MANAGEMENT SCIENCEVol. 61, No. 12, December 2015, pp. 3077–3096ISSN 0025-1909 (print) � ISSN 1526-5501 (online) http://dx.doi.org/10.1287/mnsc.2015.2203

© 2015 INFORMS

Dynamic Valuation ofDelinquent Credit-Card Accounts

Naveed ChehraziDepartment of Information, Risk, and Operations Management, McCombs School of Business,The University of Texas at Austin, Austin, Texas 78712, naveed.chehrazi@mccombs.utexas.edu

Thomas A. WeberManagement of Technology and Entrepreneurship Institute, École Polytechnique Fédérale de Lausanne,

CH-1015 Lausanne, Switzerland, thomas.weber@epfl.ch

This paper introduces a dynamic model of the stochastic repayment behavior exhibited by delinquent credit-card accounts. Based on this model, we construct a dynamic collectability score (DCS) that estimates the

account-specific probability of collecting a given portion of the outstanding debt over any given time horizon.The model integrates a variety of information sources, including historical repayment data, account-specific,and time-varying macroeconomic covariates, as well as scheduled account-treatment actions. Two model-identification methods are examined, based on maximum-likelihood estimation and the generalized method ofmoments. The latter allows for an operational-statistics approach, combining model estimation and performanceoptimization by tailoring the estimation error to business-relevant loss functions. The DCS framework is appliedto a large set of account-level repayment data. The improvements in classification and prediction performancecompared to standard bank-internal scoring methods are found to be significant.

Keywords : account valuation; consumer credit; collectability scoring; credit collections; GMM estimation;maximum-likelihood estimation; operational statistics; self-exciting point process

History : Received June 6, 2011; accepted February 9, 2015, by Noah Gans, stochastic models and simulation.Published online in Articles in Advance September 28, 2015.

1. IntroductionThe efficient collection of outstanding debt fromdefaulted credit-card accounts is mission-critical formany financial institutions. By the end of 2012,the total revolving consumer credit in the UnitedStates amounted to approximately $603 billion (Fed-eral Reserve Bank H.8, January 2013). During thatyear about $17 billion in credit-card debt was placedin collections, driven by an average delinquency rateof 2.88%.1 With such amounts at stake, even a smallimprovement in the effectiveness of debt collectionswould imply a significant bottom-line impact. Forexample, an industry-wide improvement of one per-cent in overall collection return could produce a gainin excess of $170 million, thus decreasing banks’ costof capital, which in turn would lead to more lendingand easier access to credit for consumers.2

1 The estimate is based on the charge-off and delinquency rates onloans and leases at commercial banks, reported regularly by theFederal Reserve Bank.2 According to Sufi (2009), over 80% of bank financing extended topublic firms is in the form of revolving lines of credit. Moreover,as detailed by Blanchflower and Evans (2004), credit cards play amajor role in financing small businesses.

This paper introduces a dynamic model of con-sumer repayment behavior on overdue, so-called“delinquent,”3 credit-card accounts placed in collec-tions. Both the timing and the amount of repay-ments are assumed to be random. The repaymentamounts are correlated and their distributions areobtained empirically. Each repayment influences thetiming distribution of the subsequent repayments.This stochastic feedback element captures both theaccount holder’s willingness-to-repay and ability-to-repay the outstanding balance that in turn providesvaluable insight about the actions banks should taketo maximize the expected collections revenue. Inaddition, the timing distribution of the repaymentsalso depends on the account holder’s demographic

3 Accounts with outstanding balances that are more than 30 dayspast due are considered delinquent and entered in the collectionsprocess (this definition applies to our data set in §5; the detailsare institution specific). The collections process consists of severalcollection phases often conducted by different outside collectionagencies at commission rates that increase after each phase transi-tion of the account to a different agency. The management of theoverall collections process and the ownership of the delinquentaccount generally remain with the originating bank until portfo-lios of “uncollectable” dead-reserve accounts are eventually soldoff (e.g., via auctions).

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characteristics, the account attributes, the current eco-nomic outlook, as well as planned account-treatmentactions. The model is used to construct a probabilitydistribution for the collectability (i.e., repayment like-lihood) of a delinquent account. The resulting account-specific dynamic collectability score (DCS) estimates, atany time in the collections process, the probabilityof collecting a given percentage of the outstandingbalance over a desired time horizon. The modeland scoring technique are tested using a large setof account-level repayment data and two differentmodel-identification methods—maximum-likelihoodestimation and prediction-error minimization. Theimprovements in classification and prediction per-formance compared to standard bank-internal scor-ing methods are significant. In addition, importantinsights are obtained for the pricing of delinquentdebt in terms of commission rates offered to collectionagencies.

There are four main methodological contributions.First, the model constructed in this paper represents,to the best of our knowledge, the only continuous-time stochastic approach to the prediction of repay-ment processes, leading to account-specific dynamiccollection forecasts. The forecasts are conditional onthe repayment history, observed covariates such asFICO score of the account holder, relevant macroeco-nomic data (e.g., prime rate), and a given sequenceof account-treatment actions. Second, the DCS-basedpredictions about account holder’s repayment behav-ior have—in contrast to many bank-internal scor-ing systems—intrinsic meaning in terms of concreterepayment probabilities. The flexibility of the DCS,with respect to prediction horizon, repayment thresh-old, and timing, presents a significant improvementover extant collectability scores that tend to be staticand focused on very stylized predictions such as dis-tinguishing “payers” from “non-payers.” Third, themodel lays the foundation for account-treatment opti-mization because it can be used to forward-simulaterepayment-path realizations conditional on treatmentactions. Fourth, the DCS enables the collections man-ager to obtain estimates for the expected net value ofa delinquent credit-card account and an estimate ofaccount-specific repayment activity in terms of num-ber of repayments over a given prediction horizon.

From a practical viewpoint, the results developedhere allow for a more precise prediction of repay-ment behavior. With an improved understanding ofthe sensitivity of the repayment process to covari-ates and treatment actions (in terms of the identifiedmodel parameters), it is possible to (at least locally)maximize the collection yield by optimizing over theavailable actions. In addition to producing a notice-able increase in collection yield, a rigorous modelcan help develop a realistic assessment of the risk

caused by account delinquency and improve under-writing through a better choice of account parameters.Moreover, the valuation method implied by the modelallows for an effective computation of “loss givendefault” (LGD),4 as defined by the Basel II accords(Basel Committee on Banking Supervision 2004) thatfeeds into the determination of banks’ capital-reserverequirements. The valuation method can also be usedfor pricing portfolios of dead-reserve accounts fordebt-leasing purposes. Lastly, our model enables thebank to establish effective performance-based collec-tion guidelines.

1.1. Literature ReviewWe now discuss related work in operations research,finance, economics, and applied probability.

1.1.1. Optimization of Credit Collections. Mitchnerand Peterson (1957) were among the very first tostudy the credit-collections problem from an opera-tions perspective. They noted that the empirical lossdistribution is usually bimodal,5 corresponding to theseparate repayment behavior of “skips” (contact notavailable) versus “nonskips” (contact available), andfocused on the optimal length of account treatment(which they termed “pursuit duration”) as a stop-ping problem. For this, Mitchner and Peterson wereprimarily concerned with distinguishing payers fromnon-payers, assuming full “conversion” in the caseof payment, emphasizing that “[i]n a fully rigoroustreatment, the entire sequence of intervals betweenpayments, as well as the payment amounts, shouldbe taken into account as part of the overall stochasticprocess” (p. 537).

Subsequent contributions to the credit-collectionsproblem, as laid out by Mitchner and Peterson,were sparse. Rather than focusing on collections,credit-scoring models were developed for under-writing new accounts (Bierman and Hausman 1970;for a survey, see Rosenberg and Gleit 1994) or forestimating the time-to-default (Bellotti and Crook2009). The collections problem did not appear onthe academic “radar screen” until, in 2004, theBasel II accords put a spotlight on the capital-reserve requirements induced by the in-aggregatequite substantial loss given default (LGD) in con-sumer credit (Basel Committee on Banking Supervi-sion 2004).6 Thomas et al. (2005) provide a survey of

4 The LGD is the percentage of the value of an asset that is lost inthe event of a default.5 More recently, Schuermann (2005) also emphasizes bimodality asa critical feature of recovery distributions for defaulted consumerloans.6 Altman et al. (2005) provide a collection of papers on the deter-mination of LGD. In the context of credit-card debt collections,these discussions are not fully relevant, for the collections problem

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the few notable consumer-credit models up to thatpoint. Our approach to the credit-collections problemimplements the Mitchner-Peterson program by view-ing account-repayment paths as realizations of self-exciting point processes, conditional on observationsof repayment history, account-specific and macroe-conomic covariates, as well as past and scheduledfuture account-treatment actions. In a discrete-timeframework, De Almeida Filho et al. (2010) formulatea Markov decision problem for the optimal timingof a given sequence of account-treatment actions thatis not conditioned on detailed account-specific repay-ment behavior. Although the model in our papersupports a similar goal (namely, the optimization ofaccount-treatment actions), it does strive for higherresolution through the estimation of a nonlineardynamic model in continuous time and the explicituse of account-specific features to optimize treatmentbased on action-contingent forecasts of repaymentbehavior.7

1.1.2. Corporate vs. Consumer Debt. Consumerdebt-repayment behavior differs from the corre-sponding corporate behavior in the determinants fordelinquency as well as in the relative absence ofmarket-based indicators such as bond prices (Grossand Souleles 2002, Duygan-Bump and Grant 2009).Nevertheless, the repayment process of overdue con-sumer debt bears similarities to the cash-flow streamof an index corporate default swap (CDS),8 becauseboth consist of random payments at random times.Unlike an index CDS, for which the loss distribu-tion of the constituent loans are often assumed to beindependent and identically distributed (i.i.d.) (see,e.g., Azizpour et al. 2011), in the case of delinquentconsumer loans, repayment amounts are correlated.Moreover, the valuation of delinquent consumer loansis primarily useful if it is made at the account level,since return cash flows are subject to the issuers’ col-lections practices.

This paper builds on the literature about corporatedefault risk. It extends the transform analysis devel-oped in Duffie et al. (2000) to affine jump-diffusionprocesses that also include deterministic jump terms.Errais et al. (2010) apply this type of analysis to a fam-ily of self- and cross-excited point processes to cap-ture the clustering behavior of corporate defaults. The

exists only because default has already occurred in the past. Giventhe bank’s exposure at default (EAD), it is the focus of our paperto dynamically estimate the LGD conditional on a repayment his-tory and other covariates, for a given time horizon and repaymentthreshold.7 The optimization of the credit-collections process based on ourstochastic model of the repayment process is treated in a compan-ion paper (Chehrazi et al. 2014).8 An index CDS can be viewed as a loss-insurance contract on aportfolio of corporate loans.

idea of cross-excitation was introduced in finance byJarrow and Yu (2001) through the concept of counter-party risk, where the default of one firm can affect thedefault of other firms. The results were extended byCollin-Dufresne et al. (2004) for the pricing of creditderivatives using an affine jump-diffusion model.The model presented here is related to Duffie et al.(2007) who use time-series dynamics of firm-specificand macroeconomic covariates to predict corporatedefaults. It also extends Errais et al. (2010) by includ-ing an additional term that acts as the control variablemodeling banks’ collection strategy, and thus allowsfor account-treatment optimization.

