rex yuxing du, wagner a. kamakura, & carl f. mela …mela/bio/papers/du...tomer transacts with a...

94Journal of MarketingVol. 71 (April 2007), 94–113

© 2007, American Marketing AssociationISSN: 0022-2429 (print), 1547-7185 (electronic)

Rex Yuxing Du, Wagner A. Kamakura, & Carl F. Mela

Size and Share of Customer WalletMany companies collect substantial information about their interactions with their customers. Yet information abouttheir customers’ transactions with competing firms is often sparse or nonexistent. As a result, firms are oftencompelled to manage customer relationships from an inward view of their customers. However, the empiricalanalysis in this study indicates that (1) the volume of customers’ transactions within a firm has little correlation withthe volume of their transactions with the firm’s competitors and (2) a small percentage of customers account for alarge portion of all the external transactions, suggesting the considerable potential to increase sales if thesecustomers can be correctly identified and incentivized to switch. Thus, the authors argue for a more outward viewin customer relationship management and develop a list augmentation–based approach to augment firms’ internalrecords with insights into their customers’ relationships with competing firms, including the size of each customer’swallet and the firm’s share of it.

Rex Yuxing Du is Assistant Professor of Marketing, Terry College of Busi-ness, University of Georgia (e-mail: [email protected]). Wagner A.Kamakura is Ford Motor Company Professor of Global Marketing (e-mail:[email protected]), and Carl F. Mela is Professor of Marketing (e-mail:[email protected]), Fuqua School of Business, Duke University. This articleis adapted from part of the first author’s dissertation, and he is grateful toProfessor Rick Staelin for his continued guidance and mentoring through-out the development of the dissertation. The authors also thank the Tera-data Center for Customer Relationship Management at Duke Universityfor its support.

To read and contribute to reader and author dialogue on JM, visithttp://www.marketingpower.com/jmblog.

Our study attempts to redress this limitation by develop-ing an approach to determine how much business a cus-tomer transacts with a focal firm’s competitors or, in indus-try parlance, to estimate the focal firm’s share of thecustomer’s total wallet. Such an approach could prove use-ful in CRM. For example, without information pertaining toa customer’s demand from competing firms, firms cannotdiscern a customer with high firm share and small total wal-let from a customer with low firm share and large total wal-let. Yet the marketing prescriptions for each differ consider-ably. For the former customer, the marketing prescriptionmight be to generate new primary demands (if possible),and for the latter customer, it might be more appropriate toencourage switching to the firm’s existing products.

Overview of Our ApproachA way to address the problem of not knowing how muchbusiness a customer does with the competition is through aprocedure commonly known as “list augmentation” or“database augmentation,” which overlays data obtainedfrom customer surveys or secondary sources with existingdatabases (e.g., Crosby, Johnson, and Quinn 2002;Kamakura and Wedel 2003; Kamakura et al. 2003; Wyner2001). A typical list augmentation exercise involves the fol-lowing steps: First, the focal firm (often anonymously) sur-veys a random sample of its customers, collecting informa-tion that is not available from the firm’s internal database.Second, the information from the survey is linked to infor-mation already stored in the internal database (e.g., transac-tion history, demographics) to form a sample with a com-plete set of records. Third, from this sample, predictivemodels that leverage the correlation patterns between thesurvey results and the internal data can be developed.Finally, the best-performing model is applied to the remain-der of the customer database to produce individual-levelestimates of the survey results. Compared with the costs ofdeveloping and maintaining the customer database, thecosts of implementing list augmentation are fairly small.

Central to this general framework is the development ofan effective predictive model. In this study, by augmentingthe firm’s internal database with survey information on a

With continued advances in information technology,firms are capturing an expanding array of detailedrecords of their interactions with their individual

customers. Such information has been used to generate notonly behavioral insights but also important metrics, such ascustomer lifetime value and customer equity, and it isbecoming indispensable in guiding firms’ customer rela-tionship management (CRM) initiatives (Rust, Zeithaml,and Lemon 2000). Yet this abundance of internal (within thefirm) information is often accompanied by a dearth of infor-mation on external (outside the firm) customer activities. Asa result, firms are often compelled to manage customer rela-tionships using a view of their customers that is basedmostly on internal records. However, such an inward focuscould provide misleading measures of a customer’s marketpotential. For example, customers who appear to have highvalue based on internal records may have modest growthpotential if they have only limited requirements that areserved by competing firms. In contrast, customers who havea high transactional volume with competing firms may begood targets for growth to the extent that a firm can attract alarger share of their business. Bell and colleagues (2002)indicate that this lack of individual-level, industrywide con-sumer data is a primary barrier to CRM. In the absence ofsuch information, it may be difficult for customer loyaltyprograms, cross- and up-selling applications, targeted pro-motions, and many other marketing efforts to achieve thebest return on investment.

Size and Share of Customer Wallet / 95

sample of customers’ purchases from the firm’s competi-tors, we present three models for estimating a customer’stotal purchases in a category and how large a share of thesepurchases comes from the focal firm. To our knowledge,this is the first study in the marketing literature to use listaugmentation to impute wallet size and share of wallet.

We apply our models to a proprietary data set providedby a major U.S. bank. The data set contains information formore than 34,000 customers on their uses of ten categoriesof financial products that both the bank and its competitorsoffer. We first calibrate the models on a subsample withinformation on account balances both inside and outside thebank. We then use the calibrated models to predict total andshare of requirements in each category for a validation sam-ple using only inside balances (the outside balances providethe basis for validation), thus emulating the application ofthe models to the rest of the customer database, in whichinformation about outside balances is unavailable.

Key FindingsOne of our three models stands out in terms of its predictiveperformance and managerial interpretability. It correctlypredicts 72% of the time whether a customer uses a productoffered by the focal bank’s competitors, and it offers themost accurate estimates of total and share of categoryrequirements. Furthermore, it yields insights into cus-tomers’ share decisions that can be used in guiding futureCRM initiatives. We highlight several findings, some ofwhich we conjecture may be idiosyncratic to the bank understudy (e.g., Finding 3), whereas others may be generaliz-able across firms and industries (e.g., Findings 1, 2, and 4).

1. Longer relationships are not necessarily associated withlarger share of wallet; the correlation between customertenure and share of requirements is neutral or negative infour of the ten categories we analyze. This is consistentwith the argument that relationship duration and customershare should be considered two separate dimensions of cus-tomer relationship (Reinartz and Kumar 2003).

2. Customers with high share in one category also tend to havehigh share in another category, indicating that customers’share decisions are positively correlated and, thus, thepotential for positive externalities across categories.

3. Customers’ share and total purchase decisions are some-times negatively correlated, suggesting that for some cate-gories, customers with small shares within the focal firmtend to transact a large volume outside it. These customersmight represent significant opportunities for volume growthto the extent that the focal firm can induce them to switch.

4. Customers with higher incomes tend to balance share ofrequirements across firms. This may suggest either that thefocal firm is not serving such customers well or that cus-tomers with higher incomes have incentives to allocatebusiness across firms.

To investigate the managerial value of our proposedapproach further, we conduct a series of targeting simula-tions and find that substantial lifts in targeting efficiencycan be obtained by using estimated total and share of wallet.For example, 13% of customers in the validation sample areidentified as high-potential customers because their esti-mated total wallet is in the top quintile and their estimatedshare of wallet at the focal firm is below average. These

customers account for 53% of the validation sample’s finan-cial requirements that are fulfilled outside the focal bank,suggesting considerable potential for increasing revenue tothe extent that the focal firm can induce them to switch.

We proceed as follows: The next section discusses inmore detail the value of share of category requirements inmanaging relationships with individual customers. We thenpresent the three models for estimating total and share ofcategory requirements. This is followed by our empiricalillustration, in which we present the managerial problemand data, the models’ predictive performances and parame-ter estimates, and the results from several customer-targeting simulations. Finally, we summarize and discussdirections for further research.

Share of Category Requirements asan Outward-Looking Relationship

MetricCustomer relationship management efforts are often tar-geted toward a firm’s best customers, defined as those whocontribute the most to the firm’s bottom line. Such strate-gies are effective when firms want to strengthen relation-ships with “high-value” customers and mitigate the likeli-hood that these customers will defect to competing firms.However, it could be myopic to target such customers forrelationship development because profitability measures areblind to the relationships a customer might maintain withcompetitors, and there may be little correlation between thecustomer’s growth potential and his or her current or priornet contributions. Such low correlation may predominate inindustries in which customers maintain concurrent relation-ships with multiple providers. Yet the volume of a cus-tomer’s business with the competition represents an impor-tant source of his or her potential profitability. Indeed,several researchers have proposed or shown the linkbetween customer share and profitability. For example, Gar-land (2004) examines the role of share of wallet in predict-ing customer profitability, finding that it is the singlerelationship-based measure with the most impact on cus-tomer contribution. Bowman and Narayandas (2004) andKeiningham, Perkins-Munn, and colleagues (2005) proposethat share of wallet mediates the relationship between satis-faction and profits, an effect that Bowman and Narayandas(2004) empirically confirm. Reinartz, Thomas, and Kumar(2005) find that share of wallet positively affects customertenure and profitability. Zeithaml (2000) proposes a concep-tual model in which increased share of wallet is one of fourfactors that mediate the effect of customer retention on firmprofits.

