaton rules at a much more refned fashon than before. 33

In order to estmate a complcated structural model wth Bg Data, we adopt parallel computng technques n combnaton wth the Bayesan estmaton algorthm developed by Ima, Jan and Chng (2009. Ths new method sgnfcantly reduces the computaton burden and could be used for other researchers and marketers who would lke to use structural models to solve real-world large-scale problems. There are several lmtatons of the current study that call for future work. Frst, we don t observe consumers exstng alert settngs. Some consumers may have already receved alerts to help them make fnancal decsons. In our polcy smulatons, we made bold assumptons about consumers dsutlty for readng alerts. These assumptons could be tested f we had the alerts data. The current alerts are set by consumers who mght fal to consder ther spendng dynamcs. Future feld experments are needed to test the effect of our proposed alert strategy. Second, we don t have the data about consumers decson on whether to opt-n for overdraft protecton by ATM/POS transactons. We only know that f ATM/POS transactons caused an overdraft, then the consumer must have opted-n. If no such transactons happened, we do not know the consumer s opt-n status. Had we known ths nformaton, we could have provded an nformatve pror n our the Bayesan model. The logc s that a consumer who has opted n probably has stronger needs for short term lqudty due to fluctuatons n the sze and arrval tme of ncome and expendtures. Fnally, we only model consumers non-preauthorzed spendng n the checkng account. In realty, consumers usually have multple accounts, lke savngs, credt cards and loans, wth multple fnancal nsttutons. A model to capture consumers decsons across all accounts for both short-term and long-term fnances wll provde a more complete pcture of consumers fnancal management capabltes and resources so that the bank can desgn more customzed products. References Andersen, S., Harrson, G. W., Lau, M. I., & Rutström, E. E. (2008. Elctng rsk and tme preferences. Econometrca, 76(3, 583-618. Arcdacono, P., & Mller, R. A. (2011. Condtonal choce probablty estmaton of dynamc dscrete choce models wth unobserved heterogenety. Econometrca, 79(6, 1823-1867. Arrow, K. J. 1963. Lqudty preference. Lecture VI n Lecture Notes for Economcs 285, The Economcs of Uncertanty, pp. 33-53, undated, Stanford Unversty. Bellman, Rchard 1957. Dynamc Programmng, Prnceton, NJ: Prnceton Unversty Press. Bolton, R. N. (1998. A dynamc model of the duraton of the customer s relatonshp wth a contnuous servce provder: the role of satsfacton. Marketng scence, 17(1, 45-65. Chetty, R., Looney, A., & Kroft, K. (2009. Salence and taxaton: Theory and evdence. Amercan Economc Revew, 99(4, 1145-1177. Card, D., DellaVgna, S., & Malmender, U. (2011. The role of theory n feld experments. The Journal of Economc Perspectves, 25(3, 39-62. Enav, L., & Levn, J. D. (2014. The data revoluton and economc analyss. Innovaton Polcy and the Economy, forthcomng. Evans, D., Ltan, R., & Schmalensee, R. (2011. Economc Analyss of the Effects of the Federal Reserve Board s Proposed Debt Card Interchange Fee Regulatons on Consumers and Small Busnesses. Avalable at SSRN 1769887. 34

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2. Draw µ ϑ r (populaton ( mean of ϑ from the posteror densty (normal condtonal on σϑ r 1 } I =1. µr ϑ N I =1 ϑ r 1 and { ϑ r 1 I,σ r 1 ϑ 3. Draw σϑ r (populaton varance of ϑ from the posteror densty (nverted gamma condtonal on µ ϑ r and { ( ϑ r 1 } I =1.σ ϑ r IG I µ ϑ r 2 2, I =1(ϑ r 1 2 4. For each = 1,...,I, draw ϑ r from ts posteror dstrbuton condtonal on (C d,qd,w d,µ ϑ r,σ ϑ r, whch s ( ( f ϑ C d,q d,w d, µ ϑ r,σr ϑ π (ϑ µ ϑ r,σr ϑ ρ (C d ϑ ρ (Q d ϑ ρ W d ϑ Snce there s no easy way to draw from ths posteror, we use the M-H algorthm. (a Draw ϑ r from the proposal dstrbuton q ( ϑ r 1,ϑ r (e.g., ϑ r N ( ϑ r 1,σ 2 where ϑ r s a canddate value of ϑ r. (b Compute the pseudo-lkelhood for consumer at ϑ r,.e., ρ r ( C d ϑ r (, ρ r Q d ϑ r and ( ρ W d ϑ r. Snce there s no closed form soluton to the optmal strategy profle, a lkelhood functon based on observed C t becomes nfeasble. Instead, we mplement a numercal approxmaton method to establsh a smulated lkelhood functon for estmaton. For each C t observed n the data and ts correspondng state pont Ŝ t, we use the followng steps to smulate ts densty:. Frst assume the unobserved state varables are S t = {ε t,η t, χ t,ϖ t }. Draw nr=1000 random shocks S t = {ε t,η t, χ t,ϖ t } from η t N(0,ω 2, ε t N (0,1,χ t ~EVI 24,ϖ t ~EVI;. For each balance checkng decson Q = {1, 0}and account closng decson W = {1, 0}, each random draw of S t and the observed Ŝ t, calculate the optmal consumpton by solvng the followng equatons Ct (Ŝt, S t Q,W = argmax C t = argmaxu C t ( ṽ r Q,W;Ŝ t, S t,ϑ r ( C t,q,w;ŝ t, S t,ϑ r + βê r St+1 { V } (Ŝt+1, S t+1 ;ϑ Ŝ r t, S t. Usng the calculated nr = 1000 optmal Ct (Ŝt, S t, smulate ρ r ( C d t ϑ r, the densty of the observed Ct d, usng a Gaussan kernel densty estmator. (Ths smulaton borrows an dea from Yao, Mela, Chang and Chen (2012. Moreover, and ( } ( ρ Q d t ϑ r = 1 nr exp {v χt Ct,Q = Q t,wt ; B t,r t,ψ t,y t,dl t,d t,od t,γ t,ξ t,ε t,ϖ t,ϑ r ( } η,ε,ϖ Q {1,0} exp {v χt Ct,Q,W t ; B t,r t,ψ t,y t,dl t,d t,od t,γ t,ξ t,ε t,ϖ t,ϑ r ( ρ Wt d ϑ r = 1 nr η,ε,χ ( } exp {v ϖ Ct,Q t,w = W t; B t,r t,ψ t,y t,dl t,d t,od t,γ t,ξ t,ε t, χ t,ϑ r ( } W {1,0} exp {v ϖ Ct,Q t,w; B t,r t,ψ t,y t,dl t,d t,od t,γ t,ξ t,ε t, χ t,ϑ r ( ρ O d ϑ r ( ( = ρ C d ϑ ρ r = Q d ϑ r T ( ( ρ Ct d ϑ ρ r t=1 Q d t ϑ r ( ρ W d ϑ r ( ρ Wt d ϑ r 24 Type I Extreme Value Dstrbuton 37