1.1.3. Equilibrium Models of Consumer Creditwith Default. The problems of credit scoring andindividual consumers’ default decisions (see, e.g.,Efrat 2002) have been studied, to some extent, in theeconomics literature. This work focuses on a descrip-tion of the lender-borrower interaction in the unse-cured consumer-credit market to study the welfareimplications of various regulatory policies. Chatterjeeet al. (2008) analyze an equilibrium model in whichcredit scoring is used as an index encapsulating theborrower’s reputation and creditworthiness. Ausubel(1999), Chatterjee et al. (2007), and Cole et al. (1995),among others, use game-theoretic models with asym-metric information to describe the notion of creditscoring as a screening/signaling problem, in anattempt to explain the dynamics of the market forunsecured debt. Credit scoring is also studied instatistics literature, where different classification mod-els have been used to rank consumers based on theircreditworthiness and probability of default. For exam-ple, Bierman and Hausman (1970), Boyes et al. (1989),Hand and Henley (1997), and West (2000) use dif-ferent statistical methods, including neural networks,discriminant analysis, and decision trees to distin-guish between “bad” and “good” borrowers. Bellottiand Crook (2009) include macroeconomic covariates(such as interest rate and the FTSE All-Share Index)to build a predefault score based on survival analysis.Despite several studies focusing on estimating pre-default repayment probabilities (Banasik et al. 1999,Lopes 2008), the extant literature does not considerthe estimation of post-default repayment probabilitiesthat is the focus of this paper.

1.1.4. Application and Estimation of Point Pro-cesses. Bartlett (1963) originated the systematic studyof departures of (stationary) point processes from thestandard Poisson processes (Cox and Isham 1980)using autocorrelation functions and spectral analysis.In many practical applications, such as earthquakeprediction (Ogata 1998), an important phenomenon isdata clustering that naturally lends itself to model-ing via self-exciting point processes (Hawkes 1971),

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incorporating a stochastic feedback element in theevolution of the intensity process. Relevant for ourmodel of consumer repayment behavior is the classof affine point processes (Errais et al. 2010) in whichthe stochastic differential equation for the evolutionof the intensity is linear and can accommodate jump-diffusion processes that we use to incorporate theinformation derived from covariates and account-treatment actions. Applications of self-exciting pointprocesses in the literature include clustering of air-craft hijacking events (Holden 1986), criminal behav-ior (Mohler et al. 2011), contagion in fixed-incomefinancial markets (Giesecke and Kim 2011), and con-version behavior in response to online advertising(Xu et al. 2014). The estimation of point processesusing maximum-likelihood methods was pioneeredby Rubin (1972) and Ogata (1978). Giesecke andSchwenkler (2012) examine likelihood estimators foraffine point processes when the explanatory factorsare not completely observed. We apply likelihoodestimation in conjunction with a Cramér-von Miseserror criterion (Lehmann and Romano 2005). For thecredit-collections problem, we show in §4 and §5 thatidentifying the model parameters based on the gen-eralized method of moments (GMM) can provide abetter alternative to maximum-likelihood estimation(MLE), much in the spirit of the recent advances inoperational statistics (Besbes et al. 2010).

1.2. OutlineSection 2 introduces the probabilistic structure of therepayment process, together with the main techni-cal assumptions. In §3, we determine the probabilityof repayment and derive the DCS credit-scoring sys-tem as well as a formula for the expected presentvalue of a delinquent credit-card account. Section 4is concerned with the estimation of the parametersfor the repayment and covariate processes. In §5, weapply the model and the results developed earlierand compare the predictive performance to that ofstandard bank-internal methods. In addition, we illus-trate the optimization of treatment actions. Section 6concludes.

2. ModelConsider a delinquent credit-card account placed incollections. The information that may be used to pre-dict future account repayment activity stems from avariety of sources. First, the account is characterizedby a set of attributes, including the credit limit, cur-rent outstanding balance, and annual percentage rate(APR). Second, the bank has account-holder-specificdata such as FICO score, annual household income,mortgages, personal bankruptcy events, as well ascredit limits and utilization rates for all issued creditcards (including those from other institutions). Third,

the bank observes covariates that may be relevant foran entire group of accounts such as market and priceindices, prime rate, and unemployment data. Last,the bank has access to the repayment history leadingup to the current outstanding balance; the availableinformation also includes past and scheduled futureaccount-treatment actions such as letters, calls, settle-ment offers, repayment schedules, and legal actions.

A functional model of the repayment processshould incorporate these heterogeneous sources ofinformation. For this, the stochastic sequence ofrepayments is described by a point process, whoseconditional arrival rate (intensity) is driven by pastrepayment events (a jump process), a vector oftime-varying covariates (a Markov process), and asequence of account-treatment actions (a deterministicjump process). The time-invariant covariates, whichoften include account-specific data, are used to deter-mine the parameters for the evolution of the intensity,except for its long-run steady-state that is obtainedfrom the process of time-varying covariates. The lat-ter often includes information which pertains to anentire group of accounts (e.g., in the form of macroe-conomic indicators) and is therefore referred to as agroup-covariate process. In what follows, we intro-duce the elements of the repayment process, beforedetailing the structure of the associated intensity andcovariate processes.9

2.1. Repayment ProcessA delinquent account with outstanding balanceB0 > 0, placed in the collections process at time t = 0,is credited with the repayment amounts Zi at timesTi > 0, for i ∈ 81121 0 0 09, until the balance is paid infull. We assume that the Ti are stopping times withTi <Ti+1, and that the Zi are nonnegative random vari-ables.10 For an outstanding balance Bi after the ithrepayment Zi, denote by Ri ¬Zi/Bi−1 the correspond-ing relative repayment, so

Zi = B0

i−1∏k=1

41 −Rk5Ri1 i ≥ 10 (1)

The random sequence N = 4Ti1Ri1 i ≥ 15, referred toas “repayment process,” characterizes the account’srepayment behavior. We assume that the Ri, i ≥ 1,are i.i.d. Note that notwithstanding this assumption,the absolute repayments (Zi) are correlated. For exam-ple, as soon as an outstanding balance is paid infull (Ri′ = 1, or equivalently Zi′ = Bi′−1, for some Ti′ ),

9 A summary of the notation is provided in Appendix B.10 All random variables are defined with respect to a complete prob-ability space 4ì1¦1P5 together with an information filtration � =

8¦t 2 t ∈ �+9, satisfying the usual hypotheses (Protter 2005). Ran-dom variables (except for Greek letters) are generally capitalizedwith realizations appearing in lower case.

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subsequent repayments generated by the model willbe zero while relative repayments remain i.i.d. (Zi =

Bi−1Ri = 0 ·Ri, for i > i′). Intuitively, the i.i.d. assump-tion is consistent with the “regenerative” structure ofthe repayment process, in the sense that taking intoaccount the effects of a repayment and normalizingthe remaining outstanding balance to one produces astructurally identical process. The distribution of theRi is taken as exogenous and is estimated empirically.This distribution, together with Equation (1), impliesa probability distribution for the repayment amounts4Zi1 i ≥ 15, whose fit is evaluated using the EmpiricalGeneralized Runs (EGR) test in §5.

2.2. Intensity ProcessWe denote the counting process associated with4Ti1 i ≥ 15 by Nt . This process is uniquely identified byan intensity process �t that quantifies the conditionalarrival rate of repayment events. The dynamics of theintensity process are described by a stochastic differ-ential equation (SDE). Specifically, we assume that �t

solves

d�t = �4y54��4Xt5−�t5 dt + �>

1 4y5dJt + �>

2 4y5dat1

t ∈�+1 (2)

where Xt is a Markov process with values inD ⊂ �nX (containing dynamic macroeconomic indica-tors), y ∈ �

ny+ is a constant vector (containing time-

invariant account attributes), Jt is a two-dimensionaljump process (describing the repayments), and atis a deterministic, nondecreasing, right-continuous,piecewise constant function, with values in �na

(containing scheduled account-treatment actions). Thefunctions ��4 · 5 and 4�1�11�254 · 5 are assumed to benonnegative-valued and affine.

To better understand the evolution of the inten-sity process �t , consider first the special case where4�11�25 = 0 and �� is constant. Then the jump termsin Equation (2) vanish, and the intensity process isdescribed by a deterministic law of motion,

�t = �� + 4�0 −��5e−�t1 t ∈�+1

where �0 ≥ 0 is a given initial value. In this set-ting, �t reverts to its long-run steady-state �� atthe rate �. A similar intuition applies when jumpterms are present and �� is not constant. The changein the intensity caused by the jump process Jt oraccount treatment actions at dissipates at the instan-taneous rate � as �t reverts to its long-run steady-state ��. The long-run steady state depends on thedynamic group-covariate process Xt that typicallycomprises macroeconomic observables (e.g., inflationrate, unemployment rate, CPI, and GDP). Hence, the

better the economic outlook in terms of group covari-ates, the greater is the chance of recovering the out-standing debt in the long run. In our formulation, thetendency to observe larger repayments in a boomingeconomy is mapped by the model to a probabilisticincrease in the number of repayments. Conversely, atendency to observe smaller repayment amounts in aneconomic downturn translates into a decreased likeli-hood of repayment. Therefore, without loss of gener-ality, the repayment amounts 4Zi1 i ≥ 15 are assumedto be independent of Xt .

The jump process Jt = 6Nt1Rt7> in the SDE (2) is

an equivalent representation of the repayment pro-cess N , where Nt ¬

∑i 18Ti≤t9 and Rt ¬

∑iRi18Ti≤t9. The

counting process Nt is independent of the repaymentamount Zi, reflecting the account holder’s willingness-to-repay. The mark process Rt depends on the repay-ment amounts Zi, indicating the account holder’sability-to-repay. The difference in the two dimensionsof the repayment process (Nt vs. Rt) can be appre-ciated by considering the somewhat extreme (yet nottoo unlikely) event when a relative repayment van-ishes (Ri = 0), corresponding to a bounced check. Suchan event signals the account holder’s high willingnessto repay her debt, but at the same time, it also revealsher low ability to do so.