Because scant empirical evidence exists regarding thedegree to which a customer’s purchases inside and outside afirm correlate, we exemplify this issue using informationwe obtained from a large sample of customers of a majorU.S. bank. For this sample, the bank knows each customer’sfinancial portfolio with both the bank and its competitors.We find that 81% of the sample’s internal assets (i.e., thesum of financial assets the sample customers havedeposited in the bank) come from customers who are thetop 20% in terms of internal asset (this is also known as the

96 / Journal of Marketing, April 2007

“80/20” rule). Yet these same customers account for only34% of the sample’s external assets (i.e., the sum of finan-cial assets the sample customers have deposited with thebank’s competitors). Moreover, the bank’s high-volumecustomers are often not the customers with the greatestgrowth potential; the correlation between internal assets andexternal assets is only .13. The pattern is more striking fordebt products (e.g., credit card, personal loan, mortgage);96% of the sample’s debts with the focal bank come fromcustomers who are the bank’s top 20% clientele in terms ofinternal debt; however, these customers account for only20% of the sample’s external debts. The correlationbetween internal debts and external debts is −.04. In short,we find little correlation between a customer’s internal andexternal requirements. Thus, any targeted CRM effortspredicated solely on internal information would miss manyhigh-potential customers who have large sums of assets anddebts outside the bank. This suggests that it is especiallydesirable to consider the bank’s share of a customer’s totalwallet in gauging the customer’s potential for growth.

Share of Category Requirements and Share ofWallet

To elaborate on the notion of customer share, we distin-guish between share of category requirements and share ofwallet. We define share of category requirements as theratio of (1) a customer’s requirements for a particular cate-gory of products from a focal supplier to (2) the customer’stotal requirements for products from all suppliers in thecategory (i.e., total category requirements). For a firm thatoffers multiple categories of products to its customers, wedefine the firm’s share of wallet of a customer as the shareof total requirements across all the product categories thefocal firm offers. Thus, we define share of category require-ments at the category level and share of wallet as an aggre-gate measure across all the categories in which the focalfirm competes. We use share of category requirementsbecause of its long-standing status in the literature andshare of wallet because of its popularity amongpractitioners.

Share of category requirements has long been used as ametric of brand loyalty in the context of consumer packagedgoods (Fader and Schmittlein 1993), and it is becoming animportant metric of customer relationship strength underdifferent names, depending on the industry that is using it(Malthouse and Wang 1998). For example, financial ser-vices companies call it “share of wallet,” the automobileindustry calls it “share of garage,” the fashion industry callsit “share of closet,” and the media industry calls it “share ofeyeballs.” Several articles in marketing have studied factorsthat can affect customer share. For example, Bhattacharyaand colleagues (1996) explore the relationship betweenmarketing-mix variables and brand-level share of categoryrequirements. Bowman and Narayandas (2001) assess theimpact of customer-initiated contacts on share of categoryrequirements. Keiningham, Perkins-Munn, and Evans(2003) analyze the impact of customer satisfaction on shareof wallet in a business-to-business environment. Verhoef(2003) investigates the differential effects of relationshipperceptions and marketing instruments on customer reten-

tion and customer share. In the financial services area,researchers have examined the relationship between cus-tomer characteristics and share of wallet; for example, Bau-mann, Burton, and Elliott (2005) use survey data to identifycustomer characteristics that are associated with high shareof wallet in retail banking, and Garland and Gendall (2004)use share of wallet as a factor in predicting customerbehavior.

Share of Category Requirements as a Basis forCustomer Segmentation

Several researchers have proposed the use of share of walletas a basis for segmentation. For example, Anderson andNarus (2003) propose a framework for strategically pursu-ing a customer’s business by selecting those with significantincremental share available and superior projected growthin need. Beaujean, Cremers, and Pereira (2005) propose theuse of loyalty and share of wallet as bases for customer seg-mentation. Similarly, Reinartz and Kumar (2003) proposesegmenting customers on the basis of customer tenure andshare of wallet, and Keiningham, Vavra, and colleagues(2005) propose the combination of share of wallet and cus-tomer lifetime value for the same purpose. Although thiswork is largely theoretical, subsequently, we demonstrateempirically how share of wallet as a segmentation basis canyield actionable strategic insights.

By combining customers’ share of wallet with their totalwallet and using the bank data (which we describe in moredetail subsequently), Figure 1 depicts a 5 × 5 segmentationscheme that offers an outward-looking view of the bank’scustomer base. The first dimension represents the focalbank’s share of a customer’s total financial needs (i.e.,assets plus debts) divided into the following brackets: (0,20%], (20%, 40%], (40%, 60%], (60%, 80], and (80%,100%]. The second dimension is the customer’s total finan-cial needs by quintiles. The size of each circle in the figureindicates the percentage of customers who fall into the cor-responding segment. Under this scheme, customers in dif-ferent segments can be viewed as a portfolio of assets withdifferent growth prospects.

Customers in the upper-right-hand corner of Figure 1are “ideal” from the bank’s perspective because they havelarge requirements for financial products and fulfill most ofthose requirements with the bank’s offerings. However,these customers represent a small fraction of the bank’s cus-tomer base, as is indicated by the size of their circles. Twogrowth strategies are indicated: The first is to migrate cus-tomers in the upper-left-hand corner toward the right (i.e.,gaining a larger share of their business), and the second is tomigrate those in the lower-right-hand corner upward (i.e.,increasing their category requirements). We conjecture thatthe former strategy might prove more fruitful in the shortrun because a customer’s category requirements are largelydriven by his or her intrinsic needs and are constrained byfinancial resources. In our data, the two largest cells in theupper-left-hand corner of Figure 1 account for 28% of cus-tomers. In terms of short-term growth potential in assetsand debts, customers in these cells account for 72%. Thus,we contend that customers in the upper-left-hand corner(i.e., those with a small share of wallet but a large total wal-


FIGURE 1Segmentation Based on Share of Wallet and Total Wallet

let) could be the best prospects for marketers seeking near-term growth.

Unfortunately, it is difficult to implement such anoutward-looking segmentation and targeting strategy inpractice because firms rarely have accurate customer-levelmeasures for share of wallet and total wallet. Without suchinformation, customer segments in the upper-left-hand cor-ner of Figure 1 are indistinguishable from those in the lowerright by just looking at requirements fulfilled inside thefocal firm. The main purpose of our study is to propose anapproach that helps firms estimate these two importantmeasures for their customers.

Share of Category Requirements as a Detectorfor “Silent Attrition”

Aside from being a segmentation basis, share of wallet canbe viewed as an indicator of relationship strength, whichcan be used in early detection of customer attrition.

Whereas customers who close accounts or move all busi-ness to another supplier are clearly defecting, those whosepurchases represent a smaller share of their total expendi-tures are also “defectors” (Reichheld 1996). For example,Coyles and Gokey (2002) find that 5% of customers at abank close their checking accounts annually, taking withthem 3% of the bank’s total balances. However, every year,the 35% of customers who reduced their shares with thebank significantly cost the bank 24% of its total balances.This effect was present in all 16 industries they studied(including airlines, banking, and consumer products) anddominant in two-thirds of them. This suggests that partialdefection or silent attrition (Malthouse and Wang 1998)caused by decreasing share of wallet can be more seriousthan attrition, which is detected only when a customer hasdecided to no longer use the firm’s product or service.Accordingly, marketers need to monitor share of wallet onan ongoing basis, and decreasing share of wallet should be


viewed as an early warning signal that a relationship isgradually decaying. Compared with marketing interven-tions aimed at preventing attrition, marketing efforts thatattempt to prevent share-of-wallet losses could be moreproactive and, therefore, more effective. Unfortunately,most firms’ databases do not contain up-to-date estimates ofshare of wallet for individual customers.

Share of Category Requirements and Cross-Selling

Since the early 1990s (Kamakura, Ramaswami, and Srivas-tava 1991), a growing literature has evolved on the topic ofcross-selling (Jarrar and Neely 2002; Lau, Chow, and Liu2004; Lau et al. 2003). This literature is focused on identi-fying the next product to offer a customer on the basis ofthe products previously purchased and the patterns ofpurchase incidence across all customers. More recently,researchers (Kamakura et al. 2003; Li, Sun, and Wilcox2005) have recognized that customers have multiple rela-tionships and have proposed list augmentation approachesto make cross-selling recommendations that consider thepossibility that the customer has already purchased theproduct elsewhere. To our knowledge, most (if not all)cross-selling models in the marketing literature focus onlyon the purchase incidence decision. Although cross-sellingis an important tool for developing customers, the identifi-cation of cross-selling prospects covers only one aspect(purchase incidence) of customer development, overlookingother avenues to enhance the value of a customer. This isparticularly true for industries such as banking and retail-ing, in which customers maintain concurrent relationshipswith multiple vendors. In these industries, customer rela-tionships can be developed not only by having the customermake purchases in categories in which he or she has notbought before but also by increasing the firm’s share of thecustomer’s requirements in categories in which he or shehas already made purchases. The list augmentation modelswe propose and test consider the customer’s decision toadopt a product category similar to these cross-selling mod-els, but our models also impute the total volume consumedin the product category and the share of that volume thecustomer devotes to a particular firm. In other words, weextend the cross-selling models from predicting only thepurchase incidence decision to predicting the quantity andincidence decision, thus providing a more informative esti-mate of a customer’s growth potential.