. ( { To obtan ṽ r Ŝ t, S t,ϑ, r we need ÊS r {Ṽ ( } r 1 k Ŝ k, S k;ϑ k V (Ŝ, S ;ϑ r Ŝ, S }, whch s obtaned by a weghted average of, treatng ϑ as one of the parameters when computng the k=r N weghts. In the case of ndependent kernels, for all Ŝ = {B,ψ,Y,DL,OD,Γ,Ξ }, because B,Ξ are contnuous and evolves determnstcally, ψ and OD are contnuous and evolve stochastcally, and Y,DL,Γ are dscrete so = ( } ÊS {V r B,ψ,Y,DL,OD,Γ,Ξ, S ;ϑ B r,ψ,y,dl,od,γ,ξ, S r 1 ( Ṽ k B k,ψ k,y,dl,od k,γ,ξ, S k ;ϑ k k=r N ( K hϑ ϑ r ϑ k r 1 l=r N K ( hϑ ϑ r ( Khs B B k ( ( ( f ψ k φ,g f OD k X, p Khs Ξ Ξ k ϑ l ( Khs B B l ( ( ( f ψ l φ,g f OD l X, p Khs Ξ Ξ l We repeat the same step and obtan the pseudo-lkelhood (ρ r ( O d ϑ r 1 ( condtonal on ϑ r 1. Then, we determne whether or not to accept ϑ r. The acceptance probablty, Λ, s gven by ( ( π ϑ r µ r ( ϑ Λ = mn,σr ϑ ρ r O d ϑ r ( q ϑ r,ϑ r 1 π ( ϑ r 1 µ ϑ r (,σr ϑ ρ r O d ϑ r 1 ( q ϑ r 1,ϑ r,1 where π ( denotes the pror dstrbuton. (c Repeat (a & (b for all. 5. Computaton of the pseudo-value functon, {Ṽ ( r Ŝ r, S r;ϑ r =1 (a Make one draw of the unobserved state varables S r from η N(0,ω 2, ε N ( 0,ς 2,χ ~EVI 25, and ϖ ~EVI; (b Compute the pseudo expected future value at ϑ r. Ê r S { } V (Ŝ, S ;ϑ r Ŝ r, S r = r 1 k=r N ( Ṽ k B k,ψ k,y,dl,od k,γ,ξ, S k ;ϑ k ( K hϑ ϑ r ϑ k l=r N r 1 K ( hϑ ϑ r } I ( Khs B r B k ( ( ( f ψ k φ,g f OD k X, p Khs Ξ r Ξ k ϑ l ( Khs B r B l ( ( ( f ψ l φ,g f OD l X, p Khs Ξ r Ξ l ( (c Compute Ṽ r Ŝ r, S r;ϑ r, usng the pseudo expected future values computed n (b and the optmal choces C,Q,W. Ṽ r ( Ŝ r, S r ;ϑ r where C,Q,W Ṽ r ( Ŝ r, S r ;ϑ r satsfy (d Repeat (a-c for all. 6. Go to teraton r + 1. ( { = U C,Q,W ;Ŝ r, S + βês r V = max C,Q,W U 25 Type I Extreme Value Dstrbuton } (Ŝ, S ;ϑ r Ŝ r, S r ( { } C,Q,W ;Ŝ r, S r + βês r V (Ŝ, S ;ϑ r Ŝ r, S r 38

A.3 Parallel MCMC Samplng Algorthm Table 17: Algorthm: Asymptotcally Exact Samplng va Nonparametrc Densty Product Estmaton Input: Subposteror samples, { ϑ t1 } T t 1 =1 p 1 (ϑ,...,{ϑ tm } T t M =1 p M (ϑ Output: Posteror samples (asymptotcally, as T, {ϑ } T =1 p 1...p M (ϑ p ( ϑ x N 1: Set h = 1. 10: Draw u Un f ([0,1]. 2: Draw t = {t 1,...,t M } d Un f ({1,..,T } 11: f u < wt w c then 3: Set c = t. 12: Draw ϑ t N ( 4: Draw ϑ 1 N ϑ t, h2 M I d. 13: Set c = t. 5: for = 2 to T do 14: else 6: for m = 1 to M do 15: Draw ϑ t N 7: Set t = c. 16: end f 8: Draw t m Un f ({1,..,T } 17: end for 9: Set h = (4+d 1. 18: end for ( ϑ t, h2 M I d. ( ϑ c, h2 M I d. 39