Remark 1. The arrival rate �t of the counting pro-cess Nt depends on the history of Nt itself throughboth components of J . This self-excitation feature is use-ful for modeling stochastic phenomena with cluster-ing effects, where the past arrival of events affectsthe arrival of future events. In the context of debtcollection, the frequency and amount of past repay-ments convey information about the account holder’swillingness-to-repay (Nt) and ability-to-repay (Rt).Incorporating this information into the dynamics ofthe intensity process naturally leads to a self-excitingpoint process. Viewed from this perspective, ourdynamic repayment model follows the Errais et al.(2010) form. However, a key difference between theirmodel and ours is the presence of a deterministicjump process describing scheduled account-treatmentactions. This enables us to probabilistically attributethe collections outcome over a given horizon to dif-ferent treatment actions taken in the past, establish-ing a framework for account-treatment optimization(Chehrazi et al. 2014).

2.3. Covariate ProcessWe distinguish account-specific covariates that drivethe fixed effects for any given account, and groupcovariates that determine random effects for a set (or“group”) of similar accounts.

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2.3.1. Account-Specific Covariates. The account-specific covariates are summarized by the nonnega-tive, time-invariant vector y of dimension ny . Theyinclude attributes such as credit limit, outstandingbalance, interest rate, as well as information on theaccount holder, such as FICO score, mortgage pay-ments, and status of other credit cards (current/pastdue/written off). This data is either held by the bankor made available by various credit-rating agencies,usually on a monthly basis. To separate the estima-tion of the dynamics of these covariates from theestimation of the account’s repayment behavior, weassume them to be fixed at their current values.Modeling the covariates as a time-varying processwould increase the parameter space and likely dete-riorate the out-of-sample performance as a result ofoverfitting (Hastie et al. 2009). This is also consis-tent with much of the credit-collections data observedby the authors (see §5): although account covariatessuch as FICO score show significant variation acrossaccounts/holders, they remain essentially constant forany given account over a typical collection cycle offour months. The fixed-effect covariate vector y can beadjusted periodically, resulting in a moving-horizonapproach.

The account-specific information vector y deter-mines, in large part, the account quality in terms ofits collectability (i.e., repayment likelihood). Specifi-cally, the intensity parameters �, �1, �2 depend on y.The parameter � determines how fast the intensityapproaches its long-run steady state (in the absence offurther jumps), effectively determining the memoryof the system. The parameter �1 = 6�111�127

> indicateshow past repayment events affect future repaymentevents. Its components capture the importance of theaccount holder’s willingness-to-repay (via �11) andher ability-to-repay (via �12), respectively. Lastly, theparameter �2 determines the sensitivity of the inten-sity with respect to account-treatment actions. Thelatter include bank-level actions such as moving theaccount from one collection phase (and agency) tothe next (agency), adjusting commissions paid tocollection agencies, as well as extending settlementoffers, and agency-level interventions such as sendingform letters, establishing first-party contacts, offeringrepayment plans, and filing lawsuits. The treatmentschedule is modeled by a right-continuous, piecewiseconstant, vector-valued deterministic jump process atof dimension na, each component of which describesone type of account treatment (e.g., payment-planoffers or form letters). The preformulated treatmentschedule at generates, via the SDE (2), jumps in therepayment-intensity process �t whenever an actionis carried out. Because of the exponential decay, theeffect of any given treatment intervention diminishesover time.

2.3.2. Group Covariates. Groups of similar ac-counts are influenced by time-varying group covari-ates. For example, macroeconomic covariates, suchas inflation rate, unemployment rate, CPI, GDP, andprime rate, contain relevant aggregate informationabout the economy as a whole. The vector-valuedgroup-covariate process Xt determines the long-runintensity ��4Xt5 in Equation (2) for an account group.To model the dynamics of these covariates we assumethat Xt uniquely solves the SDE

dXt =�4Xt5dt +�4Xt5dWt1 (3)

where Wt is a standard Wiener process in �nX ; � and��> are affine functions mapping �nX to �nX and�nX×nX , respectively.11 One can interpret the SDE (3)as the limit of an autoregressive conditional het-eroskedasticity (ARCH) model, when the length of thetime step in the underlying stochastic difference equa-tion is taken to zero (Nelson 1990). The ARCH-processfamily has been widely applied in the econometricsliterature to model the dynamics of financial timeseries (Bollerslev et al. 1992). Our modeling choicehere is motivated on the one hand by the high het-eroskedasticity in our observations, which is consis-tent with numerous empirical studies on financialdata (Bollerslev et al. 1994), and on the other hand bythe aforementioned asymptotic connection betweenARCH and diffusion models.

3. Performance Quantification3.1. Repayment ProbabilityGiven a positive time horizon � and the current time t,up to which information is available, we seek to deter-mine the probability that over the interval 4t1 t + �7the total repayment of an account with current out-standing balance Bt > 0 exceeds a given thresholdB ∈ 601Bt5. This amounts to computing the �-horizonrepayment probability (which—as shown next—depends only on the ratio B/Bt)

çt1 t+�4B/Bt54

= P

{ Nt+�∑i=Nt+1

Zi >B

∣∣∣∣¦t

}1 (4)

where ¦t contains all currently available information.This repayment probability constitutes our dynamiccollectability score (DCS), an explicit expression ofwhich can be obtained based on the dynamic repay-ment model in Equations (2) and (3). When passing

11 Duffie and Kan (1996) obtain sufficient conditions on � and �that guarantee the existence and uniqueness of Xt in the domainof admissible covariates, D = 8x ∈ �nX 2 �4x5�>4x5 ≥ 09, where theSDE (3) is nondegenerate. For any such � and � , Proposition A1in Appendix A provides necessary and sufficient conditions thatensure �t is well defined.

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from absolute to relative repayments as in Equa-tion (1), the right-hand side of Equation (4) becomes

P

{Bt −

Nt+�∑i=Nt+1

Zi <Bt −B∣∣∣¦t

}

= P

{Bt

Nt+�∏i=Nt+1

41 −Ri5 < Bt −B

∣∣∣∣¦t

}

= P

{ Nt+�∑i=Nt+1

Qi < log(Bt −B

Bt

) ∣∣∣∣¦t

}1

where Qi4

= log41 − Ri5 denotes the logarithm of theoutstanding balance percentage after the ith rela-tive repayment. Provided the conditional characteris-tic function (CCF) of the corresponding cumulativelogarithmic relative-repayment process Qt+� − Qt

4

=∑Nt+�

i=Nt+1 Qi is known, Lévy’s inversion theorem can beused to compute this �-horizon repayment probabil-ity. Next, the repayment CCF is derived explicitly tocompute the DCS.

3.2. Collectability ScoringDefining Xt = 6Qt1�t1X

>t 7

>, Equations (2) and (3)together specify a combined affine jump-diffusionmodel in which one of the jump factors is determin-istic. Specifically, Xt solves

dXt = �4Xt5 dt + �4Xt5 dWt + �1 dJt + �2 dat1 (5)

where Wt is a standard Wiener process in �nX , andall the remaining model parameters and variables aresuitably augmented.12

The conditional characteristic function of Qt+� −Qt isa special form of the transform that appears naturallyin affine term-structure models of interest rates, e.g.,to price zero-coupon bonds and European options(Duffie et al. 2000) or to value defaultable bondsand sovereign debt (Gibson and Sundaresan 2001,Merrick 2001). The key difference here is the pres-ence of deterministic jumps to incorporate the bank’scollection strategy and thus to optimize the accountvalue by appropriately varying this strategy. The fol-lowing result, formulated in a more general settingin Appendix A, extends a statement by Duffie et al.(2000, Proposition 1) to affine jump-diffusion pro-cesses with deterministic jump terms.

12 �4x5 = �0 + �1x in which 4�01 �15 ∈ �nX × �n

X×n

X combinesthe drift terms of Equations (2) and (3); and �4x5�>4x5 =

�0 +∑n

Xl=1 �

l1xl where

(�01 4�

l15

nX

l=1

)∈ �n

X×n

X × �nX

×nX

×nX expands

�4x5�>4x5 to a function from �nX to �n

X×n

X . Furthermore,Jt = 6Qt1Nt1Rt7 has i.i.d. jumps denoted by ãJ , and 4�11 �25 ∈

�nX

×3×�n

X×na is obtained by properly resizing 4�11�25 (see Equa-

tions (A9)–(A11) in Appendix A).

Proposition 1 (Repayment CCF). Let �m, for m ∈

8m¯1 0 0 0 1 m9, denote the known jump arrival times of as ,

for s ∈ 4t1 t+ �7, and let 4�1 �5 be the (unique) solution tothe initial-value problem

˙�s + �>

0 �s + �>

s �0�>

0 �s/2 = 01 �t+� = 01 (6)

˙�s + �>

1 �s + �>

s �1�>

1 �s/2 = 41 − Ɛ6e�>s �1ãJ 75e21

�t+� = �e11 (7)

on 6t1 t + �7, where el ∈ �nX is the l-th Euclidean basisvector, and � ∈� is given.13 Under a set of technical con-ditions, provided in Appendix A, the repayment CCF isgiven by

Ɛ6e�4Qt+�−Qt 5 �¦t7

= exp[−ã�t1 t+� −ã�>

t1 t+�Xt +

m∑m=m

¯

�>

�m�2ãa�m

]1 (8)

where ãa�m4

= a�m − a�−m

, ã�t1 t+�4

= �t+� − �t , and ã�t1 t+�4

= �t+� − �t .

Similar to Duffie et al. (2000), the aformentionedresult is obtained by noticing that Ɛ6e�4Qt+�−Qt 5 � ¦s7is a martingale for s ∈ 6t1 t + �7; hence its drift hasto be zero. This together with the affine structureof the SDE (5) provides us with ODEs (6)–(7). TheODE (7) is a generalized Riccati equation, where theterm �>

s �1�>1 �s is a shorthand description of a vec-

tor in �nX , such that 6�>s �1�

>1 �s7l ≡ �>

s �l14�

l15

>�s forl ∈ 811 0 0 0 1nX9. For special cases (e.g., when nX = 1and �1 = 0) an analytical solution to (6)–(7) can beobtained (Jódar and Navarro 1991); numerical solu-tions are available using the various standard integra-tion methods for ODEs (Butcher 2008).

Proposition 2 (Dynamic Collectability Score).The �-horizon repayment probability in Equation (4) isequal to

çt1t+� 4B/Bt5

=12 −

∫ �

−�

4e−ã�t1t+� 4jw5−ã�>t1t+� 4jw5Xt+

∑mm=m

¯�>�m

4jw5�2ãa�m−jw log41−B/Bt 55

·42�4jw55−1dw1 (9)

where 4�1 �5 solves the initial-value problem (6)–(7) for� = jw.

Example 1. Consider the case where �1 = 0, and�� is constant, i.e., no time-varying macroeconomicindicators are present. In this setting, the repay-ment timing process 4Ti1 i ≥ 15 is independent of the

13 The solution 4�s1 �s5 depends on the given boundary value 401 �5for s = t + � . The value � is in practice either complex or real, � ∈

8jw119, where j =√

−1 and w ∈ �. To keep notation simple theparametric dependence is suppressed when not needed explicitly.

