Overhaul Overdraft Fees: Creatng Prcng and Product Desgn Strateges wth Bg Data Xao Lu, Alan Montgomery, Kannan Srnvasan October 12, 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 u t = B t ˆ η t u t 4.4 Dssatsfacton and Account Closng ( (C t ; B t df (η t df B t 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 t = OD t B t C t and 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. Specfcally, we add the dssatsfacton effect to the per-perod utlty functon where U t = u t ϒ t I[B t C t < 0] In the above equaton, u t s defned n equaton 3 and ϒ s the dssatsfacton senstvty,.e., the mpact of chargng an overdraft fee on a consumer s decson to close the account. We assume that closng the account s a termnaton decson. Once a consumer chooses to close the account, her value functon (or total dscounted utlty functon equals an outsde opton 17
wth a value normalzed to 0 for dentfcaton purposes. 16 If the consumer keeps the account open, she ll receve contnuton values from future per-perod utlty functons. 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 value functon for the consumer becomes { U t + ϖ t0 + βe [V t+1 S t ] f W t = 0 V t = U t + ϖ t1 f W t = 1 where ϖ t0 and ϖ t1 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 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. 18
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 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} { 19 U t (S t + τ=t+1 β τ t U τ (S τ S t } (4
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: 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 20
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. 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. 21