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http://pubsonline.informs.org Citibank Models Credit Risk on Hybrid Mortgage Loans in Taiwan L. Douglas Smith, Canser Bilir, Vega W. Huang, Kuo-yao Hung, Mark Kaplan, To cite this article:

L. Douglas Smith, Canser Bilir, Vega W. Huang, Kuo-yao Hung, Mark Kaplan, (2005) Citibank Models Credit Risk on Hybrid Mortgage Loans in Taiwan. Interfaces 35(3):215-229. http://dx.doi.org/10.1287/inte.1050.0142 Full terms and conditions of use: http://pubsonline.informs.org/page/terms-and-conditions This article may be used only for the purposes of research, teaching, and/or private study. Commercial use or systematic downloading (by robots or other automatic processes) is prohibited without explicit Publisher approval, unless otherwise noted. For more information, contact [email protected]. The Publisher does not warrant or guarantee the article’s accuracy, completeness, merchantability, fitness for a particular purpose, or non-infringement. Descriptions of, or references to, products or publications, or inclusion of an advertisement in this article, neither constitutes nor implies a guarantee, endorsement, or support of claims made of that product, publication, or service. © 2005 INFORMS Please scroll down for article—it is on subsequent pages INFORMS is the largest professional society in the world for professionals in the fields of operations research, management science, and analytics.

For more information on INFORMS, its publications, membership, or meetings visit http://www.informs.org Vol. 35, No. 3, May–June 2005, pp. 215–229 issn0092-2102 eissn1526-551X 05 3503 0215 inf orms ® doi10.1287/inte.1050.0142 © 2005 INFORMS Citibank Models Credit Risk on Hybrid Mortgage Loans in Taiwan L. Douglas Smith, Canser Bilir Center for Business and Industrial Studies, College of Business Administration, University of Missouri–St. Louis, St. Louis, Missouri 63121 {[email protected], [email protected]} Vega W. Huang, Kuo-yao Hung, Mark Kaplan Citibank NA, Consumer Bank, 3F, Exchange Square, No. 89 Sung-Jen Road, Taipei 110, Taiwan {[email protected], [email protected], [email protected]} A new type of hybrid loan in Taiwan consists of a traditional residential mortgage with an attached line of credit.

Motivated by declines in Taiwanese property values and unexpected credit losses on all types of loans secured by residential real estate, we developed new statistical models for analyzing the credit risk on traditional mortgages, the hybrid loans, and pure equity lines of credit. Nonstationary Markovian models represent probabilities of transition among different ﬁnancial states for the three credit instruments. We used logistic and regression models to estimate the losses on defaulted loans and the utilization of credit lines. We calibrated the models with account-level data and integrated them into comprehensive forecasting models that revealed differences in risk proﬁles among the three types of credit and among different segments of each portfolio.

Key words: ﬁnancial institutions: banks; forecasting: applications.

History: This paper was refereed. T o manage assets properly, ﬁnancial institutions must analyze credit risk on portfolio segments and individual loans. Institutions must consider the potential effects of changes in economic conditions on credit losses and provide for future credit losses in their ﬁnancial statements. They need to moni- tor individual accounts and focus their collection efforts where they will be most productive. They must assess changes in ﬁnancial risk on secured loans when market values of collateral unexpectedly depre- ciate. Assessment of credit risk is part of due dili- gence when institutions buy, sell, or package assets as ﬁnancial derivatives. New credit instruments can have unique characteristics that affect the ways that loans’ risk proﬁles change over their lives. All these circumstances reinforce Zanakis et al.’s (1986) claim that ﬁnancial institutions constantly need new ana- lytical tools in their competitive business environ- ments. In this case, Citibank had introduced a new mortgage product (a hybrid consisting of a tradi- tional amortizing mortgage loan coupled with a per- sonal line of credit tied to the same collateral) ina changing foreign business environment. The bank needed a tool for estimating the related credit risk in different market segments and for estimating its exposure to credit losses under different economic scenarios. The credit manager for Asia had previously engaged us to develop analytical models for a variety of mortgage products and leases on mature portfo- lios of assets in North America. The models had been used over a dozen years in several different institu- tions to estimate reserves required for credit losses and to help in assessing risk premiums on portfolio segments considered for securitization. While recog- nizing that we faced a less mature and a more uncer- tain business environment in Taiwan, we developed and tested models with structures similar to those we had used in North America and applied them to three types of credit instruments in Taiwan (tra- ditional mortgages, home equity lines of credit, and the new hybrid loans). We used the models to gain insight on the comparative risk proﬁles of the differ- ent loan types, to validate forecasts of aggregate credit losses generated by simple extrapolative techniques 215Downloaded from informs.org by [132.174.254.97] on 25 February 2017, at 11:14 . For personal use only, all rights reserved.

Smith et al.:Citibank Models Credit Risk on Hybrid Mortgage Loans in Taiwan 216 Interfaces 35(3), pp. 215–229, © 2005 INFORMS tempered by managerial judgment, to afﬁrm ﬁndings from behavioral scoring models independently under development for problem loans, to compare credit risk in different portfolio segments, and to produce statistical evidence of changes in loss experience after issuance of new credit guidelines.

Related Research Four decades ago, Cyert et al. (1962) introduced the use of a stationary Markov chain to represent the transition of a credit through succeeding stages of delinquency and ﬁnally into default. Altman (1989) and Asquith et al. (1989) subsequently showed that the likelihood of default for an active loan changes systematically over the life of an asset. Campbell and Dietrich (1983) revealed how prepayments, delin- quencies, and defaults on home mortgages in the United States could be related to the age of the acc- ount, the ratio of outstanding principal to the prop- erty’s market value (loan-to-value ratio), interest rates, and unemployment rates. Cunningham and Capone (1990) showed differences between ﬁxed-rate and variable mortgages in their tendency toward delin- quency, prepayment, and default. Kang and Zenios (1992) described systematic changes in propensity toward prepayment and default as attributed to sea- soning and burnout effects. Lawrence et al. (1992) found, in studies of payment patterns on mobile- home mortgages, that borrower delinquency patterns were the dominant indicators of default risk and that information obtained with the loan application dimin- ished in value through time.