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relative-repayment process 4Ri1 i ≥ 15. The intensity�s , for s ∈ 6t1 t + �7, is then

�s = �� + 4�t −��5e−�4s−t5

+∑

�m∈4t1s7

�>

2 ãa�me−�4s−�m51

where the current intensity �t is known. Under theprevious assumptions, the inverse Fourier transformin Equation (9) can be obtained indirectly by calcu-lating the probability mass function of Nt+� −Nt viaevaluating the derivatives of

Ɛ6� 4Nt+�−Nt 5 �¦t7

=exp[−ã�t1 t+� −ã�>

t1 t+�Xt+

m∑m=m

¯

�>

�m�2ãa�m

]1 (10)

at � = 0, where Xt = 6Nt1�t7>, and 4�1 �5 solves the

ODE in Proposition 1 with the boundary condition4�t+�1 �t+�5= 401 log4�5e15. Thus,

�s =� − 1�

��4e−�4t+�−s5

− 1 +�4t + � − s551

�s =

log4�5

� − 1�

41 − e−�4t+�−s55

1

for all s ∈ 6t1 t + �7. The argument of the exponen-tial function in Equation (10) is therefore equal to thearea under �s when s ∈ 6t1 t + �7, where �t representsthe area generated by the fixed part of � (i.e., long-run steady-state ��), whereas �t and ��m

represent thearea contributed by the transient parts (i.e., the inten-sity’s current value �t and the jumps �2ãa�m ). Denot-ing the cumulative distribution function (CDF) of Qi

by FQ, we have

çt1 t+�4B/Bt5=

�∑n=1

P8Nt+� −Nt = n9F ∗nQ

(log

(1 −

B

Bt

))1

in which the counting process follows a Poissondistribution,

Nt+� −Nt ∼ Pois(��� +

1 − e−��

�4�t −��5

+

m∑m=m

¯

1 − e−�4t+�−�m5

��>

2 ãa�m

)1

and F ∗nQ is the n-fold convolution of FQ. In the gen-

eral setting, �t and �t can be interpreted in the sameway.14

14 The method used in this example for calculating the inverseFourier transform of Equation (9) can be extended to the generalsetting by discretizing the Qi . This approach is used in §5.2.2 toavoid numerical complications (Gibbs effect) that may arise whencomputing the DCS.

3.3. Account ValuationGiven a description of the repayment process basedon information up to the present (¦t) and theaction process a, with scheduled interventions ãa�m attimes �m ∈ 4t1 t + �7, we are now interested in deter-mining the expected �-horizon present value of anaccount, denoted by Vt1 t+�4a5.

Proposition 3 (Account Valuation). Let � be adiscount factor. The expected �-horizon present value of anaccount with outstanding balance Bt is

Vt1 t+�4a5= Bt

∫4t1 t+�7

e−�4s−t5d�t4s � a53 (11)

the cash-flow measure �t4s � a5 represents the expectedcumulative repayment over 4t1 t + s7:

�t4s �a5=1−e−ã�t+�−s1 t+�−ã�>t+�−s1 t+� Xt+

∑�m∈4t1 t+s7 �

>�−s+�m

�2ãa�m 1(12)

where 4�1 �5 is the solution to the initial-value prob-lem (6)–(7) for � = 1.

Example 2. Under the assumptions of Example 1,the cash-flow measure of Equation (12) becomes

�t4s �a5

=1−e−r��s−4r/�541−e−�s54�t−��5−∑

�m∈4t1 t+s74r/�541−e−�4t+s−�m55�>2 ãa�m 1

where r4

= Ɛ6Ri7 is the expected relative repayment.For any given discount factor �, one can then obtainthe account value by calculating the integral in Equa-tion (11) numerically. From an optimization perspec-tive, it is interesting that taking a more forceful actionat the scheduled time, or alternatively, taking a givenaction earlier, leads to a first-order stochastically dom-inant shift of the cumulative repayment distribution,resulting in a larger cash-flow measure (Kwiecinskiand Szekli 1991). The corresponding expected bene-fits in terms of account value may be offset by thehigher cost of a larger action as well as the timevalue of money of the capital expenditure requiredto finance the additional or earlier collection effort.Chehrazi et al. (2014) investigate this tradeoff in ageneral infinite-horizon framework.

4. Model IdentificationConsider now the problem of identifying the modelparameters for a group of k accounts. For eachaccount k in the portfolio K = 811 0 0 0 1 k9, account-specific data is available over the study inter-val 601hk7, with horizon hk ≥ � . The data includesthe repayment history N k = 4T k

i 1Rki 1 i ≥ 15 and a

process ak of scheduled account-treatment actions.Relevant for all accounts in the portfolio K isthe group-covariate process X whose path is alsoobserved during the study interval 601hk7, k ∈K.

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The tuple of model parameters, � = 6��3�X7, hasseparate entries for the intensity process (��) and forthe group-covariate process (�X). Because the dynam-ics of the group-covariate process do not dependon ��, the group-covariate-process parameter vector�X can be estimated based on the observed samplepath for X and independently of the observed repay-ment process N k and the parameter vector ��. Bycontrast, the estimation of �� depends on �X (fora given realization of X); however, this dependencyis only through the constraints that ensure positivityof the intensity process (see Proposition A1). Theseconstraints, together with a goodness-of-fit condition,bound the domain ä� of feasible �� but otherwisedo not affect the estimation objective (the likelihoodfunction or a GMM-loss). Hence, a sequential esti-mation procedure, identifying first �X and then ��,whereas computationally advantageous, is no lessefficient than an analytically more complex joint esti-mation of � .

4.1. Estimation of the Group-Covariate ProcessThe estimation of an affine diffusion process Xt

with parameter tuple �X = 6�1��>7 has been dis-cussed extensively in the literature on dynamic assetpricing (Singleton 2001). The methods consideredthere are based on either maximum-likelihood esti-mation (MLE) or the generalized method of moments(GMM). They are usually developed for specific finan-cial applications, given a sample of Xt (or dependentvariables such as prices and/or yields). Although inour application, both MLE and GMM can be used toobtain an estimate �∗

X of �X , GMM is often preferred,because knowledge about the CCF for affine (jump-)diffusion processes (see Proposition A2) can be usedto develop computationally tractable and asymptoti-cally efficient estimators.15

4.2. Estimation of the Repayment ProcessConditional on the parameter estimate �∗

X for themodel (3) of the group-covariate process, the identi-fication of the repayment process N = 4Ti1Ri1 i ≥ 15can be accomplished in two steps: in the first step, weestimate the distribution FR for the relative-repaymentprocess 4Ri1 i ≥ 15. In the second step, we estimate�� by fitting the intensity process �t to the observedrepayment-timing process 4Ti1 i ≥ 15. The two steps ofthe procedure are detailed in turn.

Step 1: As an estimate of FR we use the empiricalCDF

FR4r5=

∑kk=1∑Nk

hk

i=1 18Rki ≤r9

∑kk=1 N

khk

1

15 The stationarity and ergodicity of the affine (jump-)diffusion pro-cesses required for establishing consistency and asymptotic behav-ior of MLE and GMM estimators are discussed by Glasserman andKim (2010) and Zhang and Glynn (2012).

given the data 4Rki 11 ≤ i ≤Nk

hk5kk=1, taking into account

Equation (1). The i.i.d. assumption (or rather model-ing choice) in §2.1 implies the consistency and unbi-asedness of this nonparametric estimate.

Step 2: The parameter �� = 6��1�1�11�27 is esti-mated using two alternative methods: MLE andGMM; the latter is implemented via prediction-errorminimization (PEM). The operational purpose of therepayment-process model determines which methodis preferable. Although MLE is asymptotically effi-cient, using the model parameters that maximize thelikelihood of the observed data will not necessarilylead to a decision that minimizes the loss in busi-ness terms. In the presence of a modeling error,16

GMM often provides better out-of-sample results,since appropriate moment conditions can be used toconstruct loss functions that are closely related tobusiness objectives. This observation is the underlyingidea of recent investigations in operational statisticsand robust estimation where model estimation andperformance optimization are combined (Besbes et al.2010, Chehrazi and Weber 2010).

Maximum-Likelihood Estimation 4MLE5. The jointprobability density function for the arrival times4Ti11 ≤ i ≤ Nt5 of the repayment process N can bederived by establishing a link between the condi-tional density function of the interarrival times andthe intensity process (Rubin 1972).17 This leads to acompact expression for the log-likelihood function of4T k

i 11 ≤ i ≤Nkhk5,

L44T ki 11≤ i≤Nk

hk5 � 4Xs1s∈ 601hk7514Rki 11≤ i≤Nk

hk51��5

=−

∫ hk

0�ks 4��5ds+

∫ hk

0ln4�k

s−4��55dNks 1 (13)

where the estimated intensity process �ks = �k

s 4��5, inanalogy to Equation (2), is given by

d�ks = �4yk54��4Xs5−�k

s 5 ds

+ �>

1 4yk5 dJ kt + �>

2 4yk5 dakt 1 s ∈�+1

with corresponding parameter estimate �� = 6��1 �1�11 �27, for any account k ∈ K. The maximum-likelihood estimator �∗

� is then a �� that maximizesEquation (13).

Generalized Method of Moments 4GMM5. As analternative to MLE, we consider an estimator thatminimizes a loss function that is tailored to abusiness-relevant measure such as the expected

16 Modeling errors cannot be eliminated in practice, since for anyfinite amount of data there exist different models, each of whichcannot be rejected—at a given level of statistical significance.17 The likelihood function can also be obtained by a change of mea-sure (Giesecke and Schwenkler 2012).

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Figure 1 Description of the Data Set: (a) Summary Statistics; (b) Distribution of Commission Rates

0.300

0.1

0.3

0.2

Com

mis

sion

rat

e 0.4

0.5

(b)

0

0.1

0.3

0.2

0.4

0.5

0.10 0 0.10

Frequency [%]

Phase 1 Phase 2

0.20 0.300.20

{

present value of an account. For any given predic-tion horizon � , time t ≤ hk − � , and parameter esti-mate ��, Proposition 3 yields the expected presentvalue of any individual account k: V k

t1 t+�4ak � ��5

= Ɛ6∑

i e−�T k

i Zki 18t<T k

i ≤t+�9 � ¦t1 ak1 ��7. The model can

then be identified by choosing an estimate �∗�1 t that

minimizes the square deviation between model pre-diction and observed data, so

�∗

�1 t ∈ arg min��∈ä�

{∑k∈K

4V kt1 t+�4a

k� ��5− vk

t1 t+�52

}1 (14)

where vkt1 t+� =

∑i e

−�T ki Zk

i 18t<T ki ≤t+�9 is the observed �-

horizon value of the kth account. This GMM criterionis appropriate when valuation, either at the accountlevel or the portfolio level, is of interest (for examplewhen pricing uncollectable accounts, so-called “dead-reserves,” for debt leasing). As detailed in §5, MLEand GMM provide similar structural insights aboutthe repayment process, and they both exhibit a similarout-of-sample scoring performance. However, the val-uation error (see Equation (15)) of the GMM estima-tor is considerably smaller (by about 50 percent) thanthat of MLE (see §5.2.1). In applications with limiteddata, such performance improvements—as well as thebuilt-in robustness with respect to modeling error—often outweigh the loss in asymptotic efficiency, thusrendering GMM-based estimation a viable alternativeto MLE.