Summary

In light of the foregoing discussion about the advantagesand challenges of using share of wallet in CRM, the goal ofthis article is to develop a predictive model through which afirm can use its internal records, supplemented with a smallsample of external records to estimate total and share ofrequirements in all categories (and, thus, total and share ofwallet) for all customers. This enables firms to extend exist-ing customer development initiatives, such as cross-selling,and to benefit from the segmentation, targeting, and attri-tion detection strategies we discussed previously. Next, wepresent three models to achieve this aim.

1Our approach differs from cross-selling or churn analyses inso-far as we jointly forecast category ownership, total, and share ofrequirements in the context of missing data, whereas the churnmodels that Neslin and colleagues (2006) report, for example,forecast ownership only in the context of complete data.

Models for Estimating Total andShare of Category Requirements

The Modeling TaskConsider a firm that offers products in J categories, withtransaction records for N customers. The firm knows (1)

how much requirements it serves customer n incategory j (superscript 1 denotes the focal firm), and (2)

a vector of other customer characteristics (e.g., inour empirical illustration, X consists of customer incomeand length of relationship with the focal firm). The firmdoes not have information on the size of cus-tomer n’s requirements in category j served by the firm’scompetitors (superscript 0 denotes outside the focal firm).To learn about customers’ outside requirements Y0, the firmconducts a survey among a random sample, I, of its N cus-tomers. The goal of such a survey is to collect two pieces ofinformation in each product category for each customer i insample I: the self-reported inside requirements and theself-reported outside requirements By obtaining self-reported inside requirements, the firm can ensure that theyare consistent with their internally recorded counterparts.After data are cleaned to ensure accuracy, the self-reportedoutside requirements can be linked to records in the firm’scustomer database. In summary, the firm has completeinformation (Y1, Y0, and X) for only a sample (I) of cus-tomers. For the other N – I customers, information on out-side requirements, Y0, is missing. The objective is todevelop a model that uses inside requirements, Y1, and cus-tomer characteristics, X, to estimate share of categoryrequirements (S) and total category requirements (T) forthese N – I customers, thus augmenting the firm’s databasewith estimates of the customers’ relationships with compet-ing firms and, consequently, their potential for growth.

We develop three models to achieve this objective.Model A predicts whether a customer will transact in a cate-gory with the focal firm’s competitors (i.e., the outside inci-dence decision) and, if so, the transaction volume (i.e., theoutside quantity decision). Model B extends Model A bysimultaneously modeling the customers’ incidence andquantity decisions both inside and outside the focal firmacross multiple product categories, thus explicitly allowingfor the possibility that these decisions are correlated. Mod-els A and B can be used to infer the focal firm’s share of acustomer’s category requirements by dividing the cus-tomer’s inside purchases over the sum of his or her insideand (predicted) outside purchases. In contrast, Model C pre-dicts the share allocation decision directly. It makes threesimultaneous predictions: (1) whether a customer will buyin a category, regardless of the vendor (i.e., the categoryownership decision), and, if so, (2) the amount to buy in thecategory (i.e., the total decision) and (3) how large a portionfrom the focal firm (i.e., the share decision), which can be0%, 100%, or anything in between.1 In the empirical sec-

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tion of this study, we calibrate all three models with thesame data and compare them on the basis of their predictiveperformances and managerial interpretability.

Model A: Modeling Incidence and QuantityOutside the Focal Firm

In Model A, we use purchases within the focal firm, alongwith other customer data available internally, to predict thevolume transacted outside the firm. Because the decisionsabout whether to buy and how much to buy from competi-tors may be driven by different underlying processes, wepropose the following Type II Tobit regression model ofincidence and quantity conditional on incidence (Amemiya1985):

where

• is a latent variable that captures the likelihood of cus-tomer i buying in category j from a competing firm. In otherwords, this latent variable determines the incidence whethercustomer i has a relationship outside the focal firm incategory j;

•Conditional on the customer having a relationship with acompetitor, is another latent variable that determines thequantity the customer purchases from the competitor.Because the empirical distributions of conditional quantitiesare often skewed, they are assumed to be an exponentialfunction of the latent variable and thus enter the likelihoodfunction on a log-transformed scale;

•xi is a K-element vector of observed characteristics of cus-tomer i; and is an incidence indi-cator function, such that else

•εij1 and εij2 are normally distributed with the covariance

•and for identification purposes, we assume that σj1 = 1 (thiscovariance structure implies that the incidence and quantitydecisions might be correlated, which can occur if there areunobserved factors that affect both decisions); and

•α, β, γ, r, and σ are the model parameters, and there are (2 +2K + 4J + 2)J of them to be estimated.

Model A is relatively simpler than the subsequent onesin that it models the customer’s purchase decision in eachcategory independently. Notably, there are likely to beunobserved factors (which therefore cannot be included inModel A as predictors) that affect customers’ purchasesfrom both inside and outside the focal firm. If theimpact of these unobserved factors is substantial, it can leadto biased parameter estimates and, thus, to poor predictionsof total and share of category requirements. Nonetheless,we still view Model A as a highly practical solution and astrong benchmark for the other two models we propose

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2With the correlation coefficients (rj) constrained to zero and alogit link function for incidence, Model A is equivalent to a logis-tic model for the “whether-to-buy” decision coupled with a log-linear regression model for the “how-much-to-buy” decision con-ditioned on incidence. The results show that Model A and thelogistic/log-linear model with no correlations yield largely identi-cal predictions; thus, we do not discuss this null model further.

because it can be easily estimated as a Type II Tobit regres-sion, and it leads to straightforward data imputations.2

Model B: Modeling Incidence and Quantity Insideand Outside the Focal Firm

Model B generalizes Model A by jointly modeling inci-dence and quantity both inside and outside the focal firm.Modeling decisions inside and outside the firm jointlyenable potential correlations between these decisions to beconsidered. Such correlations might arise from omitted fac-tors that influence inside and outside purchases (e.g.,changes in personal status). To the extent that these correla-tions manifest, inside purchases become informative ofoutside purchases and therefore can abet the imputation ofexternal demand and customer share. Model B proceeds byassuming that customers make four decisions in each prod-uct category: (1) whether to buy from the focal firm; (2) ifso, how much; (3) whether to buy from competitors; and (4)if so, how much. Model B allows these decisions to be dri-ven by four different underlying processes that might becorrelated not only with one another but also across cate-gories because of the impact of a common set of unob-served customer-specific factors. For example, a recentdivorce (for which we do not have data) might lead toreduced purchases from all firms in all categories, suggest-ing that these decisions could be positively correlated. For-mally, we model the incidence and quantity decisions ineach category, both inside and outside the focal firm, asfollows:

Equation 2 indicates that customer i’s incidence andquantity decisions in category j are determined by fourlatent variables, The top half of Equation 2 implies thatcustomer i will buy a product in category j from the focalfirm if is greater than zero. When this happens, the cus-tomer’s requirements for the focal firm’s product are anexponential function of which implies that the condi-tional quantities of inside requirements enter the likeli-hood function log-transformed. Similarly, the bottom halfof Equation 2 states that customer i will buy a product incategory j from other firms if is greater than zero, and ifso, the customer’s requirements served outside the focalfirm are an exponential function of

As indicated previously, we attempt to capture theimpact of unobserved customer-specific factors that simul-taneously affect customers’ incidence and quantity deci-sions made across product categories and inside and outside

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the focal firm. Formally, we adopt the following structureon the four latent variables and

where

•xi is defined as previously, a K-element vector of observedcharacteristics of customer i;

•zi is a P-element vector that captures unobserved, individual-specific factors that affect the incidence and quantity deci-sions of customer i; each element of zi is assumed to be i.i.d.standard normal; εij1, εij2, εij3, and εij4 are the stochastic com-ponents in each decision that are normally distributed withvariances and respectively; for identifica-tion purposes, it is assumed that σj1 = σj3 = 1; and

•α, β, γ, and σ are the model parameters; P, the dimensionalityof unobserved customer factors, is to be determined empiri-cally; given K, P, and J, ([1 + K + P] × 4 + 2)J parametersneed to be estimated.

The foregoing structure provides a parsimonious repre-sentation of the correlation pattern between the 4 × J (i.e.,the number of decisions in each category times the numberof categories) latent variables, Substantial parsimonywill be gained when J is large and decisions across thesecategories are interrelated. As with Model A, a caveat ofModel B is that its parameter estimates cannot be directlyinterpreted to learn about how customers decide share allo-cations across vendors. Accordingly, in Model C, we modelcustomers’ share decisions explicitly.

Model C: Modeling Category Ownership, Total,and the Focal Firm’s Share

In Model C, we develop an approach that can be used topredict a customer’s category demand and the focal firm’sshare directly when only inside purchases are recorded. Ourmotivation for developing such a model is twofold. First,we want to understand how observed customer characteris-tics affect the total and share-of-category-requirementsdecisions. This is important because different customercharacteristics may be related to a customer’s categorydemand and share allocation in different ways, and mar-keters who are interested in understanding these relation-ships can use such insights to devise customer developmentstrategies accordingly. The second reason for modeling totaland share of category requirements explicitly is that such aspecification aligns itself well with the choice-modeling lit-erature that decomposes consumer purchase decisions into“incidence, quantity, and choice” (Gupta 1988), in whichconsumers decide whether to buy in a category and, if so,how much and, finally, which brands to choose. In ModelC, we attempt to decompose customers’ category require-ment decisions in a similar way—namely, “ownership,total, and share.” Of note, our approach and context is dif-ferent from that of Gupta (1988) and others insofar as (1)the goal is to impute these decisions with incomplete infor-mation, (2) we consider these decisions made across multi-

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η α

ij j i j i j ij

ij j i

x z

x

= + + +

= +

′ ′

′′ ′

′ ′

β γ ε

η α β γ ε

j i j ij

ij j i j i j i

z

x z

2 2 2

3 3 3 3

+ +

= + + +*jj

ij j i j i j ijx z

3

4 4 4 4 4η α β γ ε* ,= + + +′ ′

ηij4* :ηij3

* ,ηij2* ,ηij1

* ,

3White tests for heteroskedasticity on the log-transformednonzero balances show that the residuals of the model are not afunction of the predicted log total balances, which indicates thatthe homoskedasticity assumption with regard to the stochasticcomponents conforms to the data.

ple categories, and (3) the choice and share decisionsrequire different treatments.