Zipkin (1993) used Markov chains in evaluat- ing mortgage-backed securities. Smith and Lawrence (1995) showed how multinomial logit models or, alternatively, normalized probabilities from nonlinear regression models could be used to create nonstation- ary Markov chains that capture systematic changes in transition probabilities as credits mature. Smith et al. (1996) constructed a comprehensive model for US home mortgages that predicts incidence of prepay- ment, delinquency, default, and losses on defaulted loans. Rosenberg and Gleit (1994) summarized quan- titative methods used in credit management, with particular attention to developing credit scores based on borrower characteristics and credit history. Weextended and integrated these various concepts to accommodate the new credit instrument that Citibank issued in Taiwan, where general real estate prices declined in recent years and where the foreclosure pro- cess is long compared with practice in North America. The Modeling Structure We structured three models for the different types of loans around a nonstationary Markov chain that pre- dicts, at monthly intervals, the likelihoods that loans will be in alternative ﬁnancial states. For standard mortgages with regular amortization schedules, we deﬁned the alternative ﬁnancial states of the credit as follows:

(1) Current, with payments up to date, (2) Delinquent by one to 29 days, (3) Delinquent by 30 to 59 days, (4) Delinquent by 60 to 89 days, (5) Delinquent by 90 to 119 days, (6) Delinquent by 120 to 149 days, (7) Delinquent by 150 to 179 days, (8) In default (delinquent by 180 or more days) and subject to potential loss, or (9) Paid off in full with no loss.

These states are the stages of progression toward default upon which analysts base standard roll-rate models of risk for various types of credit. Smith and Lawrence (1995) and Smith et al. (1996) also used these state deﬁnitions in more complex models for ﬁrst and second mortgages.

The hybrid loan with a complementary line of credit calls for a different approach. With a tradi- tional home mortgage, the borrower makes payments according to a ﬁxed schedule and (with stable or rising housing prices) systematically increases the amount of equity held in the property. With the hybrid loan, the borrower has no set pattern for uti- lization of the complementary credit line. The bor- rower may gradually draw upon the account up to its credit limit to purchase other goods or services or to meet outstanding credit obligations. Alternatively, he or she may use the credit line in one period and com- pletely pay off the principal in the following period while the mortgage component of the loan amortizes.

Thus, the hybrid loan, while secured by real estate, may have some attributes and usage patterns similarDownloaded from informs.org by [132.174.254.97] on 25 February 2017, at 11:14 . For personal use only, all rights reserved.

Smith et al.:Citibank Models Credit Risk on Hybrid Mortgage Loans in TaiwanInterfaces 35(3), pp. 215–229, © 2005 INFORMS 217 Initial state One month oldxmonths oldx+ 1 months old . . . Unutilized Unutilized Low utilization Medium utilization High utilization and current Over limit and current High utilization and delinquent Over limit and delinquent Paid off UnutilizedUnutilized Low utilization Medium utilization High utilization and current Over limit and current High utilization and delinquent Over limit and delinquent Paid off Unutilized Low utilization Medium utilization High utilization and current Over limit and current High utilization and delinquent Over limit and delinquent Paid off Default No loss Loss Percent of commitment lost . . . Figure 1: Possible state transitions include default and prepayment outcomes. to a mortgage, others similar to consumer credit, and others akin to unsecured credit lines. To model the hybrid loans (with the amortizing components and complementary credit lines), the bank therefore had to integrate the two loans tied to the same collateral into a single record that contained information about the property, terms of the combined loan, character- istics of the neighborhood in which the property was located, characteristics of the borrower at the time of loan application, credit-line utilization, monthly payments, and any associated credit losses (write- offs and recoveries). Drawing on previous experi- ence in modeling independent equity-line loans in the United States, and after analyzing the rates of default and prepayment for different levels of credit utiliza- tion on the hybrid loans, we deﬁned nine alternative ﬁnancial states (with possible transitions as depictedin Figure 1) for the hybrid loans and for pure credit lines without an amortizing component:

(1) Unutilized (with an open credit line and zero balance), (2) Low utilization (with a balance over zero per- cent and up to a designated low percentage of the credit line used), (3) Medium utilization (with greater than low uti- lization and up to a designated medium percentage of the credit line used), (4) High utilization, current (with greater than medium utilization and up to the credit limit with less than two payments overdue), (5) Overutilized, current (with a balance over the current credit limit and less than two payments overdue),Downloaded from informs.org by [132.174.254.97] on 25 February 2017, at 11:14 . For personal use only, all rights reserved.

Smith et al.:Citibank Models Credit Risk on Hybrid Mortgage Loans in Taiwan 218 Interfaces 35(3), pp. 215–229, © 2005 INFORMS (6) High utilization, delinquent (with greater than medium utilization and up to the credit limit with two or more payments overdue), (7) Overutilized, delinquent (with a balance over the credit limit and two or more payments overdue), (8) Defaulted (delinquent by six or more payment cycles or written off), and (9) Paid off (account paid in full and closed by the borrower as a good credit).

We did not deﬁne separate states for delinquent loans with low-to-medium utilization because the few that existed did not exhibit the same propensity to default as delinquent loans with high utilization of the credit line. This behavior could be expected because borrowers could usually draw further on their credit lines to make payments, thus avoiding further delinquency. Loans can move into the overuti- lized state if interest is imposed when borrowers are at their credit limits, when borrowers inadvertently draw down beyond their credit limits, or when the credit limit is restricted by the bank for other reasons.

Between September 1998 and November 2001, for example, there were a total of 450,302 state transitions (Table 1) with initial states distributed as follows:

—15.89 percent from State 1: unutilized, —12.65 percent from State 2: low utilization, —39.63 percent from State 3: medium utilization, —26.43 percent from State 4: high utilization and current, —3.87 percent from State 5: overutilized and current, —1.41 percent from State 6: high utilization and delinquent, and 2. Low 3. Medium 4. High utilization, 5. Over limit, 6. High utilization, 7. Over limit, From/to 1. Unutilized utilization utilization Current Current Delinquent Delinquent 8. Defaulted 9. Paid off 1. Unutilized 85 62 9 07 2 94 0 56 0 13 0 00 0 00 0 01 1 67 2. Low utilization 9 71 73 63 14 53 0 75 0 16 0 00 0 00 0 01 1 22 3. Medium utilization 1 22 3 56 82 50 10 72 0 60 0 10 0 00 0 02 1 27 4. High utilization, 0 39 0 50 13 99 81 35 0 46 1 78 0 01 0 01 1 51 Current 5. Over limit, Current 0 59 0 75 4 06 0 41 91 69 0 00 0 96 0 02 1 51 6. High utilization, 0 06 0 03 2 25 17 43 0 00 69 11 0 28 9 08 1 76 Delinquent 7. Over limit, 0 00 0 00 0 34 0 34 9 26 0 34 77 02 11 66 1 03 Delinquent Table 1: Average percentage rates of transition depend on current states. —0.13 percent from State 7: overutilized and delinquent.