Remark 2. Both MLE and GMM require observ-ing Xt over the entire study interval. In practice,some group covariates may only be observed at dis-crete time instances. In such a case, one can constructthe relevant continuous sample path by interpolatingbetween the observed values. Alternatively, one canwork with an intensity process åt specified under theinternal history of N by taking the expectation of �t

with respect to X, thus removing the dependence of �t

on the sample path of X (Brémaud 1981, Theorem 14).

This expectation can be evaluated explicitly by adapt-ing Proposition A2.18

5. ApplicationsWe now employ the framework developed thus farto estimate the collectability and the value of delin-quent credit-card accounts. For this, we use a largedata set of account-specific credit-collections data andidentify our model using both MLE and GMM esti-mation methods, as detailed in §5.1. We then com-pare the two methods and highlight the operationalgains that can be obtained from GMM when adapt-ing it to business-relevant objectives (§5.2.1). The DCSconstructed in Proposition 2 is used to determine thepayback probability and value accounts (§5.2.2). Theout-of-sample prediction performance of DCS at dif-ferent instances of the collections process is illustratedby Receiver Operating Characteristic (ROC) curves,and compared to a bank-internal scoring (BIS) system(§5.2.3). Finally, the model is used to optimize com-mission rates for debt-leasing agencies in a typicaltwo-phase external collections process (§5.2.4).

5.1. Data Set and ModelThe available data set contains approximately 6,600delinquent accounts with a total outstanding balanceof $6 million, placed in collections between 2004 and2006; see Figure 1(a) for summary statistics. To evalu-ate the prediction performance of the model we limitaccount-specific covariates y to two attributes: out-standing balance at placement and predefault FICOscore.19 We consider two high-level collection actions:assigning an account to a new collection agency(effectively starting a new collection phase) and set-ting commission rates. In our data set, an account’s

18 Specifically, åt solves dåt = �4y54��4xt5 − åt5 dt + �>

1 4y5dJt +

�>

2 4y5dat , where xt4= Ɛ6Xt �¦07 solves dxt =�4xt5 dt.

19 Bank-internal data sets usually contain additional attributes suchas credit limit, interest rate, utilization rate, and past-due/personal-bankruptcy events. In our implementation, both the placement bal-ance and the predefault FICO score are linearly mapped to 60117.

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repayment behavior is tracked over two collectionphases. To avoid conflict of interest, e.g., agencies’artificially delaying collections, a commission rate isincreased only when an account is reassigned to adifferent agency. Hence, the process of scheduledaccount-treatment actions a includes at most twointerventions (m = 2): ãa�1

= 611 c17> where �1 = 0 is

the placement time for phase 1; and ãa�2= 611 c2 −c17

>

where �2 denotes the placement time for phase 2 (ifapplicable). The distributions of account-specific com-mission rates c1 and c2 for phases 1 and 2, respec-tively, are summarized in Figure 1(b); some accountswere transitioned to the second phase (shaded dark),on average after 120 days of nonpayment in the firstphase. The dashed lines illustrate the average com-mission rates for the two collection phases.

In addition to the account-specific covariates dis-cussed earlier, our data set includes a BIS derivedfrom quasi-static regression models (e.g., Crook et al.2007) and heuristic data-mining methods; it providesa natural benchmark. Since the BIS is often pre-computed at the outset for later use, we enhanceits dynamic performance using Bayesian updating,termed “updated BIS” (uBIS), based on the observedrepayment history for each account.

5.1.1. Repayment-Process Discretization. The re-payment process 4T k

i 1Zki 11 ≤ i ≤ Nk

hk5 of account k

is approximated by lump-sum payments Zki at the

collection-report times T ki , where Zk

i corresponds tothe sum of all repayments between T k

i−1 and T ki . Fig-

ure A.1 in Appendix A shows the empirical CDFof the relative repayments, FR. Pearson’s chi-squaredtest rejects the hypothesis that the Rk

i are indepen-dent at the account level (i.e., for a fixed k). In theavailable data set, the average number of paymentsis small (approximately 2). Hence, the hypothesis thatthe Rk

i are identically distributed cannot be tested atthe account level. At the aggregate level, when pool-ing the relative-repayment sequences for all k ∈K, theEmpirical Generalized Runs test does not reject thehypothesis that the Rk

i are i.i.d.20

5.1.2. Account Stratification. The data set can besplit into random “training” and “testing” subgroupsby stratifying either across time or across accounts.In the former version, the model is trained over theinitial portion (e.g., 60 percent) of the study inter-val and the out-of-sample performance is determinedover the remaining length. This mode of analysis isattractive when a decision maker is interested in theoptimal account-specific timing and design of a set-tlement offer. In the latter version, the data set is

20 The corresponding values of the EGR test statistics T11n4�15,T11n4�25, T�1n4�15, and T�1n4�25, as defined in Cho and White(2011), are 0.0064, 0.0023, 0.597, and 0.1846, respectively; all aremajorized by the corresponding critical values at the 1% level.

divided into training and testing subgroups acrossaccounts, so parameters can be identified over the fulllength of the collections process. This is useful whena decision maker is interested in valuing a portfolioof accounts (at the aggregate level, including over-lapping generations). The latter version is adoptedhere for account-valuation and collection-action opti-mization. Thus, 60% of accounts (3,757) were ran-domly selected for training and the remaining 40% ofaccounts (2,504) for testing. In these two subgroups,at least one payment was made by 83% (3,115) and84% (2,096) of all accounts, respectively.21

5.1.3. Parameter Estimation. The average rate onone-month negotiable certificates of deposit (CD),taken from the Federal Reserve Bank’s H.15 peri-odic reports, serves as group-covariate process Xt . Asnoted in §5.1, the testing and training subgroups aretracked over the same study period: 2004 to 2006. Toavoid the bias that would result from including study-period samples of X in the estimation of the intensityparameter �� (they should remain unknown for thetesting subgroup), only observed covariates prior tot = 0 (i.e., 2004) are taken for the estimation of �X ,whereas åt (see Remark 2) is used to estimate ��.22

Given �∗X , the intensity parameter �� is estimated by

both MLE and GMM methods. Tables 1 and 2 sum-marize the results and also provide the (asymptotic)standard error (SE) for each estimator.23 The valid-ity of the model is confirmed by the Cramér-vonMises (CVM) goodness-of-fit test (see Appendix A fordetails).24 The CVM-test rejects the model when �1 isforced to zero, which implies that the intensity pro-cess is indeed sensitive to the jump process Jt , thusjustifying empirically the use of a self-exciting pointprocess as a model for the stochastic repayments.

5.1.4. Interpretation. The first three componentsof �� = 6�1�11�21��7 are affine functions of theaccount-specific covariate vector y, whereas the lastcomponent is an affine function of the group covari-ate Xt . One can compare MLE and GMM estimates

21 The treatment of data censoring owing to the fact that full repay-ment may be observed infrequently in typical repayment data isdiscussed in Appendix A.22 Based on 420 monthly samples for one-month negotiable CDs,from 1970 to 2004, an estimate of 4�01�15 is obtained: 4�∗

01 �∗

15 =

43070 × 10−51−60089 × 10−45 with the standard error of 43075 ×

10−5160828 × 10−45 calculated according to Hansen (1982).23 In our implementation �0− = å0− = 0. This corresponds to thenotion that if a delinquent account is not placed in collection, therewill be no repayment. The SE of the MLE estimates are obtainedby approximating the Fisher information (Berndt et al. 1974). TheGMM estimates are derived by taking �= 0, t = 0, and � = 120 days(see Equation (14)), with SE derived according to Hansen (1982).24 The values of the CVM-test for the MLE and GMM parameterestimates in the testing subgroup are 00403 and 00441, respectively.Both are majorized by the 5%-level critical value of 00461.

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Table 1 Parameter Estimates for MLE (Including Standard Error); 0 ≤ 0† ≤ 10−3

�∗

1 �∗

2 �∗

�

y �∗ �∗

11 �∗

12 �∗

21 �∗

22 �∗

�10 �∗

�11

1 4×10−25 00522 00400 00449 0† 00778 00288 −602584000665 4001265 4003005 4000865 4006175 4000475 4102645

B0 4×10−25 00291 10356 −00370 0† −00213 N/A N/A4000865 4001855 4004705 4001315 4009855

FICO 4×10−25 00913 10311 −00079 0† 120584 N/A N/A4001025 4002115 4005145 4001845 4103995

Table 2 Parameter Estimates for GMM (Including Standard Error); 0 ≤ 0† ≤ 10−3

�∗

1 �∗

2 �∗

�

y �∗ �∗

11 �∗

12 �∗

21 �∗

22 �∗

�10 �∗

�11

1 4×10−25 10976 00777 00442 0† 0† 10163 −2502664900325 41406075 42404415 4607385 41302465 4201115 46509605

B0 4×10−25 00128 30596 −00442 0† −0† N/A N/A41300585 41504405 42800195 4205525 4804875

FICO 4×10−25 00262 −00776 −0† 0† 130363 N/A N/A41308885 41209455 41201835 42000765 45802025

by examining the sign and magnitude of the coeffi-cients for each affine function. The sign determinesthe directional dependence of the considered func-tion with respect to the changes in the correspondingcovariate. For example, the sign of �∗

�11 is negative forboth GMM and MLE. This suggests that as the inter-est rate increases, �∗

�decreases resulting in a smaller

repayment probability and thus a lower accountvalue. Similarly, the sign of the coefficient of �∗ for theFICO score is positive for both estimation methods.Thus, with increasing creditworthiness the inertia ofthe intensity process decreases, effectively shorteningthe memory of the repayment process. In other words,for a larger FICO score the past becomes less predic-tive of the future. Indeed, if a large FICO score wasa good indicator for repayment behavior, the accountshould not have ended up in collections. The valueof �∗ also increases with the outstanding balance B0

for both GMM and MLE estimates. This means that ahigher placement balance also decreases the memoryof the repayment intensity. When the balance is high,the current financial status becomes a better predictorthan the account holder’s past financial status.