Consequently, Model C assumes that a customer facesdecisions of ownership (whether to own a product cate-gory), total (the total category requirements if he or shedecides to own), and share (the share of the total require-ments, if any, to be served by the focal firm). Formally,

Equation 4 posits that customer i’s purchase decisionsin category j are governed by three latent variables, Thetop portion of Equation 4 states that the customer will buy aproduct in the category only if is greater than zero (i.e.,category ownership), and when this happens, the customer’stotal requirements Tij are an exponential function of which implies that the conditional quantities of totalrequirements Tij enter the likelihood function log-transformed. The bottom portion of Equation 4 implies thatthe customer will allocate some of his or her purchases tothe focal firm if is greater than zero. When this happens,the customer might have all his or her requirements servedby the focal firm (if is greater than or equal to one) or ashare of them that is equal to Thus, we model share ofcategory requirements as a distribution truncated at 0 and 1.Furthermore, we assume that the three latent variables

and are functions of a common set of factors,observed and unobserved, with a structure as follows:

where

•xi and zi are defined as previously, denoting K observed char-acteristics and P unobserved factors, respectively, associatedwith customer i, with each element of zi assumed to be i.i.d.standard normal; zi can also be interpreted in the terminologyof factor analysis as customer i’s scores on P latent factors, inwhich γj are the factor loadings (which we demonstrate howto interpret in the “Results” section); εij1, εij2, and εij3 are thestochastic components in each decision, normally distributedwith variances and respectively;3 for identifica-tion purposes, we assumed that σj1 = 1; and

•α, β, γ, and σ are the model parameters; P, the dimensionalityof unobserved customer factors, is to be determined empiri-cally; given K, P, and J, ([1 + K + P] × 3 + 2)J parametersneed to be estimated.

We denote the variance–covariance matrix of the latentvector as Λ(3J × 3J), arising from 3 decisions by J cate-ηi

*

σ j32 ,σ j2

2 ,σ j12 ,

( ) *

*

5 1 1 1 1 1

2 2

η α β γ ε

η α

ij j i j i j ij

ij j i

x z

x

= + + +

= +

′ ′

′′ ′

′ ′

β γ ε

η α β γ ε

j i j ij

ij j i j i j i

z

x z

2 2 2

3 3 3 3

+ +

= + + +*jj3 ,

ηij3*ηij2

* ,ηij1

* ,

ηij3* .

ηij3*

ηij3*

ηij2* ,

ηij1*

ηij* .

( ) , exp( );* *4 01 2if then T else Tij ij ij iη η> = jj

ij ij ijif then S else if

=

≤ = ≥

0

0 0 13 3η η* *, ; ,, ;

.*

then S

else S

ij

ij ij

=

=

1

3η


4We do not have data for additional behavioral variables, suchas the number of branch visits and checks withdrawn. However,the variables we do employ (income and tenure) generalizebeyond the banking industry. Adding industry-specific variables aspredictors would serve to increase the effectiveness of ourapproach.

gories. Thus, for the jth and gth categories (j and g in J) andthe hth and lth decisions (ownership, total, or share), theforegoing structure implies that the variance and covariancebetween and can be expressed as follows:

Equation 6 depicts the role of γ, the 3J × P factor loadings,in reducing a large 3J × 3J variance–covariance matrix intoa smaller 3J × P factor space as Λ = γ × γ′ + Σ, where Σ is adiagonal matrix with as entries. As such, the structuredepicted in Equation 5 provides a parsimonious representa-tion of the nature and strength of the correlations between3 × J decisions, namely, ownership, total, and share deci-sions made by each customer in J product categories. More-over, the estimates of factor loadings lend themselves tosubstantive interpretation and graphical display, which weillustrate subsequently in our application. Detailed proce-dures for estimating this model and for imputation appear inAppendixes A and B, respectively.

Data for Empirical IllustrationData CollectionA major U.S. bank provided the data we used for ourempirical illustration. This bank provided us with informa-tion for 34,142 of their customers regarding their balancesoutside the bank in ten categories includingnoninterest checking, interest checking, savings, certificatesof deposit (CDs), personal managed investments, car loan,personal loan, line of credit, credit card loan, and mortgage.A third-party research firm compiled these data in an ongo-ing market audit study. Each observation from this auditdetails when the information was collected, and only house-holds that had at least one account with positive balancewith the bank were included to ensure that these customerswere active. For the periods during which the market auditswere conducted (i.e., from 1999 Q3 to 2002 Q2, withapproximately 4500 households each quarter), this informa-tion on customer external relationship was aligned withrecords on the balances these households had inside thebank in the ten categories thus providing aunique data set that includes both total (T1, …, T10) andshare of (S1, …, S10) category requirements.

The data set provided by the bank also contains twoobserved customer characteristics—annual householdincome and customer tenure—that we use as predictorvariables (x1 and x2, respectively) in the empirical illustra-tion of our proposed models.4 Household income is pretaxand includes income from both salaries and interest andinvestment returns. Income-related information is garneredfrom multiple sources, including Acxiom and customerself-reports (e.g., loan applications), and it is updated peri-

( , ),Y Y11

101...,

( ),Y Y10

100, ...,

σ jh2

( ) var( ) cov( ,* *6 2η γ γ σ η ηijh jh jh jh ijh i= ′ + or gg jh gl l* ) .= ′γ γ

ηigl*ηijh

*

5Many firms may not be able to afford a large-scale survey oncustomers’ external transactions. In such cases, because of its effi-ciency, K-fold cross-validation could be more preferable than theholdout sample cross-validation.

odically through a proprietary “householding” process.Given its skewed nature, household income is log-transformed. Customer tenure is measured in the number ofyears since the first account was opened at the focal bank.

Because most firms in other industries would not havethe benefit of market audit studies conducted by a third-party research supplier, they would need to conduct a sur-vey (often anonymously) on their own with a sample oftheir customers to obtain data on the volume of businessthese customers do with competitors. For this sample ofcustomers, with complete information on their relationshipswith the firm and its competitors, our proposed models canbe calibrated and then used to impute total and share ofcategory requirements for all customers. To improve theaccuracy of the imputation results, it is desirable to checkfor self-reporting errors in the survey data. Because firmscannot obtain transactional data from their competitors, it isdifficult to check directly the error rates in the sample cus-tomers’ self-reported data on the business they transact out-side the focal firm. An indirect approach is to collect self-reported data on the business the sample customers transactinside the focal firm and to ensure that these data align withinternal records. Accordingly, we compared the accountbalances inside the bank, which we obtained from internalrecords, with the customers’ self-reports, which weobtained through the market audit study. We found littlesystematic discrepancies; the mean reporting error (marketaudit reports – internal records) is not significantly differentfrom zero in nine of the ten categories, except for that ofsavings. Therefore, we presume that the balances outsidethe bank from the market audit study also exhibit little sys-tematic error.

Study Design

We apportion our data randomly into a calibration samplecomposed of 23,957 customers and a validation sample of10,185 customers. This apportionment is intended to reflectthe data available to firms in practice, in which they havecomplete data for only a subset of their customers (analo-gous to the calibration sample) and incomplete data for therest of their customer base (analogous to the validation sam-ple).5 Accordingly, for the calibration sample, we use all theinformation available, including account balances inside thefocal bank (Y1), household income (x1), and customertenure (x2), as well as account balances outside the focalbank (Y0). The validation sample mimics the remainder ofthe customer base for whom the focal bank does not knowbalances the customers might keep with competitors. Wefirst use the calibration sample to fit the proposed models.Then, in the validation sample, we apply the calibratedmodels to the internal data (i.e., inside balances, householdincome, and customer tenure) to impute the outside bal-ances and therefore predict total (T) and share of (S) cate-gory requirements. Finally, we evaluate the models’ predic-tive performances in the validation sample by comparing


TABLE 1Structure of the Data Set Used in Empirical Illustration

Inside the Bank Outside the Bank

Calibration Sample with 23,957CustomersTotal and share of category require-

ments available by combining inter-nal balances with external balancesfrom a market audit study conductedby a third-party research firm

Internal balances in ten accounts

Household income

Length of relationship with the bank

External balances in ten accounts fromthe market audit study

(Thus, total and share of categoryrequirements observed)

Validation Sample with 10,185 CustomersMimicking the remainder of the cus-

tomer base in which information ontotal and share of category require-ments is unavailable

Internal balances in ten accounts

Household income

Length of relationship with the bank

Total and share of category requirementsto be imputed

(Performance evaluated in relation toavailable data from the market audit

study)

the imputed T and S against their observed counterparts.Table 1 summarizes this design.