Accounts tend to persist in their current states from one month to the next, or to make transitions to the adjacent states (that is, to change incrementally).

Accounts are more likely to return to the state with an unutilized line of credit (reverting to the pure mort- gage state) if their utilization is low. Most transitions to the defaulted state occur in accounts that are delin- quent with high utilization or overutilized and delin- quent. Transitions to the early paid-off state are not so strongly related to the previous state. As expected, loans with unutilized lines of credit have higher than average propensities to shift to the paid-off state—so, however, have loans with high utilization and delin- quency. Usually borrowers default because they lack the ﬁnancial resources to make the payments (inabil- ity to pay), although they may occasionally decide to abandon properties with negative net equity, leav- ing them in the hands of the lender. Prepayment may occur for a variety of reasons. People relocate and sell their homes in the process. Borrowers with ﬁnancial resources pay off their obligations to avoid interest charges. Others reﬁnance their loans with the same or other institutions when they can obtain more favor- able interest rates. The bank occasionally rewrites loans for borrowers in difﬁculty, with the expectation that they will be able to meet their future obligations.

In aggregate, the model predicts the expected num- ber of accounts in the portfolio that will fall into each of the ﬁnancial states at the beginning (or end) of each month in the forecasting horizon. It also accu- mulates the expected value of the associated principal balances. At the beginning of the forecast period, theDownloaded from informs.org by [132.174.254.97] on 25 February 2017, at 11:14 . For personal use only, all rights reserved.

Smith et al.:Citibank Models Credit Risk on Hybrid Mortgage Loans in TaiwanInterfaces 35(3), pp. 215–229, © 2005 INFORMS 219 state of an active account (including its principal bal- ance) is known with certainty. Over the forecasting horizon, the model adjusts the conditional transi- tion probabilities for each individual loan as the cor- responding conditional account balances, economic conditions, and values of the security for the loans are forecasted to change. For the hybrid loan, the account balance consists of an amortizing component and a complementary balance on the credit line.

The comprehensive forecasting model has six basic components: (1) a component that estimates the prob- abilities of transition among the alternative states from one month to the next and thence the likeli- hoods that the account is in each of the alternative states over the forecasting horizon, (2) a component for estimating credit utilization each month, (3) a com- ponent that estimates the likelihood of a loss if the borrower were to default in a given period, (4) a com- ponent that estimates the percentage of the loan com- mitment that the bank will lose if default occurs at the corresponding time, (5) a component for distributing those losses to account for loss recognition over the months following default, and (6) a component that estimates the updated gross appraised value as a per- centage of estimated market value at time of default (for use in the rule-based loss accrual). (The mathe- matical structure of the model and the procedures for calibrating the conditional probabilities are presented in the appendix.) State Probabilities and Transitions We need a minimum of 63 equations (for seven pos- sible initial states and nine possible ending states) to determine how the transition probabilities differ among accounts and change through time. In fact, we used 126 estimating equations to take advan- tage of additional information available at the begin- ning of the forecasting horizon to reﬁne our estimates of the transition probabilities for the ﬁrst period.

The explanatory variables included economic factors, loan characteristics, collateral characteristics, indica- tor variables for geographical region to capture differ- ences in real estate markets, and indicator variables for January and February to adjust for the effects of the Chinese New Year. (Other monthly seasonal indicator variables and indicator variables for thenumber of calendar days in the month were also statistically signiﬁcant in some equations. Risk man- agers, however, found the resulting saw-tooth pat- terns in plots of the transition probabilities to be somewhat distracting. Because the reﬁned seasonal- ity did not affect the ﬁnal results materially, they preferred a simpler version of the model which incor- porated only the January and February seasonal indi- cators.) The geographic variables allow for differences among regional economies, business practices, and housing markets that the economic variables do not capture. In addition, for the ﬁrst month in the fore- casting horizon, we can consider the speciﬁc patterns of historical account states (known in the industry as behavioral score). As a surrogate for the behav- ioral score, we employ a forecasted next state, which is an index variable between 1 and 8 (representing the states from unutilized through defaulted). We derive the forecasted next state from an exponentially smoothed average with trend (truncated at 0 or 8 if necessary). We include it among the independent variables (along with an interaction term formed with the complement of an indicator of whether the loan is in its ﬁrst year). This augmented set of explanatory variables gives rise to the other set of 63 equations that we employ for calculating the transition proba- bilities for the ﬁrst period of the forecasting horizon.

The speciﬁc variables in each group (and their cor- responding names in the model’s equations) are the following:

—Current Financial State of Loan —Implicit variable, as separate sets of transi- tion equations are developed for transitions from each state (with a diminishing number of explanatory variables as delinquency increases and sample sizes decrease accordingly).

—Indicator of whether a credit restriction has been imposed on the account (crrestr).

—Loan Characteristics —Proportion of loan term in months that has expired (matprp).

—Interest rate on loan (intrate).

—Original loan-to-value ratio for the mortgage loan (origlv).

—Indicator of exceptions to usual guidelines and business practices when issuing credit (gptype).Downloaded from informs.org by [132.174.254.97] on 25 February 2017, at 11:14 . For personal use only, all rights reserved.

Smith et al.:Citibank Models Credit Risk on Hybrid Mortgage Loans in Taiwan 220 Interfaces 35(3), pp. 215–229, © 2005 INFORMS —Whether the account type is under surveillance and prone to early-default classiﬁcation (accdef).

—Borrower Characteristics —Indicator of whether borrower has college edu- cation (colleduc).

—Indicator of critical credit-qualifying deviation (critdev).

—Indicator of employment in occupations with historically high delinquency rates (unfavocc).

—Geographic Indicators —Central and south (with north as base).

—Reputational quality of retail market (prim- area).

—Collateral Type —Indicator of townhouse or villa, or other type of collateral (thsvilla, othcoll).

—Combination of collateral type and property location (primthse, primoth, athse).

—Indicator of whether age of building exceeds 30 years in Taipei City or 20 years elsewhere (oldbldg).

—Economic and Market Variables —Estimated loan-to-value (estlv) computed as the ratio of the balance on the amortizing component of the loan, relative to indexed market value of the collateral.