Although the signs of MLE and GMM estimatestend to be consistent, their magnitudes may differ, inpart because coefficients can offset each other. Insteadof directly comparing the latter, we therefore com-pare the values of the affine functions for a (ficti-tious) account with average covariate vector yavg forthe considered account segment (low pre-default bal-ance, high FICO score). Panel (a) of Table 3 shows that

for both MLE and GMM there is a large difference inthe sensitivities of the repayment-arrival intensity inEquation (2) to the different components of the repay-ment process Jt = 6Nt1Rt7

>. Specifically, �∗114yavg5 �

�∗124yavg5, so to predict repayment behavior in this

segment the willingness-to-repay (Nt) proves signif-icantly more important than ability-to-repay (Rt).25

This suggests that testing actions with the potential toboost an account holder’s willingness-to-repay, suchas filing a lawsuit or offering credit reinstatementupon full repayment, may constitute useful businessexperiments for this segment. For both MLE andGMM, the account-treatment sensitivities vary acrossactions (�∗

214yavg5 � �∗224yavg5): for the segment under

consideration the mere placement of an account witha new agency proves less important than an increasein the commission rate. The repayment intensity �t

in the SDE (2), and therefore also åt (see Remark 2),is sensitive to a change of the commission rate; thissensitivity increases in the FICO score and diminisheswith greater placement balance B0 (see the coefficientsof �22 in Tables 1 and 2).

5.2. Performance AnalysisIn what follows, we compare the performance of themodel for MLE against GMM. We then illustrate thecomputation of the dynamic collectability score as

25 Because the accounts have low balance and high pre-defaultFICO scores, the group is considered “highly collectable.” This maynot hold for other segments, e.g., for low-FICO-score/high-balanceaccounts.

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Table 3 MLE vs. GMM: (a) Average Model Parameters;(b) Relative Valuation Error

Panel (a)

�∗

14yavg5 �∗

24yavg5

�∗4yavg5 �∗

114yavg5 �∗

124yavg5 �∗

214yavg5 �∗

224yavg5

MLE 00010 00015 00003 0† 00056GMM 00021 00021 00002 0† 00052

Panel (b)

Relative valuation error (%)

Training MLE 1108GMM 509

Testing MLE 1107GMM 507

the backbone of the method. A comparison of the pre-diction performance for the two model-identificationmethods follows before we show how to use it foroptimizing account treatment by setting commissionrates.

5.2.1. MLE vs. GMM. The two identificationmethods yield similar intuition about repaymentbehavior in the segment K. As shown in §5.2.3,their prediction performance is also similar. How-ever, significant performance differences may arise forbusiness-relevant objectives that are closely relatedto the PEM moment conditions. For example, whenaggregating the expected values of different accountsin a segment, one obtains a business-relevant objec-tive (“segment value”), for which the bank seeks accu-rate forecasts. The corresponding relative valuationerror (RVE) is

RVE =

∣∣∣∣∑k

k=1 Vkt1 t+�4a

k5−∑k

k=1 vkt1 t+�

∑kk=1 v

kt1 t+�

∣∣∣∣0 (15)

As shown in panel (b) of Table 3 (for collectionphase 1, with �= 0, t = 0, and � = 120 days), the RVEfor the GMM method, implemented via PEM, clearlyoutperforms the RVE for MLE. The main reason for

Figure 2 Cumulative Payback Probability çt1 t+� 4B/Bt 5 for a Sample Account (B0 = $884013, FICO= 555) Together with Account Value Vt1 t+� 4a5/Bt

(Dashed Line), for � ≤ 42 Days

350.2

0.4

0.6

Cum

ulat

ive

paym

ent

0.8

1.0(a)

45 50 55

Days in collection

60 65 70 7540

0.05

0.05

0.1

0.1

0.15

0.15

0.15

0.2

0.2

0.25

0.25

0.30.35

0.10.05 0.1

70

0.4

0.5

0.7

0.6

Cum

ulat

ive

paym

ent

0.8

0.9

1.0

80 85 90

Days in collection

95 100 105 11075

0.05

0.05

0.1

0.15

0.1

0.1

0.15

0.2

0.2

0.2

0.3

0.35

0.30.35

0.25

0.25

0.25

0.4

0.4 0.450.5

0.150.05 0.15

(b)

the performance difference is the presence of model-ing errors (see Footnote 16). As noted at the end of§5.1, the repayment-process model responds in a nat-ural way to the various outside forces, such as repay-ment events or treatment actions. The linear intensitydynamics and the assumed i.i.d. relative repaymentamounts should be considered a reasonable first-orderapproximation of reality.

5.2.2. Payback Probability. The DCS summarizesthe probability of receiving an aggregate repaymentin excess of a given threshold B over a given inter-val 4t1 t+�7. Figure 2 depicts this repayment probabil-ity in shades of grey with corresponding iso-quantilelines for a sample account (in the testing subgroup).

The DCS varies with time t. In the study period6011127 days, the account in Figure 2 made two repay-ments, $160.77 (18.2%) at T1 = 22 days and $150 (20%)at T2 = 53 days. In Figure 2(a), the payback probabilityçt1 t+15 days440%5 for repayment of at least 40% of theremaining outstanding balance over the next � = 15days, conditional on the information available up tothe current time, t = 35 days, is approximately 10%. InFigure 2(b), this probability, based on information ofup to t = 70 days, has increased to approximately 15%.This is primarily as a result of receiving a repayment(at T2 = 53) that updates both the account holder’swillingness-to-repay and ability-to-repay. The dottedlines indicate the expected values Vt1 t+� , normalized tothe interval 60117, for t ∈ 8351709 days and � ≤ 42 days.The DCS not only incorporates real-time information,it is also flexible with respect to variations of predic-tion horizon and payback threshold, the desired val-ues of which can be readily substituted. Thus, at eachtime in the collections process the collections man-ager obtains a dynamic account valuation based on allavailable information.

5.2.3. Prediction Performance. The predictive ac-curacy of the DCS, for both MLE- and GMM-identifiedmodel parameters, is illustrated by the ROC curvesin Figure 3, where account value is used for rank-ing based on collectability. The ROC curves for the

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Figure 3 Scoring Performance: (a)–(c) ROC Curves at Different Times in the Collections Process; (d) AUC

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

False-positive

00.10.20.30.40.50.60.70.80.91.0

(c) (d)

Tru

e-po

sitiv

e

Week 16

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00

0.10.20.30.40.50.60.70.80.91.0

DCS (MLE)DCS (GMM)uBISBIS

False-positive

Tru

e-po

sitiv

e

Week 0(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

False-positive

00.10.20.30.40.50.60.70.80.91.0

Tru

e-po

sitiv

e

Week 8(b)

DCS (MLE)DCS (GMM)uBISBIS

DCS (MLE)DCS (GMM)uBISBIS

BIS (dotted) and the uBIS (dashed) serve as baselinereference. The true-positive rate corresponds to thepercentage of payers who are identified correctly byeach model. A “payer” denotes an account with atleast one observed payment in the interval 4t1 t + �7with � = 42 days. As shown in Figure 3, the DCS per-forms significantly better than both BIS and uBIS, eventhough it uses considerably less information. The tablein Figure 3(d) provides the area under the ROC curves(AUC) for each classifier, equivalent to the probabil-ity that a randomly chosen positive instance is rankedabove a randomly chosen negative instance.

5.2.4. Treatment Optimization. Because the dyna-mic repayment model includes account-treatmentactions, it is possible to use the account valuation

Figure 4 (a) Net Revenue and (b) Expected Gain from Commission Increase, as a Function of FICO for B0 ∈ 8$5001$1,5009, Given an InitialCommission Structure of 4c11 c25= 415%125%5

B

BB �c

B �cB �cB �c

in Proposition 3, as a function of the action pro-cess at = 6at111 at127

>, for treatment optimization. Morespecifically, the bank’s net revenues depend on theaccount-specific commission rates cm in the collectionphases m ∈ 81129. We limit attention here to the opti-mization of the commission rate (i.e., action at12), rep-resented by 4c11 c25.

A typical account in the data set has at time �1 = 0 aconstant commission rate of c1 = 15% in phase 1 andis moved after �2 = 120 days without full payment tophase 2 of the collections process at c2 = 25% commis-sion. We compare these modal rates with optimizedmodel predictions, given a capital cost of � = 10%.The net value, over the time interval 4t1 t + �7 witht = 0 and � = 360 days, as a function of FICO score

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Figure 5 (a) Net Revenue as a Function of the Commission Structure 4c11 c25; (b) Unit Revenue V01 � 4a5/B0 (Dashed Line) and Marginal Net Revenue(Solid Line) for a Symmetric Commission Structure 4c1 c5 in Terms of the Equivalent Price p = 1 − c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00

0.2

0.4

0.6

0.8

1.0

p = 1 – c

0

0.2

0.4

0.6

0.8

1.0

0 0.2 0.4 0.6 0.8 1.0

c1

c 2

0.3

0.440.40.42

0.460.48

0.5

0.5

0.40.42

0.440.46

0.48

0.480.46

0.440.42

0.4

0.30.3

0.2

0.10.10.1

0.2 0.2

0.515

(a) (b)

Net revenue

and outstanding balance, is shown in Figure 4(a);one observes that, per outstanding dollar, accountswith lower balances are more valuable. Furthermore,account holders with higher pre-default FICO scoresare more likely to pay, so accounts with both a highFICO score and a low outstanding balance are amongthe most collectable (on a per-dollar basis). Figure 4(b)illustrates the marginal gain/loss when the commis-sion rates go up, for each phase m ∈ 81129. All of thecurves are unimodal with a peak in the neighborhoodof FICO = 550 (not shown for two curves). Thus, themarginal net yield increases in the FICO score untilaccount quality is sufficiently high, so the price paidfor additional collection effort no longer warrants thedifference in collection yield.

Figure 5(a) shows the net revenue of an account(B0 = $1,000, FICO = 600) as a function of the com-mission rates c2 ≥ c1, with maximum at 4c∗

11 c∗25 =

430%130%5. The symmetry of the commission struc-ture follows from the underlying simplification thatthe agencies are homogenous. Indeed, to avoid ini-tially overpaying for collection effort and a prema-ture termination of the process, necessarily c∗

2 ≥ c∗1 .

Because the two collection agencies are homogenous,the phase 2 commission c2 cannot be strictly largerthan the phase 1 commission c1 at the optimum.26

Based on this, it is possible to optimally price anaccount for leasing it out to collection agencies. Letc = c1 = c2 denote a symmetric commission structure.The optimal commission rate c∗ can be obtained anal-ogously to the standard monopoly pricing rule (see,e.g., Tirole 1988) by viewing p = 1 − c as the price(i.e., unit revenue) for an account leased to a collec-tion agency. Because the outstanding balance is sunkat the time of placement, the opportunity cost of col-lection is zero, and the realized value (i.e., revenue)

26 In practice, the agency in the second phase is often more special-ized (e.g., a law firm) and therefore requires a higher commissionrate because of its higher collection cost.

for the bank becomes 41 − c5V01 �4a5. Given the priceelasticity of the collection value,

�4c5= −1 − c

V01 �4a5

¡V01 �4a5

¡41 − c5= −

p

V01 �4a5

¡V01 �4a5

¡p1

we obtain that�4c∗5= 1 (16)

at the optimum. It is therefore best for the bank tocharge a commission rate that sets the price elas-ticity of the collection value equal to 1. Figure 5(b)illustrates the monopoly pricing rule (16), implying arevenue-maximizing price of p∗ = 1 − c∗ = 70% in theexample.

