MORTGAGE LENDING The Competing Framework for Mortgages: Modeling the Interaction of Prepayment and

by Arden Hall and Kyle G. Lundstedt his article discusses how prepayment and default constitute competing risks in mortgage lending, provides Texamples of the importance of using a combined approach when evaluating the of whole loans and MBS, and concludes with practical implications of using the competing risks framework. hough it may seem apt, Inside Mortgage (IMF, June tions. The IMF estimated that the phrase “competing 10, 2005) noted the following fact: the 50 largest financial services Trisks” in the title of this “During the first three months of holding companies held a com- article does not refer to the annual the year, non-prime lenders bined $1.01 trillion in whole loans budget battle between various churned out an estimated $184 bil- during that same first quarter of functions within lion in new loans. Putting that into 2005. Moreover, according to large financial institutions. Rather, perspective, more than one out of Inside MBS & ABS (July 17, 2005), it is a framework for modeling the every four loans—or 28.5%—of all a weekly newsletter published by impact of separate causes for attri- new mortgages made during the the IMF, Fannie Mae and Freddie tion. In the mortgage world, these first quarter of the year went to Mac bought $212 billion in non- are the separate, but interdepen- borrowers in the subprime and Alt conforming MBS during 2004, dent, risks of prepayment and A categories.” Moreover, SMR assuming much of the risk default. For prime mortgages Research estimates for the underlying mortgages.1 (whole loans) and for mortgage- over $700 billion in junior liens Thus, the increasing in backed securities (MBS), prepay- outstanding at the end of 2004 the system is held by a large num- ment risk has long dominated the (Home Equity Loans: 2005 Outlook). ber of institutions, and even the issue of credit risk. Historically, in Therefore, loans with greater credit GSEs now must learn to assess the secondary market, the three risk represent a significant and the greater credit risk from non- government-sponsored enterprises increasing portion of the primary conforming products. (GSEs) guaranteed the credit risk mortgage market. These recent developments of most conforming mortgage A large number of these increase the importance of default loans, which represented the bulk loans, and the associated credit risk vis-à-vis prepayment risk for of the primary market. risk, are held on the balance mortgage lenders, whether they However, a recent issue of sheets of large financial institu- are portfolio lenders or buyers of © 2005 by RMA. Arden Hall is senior vice president in Wells Fargo’s Consumer Credit Group. He leads the Modeling and Analytics Group within Strategic Risk Management. He has worked for of America and the Federal Home Loan Bank of San Francisco and has an economics Ph.D. from U.C. Berkeley. Kyle Lundstedt directs the credit risk modeling effort at Andrew Davidson & Co, Inc. (ADCo). He has worked for LoanPerformance and KMV and has a finance Ph.D. from U.C. Berkeley. This article represents the views of the authors and does not necessarily reflect the views of Wells Fargo or ADCo. 54 The RMA Journal September 2005 The Competing Risks Framework for Mortgages: Modeling the Interaction of Prepayment and Default