—Total credit exposure (including mortgage loan and credit line) relative to estimated market value of the collateral (totlv).

—Difference between interest rate on loan and average interest rate on new loans from the bank (intdif).

—Regional unemployment rate (unemrate).

—Seasonal Indicators —Indicators for January and February to capture effects of the Chinese New Year (Jan, Feb).

—Historical Payment Trends (employed for ﬁrst- period transitions only) —Indicator of whether in ﬁrst year of loan (frstyr).

—Exponentially smoothed prediction of ﬁnan- cial state score if not in ﬁrst year of loan fyrscr= 1−frstyr ∗fcstscr .

The coefﬁcients of several of the components of the normalized logistic models (for ﬁrst-period tran- sitions, including predicted state score) are also presented in the appendix (Table A1). Positivecoefﬁcients point to increases in the likelihood of transition to the respective states. The number of data points for ﬁtting the models ranged from 583 for loans over the credit limit and delinquent on home mortgage payments to 71,566 for mortgages with unutilized lines of credit. This results in our ﬁt- ting models with more explanatory variables for the lower-indexed ﬁnancial states. We use a complemen- tary set of logistic functions (without the behavioral score surrogate) to estimate transition probabilities in subsequent months.

We assess the net impact of changes in the value of an independent variable by altering the charac- teristics of a loan, its collateral, or economic condi- tions (through changes in housing appreciation rates or initial market value) accordingly and driving the loan through the forecasting recursion. We report the results of such a simulation in a series of reports and plots of state probabilities, transition probabili- ties, default rates, and pay-off rates.

As in other studies of credit risk tied to real estate, we found that falling market values resulted in increased delinquencies and defaults on loans and in larger losses following default. Even after account- ing for general change in market value, the forecasted states using the credit-score surrogate, the loan char- acteristics (such as original loan-to-value ratio and whether a credit restriction had been imposed on the account), and characteristics of the collateral (such as town house versus apartment), we found that the geographic indicator variables retain statistical signif- icance and seem therefore to capture other character- istics of the local markets.

In North American studies, we and other analysts found that the likelihood of prepayment is sensitive to the difference in interest rate on the loan and prevail- ing rates in the marketplace. For this study, the bank ofﬁcials in Taiwan were unable to produce reliable measures of competitive interest rates on a consis- tent basis. We therefore used the average rate for new loans issued by the institution itself as the competitive benchmark. The collection of variables that may con- tain information about incentive and ability to pre- pay a loan include totlv, crrestr, estlv, intrate, and intdif (deﬁned above). Interest rates on a new loan are affected by prevailing rates at the time of initia- tion and also by the creditworthiness of the borrower.Downloaded from informs.org by [132.174.254.97] on 25 February 2017, at 11:14 . For personal use only, all rights reserved.

Smith et al.:Citibank Models Credit Risk on Hybrid Mortgage Loans in TaiwanInterfaces 35(3), pp. 215–229, © 2005 INFORMS 221 A high current interest rate on a loan would pro- vide the borrower with an incentive to reﬁnance when interest rates drop, but with risk-based pricing (that results in higher interest rates for borrowers judged to be higher risks), it may also indicate that the bor- rower may have trouble qualifying for a new loan.

A credit restriction (crest) would similarly indicate a need for the borrower to pay down loans, but the borrower may have difﬁculty in doing so. These vari- ables are statistically signiﬁcant as a block and indi- vidually in several of the logistic functions involving transition to the paid-off state. It is difﬁcult, how- ever, to employ them individually for stress testing.

When, for example, we imposed a one percent drop in future interest rates in simulations of the forecasting model, the projected aggregate prepayment rates for the hybrid loans increased only from 19 to 20 percent over 24 months. We had observed a greater impact of a corresponding change when testing similar scenar- ios for mortgage loans in North America. The bank’s lack of competitive interest-rate information and short account history with risk-based pricing (causing a confounding of information about incentive to pre- pay and ability to prepay) precluded our using the model to evaluate risk in connection with future inter- est rate changes. We therefore made projections for the Taiwanese hybrid loans under a nominally stable interest-rate scenario. We geared economic scenarios to assumptions about the future changes in housing prices (which themselves are admittedly affected by prevailing interest rates).

Utilization of the Credit Line We estimated the degree of utilization of the line of credit for each of the six nonterminal states with nonzero utilization. We employed regression mod- els that use the previous ﬁnancial state and utiliza- tion level as explanatory variables. Utilization within a ﬁnancial state is higher on average if the bor- rower made a transition from a high-utilization state and from higher-utilization levels within the previ- ous state. In other words, a separate utilization esti- mate for the credit line is produced, contingent on the loan’s being in each of the active states. The expected utilization of the credit line at a point of time is computed as the sum of the product of the contin- gent utilizations and the respective probabilities asdetermined from the state probabilities. We deﬁne the total balance of the outstanding credit as the sum of the mortgage balance and the credit-line balance. We deﬁne the total credit exposureC t as the sum of the mortgage balance and the larger of the credit-line balance and credit-line commitment. Realization of Loss The bank may realize a loss when a transition to State 8 occurs. The resulting expected loss in periodt is expressed as the product of the probability that a transition occurs to State 8 during periodt, the prob- ability of nonzero loss on default, and the expected proportion of the total credit exposure (usually credit- line commitment plus balance on the pure mortgage component of the loan) that will materialize as the net credit loss after recoveries.

Likelihood of Loss on Default We estimated the probability of incurring a loss in the event of default by using a logistic function contain- ing variables similar to those used for adjusting the state-transition probabilities. First, we used a regres- sion model to estimate the total outstanding balance on default. Then, we used a derivative of this vari- able (totlv) in the logistic model for likelihood of loss on default. To allow sufﬁcient time for accrual of losses and recoveries, we calibrated this model using 479 loans that defaulted in the years 1998 through 2001 (with a minimum of 24 months of loss and recov- ery data through January 2003). Overall, the model was statistically signiﬁcant at the 0.0001 level. The variable reﬂecting the combination of the value of the collateral and outstanding loan commitment was marginally signiﬁcant at the 0.0001 level; the vari- able reﬂecting the estimated outstanding loan bal- ance at default relative to the value of the collateral was marginally signiﬁcant at the 0.05 level. These were the dominant variables in estimating the like- lihood of loss on default. Indicators of geographical location and property characteristics added explana- tory power in the expected direction. Regional unem- ployment rate and interest rate on the loan failed to provide predictive power in the expected direction.