6. ConclusionThe dynamic collectability score (DCS) method devel-oped in this paper can be viewed as the backbone ofcollections optimization. In contrast to standard bank-internal scores, the DCS reflects the actual repaymentprobability conditional on a repayment threshold, acollection horizon, and scheduled account-treatmentactions. The DCS is therefore a very flexible mea-sure which can be tailored for a range of applications,such as the following six examples. First, it can beused in its base version as a forecasting and scor-ing tool, which by itself outperforms BIS methodsin terms of the classic “payer” versus “non-payer”classification, as shown in §5. Second, the DCS fig-ures prominently in the valuation formula of Propo-sition 3. Third, because the DCS is determined con-ditional on an action sequence, it can be used tooptimize the collections process. The key differencefrom standard methods is that our model allowsfor a simple forward simulation of repayment pro-cesses. The details of the associated stochastic opti-mization problem are discussed in a companion paper(Chehrazi et al. 2014). Fourth, the DCS valuation for-mula can be used to optimize compensation for col-lection agencies in terms of their optimal commissionrate, at the account level. If the commission is tied to

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the DCS, then, because of the increased efficiency inallocating effort, both the collection agency and thecredit-card issuing bank stand to gain. Fifth, becauseof the dynamic nature of the DCS, it is instrumentalin the decision of when and how to settle a credit-card account. Last but not least, more resolution andreliability in the valuation of delinquent accounts alsoimprove the life-cycle assessment of account value,and thus ultimately enhance the quality of a bank’sunderwriting decisions. Hence, by backward induc-tion, a functioning DCS method can be expected topositively impact the entire account life cycle.

AcknowledgmentsThe authors thank Darrell Duffie, Damir Filipovic, KayGiesecke, Peter Glynn, Robert Phillips, Ken Singleton, AssafZeevi, as well as the credit-collection and risk teams atAmerican Express, Citicorp and J. P. Morgan Chase for help-ful comments and discussions.

Appendix A. Analytical Details

Feasibility DomainLet �4x5 ≡ �0 + �1x with 4�01�15 ∈ �nX × �nX×nX , �4x5 ≡

èv4x5 with è ∈ �nX×nX nonsingular (Duffie and Kan 1996),and v4x5 ≡ diag 4u1/2

1 4x51 0 0 0 1u1/2nX 4x55 with

ul4x5≡ ul10 +u>

l11x1

where ul10 ∈ � and ul11 ∈ �nX , for all l ∈ 811 0 0 0 1nX9. Thedomain D of Xt can be described as a polyhedron,

D= 8x ∈�nX 2 min8u14x51 0 0 0 1unX4x59≥ 090

Setting ��4x5≡ ��10 +�>�11x, and denoting

u0 =

u110

000

unX 10

and u1 =

u>111

000

u>nX 11

1

the following result provides conditions that ensure the pos-itivity of the intensity process �t .

Proposition A1 (Positivity of Intensity). Given initialdata 4�01X05 in �++ ×D, the intensity process �t is positive27 ifand only if there exists a vector � ∈ �nX

+ such that ��10 ≥ �>u0and �>

�11 = �>u1.

Proof. To ensure the positivity of �t it is necessary andsufficient to ensure that d�t/dt is positive at the boundary,where �t = 0. The latter holds if and only if ��4x5 ≥ 0 onD, given that �1�11�2 ≥ 0 (as required by the model). Wetherefore need to show that

��4x5= ��10 +�>

�11x ≥ 01 ∀x ∈D= 8x ∈�nX 2 u0 +u1x ≥ 091

if and only ifG= 8� ∈�nX+ 2 ��10 ≥ �>u0 and �>

�11 = �>u19 6= �.

27 Positivity is required only almost everywhere, i.e., everywhereexcept for a zero-measure set of times.

Figure A.1 Empirical CDF of R

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00

0.2

0.4

0.6

0.8

1.0

R

CD

F

Sufficiency. If there exists � ∈G, then

�4x5= ��10 +�>

�11x = ��10 +�>u1x ≥ �>(u0 + u1x

)≥ 01

for any x ∈D.Necessity. Assume 4��101��115 is such that ��4x5 ≥ 0 for

any x ∈ D. We now show that G is nonempty. To simplifythe exposition, we restrict attention to the case where u1has full rank.28 The following intermediate result impliesthe nonemptiness of G.

Claim: There exists � ∈�nX+ such that �>

�11 = �>u1.

Proof. Since u1 has full rank, there exists � ∈ �nX suchthat �>

�11 = �>u1. It is therefore enough to show that � ≥ 0.If this is not true, then there is a unit vector el in �nX with�>el < 0. Using the Gram-Schmidt process if necessary, it ispossible to find a vector xl ∈�nX such that u1xl = el. For anyscalar � > 0 and x ∈ D, one can then conclude x + �xl ∈ Dwhile at the same time for � large enough ��4x + �xl5 =

��10 + �>�11x + ��>el becomes negative. This yields a con-

tradiction, and thus establishes the validity of the claim.

Since u1 has full rank, there exists x ∈ D such thatu0 + u1x = 0. Using the claim, there exists � ∈ �nX

+ suchthat �>

�11 = �>u1, whence

0 ≤ ��4x5= ��10 +�>

�11x = ��10 +�>u1x = ��10 −�>u00

Thus, one also obtains ��10 ≥ �>u0 that implies that � ∈G 6=

�, completing our proof. �

Affine Jump-Diffusion Processes withDeterministic JumpsConsider the general case

dXt = �4Xt5 dt + �4Xt5 dWt + �1 dJt + �2 dat1 (A1)

where Wt is a standard Wiener process in �nX ; �4x5= �0 +

�1x in which 4�01 �15 ∈ �nX × �n

X×n

X ; �4x5�>4x5 = �0 +∑nX

l=1 �l1xl where

(�01 4�

l15

nX

l=1

)∈ �n

X×n

X × �nX

×nX

×nX ; J is an

�nJ -dimensional pure jump process whose jumps ãJ are

i.i.d., and its arrival intensity is an affine function of X,of the form �4Xt5 = �0 + �>

1 Xt with 4�01 �15 ∈ � × �nX ; a

is a deterministic, right-continuous jump process, whosejumps are at fixed, increasing arrival times �m, for m ≥ 1;and 4�11 �25 ∈ �n

X×n

J × �nX

×na . The following proposition

28 A complete proof is available upon request. Important for theresults in this paper is the sufficiency of G 6= � for the positivityof �t . The necessity provides a guarantee that the nonemptinessof G constitutes a minimal requirement for the admissibility of theintensity process.

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provides a semianalytic expression for the transform

¬4t1 t+�74�01 �11 �1 X54= Ɛ4e−

∫ t+�t �4Xs 5 ds+�>Xt+� �¦t51 (A2)

where � ∈�nX , �4x5= �0 + �>

1 x with �0 ∈�, �1 ∈�nX .

Proposition A2 (Affine Transform). Let �m, m ∈

8m¯1 0 0 0 1 m9, denote the jump arrival times of as , for s ∈ 4t1 t+�7,

and let 4�1 �5 be the (unique) solution to the initial-value problem

˙�s + �>

0 �s + �>

s �0�>

0 �s/2 = �0 + 41 − Ɛ6e�>s �1ãJ 75 �01

�t+� = 01 (A3)

˙�s + �>

1 �s + �>

s �1�>

1 �s/2 = �1 + 41 − Ɛ6e�>s �1ãJ 75 �11

�t+� = �1 (A4)

on 6t1 t + �7. If the conditions(i) Ɛ6

∫ t+�

0 ��s�ds7 <�, with �s4= �4Xs5ës4Ɛ6e

�>s �1ãJ 7− 15,

(ii) Ɛ64∫ t+�

0 ��s�2 ds51/27 <�, with �s

4= ës�

>s �4Xs5,

(iii) Ɛ6�ët+��7 <�, with ës4= exp6�s + �>

s Xs −∫ s

0 �4Xs5ds7,hold, then

¬4t1 t+�74�01 �11 �1 X5= exp[�t + �>

t Xt +

m∑m=m

¯

�>

�m�2ãa�m

]1

where ãas4= as − as− .

Proof. Using the law of iterated expectations, Equa-tion (A2) becomes

Ɛ6e−∫ t+�t �4Xs 5 ds+�>Xt+� �¦t7

= Ɛ6e−∫ �mt �4Xs 5 dsƐ6e

−∫ t+��m

�4Xs 5 ds+�>Xt+��¦�m

7 �¦t70

For s ∈ 6�m1 t + �7, the process as is constant, so Xs exhibitsaffine jump-diffusion dynamics. Thus, by Proposition 1 ofDuffie et al. (2000),29

Ɛ6e−∫ �mt �4Xs 5 dsƐ6e

−∫ t+��m

�4Xs 5 ds+�>Xt+��¦�m

7 �¦t7

= Ɛ6e−∫ �mt �4Xs 5 ds+�4�m5+�>

�mX�m �¦t7 (A5)

where for s ∈ 6�m1 t + �7:

˙�s +�>

0 �s +�>

s �0�>

0 �s/2= �0 +41−Ɛ6e�>s �1ãJ 75�01 �t+� =01

˙�s +�>

1 �s +�>

s �1�>

1 �s/2= �1 +41−Ɛ6e�>s �1ãJ 75�11 �t+� = � 0

Rewriting Equation (A5) in the form

Ɛ6e−∫ �mt �4Xs 5 ds+�

�m+�>

�mX�m �¦t7

= Ɛ6e−∫ �mt �4Xs 5 ds+�

�m+�>

�m4X

�m−�2ãa�m

5+�>

�m�2ãa�m �¦t7

= e��m

+�>

�m�2ãa�m Ɛ6e

−∫ �mt �4Xs 5 ds+�>

�m4X

�m−�2ãa�m

5�¦t71

one obtains that the effect of the deterministic jump fac-tor a is decoupled from the dynamics of Xs for s ∈

6�m−11 �m7. Rewriting −∫ �mt �4Xs5 ds as −

∫ �m−1t �4Xs5 ds −∫ �m

�m−1�4Xs5 ds, by using the law of iterated expectations and

again Proposition 1 by Duffie et al. (2000), one obtains

Ɛ6e−∫ t+�t �4Xs 5ds+�>Xt+� �¦t7

=e��m

+�>

�m�2ãa�m Ɛ6e

−∫ �m−1t �4Xs 5ds+�

�m−1+�>

�m−1X�m−1 �¦t71 (A6)

29 The result remains valid for 4�01 �15 ∈�×�nX .

where 4�s1 �s5, for all s ∈ 6�m−11 �m7, is determined as solu-tion of the initial-value problem