MBS. However, the presence of model is simply a model tive variables, such as the current prepayment risk limits the appli- designed to predict the probabili- loan-to-value (LTV) ratio or the cability of traditional approaches ty of attrition given that the sub- FICO score, affect both prepay- to default modeling. The compet- ject has not yet left. For prepay- ment and default. For example, ing risks framework for modeling ment, this means predicting the increased current LTVs or prepayment and default confers probability of prepayment in a decreased FICOs likely increase important advantages relative to given month for all borrowers who default for mortgages; however, these traditional approaches, par- have not yet prepaid. This these same variables may ticularly in the context of valua- methodology sees heavy use in decrease prepayment likelihood. tion, risk management, and capital prepayment modeling.3 Hazard Significant academic and reg- allocation. models for prepayment commonly ulatory literature applies the com- include age and current rate levels peting risks framework to the haz- The Need for Hazard Models as explanatory variables. ards of prepayment and default.5 As with other types of con- While the hazard modeling is However, industry use of compet- sumer assets, it is important to well understood by investors and ing risks models for mortgage pre- address the timing of the default by Wall Street, technique has less payment and default is still in its event, and to account for static commonly been applied to mort- infancy. predictive variables. But mort- gage default. Hazard models, how- gages are unique in offering the ever, are frequently used in assess- Understanding How Prepayment borrower an important and valu- ing the risk of default or bankrupt- and Default Affect One Another able to prepay the loan cy for corporate bonds.4 In the case When there is more than one early. Other consumer loans are of mortgages, a hazard (hazard) affecting survival, prepayable, but only mortgages would predict the probability that they compete. Mortgages face the offer a significant financial reward the mortgage defaults in a particu- risk of attrition either from pre- for careful use of the option. This lar month, given that it has not yet payment or from default. poses a particular challenge for defaulted or prepaid. Such a model Simultaneous estimation of these default modelers.2 Consumers fre- typically would include age, cur- produces a competing quently use the prepayment rent house-price levels, and bor- risks model.6 competing risks haz- option when it is to their advan- rower FICO scores as explanatory ard models, like traditional pre- tage. When rates reached variables. payment models, have different 30-year lows in 2003, the monthly One might expect that a implications, depending on the prepayment rate for prime mort- default hazard model for mort- projected economic scenario. gages reached nearly 7%. The gages could simply be estimated Thus, looking at results in differ- average lives of mortgages vary and then used with an existing ent scenarios is the easiest way to enormously due to differing pre- prepayment model. As it turns understand the interaction of the payment rates and lead to signifi- out, however, the two hazards of prepayment and default hazards cantly different cumulative losses prepayment and default compete in a competing risks framework. for mortgages with similar credit with each other in a way that Consider Figures 1 through 6, characteristics. As a result, build- requires simultaneous develop- which depict a representative ing an accurate life-of-loan loss ment and estimation of the com- competing risks model applied to model for mortgages is very diffi- peting risks. The underlying logic a hypothetical pool of loans.7 The cult. is straightforward: Loans that have results are illustrative only; how- Prepayment modelers take a prepaid cannot default, and vice ever, the relative prepayment and different approach (as illustrated versa. As a result, any forecast of default behavior is reasonable for by the fact that the life-of-loan cumulative defaults must be built credit-sensitive mortgages. In prepayment rate is a concept up from monthly predictions of each graph, the age of the loan is unheard of in the industry), using both prepayment and default. varied along the x-axis, while the what are called hazard models. A Moreover, some observed predic- other variables, such as the LTV

55 The Competing Risks Framework for Mortgages: Modeling the Interaction of Prepayment and Default

Figure 1 is low. In the rising-rate scenario, Conditional Prepay and Default Incidence 4% however, the increased duration of the pool leaves more opportunity Prepay 3% for default; hence, the cumulative default rate is much higher for the 2% same . The key point of Figure 2 is 1% the inverse relationship between Default prepayment and default. When 0% rates fall, more loans prepay but 0 24 48 72 96 120 fewer default, so overall attrition Age in Months is less than it would have been had the number of defaults been ratio and FICO scores, are held conditional prepayment rate falls. unchanged. Similarly, when rates constant. Similarly, several credit-related rise, attrition slows because of Figure 1 gives hypothetical variables, such as FICO and LTV, slower prepayment, but the conditional prepayment and affect the conditional probability impact is reduced by additional default rates by age. These condi- of default. defaults. The effects of prepay- tional rates correspond to an The first graph of Figure 2 ment and default on overall attri- “intensity” rate or “flow” of pre- shows the cumulative prepay- tion are negatively correlated. payment and default. Thus, a ments and default rates in a sce- Changes in prepayments driv- mortgage still on the books after nario where interest rates increase en by interest rates not only affect 23 months has roughly a 3% 100 basis points over one year. the magnitude (cumulative chance of prepaying in month 24 The survival curve shows the pro- defaults), but also timing (default and a 0.5% chance of defaulting. portion of the original pool incidence), as shown in Figure 3. Let’s now consider default remaining, revealing the level of The two cumulative default and prepayment estimates from attrition. The second graph shows curves in the left graph are taken our competing risks model in a the same pool subjected to an from the earlier pair of graphs in variety of future scenarios. Pre- decline. Figure 2. The right graph shows payment in our representative The cumulative prepayment the unconditional probability of model is strongly affected by and default rates obviously are default for the two interest rate interest rates. If the mortgage rate significantly different. In the scenarios. That is, the right graph falls below the level current when falling-rate scenario, fast prepay- shows the proportion of the origi- the mortgage was originated, the ment rates leave far fewer oppor- nal pool that defaults in each conditional prepayment rate rises; tunities for loans to default; month. The curves initially rise if the mortgage rate rises, then the hence, the cumulative default rate

Figure 2 Cumulative Prepay, Default, and Survival Cumulative Prepay, Default, and Survival Decline in Mortgage Rates Rise in Mortgage Rates 100% 100% 80% 80%