This may be due to their relation to risk assessment when initiating or rewriting a loan, or to their rela- tion to property values, which are reﬂected also in theDownloaded from informs.org by [132.174.254.97] on 25 February 2017, at 11:14 . For personal use only, all rights reserved.

Smith et al.:Citibank Models Credit Risk on Hybrid Mortgage Loans in Taiwan 222 Interfaces 35(3), pp. 215–229, © 2005 INFORMS aforementioned variable expressing loan commitment relative to property value. Default was more likely, other things equal, when the surrogate credit score predicted that the loan would be in the higher usage and delinquent states.

Estimating Loss Severity by Allocating the Expected Net Credit Loss We estimated the expected net credit loss (NCL) as a percentage of loan commitment from data for 373 loans on which the bank incurred losses. Other things being equal, the bank suffered higher percent- age losses on loans with high estimated loan-to-value ratios at time of default, with tempering of the esti- mates according to age of the loan, its interest rate, whether or not the Chinese New Year occurred in the month, and by miscellaneous attributes of collat- eral. The timing of loss recognition is another issue.

The foreclosure and sale of properties can take con- siderable time. Losses are therefore further distributed among the months following default, in accordance with historical patterns of loss recognition. The net result shows loss rates on a new loan increasing in early years and then decreasing (Figure 2), but to a lesser extent for the hybrid loans than on traditional mortgages.

Estimating Loss by Using the Write-Off Rule The recognition of losses is, to some extent, a man- agerial decision and is subject to periodic review.

In practice, the bank takes a series of write-downs according to successive appraisals of the value of the collateral. Depending on expectations regarding the supply and demand for housing units, the amount 0 0.001 0.002 0.003 0.004 0.005 0.006 0 20 40 60 80 100 120 140 Age (months) Default risk Figure 2: Risk for the hybrid loan increases and then tapers off gradually. the bank loses on a property is the difference between the outstanding balance on the hybrid loan (mortgage principal plus credit balance) and the expected recov- ery from sale of the property (a chosen percentage of the updated gross appraised value). The percentage reﬂects commissions, other costs, and expected sales price relative to appraised value. Because the write- off rules may change, we developed an alternative model for estimating realized loss, which we call the rule-based write-off. At ﬁve review points (at default, and at succeeding six-month intervals) the bank real- izes the loss according to the applicable rule. To use the write-off rule as an alternative method of fore- casting credit losses, one must estimate the updated appraised value of the collateral on the one hand and the total outstanding balance on the loan at the time of default on the other hand. For the former, we use a regression model to estimate the ratio of the updated market value to the indexed market value (the origi- nal appraised value adjusted for the regional housing price index) of the home at the time of default. We estimate the logarithm of the updated appraisal ratio as a function of:

—the age of the loan, —the loan commitment relative to estimated mar- ket value, —the interest rate, —regional geographic variables, —whether the loan was issued under new strategic guidelines, —whether it was a reﬁnancing that was considered to be risky, —whether the collateral is a town house or villa,Downloaded from informs.org by [132.174.254.97] on 25 February 2017, at 11:14 . For personal use only, all rights reserved.

Smith et al.:Citibank Models Credit Risk on Hybrid Mortgage Loans in TaiwanInterfaces 35(3), pp. 215–229, © 2005 INFORMS 223 —whether the bank had overlooked a critical devi- ation on a credit report when issuing the loan, —whether it is the ﬁrst updated appraisal follow- ing default, —whether it is the second updated appraisal fol- lowing default, and —how many months had passed since default.

We apply the inverse of the logarithm of the updated appraisal ratio to the indexed market value to produce the estimate of the updated appraised value at chosen loss-recognition points (for example, using ﬁve points in time, at the time of default and at 180-day intervals thereafter). We parameterize the model in accordance with the projected write-down policy. Speciﬁcally, the analyst indicates the number of points in time at which the write-offs are to occur and the percentage of updated appraised value to be assumed as recoverable on disposal of the asset. With the write-off rule, we provide an alternative to the net Type of model Component (Calibration procedure) Types of variables used State transition probabilities (126 estimating equations)Normalized logistics (SAS LOGISTIC) or multinomial logit (SAS CATMOD) or truncated and normalized regression (SAS REG)Economic variables, loan characteristics, collateral characteristics, borrower characteristics, geographic indicators, seasonal indicators, and historical payment pattern Expected utilization (six estimating equations for each of the states with a utilized credit line)Regression (SAS REG) Previous ﬁnancial state, previous utilization level Probability of experiencing a loss on a defaulted loanLogistic (SAS LOGISTIC) Economic variables, loan characteristics, collateral characteristics, geographic indicators, borrower characteristics Total loss exposure relative to market value of the property on defaultRegression (SAS REG) Loan characteristics, economic variables, geographical characteristics, collateral characteristics Percentage of loan exposure lost if loss occursRegression (SAS REG) Loan characteristics, economic variables, geographical characteristics, collateral characteristics, borrower characteristics Ratio of updated appraised value to indexed market value at the time of defaultRegression (SAS REG) Loan characteristics, economic variables, geographical characteristics, collateral characteristics, borrower characteristics Likelihoods that reappraisal and associated write-downs occur (four estimating equations)Logistic (SAS LOGISTIC) Loan characteristics, economic variables, geographical characteristics, collateral characteristics, borrower characteristics Commitment Original commitment used to avoid the need to forecast change in commitmentFixed at original value Table 2: The comprehensive model has seven major components with a ﬁxed loan commitment. credit loss allocation and allow for changes in busi- ness policy in recognizing losses. In this instance, we developed the model for the updated appraised value from 525 updates on properties securing hybrid loans through 24 months following default.

To estimate the total expected loss associated with the default of loans in periodtfrom the write-off rule, we estimate the loss that will be recognized for indi- vidual loans at various review points and accumulate those amounts across accounts to determine the total default risk associated with the portfolio.

The Integrated Model We integrated these basic constituents (Table 2) into a comprehensive recursive model that considers, for every loan in the portfolio, the probabilities of being in the nine alternative states at monthly intervals over the forecasting horizon, the expected levels of utilization conditioned upon the ﬁnancial state, theDownloaded from informs.org by [132.174.254.97] on 25 February 2017, at 11:14 . For personal use only, all rights reserved.