Summary Statistics

Table 2 reports several summary statistics of the data forboth the calibration and the validation samples. Not surpris-ingly, the two subsamples are similar. Column 2 shows theaverage category ownership of each type of product. Col-umn 3 shows the average total balances conditional on own-ership. Among customers who have positive total balances,Column 4 shows the percentage of customers who havezero balance with the focal bank (thus, share = 0). Column5 reports the percentage of customers who have positivebalances both inside and outside the bank (thus, 0 < share <1). Approximately 60% of the bank’s customers have finan-cial assets inside and outside the bank. Similarly, approxi-mately 50% of the bank’s customers have financial debtsboth inside and outside the bank. Column 6 shows the per-centage of customers who are exclusive customers of thefocal bank (thus, share = 1). Approximately one in five ofthe bank’s customers have 100% of their requirements forfinancial assets served by the bank. Fewer than one in tendo so when it comes to fulfilling requirements for financialdebts. Together, Columns 4–6 suggest the importance oftreating the distribution of share decisions as truncated atboth 0 and 1.

Finally, Column 7 shows the bank’s average share ofrequirements in each category. The bank’s deposit products(interest and noninterest checking, savings, and CDs) per-form best in terms of obtaining a large share of customers’business. Loan products, except for line of credit, havemuch smaller shares than deposit products. The bank per-forms most poorly with respect to attracting customers’investment dollars. Taking this information as a whole, thebank has only approximately 20% of its existing customers’total wallet, reflecting high competitive intensity in many ofthe bank’s categories. This competition amplifies the needto use accurate estimates of share of category requirements

to ascertain potential targets across the bank’s spectrum ofproduct categories.

The other two variables included in our empirical illus-tration are the customers’ household income and tenurewith the focal bank. The mean, median, and standard devia-tion of income are $54,566, $48,086, and $36,198, respec-tively, for the calibration sample and $54,859, $48,776, and$35,684, respectively, for the validation sample. The mean,median, and standard deviation of customer tenure (inyears) are 9.4, 10, and 3.8, respectively, for the calibrationsample and 9.2, 10, and 3.9, respectively, for the validationsample.

ResultsModel ComparisonWe begin our analysis by comparing the predictive perfor-mances of Models A, B, and C in the validation sample. Asa basis for comparison, we also consider a log-linear regres-sion of the form ln(Y0

ij + 1) = αj + xi′βj + Ind(Y1i)′γj0 +

ln(Y1i)′γj1 + εij, because this model can be

readily estimated with ordinary least squares.We consider the ability of each model to predict total

and share of category requirements for each separate cate-gory, as well as for all assets together, all debts together,and all categories as a whole (i.e., share of wallet and totalwallet). We consider three dimensions: outside productownership, total category requirements (wallet size), andshare of category requirements (share of wallet). As Table3, Panels A and B, indicate, the log-linear model’s perfor-mance on these dimensions is dominated by the other mod-els. Given that Model A, similar to the log-linear model, canbe estimated using off-the-shelf software and dominates thelatter, we refrain from further discussion of the log-linearmodel. We organize our discussion by each dimension.

Predicting outside product ownership. Table 3, Panel A,compares the models’ abilities to predict outside product

ε σij jN~ ( , ),0 2


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tag

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l ass

ets

99.8

99.7

95,1

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51.5

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chec

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52.0

51.7

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159,

420

38.9

38.4

17.3

16.9

43.8

44.6

48.4

49.9

chec

king

(20,

862)

(17,

317)

Sav

ings

79.5

79.4

19,9

1218

,800

43.2

42.8

23.6

24.6

33.2

32.6

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(231

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87.3

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79,8

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48.4

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loan

46.3

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b Sta

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6A hit occurs either when the customer has positive externalbalances and the model predicts that the likelihood of this occur-rence will exceed 50% or when the customer has zero balancesoutside the bank and the model predicts that the likelihood of posi-tive external balances will be below 50%. Depending on the rela-tive costs of misclassification (false positive versus false negative),the threshold (50%) can be adjusted upward or downward.

7We also calculated the MAD of the log-transformed predictedtotal category requirements, and the performance of Model Cremained much better than the other two models.

ownership in terms of the hit rate (i.e., the percentage oftrue positives and negatives), the percentage of false posi-tives, and the percentage of false negatives.6 The resultsshow that all three models perform equally well in terms ofpredicting whether a customer has positive balances outsidethe bank in a particular category; hit rates, false-positiverates, and false-negative rates average approximately72~73%, 11~12%, and 16~17%, respectively.

Predicting total category requirements. To compare themodels’ performance in predicting total category require-ments, we calculate the mean absolute deviation (MAD) ofthe predicted total category requirements.7 To reflect thevariability inherent in our data, we also calculate the MADfor a naive model, in which the predicted total categoryrequirements are simply the account balances inside thebank plus the average account balances outside the bank.Table 3, Panel B (Columns 2–6), reports the naive model’sMAD and the percentage improvement (reduction) in MADrelative to the naive model for the various predictive models(which is equal to 1 − MAD_Model/MAD_Naive). Theresults show that Model C performs much better than Mod-els A and B in predicting the exact sizes of total categoryrequirements (or, equivalently, requirements served outsidethe focal bank). This is the case for total wallet, total assets,and total debts, as well as for all ten individual productcategories.

Predicting share of category requirements. For thiscomparison, we calculate the MAD of the predicted shareof category requirements. By definition, for a category witha zero account balance in the focal bank, share of require-ments for that category is either zero (when balance outsidethe bank in the category is greater than zero) or not defined(when balance outside the bank in the category is alsozero); consequently, accuracy in predicting share of require-ments for a particular category is relevant only with respectto observations for which there is a positive account balancein that category in the focal bank. Again, as a reference, weuse a naive model in which predicted share of categoryrequirements is equal to account balances inside the bankdivided by the sum of account balances inside the bank andthe average account balances outside the bank. Table 3,Panel B (Columns 7–11), reports the MAD of the naivemodel, along with the percentage improvement (reduction)in MAD relative to the naive model for the various predic-tive models. Again, Model C outperforms Models A and Bin predicting share of category requirements, which is thecase for total wallet, total assets, and total debts, as well asfor nine of the ten individual product categories. To our sur-

8We also estimated Model C with customer income and tenureomitted. The results indicate that most of the model’s predictivepower arises from internal transactions rather than demographicvariables, such as income or tenure.

prise, in eight of the ten categories, the simpler Model Adoes better than the more sophisticated Model B.

Taking the model comparison results reported in Table3, Panels A and B, as a whole, we conclude that if the goalis simply to predict whether a customer uses a productoffered by the competition (i.e., the cross-selling predictionthat Kamakura et al. [2003] propose), any of the three mod-els would be sufficient. Conversely, if the goal is to predictthe sizes of total and share of category requirements, ModelC is more accurate than the other models. Given that (1)total and share of category requirements are central toassessing customer potential, (2) Model C’s parameter esti-mates can be readily interpreted to gain insights into cus-tomers’ total and share decisions, and (3) Model C providesthe best prediction overall, we believe that Model C has thegreatest utility for imputing total and share of categoryrequirements.8 As such, to conserve space, we subsequentlyfocus on the estimation results for Model C. (Parameterestimates for Models A and B are available on request.)

Estimation Results for Model C

Estimation of Model C from the calibration sample for P =0–3 latent factors leads us to choose the model with P = 2.The Bayesian information criteria (BICs) for Model C withP = 0, 1, 2, and 3 and for the other two models appear inTable 4. Parameter estimates of the two-factor (P = 2)Model C appear in Table 5, Panel A (which includes theintercepts, α, and the coefficients on customer income andtenure, β) and Panel B (which reports the factor loadings,γ).

Income. Table 5, Panel A, indicates that customerincome positively and significantly correlates with bothownership and total decisions across the product categories.Customers with higher incomes are more likely to ownassets and debts (e.g., β1,income = .49 for savings, p < .01)and have higher requirements when they own them (e.g.,β2,income = .86 for savings, p < .01). Customers with higherincomes also have a significantly smaller share of theirrequirements for financial products with the bank understudy (e.g., β3,income = −.54 for savings, p < .01). We con-jecture that there could be many causes for this result. First,with greater requirements for financial products (Asher2001; Barr 2004) and fewer restrictions (Sharir 1974; Zeit-haml 1985), high-income customers stand to gain more byspreading their “nest eggs” across more financial institu-tions. Second, competition may be more intense for thehigh-income households’ wallets (Bielski 2004), givingthem more incentives and opportunities to fulfill theirrequirements at multiple competing institutions. Third, cus-tomer loyalty and satisfaction have been shown to be nega-tively linked to or moderated by income (Cooil et al. 2007;Homburg and Menon 2003; Korgaonkar, Lund, and Price1985; Zeithaml 1985). Finally, the bank under study may becomparatively less preferred by its higher-income clientele,for example, because of a weak market position relative to


TAB

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1628


TABLE 4Fit Measures for the Predictive Models in the Calibration Sample

Model A

Number of NotFactors Applicable P = 0 P = 1 P = 2a P = 3 P = 0 P = 1 P = 2a P = 3

Number of parameters 000,460 000,140 000,180 000,220 000,260 000,110 000,140 000,170 000,200

BIC 529,695 895,825 878,446 870,232 870,528 838,238 819,865 811,907 811,961aThe smallest BIC and, thus, the number of factors chosen.