˙�s +�>

0 �s +�>

s �0�>

0 �s/2= �0 +41−Ɛ6e�>s �1ãJ 75�01 ��m

=01

˙�s +�>

1 �s +�>

s �1�>

1 �s/2= �1 +41−Ɛ6e�>s �1ãJ 75�11 ��m

= ��m0

Since the ODEs governing the dynamics of �s and �s on6�m1 t + �7 and �s and �s on 6�m−11 �m7 are identical, Equa-tion (A6) simplifies to

Ɛ6e−∫ t+�t �4Xs 5 ds+�>Xt+� �¦t7

= e��m−1

+∑m

m=m−1 �>

�m�2ãa�m

× Ɛ6e−∫ �m−1t �4Xs 5 ds+�>

�m−14X

�m−1−�2ãa�m−1

5�¦t71

where 4�s1 �s5 solves the initial-value problem

˙�s + �>

0 �s + �>

s �0�>

0 �s/2 = �0 + 41 − Ɛ6e�>s �1ãJ 75�01

�t+� = 01 (A7)

˙�s + �>

1 �s + �>

s �1�>

1 �s/2 = �1 + 41 − Ɛ6e�>s �1ãJ 75�11

�t+� = �1 (A8)

for all s ∈ 6�m−11 t+ �7. Continuing the extension toward theleft in the same manner yields

Ɛ6e−∫ t+�t �4Xs 5 ds+�>Xt+� �¦t7= e

�t+�>t Xt+

∑mm=m

¯�>

�m�2ãa�m 1

where 4�1 �5 solves the initial-value problem (A7)–(A8) on6t1 t + �7. �

Proof of Proposition 1. The proof follows from theproof of Proposition A2. Specifically, let Xt = 6Qt1�t1X

>t 7

>,where

dQt = dQt1

d�t = �4��4Xt5−�t5 dt + �>

1 dJt + �>

2 dat1

dXt = �4Xt5 dt +�4Xt5 dWt 0

Then (referring to Table B.1 in Appendix B for any extranotational definition) it is

�0 =

0���10

�0

∈�2+nX 1

�1 =

0 0 0

0 −� ��>�11

04nX×25 �1

∈�42+nX 5×42+nX 51 (A9)

�0 = 0, �1 = e2 with el the lth standard Euclidean basis vec-tor of �2+nX , and

�1 =

1 0 00 �11 �12

0nX×3

∈�42+nX 5×31 �2 =

01×na

�>2

0nX×na

∈�42+nX 5×na 0

(A10)

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Chehrazi and Weber: Dynamic Valuation of Delinquent Credit-Card Accounts3094 Management Science 61(12), pp. 3077–3096, © 2015 INFORMS

Futhermore, ãJ = 6Q111R7> and �4x5 = èv4x5, withè1 v4x5 ∈�42+nX 5×42+nX 5 and

è=

042×25 042×nX 5

04nX×25 è

1 v4x5=

042×25 042×nX 5

04nX×25 v4x5

1

(A11)where è1v4x5 ∈ �nX×nX are detailed at the outset of thisappendix. Hence, one obtains

Ɛ6e�4Qt+�−Qt 5 �¦t7

=¬4t1 t+�740101�e11X5·¬4t1 t740101−�e11X5

=exp[−ã�t1 t+� −ã�>

t1 t+� Xt +

m∑m=m

¯

�>

�m�2ãa�m

]1

which concludes the proof. �

Proof of Proposition 2. Using the CCF of Qt+� − Qt ,provided by Proposition 1, the Fourier transform of the CDFof Qt+� −Qt can be written as

��d4w5+1jw

·

(exp

[−ã�t1 t+� 4−jw5−ã�>

t1 t+� 4−jw5Xt

+

m∑m=m

¯

�>

�m4−jw5�2ãa�m

])1

where �d4 · 5 denotes the Dirac delta function (a distribu-tion). Taking the inverse Fourier transform of the previousequation at log41 −B/Bt5 yields the result. �

Proof of Proposition 3. Let � > 0 be a discount factor.The value of an account over 4t1 t + �7 is defined as theexpected sum of discounted payments,

Vt1 t+� 4a5 = Ɛ

[ ∑i2 Ti∈4t1 t+�7

e−�4Ti−t5Zi

∣∣∣∣¦t1 a

]

= Ɛ

[∫4t1 t+�7×60117

e−�4s−t5rBs−N4ds1 dr5

∣∣∣∣¦t1 a

]1

where N4ds1 dr5 is the equivalent random-measure repre-sentation of N = 4Ti1Ri1 i ≥ 15 and Bs = B0 −

∑i Zi18Ti≤s9

denotes the remaining outstanding balance at s ∈ 4t1 t + �7.Using Fubini’s theorem, we obtain that

Ɛ

[∫4t1 t+�7

e−�4s−t5∫60117

rBs−N4ds1 dr5

∣∣∣∣¦t1 a

]

=

∫4t1 t+�7

e−�4s−t5Ɛ

[∫60117

rBs−N4ds1 dr5

∣∣∣∣¦t1 a

]0

Furthermore,

∫4t1 t+s7

Ɛ

[∫60117

rBs−N4ds1 dr5

∣∣∣∣¦t1 a

]

= Ɛ

[∫4t1 t+s7×60117

rBs−N4ds1 dr5

∣∣∣∣¦t1 a

]

= Ɛ

[ ∑Ti∈4t1 t+s7

Zi

∣∣∣∣¦t1 a

]

= Bt41 − Ɛ6exp4Qt+s −Qt5 �¦t1 a750

By Proposition 1, it is

Ɛ6exp4Qt+s −Qt5 �¦t1a7

=exp(−ã�t+�−s1t+� −ã�>

t+�−s1t+� Xt +∑

�m∈4t1t+s7

�>

�−s+�m�>

2 ãa�i

)1

where 4�1 �5 solves the initial-value problem

˙�s + �>

0 �s + �>

s �0�>

0 �s/2 = 01 �t+� = 01

˙�s + �>

1 �s + �>

s �1�>

1 �s/2 = 41 − Ɛ6e�>s �1ãJ 75e21 �t+� = e11

on 6t1 t+�7. Denoting 1−Ɛ6exp4Qt+s −Qt5 �¦t1 a7 by �t4s � a5,which is a nondecreasing right-continuous function, theaccount value is given by Equation (11). �

Model IdentificationIn our implementation of the model, in addition to the fea-sibility constraints of Proposition A1, a goodness-of-fit con-straint is imposed to bound the parameter domain. Thisconstraint relies on the fact that any point process Nt (withN� = �) can be transformed into a standard Poisson pro-cess using an appropriate change of time (Papangelou 1972).By imposing an upper bound for the distance of the trans-formed point process to a standard Poisson process one canbound ä�. In addition to facilitating the numerical imple-mentation, constraining the parameter space is necessary,since differentiating Equation (13) with respect to � and sub-sequently taking the limit for �→ � yields

lim�→�

�2 ¡

¡�L44T k

i 11 ≤ i ≤Nkhk5 � 4Xs1 s ∈ 601hk751

4Rki 11 ≤ i ≤Nk

hk51 ��5 > 01

so that for � large enough the likelihood function isstrictly increasing. In our implementation, we use theCramér-von Mises goodness-of-fit measure, CVM44T k

i 11 ≤ i ≤ Nk

hk5kk=11 ��5, as proximity metric (Lehmann and

Romano 2005).

Censored-Data TreatmentIn our data set, a significant portion (32%) of accounts donot pay in full; for those the last interarrival times aregenerally censored over the given finite study period. Thistruncation phenomenon introduces a noticeable spike in theempirical distribution of payment interarrival times Sk

i =

T ki −T k

i−1 as accounts are moved to the next collection phasefollowing some defined period of inactivity. Consequently,any �� (for which the intensity process is positive) fails tosatisfy the Cramér-von Mises goodness-of-fit measure at the5% level. This problem is resolved in our implementation byvirtue of the fact that under the true intensity process �t4��5the Sk

i 4��5 are i.i.d. exponentially distributed. Thus, becauseexponential random variables are memoryless, the censoredportion of the last interarrival payment time of each accountcan be reconstructed by a draw of an exponential randomvariable. In our implementation, �� is considered feasible ifboth the corresponding intensity process is positive and theCVM test does not fail (with censored data reconstructed asbefore).

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Chehrazi and Weber: Dynamic Valuation of Delinquent Credit-Card AccountsManagement Science 61(12), pp. 3077–3096, © 2015 INFORMS 3095

Appendix B. Notation

Table B.1 Summary of Main Notation, in the Order of Appearance

Symbol Description Range

t Current time �+

i Index of repayment event �Ti Arrival time of ith repayment �+

Zi Amount of ith repayment �+

Bt 3 Bi Outstanding balance (at time t ; at t = Ti ) �+

Ri4= Zi/Bi−1 Relative repayment at t = Ti 60117

FR Distribution of relative repayment Ri ��

r Expected relative repayment (r 4= Ɛ6Ri 7) 60117

Nt4=∑

i 18Ti≤t9 Repayment counting process (willingness-to-repay) �Rt

4=∑

i Ri18Ti≤t9 Cumulative relative repayment process (ability-to-repay) �+

Qi4= log41 −Ri 5 Normalized logarithmic balance at time t = Ti �−

FQ Distribution of normalized logarithmic balance Qi ��

Qt4=∑

i Qi18Ti≤t9 Cumulative normalized logarithmic balance process �−

Jt4= 6Nt 1Rt 7

> Repayment process ��+

at Account-treatment schedule �na+

m Index of account-treatment action �+

�m Time of mth scheduled treatment action �+

�t Intensity process �+

y Account-specific covariates �ny

��4x54= ��10 + �>

�11x Long-run steady-state intensity �+

�4y 54= �0 + �1y Mean reversion rate for intensity process �+

�14y 54= 6�114y 51 �124y 57

> Sensitivity of �t to Jt �2+

�24y 5 Sensitivity of �t to at �na+

Wt Wiener process �nX

Xt Group-covariate process �nX

�4x54= �0 +�1x Drift term of group covariates �nX

��>4x54= �0 +

∑nXl=1 �

l1xl Diffusion term of group covariates �nX×nX

+

� 4= 8¦t 2 t ∈�+9 Information filtration

¦t Available information at t �el Unit vector in the standard Euclidean base of �nX 8e11 0 0 0 1 enX 9el Unit vector in the standard Euclidean base of �n

X 8e11 0 0 0 1 enX9

j Imaginary unit 8√

−19s Generic time index �+

4�s1 �s5 Solution of the ODE (6)–(7) �2

� Prediction horizon �++

�t 4s � a5 Cash-flow measure 60117� Discount factor �+

Vt1 t+� Expected � -horizon present value �+

vt1 t+� Realized � -horizon present value �+

çt1 t+� Repayment probability on 4t1 t + � 7 (DCS) 60117K 4

= 811 0 0 0 1 k9 Account portfolio 2�

k Index of account (in portfolio K= 811 0 0 0 1 k9) Khk Study horizon for account k �++

�4= 6��3 �X 7 Model-parameter tuple ä� ×äX

� Price elasticity of collection value �

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