Smith et al.:Citibank Models Credit Risk on Hybrid Mortgage Loans in Taiwan 224 Interfaces 35(3), pp. 215–229, © 2005 INFORMS likelihood of loss if default occurs, the magnitude of losses on defaulted loans, and expected timing of loss recognition and recovery following default. We aggre- gate the projections for individual loans to produce corresponding data for the portfolio as a whole or for speciﬁed segments. We thus produce monthly projec- tions of the numbers and percentages of accounts in the beginning portfolio that are in each of the nine alternative ﬁnancial states at the beginning of each month in the forecasting horizon; the percentages of active accounts that are in each of the seven nonter- minal states, total expected losses, and the expected loss in dollars in each geographical area. We also produce cumulative summaries for ﬁve chosen peri- ods. In a periodic planning cycle, one would recali- brate the coefﬁcients of the transition equations using data through the most recent month. One would recalibrate the severity model using loss experience for loans that entered the defaulted state at least 24 months ago. This allows enough time for the bank to accumulate the great majority of recoveries from its collection efforts. By running forecasts for differ- ent subsets of the portfolio, we can compare their risk proﬁles.

The resulting risk proﬁle (the product of likelihood of defaulting and severity of default) for a hybrid loan as it ages (Figure 2) is similar to the general pat- tern reported in research on standard ﬁrst mortgages in North America (Smith et al. 1996) and similar to the general pattern derived for standard Taiwanese mortgages. Risk increases at ﬁrst and then decreases as the mortgage component of the loan is amortized and as the collateral increases in value (assuming, in this case, that the value of the collateral appreciates at the rate of three percent per year). This season- ing effect, however, appears to be more muted for the hybrid product than for traditional mortgages. Risk seems to taper off more slowly beyond the fourth year. This behavior is perhaps to be expected, as the borrower can draw on the complementary credit line and increase the loan-to-value ratio at will.

Out-of-Sample Tests of Forecasting Accuracy We constructed the models based on state transi- tions from August 1998 through November 2001. Toproduce an out-of-sample test of the comparative forecasting and discriminating power of the model, we applied the model to accounts active on Novem- ber 30, 2001 and predicted their status each month through October 2002. We are interested in three types of forecasting accuracy. First, we want to predict the total number of loans in each ﬁnancial state at the end of each month (with special attention to defaulted loans and the resulting losses). Second, we want to be able to identify the speciﬁc loans that are most likely to default. Third, we want to be able to forecast the aggregate net credit losses after recoveries.

The forecasted default rate over the 11-month hori- zon was predicted to be 2.9 percent, quite close to the actual rate of 3.1 percent. As reported by Smith et al. (1996), much of the explanatory power in the model, particularly when predicting defaults, comes from the ﬁnancial states of the individual loans at the beginning of the forecasting horizon. To assess the incremental value of other information and modify- ing the transition probabilities accordingly, we created a version of the forecasting model that employed sta- tionary transition probabilities based on average rates of transition for the same periods used to calibrate the logistic equations. When we used that model, the predicted default rate was 2.5 percent over the 11 months, lower than the 2.9 percent we predicted with the model using nonstationary transition proba- bilities and lower again than the actual default rates of 3.1 percent (Table 3).

Accounts in the test period were paid off at a dra- matically faster rate than forecasted (39 percent actual versus eight percent forecasted). This turned out to be caused by a boom in reﬁnancing caused by a drop in interest rates and more aggressive lending on the part of competing institutions during 2002–2003. The bank had deliberately followed a cautious strategy amidst the reﬁnancing boom. While we predicted defaults more accurately with the additional information, iron- ically, we predicted prepayments more accurately by using average transition rates.

The forecasted distributions of the states of loans that remain active (not shown) were quite close to actual values, although aggregate delinquencies (on highly utilized and overutilized accounts) crept up at a higher rate than projected. This rate may arise partly because such accounts were less attractive risks forDownloaded from informs.org by [132.174.254.97] on 25 February 2017, at 11:14 . For personal use only, all rights reserved.

Smith et al.:Citibank Models Credit Risk on Hybrid Mortgage Loans in TaiwanInterfaces 35(3), pp. 215–229, © 2005 INFORMS 225 Using nonstationary transition Using stationary probabilities (126 equations) transition probabilities Forecasted Actual Forecasted Months percent percent Percent percent Percent ahead default default deviation defaulted deviation 10 31 0 39−21 20 34−13 2 20 71 0 85−15 80 63−25 2 31 05 1 20−12 60 90−25 3 41 34 1 49−9 91 14−23 3 51 60 1 70−6 21 37−19 9 61 84 1 98−7 31 58−20 2 72 06 2 26−9 21 78−21 3 82 27 2 45−7 41 98−19 1 92 47 2 67−7 42 17−18 6 10 2 67 2 87−7 22 35−18 0 11 2 86 3 05−6 52 54−17 0 Table 3: Forecasts are more accurate with nonstationary probabilities.

reﬁnancing and partly because of calibration errors for a new ﬁnancial product. The median age of the loan for the transitions used to calibrate the models was 26 months.

We looked at how well the model identiﬁes the loans that are most likely to default by sorting the loans in declining order of their forecasted proba- bilities of being in the defaulted state at the end of October 2002 and comparing the predicted versus actual default rates for groups of 1,000 accounts. Both the predicted and actual default rates show systematic Predicted Actual Predicted Actual Cumulative marginal marginal marginal marginal number of default default prepayment prepayment accounts percent percent percent percent 1,000 11 917 019 548 1 2,000 2 73 315 142 6 3,000 2 72 314 533 9 4,000 1 30 614 533 4 5,000 1 21 314 041 7 6,000 1 21 113 940 2 7,000 0 70 213 936 8 8,000 0 30 213 939 4 8,742 0 30 112 632 6 Table 4: Discriminating power was strong for defaults and weak for pre- payments.Model Average Cumulative predicted Actual roll-rate Actual Behavioral number of default default predicted default score actual accounts percent percent default percent percent default percent 1,000 3 93 42 72 53 6 2,000 1 51 12 32 00 9 3,000 0 91 01 20 70 8 4,000 0 40 10 90 50 3 4,966 0 10 10 30 00 1 Table 5: Discriminating ability is superior to that of average roll-rate extrapolations and very similar to that of bank’s behavioral score for problem loans.

decreases, revealing quite good discriminating power for defaults (Table 4). When we performed a simi- lar analysis for loans paid off, it was evident that we were not so successful in discriminating among loans likely to be prepaid (Table 4). Although both pre- dicted and actual pay-off rates are 50 percent higher for the groups of loans considered most likely to be paid off than for the groups of loans considered least likely to be paid off, the model’s discriminating power for loans likely to be prepaid is weak in the interme- diate categories.