Model CModel B

TABLE 5Model C Parameter Estimates

A: Observed Customer Characteristics (Number of Parameters = 90)

Total ShareOwnership (Conditional on Ownership) (Conditional on Ownership)

αα1 ββ1 ββ1 αα2 ββ2 ββ2 αα3 ββ3 ββ3Category Intercept Incomea Tenure Intercept Income Tenure Intercept Income Tenure

Noninterest checking .78 .07 –.09 8.01 .63 .05 1.04 –.61 .56Interest checking .16 1.12 .37 7.88 .76 .22 .46 –.65 .75Savings .91 .49 .06 8.39 .86 .38 .42 –.54 .27CDs –.96 .20 .24 9.59 .24 .29 –.03 –.76 .43Investments .14 .79 .09 9.99 1.06 .49 –2.51 –.26 .41Car loan –.09 .51 –.20 9.07 .33 –.02 –3.64 –.39 –.07Personal loan –.70 .28 –.20 9.09 .29 .02 –4.75 –1.00 1.68Line of credit –.96 .41 .11 7.88 .57 .25 1.18 –1.99 2.01Credit card .45 .24 –.08 7.71 .41 .00 –.31 –.14 .19Mortgage .03 .78 .11 10.79 .51 –.15 –10.53 –.40 –.61

B: Factor Loadings and Variances (Number of Parameters = 80)

Total ShareOwnership Decision (Conditional on Ownership) (Conditional on Ownership)

γγ1,1 γγ2,1 γγ1,2 γγ2,2 γγ1,3 γγ2,3Factor 1 Factor 2 Factor 1 Factor 2 Factor 1 Factor 2

Category Loading Loading Loading Loading σ2 SD Loading Loading σ3 SD

Noninterest checking –2.76 –.30 –.01 .00 1.05 .16 –1.20 1.52Interest checking 4.01 1.15 .67 .14 1.04 .75 –1.75 1.23Savings .16 .27 .50 .12 1.48 .31 –1.45 .70CDs .34 .15 .40 .09 1.28 .49 –2.01 2.09Investments .10 .16 .42 .13 1.53 .49 –.50 1.94Car loan –.25 –.17 –.03 .00 .97 –.11 .81 4.03Personal loan –.21 –.17 –.02 –.04 1.19 1.36 –.49 5.68Line of credit –.03 –.18 .09 .07 1.51 .95 –4.12 3.43Credit card –.21 –.22 –.12 –.15 1.12 .09 –.02 1.11Mortgage –.14 –.23 –.02 –.09 1.36 1.97 3.43 15.66aIncome was log-transformed.Notes: Parameter estimates that are significant at p < .01 are in bold.

other nonretail bank providers. In any case, the bank shouldattend to its lower share of its higher-income customers’wallets and determine whether it should use better-targetedmarketing treatments to redress this unfavorable position.

Tenure. The relationship between customer tenure andthe customer ownership, total, and share decisions is less

clear. Longer tenures are associated with a larger share ofcustomers’ checking, CD, investment, personal loan, andline-of-credit balances (as reflected in the positive and sig-nificant β3,tenure). In contrast, there is no significant correla-tion between customers’ tenure and the bank’s share of theirsavings, car loan, and credit card balances (as reflected in


the insignificant β3,tenure). For customers with longertenures, the bank has a smaller share of their mortgages(β3,tenure = −.61, p < .01). The lack of a positive relationshipbetween customer tenure and customer share in certaincategories (Cooil et al. [2007] also find this for anotherfinancial institution) may deserve the bank’s close attentionbecause it may imply that lengthening the relationship withcustomers does not necessarily translate into a larger shareof the customers’ business in these categories. To pinpointthe dynamics between customer tenure and customer share(Keiningham, Vavra, et al. 2005), however, the bank needsto collect longitudinal data on customer share (instead ofusing only cross-sectional data).

Factor loadings. Of the 60 estimated factor loadingsthat appear in Table 5, Panel B, 52 are significant at the p <.01 level, indicating that the latent factor structure capturesunobserved individual-specific factors that have caused thecorrelations among ownership, total, and share decisions inthe various product categories. To visualize these correla-tions, Figure 2, Panels A and B, plots the factor loadingsfrom Table 5, Panel B. The rendering of correlation patternsyields managerial insights into customers’ ownership, total-wallet, and share-of-wallet decisions among the bank’s ser-vices. For example, a positive correlation between two own-ership decisions means that a customer who is more likelyto own one service is also more likely to own another. InFigure 2, Panels A and B, any two vectors pointing in thesame (or opposite) direction indicate that decisions thesevectors represent are positively (or negatively) correlated;two vectors that are orthogonal to each other imply that thecorresponding decisions are independent of each other. Westandardized all the factor loadings plotted in the figure sothat the length of the vectors and the angle between themdirectly reflect the strength of correlation between the deci-sions they represent.

Figure 2, Panel A, captures correlations between theownership and the total-requirements decisions. Severalpoints are noteworthy:

•There is an assets-versus-debts dimension in terms of thetotal-requirements decisions. Interest checking, savings, CD,and investment decisions point in one direction, and creditcard, mortgage, and personal loan decisions point in the otherdirection. Thus, all else being equal, customers with morefinancial assets have fewer debts (except for line of credit).This also provides face validity for the correlation results;

•In terms of the ownership decision, customers who own anoninterest checking and/or a debt account are less likely toown an interest checking account. Ownership of interestchecking, savings, CD, and investment accounts are posi-tively correlated, thus reflecting some customers’ propensi-ties to save. Similarly, customers with one type of debt aremore likely to have other types of debt and are less likely tohave financial assets (except for noninterest checking); and

•The same factors that induce customers to assume debt leadthem to be deeper in debt (vectors representing debt owner-ship and total debt point in the same direction). A similarrelationship exists for assets.

More germane to targeting customers with high growthpotential, the factor loadings plotted in Figure 2, Panel B,depict the correlations between total-requirements and

share-of-requirements decisions across various productcategories, suggesting the following:

•Typically, share decisions are positively correlated acrossproduct categories, except for car loans. Thus, customers whodo the bulk of transactions with the bank in one category alsotend to do so in others, suggesting the potential for positiveexternalities across categories;

•In five of the ten categories (i.e., interest checking, savings,CDs, investments, and car loans), the bank has been efficientin targeting in the sense that the bank has larger shares of cus-tomers with larger requirements;

•However, for line of credit, total and share of requirementsare independent. This implies that the bank might not be tar-geting high-volume customers to promote its line of credit;and

•For personal loans, credit card, and mortgage, share of cate-gory requirements and total category requirements are nega-tively correlated. This suggests that the focal bank has a lowshare among high-requirement customers in these three cate-gories. Thus, these customers represent potentially attractivetargets.

Testing the Targeting Efficiency of OurRecommended Approach

To ascertain the efficacy of Model C for targeting customerswith large total requirements but small current shares, weassess Model C’s ability to identify customers whose totalassets (i.e., sum of checking, savings, CDs, and invest-ments) are in the top quintile but the bank’s shares are lowerthan average. Such customers could be denoted as being“platinum” in growth potential under Rust, Zeithaml, andLemon’s (2000) “customer pyramid” framework. Weexplore Model C’s ability to identify these customers byapplying the calibrated model to the validation sample andpredicting who is in this platinum-potential segment. Wethen compare these predictions with the actual data.

Although customers we predicted to have platinumpotential (i.e., the top quintile in total assets but lower thanaverage shares with the focal bank) constitute only 12% ofthe validation sample, they account for 37% of the valida-tion sample’s assets outside the bank. This leads to a lift of37%/12% = 3.1 for Model C. Comparable lifts for ModelsA and B are 2.6 and 2.5, respectively. Using the observeddata to identify the platinum-potential segment (i.e., perfecthindsight and maximum lift achievable) yields a lift of 4.7.Together, these results suggest that our list augmentationapproach enables firms to target customers with high poten-tial for short-term growth and that Model C performs betterthan Models A and B at this task.

Similar targeting exercises can also be performed fortotal debts or on a category-by-category basis. For example,compared with other categories, the focal bank has thesmallest share of its customers’ investment dollars (seeTable 2). Model C suggests that if the bank targets cus-tomers whose requirements for investment products are pre-dicted to be in the top quintile and whose share allocated tothe bank is predicted to be below average, the bank will beaddressing 18% of its customer base that account for only1% of the investment dollars already inside the bank but


B: Share and Total Decisions

FIGURE 2Factor Loadings

A: Ownership and Total Decisions

Decisions: ownership (_o): total (_t)

Assetsnon = noninterest checking

int = interest checkingsav = savingscds = CDs

invest = investments

Debtscar = car loan

pl = personal loanloc = line of creditcc = credit card

mor = mortgage

Decisions: share (_s): total (_t)

Assetsnon = noninterest checking

int = interest checkingsav = savingscds = CDs

invest = investments

Debtscar = car loan

pl = personal loanloc = line of creditcc = credit card

mor = mortgage


48% of the investment dollars that are currently outside thebank.

Finally, consider the segmentation scheme that Figure 1illustrates. Using estimated share of wallet and total walletto segment its customer base, the bank can target the seg-ment on the upper-left-hand corner—namely, those withtotal wallet in the top quintile and below-average share ofwallet. Our approach based on Model C yields 13% of thevalidation sample customers for targeting. These predictedtargets account for 51% of the validation sample’s financialrequirements that are currently served outside the focalbank. Again, this suggests substantial targeting efficiency(with a lift of 51%/13% = 3.9).

Concluding RemarksFirms lack individual-level, industrywide customer databecause they seldom have information about their cus-tomers’ relationships with competitors. As a result, manyCRM initiatives ignore customers’ transactions with com-peting firms, which we believe reflects a tendency of enter-prises taking the new customer-centric paradigm of market-ing to an inward-looking extreme. We provide directevidence that transaction levels inside a firm alone arelargely uninformative with respect to a customer’s transac-tion levels outside the firm. This highlights the risks ofgauging a customer’s potential value by relying solely onprofitability recorded in the internal database. Both internaland external data are necessary to distinguish customerswith large total market potential and small share from thosewith small total market potential and large share. Althoughthese two types of customers are indistinguishable on thebasis of internally recorded transactions alone, they call fordifferent relationship development strategies.

In general, it is infeasible to obtain transaction recordsfrom competing firms, but firms can obtain external rela-tionship data for a small sample of their customers throughcustomer surveys or other secondary sources. We present alist augmentation approach that enables firms to combineinternal records with these external data collected for a sam-ple of customers to calibrate a predictive model that canthen be used to estimate total and share of requirements forall customers in all categories. In our empirical illustration,we apply three such models to a proprietary data set pro-vided by a major U.S. bank, containing information formore than 34,000 customers’ holdings of financial productsin ten categories both inside and outside the bank. Thesedata enable us to test the performance of these models inimputing wallet size and share of wallet only on the basis ofinternal data. We demonstrate that firms can use ourapproach to segment customers on the basis of share of wal-let and total wallet and to discriminate between high-share,small-wallet customers and low-share, large-walletcustomers.

One of our three proposed models stands out in severalways: (1) the parsimony of the parameter space, (2) betterperformance in predicting the sizes of total and sharevariables and in targeting customers with large wallet butlow share, and (3) the ease in interpreting the parameterestimates for behavioral insights (e.g., how customers’

share decisions are correlated across product categories,how these decisions are correlated with total decisions andother customer characteristics). In addition, this modelextends extant approaches for imputing data missingbecause of subsampling (Kamakura and Wedel 2000; Littleand Rubin 2002; Schafer 1997) to the context in which mul-tiple unobserved customer decisions—which categories toselect, how much to expend in these categories, and howlarge a share for the focal vendor in each category—aresimultaneously imputed from the observed joint outcomesof these decisions (i.e., how much, if any, to expend on thefocal vendor’s offerings).

Our study also leads to some notable behavioral find-ings on customers’ share decisions. For example, we findthat the bank under study has a smaller share of its higher-income customers’ wallets. Thus, the bank may want toinvest more in its high-income clientele to gain a largershare of their business. In addition, longer customer tenureis not necessarily associated with larger customer share,which runs counter to the conventional wisdom that thelonger a customer stays with a company, the more he or shebuys from the company. Related to tenure, another avenuefor further exploration is to ascertain whether a decreasingshare of wallet can be used as an early warning signal toprevent customer attrition, which inevitably entails examin-ing longitudinal share-of-wallet movements. In general, webelieve that variations in share of requirements across cus-tomers, product categories, and periods should prove to be afertile ground for response modeling and for evaluating theimpacts of different marketing instruments and competitiveactions. Insights from these studies can be used to identifythe lead indicators and drivers behind such variations,which can then be used to optimize allocation of marketingresources in CRM.

Although we illustrate our approach using data from amajor consumer bank, caution should be exercised inextrapolating our specific findings to other institutions andindustries. Nonetheless, the modeling framework we devel-oped herein can be readily adapted to any industry in whichconsumers routinely fulfill their category requirements bypurchasing various products/services simultaneously frommultiple competing suppliers (e.g., retailing, direct market-ing). We believe that share of category requirements canpotentially play a more prominent role in improving theunderstanding and management of customer relationshipsin a broad range of industries, and our modeling frameworkcould be viewed as a step toward achieving this goal bygenerating insights into customers’ allocation of their busi-ness across vendors and by augmenting firms’ customerdatabases with reasonably accurate estimates of total andshare of category requirements for each individualcustomer.

Appendix AModel Estimation

In Appendix A, we focus on the estimator for Model C.First, let λijk � αjk + x′iβjk + z′iγjk for k = 1, 2, or 3. For allpossible scenarios of Tij and Sij, we have the following den-sities conditional on zi:


When Tij = 0, f(Tij, Sij) = f(εij1 ≤ –λij1), or formally,

when Tij > 0 and Sij = 1, f(Tij, Sij) = f[εij1 > –λij1, εij2 =ln(Tij) – λij2, εij3 ≥ 1 – λij3], or

when Tij > 0 and Sij = 0, f(Tij, Sij) = f[εij1 > –λij1, εij2 =ln(Tij) – λij2, εij3 ≤ –λij3], or

when Tij > 0 and 1 > Sij > 0, f(Tij, Sij) = f[εij1 > –λij1, εij2 =ln(Tij) – λij2, εij3 = Sij – λij3], or

The terms Φ(⋅) and ϕ(⋅) represent cumulative and probabil-ity density functions of the standard normal distribution,respectively. Because zi is unobservable from a modeler’sperspective, we need to integrate it out when deriving thelikelihood contribution of individual i:

where , , , and denote the product over four dif-ferent types of observations across all product categories;an observation belongs to Type 1 if Tij = 0, Type 2 if Tij > 0and Sij = 1, Type 3 if Tij > 0 and Sij = 0, and Type 4 if Tij >0 and 1 > Sij > 0.

For a given P, the dimensionality of the unobserved cus-tomer characteristics, we estimate Model C by maximizingthe sample likelihood function, which is defined as theproduct of the individual likelihood functions in EquationA5 across all i’s in the calibration subsample of I. We usesimulation to evaluate the integrals, which gives rise to asimulated maximum likelihood estimator in which the indi-vidual likelihood contributions in Equation A5 can beapproximated as

Π4Π3Π2Π1

( ) ( ) ( )[ ] [ ]A L Prob z Prob z Probi ij i ij i i5 11

22

= ∏∫ ∏ jj i

ij i ipp

P

i

z

Prob z z

[ ]

[ ]

( )

( ) ( )

33

44 1

1

∏

∏ ∏=

⋅ϕ dz ⋅⋅ ⋅ dziP ,

( ) ( , | ; , ) ( )ln(

A f T S z xT

ij ij i i ijj

41

12

Θ Φ= ×λσ

ϕ iij ij

j

j

ij ij

j

S

) −⎡

⎣⎢⎢

⎤

⎦⎥⎥

×−⎛

⎝⎜

⎞

⎠

λσ

σϕ

λσ

2

2

3

3

3

1⎟⎟ ≡ Prob zij i[ ]( ).4

( ) ( , | ; , ) ( )ln(

A f T S z xT

ij ij i i ijj

31

12

Θ Φ= ×λσ

ϕ iij ij

j

ij

jijProb

) −⎡

⎣⎢⎢

⎤

⎦⎥⎥

×−⎛

⎝⎜

⎞

⎠⎟ ≡

λσ

λσ

2

2

3

3

Φ [[ ]( );3 zi and

( ) ( , | ; , ) ( )ln(

A f T S z xT

ij ij i i ijj

21

12

Θ Φ= ×λσ

ϕ iij ij

j

ij

jiProb

) −⎡

⎣⎢⎢

⎤

⎦⎥⎥

×−⎛

⎝⎜

⎞

⎠⎟ ≡

λσ

λσ

2

2

3

3

1Φ jj iz[ ]( );2

( ) ( , | ; , ) ( ) [A f T S z x Probij ij i i ij ij1 1 1Θ Φ= − ≡λ ]]( );zi

where is the rth draw from a P-dimensional multivariatestandard normal distribution and R is the total number ofdraws taken. An appealing aspect of the simulated maxi-mum likelihood estimator is that the simulated likelihoodfunction in Equation A6 is twice differentiable, simplifyinglikelihood maximization with gradient-based search algo-rithms. We use a unique Halton sequence to simulate eachdimension of zi and assign a different set of draws to eachcustomer. In determining the appropriate R, we adopt thefollowing heuristic: For any given P, we begin with 50Pdraws and then double the number of draws until no signifi-cant improvements can be obtained in estimator efficiency(determined by comparing the standard errors of parameterestimates based on different numbers of draws, as well asthe associated likelihoods).

This discussion indicates how our model can be esti-mated for a given P. To determine the number of factors, weuse the BIC. More specifically, we begin with P = 1, thenP = 2, and so on, until the resulting BIC stops decreasing.

Appendix BImputation Procedure

After calibrating Model C, we are interested in makinginferences about the unobserved customer heterogeneities;that is, we impute zi, the latent factor scores of each indi-vidual customer, by combining model parameter estimatesand information available for individual i. Depending on theavailability of information at the individual level, the for-mula for imputing zi varies. We detail the imputation proce-dures for two scenarios of data availability.

In Scenario 1, both and are observed (or, equiva-lently, both and are observed). Recall that

for k = 1, 2, 3, or 4, as defined in AppendixA. We determine the marginal density function for and zi as follows:

Because and are observed, this density function canbe calculated for any zi, which enables us to impute cus-

SijTij

( ) ( , , , , , ; , )B f T S T S z xi i iJ iJ i i1 1 1 ... Θ�

= ff T S z x z

Pr

ij ij i ij

J

ipp

P

( , | ; , ) ( )Θ�

= =∏ ∏

=

1 1

ϕ

oob z Prob z

Prob z

ij i ij i

ij i

[ ] [ ]

[ ]

( ) ( )

( )

11

22

33

∏ ∏

∏∏ ∏

∏=

≡

Prob z

z Q z

ij i

ipp

P

i i

[ ]( )

( ) ( ).