The bank was also interested in whether the model was in agreement with projections from a sepa- rate behavioral scoring model devised speciﬁcally for ranking potential problem loans according to their likelihood of default over a six-month period. The bank was developing behavioral scores for loans with at least a minor delinquency to identify those that most require the attention of the collections depart- ment. The scores were based on many of the variables used in our model and on additional information about the status of other ﬁnancial accounts. We com- pared the discriminating data for defaults for (1) our model, (2) our model using average transition prob- abilities instead of the logistic equations, and (3) the bank’s behavioral score (Table 5). The discriminating ability of the hybrid mortgage model and the bank’s behavioral scoring model were very similar for the subset of loans for which the bank had produced behavioral scores. Both the hybrid mortgage model and the behavioral scoring model were signiﬁcantly better than using average transition probabilities on the current state of the loan.Downloaded from informs.org by [132.174.254.97] on 25 February 2017, at 11:14 . For personal use only, all rights reserved.

Smith et al.:Citibank Models Credit Risk on Hybrid Mortgage Loans in Taiwan 226 Interfaces 35(3), pp. 215–229, © 2005 INFORMS Finally, we considered the projections of credit loss made with the two approaches. Banks’ aggres- siveness in recognizing loss and timing recoveries on recognized losses can be highly variable because of business practice, economic conditions, and legal constraints. The NCL allocation resulted in a cumu- lative forecasted loss that was 15 percent higher than that based on the loss rule over the 11-month period.

Considering the great amount of uncertainty involved in projecting aggregate losses in this business envi- ronment and the considerable difference in the two approaches, this degree of accord in total projected loss was quite satisfying. The timing of loss recogni- tion under the two approaches was, however, quite different, with an apparent delay in recognizing losses for the NCL allocation. The allocation by loss rule was better in this regard. Only 31 percent (the percentage derived from historical data) of the NCL was recog- nized in the ﬁrst month following default under NCL allocation. The bank, in recent years, has apparently been more prompt in recognizing potential loss. We should bring the allocation of NCL following default into alignment with more recent practice by adjusting the percentages used to distribute the losses among periods following default. We left them intact for this paper, however, to avoid contaminating our out-of- sample tests.

Discussion and Conclusion Forecasting with statistical models using account- level data can improve accuracy and enrich insight.

It allows the production of contingency forecasts and assessment of risks for portfolio segments and helps to reveal business dynamics. Plots of projected state probabilities and losses on default reveal different risk proﬁles for traditional mortgages, hybrid loans, and home-equity lines of credit as an account ages.

Banks typically use a number of models for dif- ferent aspects of credit management, often separately analyzing default risk in aggregate and prepayment rates in aggregate and using separate behavioral scor- ing models to identify speciﬁc loans that are most likely to default. Our integrated model makes a major contribution in accounting for interactions among these individual elements while providing perspec- tive on changes in portfolio performance. The modelscan be augmented with indicator (dummy) variables to help managers to assess the impact of interventive strategies, such as changes in credit policy (for issuing new loans or restructuring problem loans) or market- ing strategy. In this case, for example, we augmented the model for regular mortgages to include indica- tor variables at times in 2002 when the bank issued new credit-policy bulletins that would affect collec- tions or credit-granting decisions. The bank credit- strategy variables were highly statistically signiﬁcant, and assumptions about their continuing effect had a substantial impact on forecasts generated from January 2003 on.

The forecasts made by our approach are an infor- mative complement to forecasts derived with other methods. By using average transition probabilities in place of the those adjusted with the logistic func- tions, one can generate results comparable to those that would be obtained from aggregate roll-rate mod- els and compare them for base-line purposes. Then, one can use the comprehensive model to estimate the likely change in portfolio performance as market val- ues change and the portfolio seasons.

To facilitate monthly updating, we embedded the model in a forecasting system that automatically regenerates equations and prints them to a ﬁle that is imported to the forecasting program. The users can alter the parameters that serve to deﬁne low and medium levels of account utilization and select different levels of statistical signiﬁcance for retain- ing variables in the model. They can experiment with different model conﬁgurations and include new explanatory variables, such as indicator variables that coincide with documented changes in business practice.

Our model, in this instance, was quite effective at estimating default risk and identifying the speciﬁc loans that were most likely to default. Its discriminat- ing ability for problem loans was comparable to other behavioral scoring techniques developed separately with a much narrower focus. It revealed statistically signiﬁcant changes in portfolio performance follow- ing changes in business practices. Hidden factors in the Asian business environment wreaked havoc, how- ever, with its estimation of prepayment rates. Lore had it that competitive banks were much more eager than Citibank to capture reﬁnancing business in theDownloaded from informs.org by [132.174.254.97] on 25 February 2017, at 11:14 . For personal use only, all rights reserved.

Smith et al.:Citibank Models Credit Risk on Hybrid Mortgage Loans in TaiwanInterfaces 35(3), pp. 215–229, © 2005 INFORMS 227 midst of low interest rates at that time. Some insti- tutions might also have been qualifying borrowers under governmental programs for new home owners even though they were reﬁnancing existing homes.

We would need further research and market intelli- gence to deal with the prepayment issue.

Appendix. Mathematical Representation of the Model for Hybrid Mortgage Loans To represent the model mathematically, we deﬁne P j t =probability that the loan is in statejat the beginning of periodt, p kj t =probability that a loan that is in statekat the beginning of periodtwill make a transition to statejfor the beginning of periodt+1, and U j t =expected proportion of credit line utilized if the loan is in statejat the beginning of periodt.

Then, allowing for nonstationary transitional prob- abilities, we express the likelihood that a loan is in statejat timet+1 as a Markov chain.

Approximation of Multinomial Logistic Equations for the Transition Probabilities A natural approach for estimating the transition prob- abilities is to employ a multinomial logistic structure and to use maximum likelihood procedures to pro- duce seven sets of eight logistic functions of the form log p kj t /p k9 t =f kj X t for computing the probabilities of transition to end- ing statesj=1 2 9. The logistic expressions involveX t , a vector of explanatory variables (some continuous, others categorical) that are applicable to periodt.