44

1

ϕ

Sij,Tij,Πk ij k iProb z[ ]( )

Yij0Yij

1SijTij

zir

( ) ( ) ( )[ ] [ ]A LR

Prob z Prob zi ij ir

ij ir

r

61

11

22

≈ ∏ ∏= 11

33

44

R

ij ir

ij irProb z Prob z

∑

∏ ∏[ ] [ ]( ) ( ),


tomer i’s latent factor scores by maximizing Qi(zi) over zi.Formally, (Til, Sil, ..., TiJ, SiJ; xi, = arg max[Qi(zi)].

In Scenario 2, only is observed, whereas andare unobserved. Again, the marginal density function

for and zi is

When is available for , f(Y1il, ..., Y

1iJ, zi;

xi, is defined as

Thus, when only is observed, the central task inimputing zi is to calculate conditional density f(Y1

ij|zi; xi,Depending on whether or different for-

mulas are needed to calculate Whenit can result from two kinds of situations: (1) Tij = 0

or (2) Tij > 0 and Sij = 0. Formally, when

where When it can also result from two kinds of

situations: (1) or (2) Tij × Sij =Formally, when

Φ Φ( )ln( )

λσ

ϕλ

σijj

ij ij

j

Y1

2

12

2

1×−⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎧⎨⎪

⎩⎪

λλ

σ σ

ϕ

ij

j jS

ijY

S

3

3 21 0

1

1 1−⎛

⎝⎜

⎞

⎠⎟ +

⎛

⎝⎜

> >∫

ln⎜⎜

⎞

⎠⎟⎟−

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

−λ

σ σϕ

λ

σ

ij

j j

ijS2

2 3

31

jjij iS U z

31

⎛

⎝⎜

⎞

⎠⎟

⎫⎬⎪

⎭⎪≡d [ ]( ).

f T Y S z x f TY

SSij ij ij i i ij

ij( , | ; , ˆ ) ( |= = + =1

1

1 Θ iij

S

i i ij i iS z x f S S z x

1 0> >∫

= =, ; , ˆ ) ( | ; ,Θ dˆ )Θ S =

( ) ( | ; , ˆ ) ( , |B f Y z x f T Y S zij i i ij ij ij i5 11 1Θ = = = ;; , ˆ )

( ,

| ; ,

x

f T S Y S

z x

i

ij ij ij ij

i i

Θ

+ × = >

>

1 1

0 ˆ̂ )Θ =

Yij1 0> ,Y Sij ij

1 1 0and > > .T Y Sij ij ij= =1 1and

Yij1 0> ,

λ α β γijk jk i jk i jkx z k≡ + ′ + ′ =for 1, 2, or 3.

( ) ( | ; , ˆ )

( | ; , ˆ )

B f Y z x

f T z x

ij i i

ij i i

4

0

1 Θ

Θ= = + ff T S z xij ij i i

ij ij

( , | ; , ˆ )

( ) ( )

> =

= − +

0 0

1 1

Θ

Φ Φλ λ ××−⎛

⎝⎜

⎞

⎠⎟ ≡Φ

λ

σij

jij iU z

3

30[ ]( ),

Yij1 0= ,

Yij1 0= ,

f Y z xij i i( | ; , ˆ ).1 ΘYij

1 0> ,Yij1 0=ˆ ).Θ

Yij1

( ) ( , ; , ˆ )

argmax

B Z Y Y z xi i iJ i i3 11 1� , ..., Θ

= ff Y Y z x

f

i iJ i i( , , , ; , ˆ )

argmax

11 1... Θ⎡⎣ ⎤⎦

= (( | ; , ˆ ) ( )Y z x Zij i ij

J

ipp

P1

1 1

Θ= =∏ ∏

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

ϕ ..

ˆ )Θ∀jf Y z xij i i( | ; , ˆ )1 Θ

( ) ( , ; , ˆ ) ( |B f Y Y z x f Y zi iJ i i ij2 11 1 1, ..., Θ = ii i

j

J

ipp

P

x

Z

; , ˆ )

( ).

Θ=

=

∏

∏×

1

1

ϕ

Yij1

Yij0

Sij,Tij,Yij1

Θ�)Zi�

Combining this, we obtain the following:

where and denote the product over two differenttypes of observed across all product categories; anobservation belongs to Type I if and to Type IIif To impute the latent factor scores when only is observed, we can maximize Vi(zi) over zi. Formally,

(Y1il, ..., Y

1iJ; xi, = arg max[Vi(zi)].

We now discuss the prediction of and Given the estimated model parameters, and the imputed latentfactor scores, we can predict customer i’s total categoryrequirements, and shares of category requirements thatare served by the focal firm, conditional on (1) observed levels of category requirements that are served by the focalfirm, and (2) xi, observable customer characteristics.(Instead of plugging in an alternative is to integrate ziout, which in practice leads to predictions that are of no sig-nificant differences but could take significantly more timeto implement when the dimensionality of zi is high.) Therest of this section demonstrates the formulas for makingthese predictions.

For categories in which we can predict, forexample, the expected value of Tij conditional on Tij > 0,

and the expected value of Sij conditional on Tij > 0,

Similarly, for categories in which , we can makethe foregoing as follows: The expected value of Tij condi-tional on Tij > 0 is

( ) ( | , , ; , )B E T T Y Y z x

Y

ij ij ij i i

ij

9 0 01> = >

=

� �Θ

11

2

12

21

1⎛

⎝⎜

⎞

⎠⎟

⎧⎨⎪

⎩⎪

( ) −⎡

⎣⎢⎢

⎤

⎦⎥⎥σ

ϕλ

σj

ij ij

j

YlnΦΦ

λσ

σϕ

λ

ij

j

ij

j

ij

Y

S

Y

S

3

3

1

2

1

1

1

−⎛

⎝⎜

⎞

⎠⎟

+

⎛

⎝⎜

⎞

⎠⎟ −ln iij

j jS

ijS

2

2 31 0

1

σ σ

ϕλ

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

−

> >∫

33

3 2

121

σ σϕ

λ

σj j

ij ij

j

SY⎛

⎝⎜

⎞

⎠⎟

⎫⎬⎪

⎭⎪

⎧⎨⎪

⎩⎪

( ) −d

ln

22

⎡

⎣⎢⎢

⎤

⎦⎥⎥

Yij1 0>

( ) ( | , , ; , ) .B E S T Y z xij ij ij i i8 0 0 01> = =� �Θ

( ) ( | , , ; , ) expB E T T Y z xij ij ij i i i7 0 01> = =� �Θ λ jjj

22

2

2+

⎛

⎝⎜

⎞

⎠⎟

σ,

Yij1 0= ,

zi�,

Yij1,

Sij,Tij,

zi�,

Θ�,Sij.Tij

ˆ )ΘZi�

Yij1Yij

1 0> .Yij

1 0=Yij

1Π1Π0

( ) ( , ; , ) ( |B f Y Y z x f Y zi iJ i i ij6 11 1 1, ..., Θ� = ii i

j

J

ipp

P

ij i i

x

Z U z U

; , )

( ) ( )[ ]

Θ�

=

=

∏

∏ ∏=

1

10

0

ϕ jj i ipp

P

i iz Z V Z[ ]( ) ( ) ( ),11 1∏ ∏

=

≡ϕ


and the expected value of Sij conditional on Tij > 0 is

( ) ( | , , ; , )B E S T Y Y z xij ij ij i i10 0 0

1

1> = >

= ×

� �Θ

11 1

2

12

2

3

3σϕ

λ

σλσj

ij ij

j

ij

j

Yln( ) −⎡

⎣⎢⎢

⎤

⎦⎥⎥

−⎛

⎝⎜Φ

⎞⎞

⎠⎟

⎧⎨⎪

⎩⎪

3

3 21 0

1 1−⎛

⎝⎜

⎞

⎠⎟ +

> >∫Φ

λσ σ

ϕ

ij

j jS

lnnY

S S

ijij

j j

1

2

2 3

1

⎛

⎝⎜

⎞

⎠⎟ −

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

λ

σ σϕ

−−⎛

⎝⎜

⎞

⎠⎟

⎫⎬⎪

⎭⎪

λσ

ij

j

S3

3

d ,

+ ×

⎛

⎝⎜

⎞

⎠⎟ −

⎡

⎣

⎢⎢⎢S

Y

S

j

ijij

j

1

2

1

2

2σϕ

λ

σ

ln

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

−⎛

⎝⎜

⎞

⎠⎟

> >∫

1 03

3

3

1

d

Sj

ij

j

S

σ

ϕλ

σSS

Y

j

ij ij

j

⎫⎬⎪

⎭⎪

⎧⎨⎪

⎩⎪

( ) −⎡

⎣⎢⎢

⎤

⎦⎥⎥

1

2

12

2σϕ

λ

σ

λ

ln

Φ iij

j j

ijij

j

Y

S3

3 2

1

2

2

1 1−⎛

⎝⎜

⎞

⎠⎟ +

⎛

⎝⎜

⎞

⎠⎟ −

⎡

σ σϕ

λ

σ

ln

⎣⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

−⎛

⎝⎜

> >∫

1 0

3

3

3

1

S

j

ij

j

S

σϕ

λσ

⎞⎞

⎠⎟

⎫⎬⎪

⎭⎪dS .

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