We found that an approximation to the multinomial logistic equations produces virtually identical results and is much more efﬁcient computationally. We pro- duce seven sets of nine logistics functions of the form log p kj t /p ko t =f kj X t and then normalize the resulting probabilities for each set so that they add to 1.0. In this case,p ko t refers to the likelihood of being in any state other than statej. Some transitions have probabilities verynear zero, and we set them to zero (or very small ﬁxed values) accordingly if we have insufﬁcient data points for constructing an estimating equation. State 8 (defaulted) and State 9 (paid off) are absorbing states, thus renderingp 88=1 andp 99=1.

State Probabilities P j t+1 = 7 k=1 Pk t ∗p kj t forj=1 2 9 (1) Utilization of the Credit Line We designate the expected utilization of the credit line at the beginning of periodtasU t and compute it as U t = 7 j=1 Pj t ∗U j t (2) Expected Credit Loss by Allocation of NCL The expected loss due to a default occurring on the account in periodtis expressed as E LOSS t = P 8 t+1 −P 8 t ∗ t∗ t/100 ∗C t (3a) where E(LOSS t) is the expected loss in periodt; tis the likelihood of incurring a loss if the loan goes into default during periodt; tis the percent of the credit exposureC t that the bank can expect to lose if a loss occurs from a default in periodt. Loss associated with default in monthtbut recognizedfmonths later (in accordance with historical patterns of loss recog- nition) is E DLOSS t+f =E LOSS t ∗ tf (3b) where tfis the net proportion of credit loss (write-off less recovery) that is to be realizedfmonths follow- ing default. The sum of the tf values is 1.0. Values at the end of the recovery interval may, however, be negative, as recoveries dominate write-offs.

Realization of Loss by Write-Down Rules At ﬁve review points (at default and at succeeding six-month intervals), the bank realizes loss by write- down rules that are expressed mathematically as E DLOSS t+f = P 8 t+1 −P 8 t ∗Max 0 C t −r f∗UPDAPPR t+f − DLOSS (4)Downloaded from informs.org by [132.174.254.97] on 25 February 2017, at 11:14 . For personal use only, all rights reserved.

Smith et al.:Citibank Models Credit Risk on Hybrid Mortgage Loans in Taiwan 228 Interfaces 35(3), pp. 215–229, © 2005 INFORMS Unutiliz Unutil Lowutil Lowutil Lowutil Medutil Medutil Medutil HighCurr HighCurr HighCurr OverCurr OverCurr OverCurr HighDelq HighDelq HighDelq HighDelq Overdelq Overdelq Overdelq Variable to to to to to to to to to to to to to to to to to to to to to name Lowutil Paidoff Unutil Medutil Paidoff Lowutil HighCurr Paidoff Medutil HighDelq Paidoff Medutil Overdelq Paidoff Medutil HighCurr Default Paidoff OverCurr Default Paidoff INTERCEPT−7 031−4 113 5 7645−13 534−3 898 9 4411−26 035−1 815 21 4639−40 827−4 191 6 7212−45 206−0 549 9 228 10 6687−23 225−0 020 1 342−16 640 0 447 MATPRP 0 454 d 2 070 d −0 419 d 5 210 d 0 695 a 0 188 d UNEMRATE 0 098 d−0 005 d 0 077 c 0 089 d 0 087 d 0 113 d TOTLV 0 300 a−0 172 d−10 942 d 8 455 d −3 571 d 1 388 d 0 144 a 0 853 d−0 032 d 1 285 d 2 392 d 2 587 b−2 596 b −0 648 a −2 364 a ESTLV 10 521 d−8 417 d−5 442 d 1 881 d−0 430 d−3 183 d−0 379 d −0 079 d −1 494 d −4 824 d 2 102 b −0 768 b −1 587 a INTRATE 0 181 d−0 004 d 0 229 d 0 324 a−0 105 b 0 554 d 0 278 a−0 056 a 0 534 d−0 001 d 1 179 d INTDIF−0 201 d 0 018 d−0 273 d−0 220 d −0 133 d−0 200 d−0 862 d ORIGLV−1 307 d−0 106 d 1 033 d 2 613 d 4 049 d JAN 0 153 b FEB 0 143 a−0 208 d CENTRAL 0 227 d 0 322 d 0 120 b 0 118 b−0 128 d 0 632 c 0 334 d 0 010 d SOUTH−0 053 d −0 181 d−0 288 a −0 301 d −0 089 c 0 323 d−0 085 d −0 238 a GPTYPE 0 273 d 0 124 b 0 376 a PRIMAREA−0 145 c −0 116 b −0 078 b 0 061 d PRIMTHSE−0 095 d 0 270 a −0 222 b 0 546 a 0 051 d 0 838 b 0 550 a PRIMOTH−1 199 a 0 175 a−1 191 b 0 445 a 0 028 d OTHCOLL 0 150 a−0 038 d −0 161 b COLLEDUC−0 026 d 0 227 d ATHSE OLDBLDG−0 158 c 0 301 d −0 311 d 0 199 d UNFAVOCC−0 046 d 0 088 a −0 129 b 0 104 d−0 630 b THSVILLA−0 021 d CRITDEV−0 052 d CRRESTR−2 484 d19 224 d−6 069 d−0 706 b 6 106 d−7 943 d−6 062 d 6 823 d−18 511 d 0 682 b 20 020 d −8 926 d 2 093 d 5 670 d−9 3563 d−8 752 d 0 530 d 4 655 d −2 040 c 0 880 b FRSTYR 3 319 d−0 206 d−8 159 d 8 361 d−4 916 d−10 428 d 18 024 d−6 148 d−22 382 d 30 454 d−0 212 d −9 779 d 23 964 d−5 190 d−12 0328 d−10 996 d 20 462 d−5 909 d FYRFCSR 2 512 d −4 321 d 3 682 d−2 669 d−3 625 d 5 611 d−2 748 d−5 968 d 7 298 d−0 079 d −2 119 d 5 206 d−1 425 d−2 3231 d−2 077 d 3 316 d−1 906 d −0 506 d 2 376 d−1 100 d Table A1: Coefﬁcients of selected logistic functions for transition probabilities are consistent with expected impact of factors.

Smith et al.:Citibank Models Credit Risk on Hybrid Mortgage Loans in TaiwanInterfaces 35(3), pp. 215–229, © 2005 INFORMS 229 wherer fis the proportion of the updated appraised valuefmonths following default that is expected to be recovered when sold and DLOSS is the cumu- lative write-off that would have been expected to be realized on the loan prior to periodt+fif default had occurred in periodt.

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