60% Default + Prepay 60% Default + Prepay Prepay Prepay 40% Survival 40% Survival 20% 20% 0% 0% 0 24 48 72 96 120 0 24 48 72 96 120 Age in Months Age in Months

56 The RMA Journal September 2005 The Competing Risks Framework for Mortgages: Modeling the Interaction of Prepayment and Default

Figure 3 Cumulative Default Incidence Monthly Default Incidence 25% 0.4% 20% 0.3% 15% Rates Rise Rates Rise 0.2% 10% Rates Fall 0.1% 5% Rates Fall 0% 0.0% 0 24487296120024487296120 Age in Months Age in Months because of seasoning, but then fall both lower prepayments and high- example, a pool may have an aver- because the default rates are er defaults, while the second pool age coupon of 6% and an average being applied to a successively has attributes (lower LTV, higher FICO of 700; however, the indi- smaller pool of mortgages. When coupon rates, higher FICO score, vidual loans in the pool likely will rates rise, not only are there more etc.) that lead to higher prepay- have coupons and FICOs that defaults each month, but the peak ments and lower defaults. Figure vary significantly from the pool in defaults occurs later. 4 illustrates the effects of combin- averages. This heterogeneity has a Recall that Figure 3 held col- ing interest rate changes with significant impact on pool behav- lateral quality constant, given our these changes in the underlying ior. hypothetical pool. However, a collateral characteristics. Imagine two pools of loans principal benefit of hazard models Combining low prepayment/ with identical average characteris- is the ability to isolate the effect of high default characteristics with tics. However, the homogeneous pool composition (static character- rising interest rates sends the pool is comprised of many identi- istics, such as documentation type cumulative default rate skyrocket- cal loans whose individual charac- or original FICO score) from the ing. Conversely, combining high teristics are identical to the pool’s effect of macro risk factors prepayment/low default character- average characteristics. In con- (dynamic effects driven by interest istics with falling interest rates trast, the heterogeneous pool has rates or housing prices). reduces default rates dramatically. the same average characteristics as Imagine, then, two portfolios the homogeneous pool, but is or underlying collateral pools with The Impact of Heterogeneity comprised of many loans with dif- different characteristics. The first A pool of mortgages can be fering individual characteristics. pool has collateral characteristics expected to have a range of values As Figure 5 illustrates, the (higher LTV, lower coupon rate, for the characteristics that deter- heterogenous pool has a survival low FICO score, etc.) that lead to mine prepayment and default. For curve that is initially steeper and later flatter than a homogeneous Figure 4 Cumulative Default Incidence pool, all of whose loans have aver- 30% age characteristics. 25% Heterogeneity also will cause Low Prepay High Default Collateral with Rising Rates more lifetime defaults and a dif- 20% ferent time pattern of defaults 15% than would be inferred by analyz- ing a homogeneous pool with 10% average characteristics, as illustrat- 5% High Prepay Low Default Collateral with Falling Rates ed in Figure 6. This behavior is easily under- 0% 024487296120stood. With a mix of loans of dif- Age in Months ferent characteristics, the hetero- 57 The Competing Risks Framework for Mortgages: Modeling the Interaction of Prepayment and Default

competing risks approach natural- Figure 5 Joint Survival Rates 100% ly provides not only cumulative Homogeneous Pool losses, but also the time pattern for losses. This is important in 75% assessing the profitability of mort- gages: Losses taken early have 50% higher present values and reduce profitability more than losses 25% taken later. Heterogeneous Pool In addition, competing risks 0% models, when combined with a 0 24 48 72 96 120 projected economic scenario, may Age in Months yield greater accuracy than tradi- tional loss-reserving approaches. geneous pool has some loans that duce significant errors. Some- Of course, this requires the fore- are more likely than average to times, however, disaggregating casting of economic variables like prepay. These loans cause the pools can provide no benefit interest rates and house prices, heterogeneous pool initially to because loan level detail on pre- and economists’ success in fore- prepay faster than the homoge- dictive variables is not available. casting these variables has been neous pool. However, once many In that case, there are statistical limited.9 At a minimum, however, of these loans have attrited, the techniques available to eliminate competing risks models can pro- 8 surviving loans from the heteroge- the bias illustrated in the graphs. vide accurate cumulative loss esti- neous pool are those less likely mates given the economic sce- than average to prepay. Over time, The Practical Implications of nario. Thus, an analyst can try out the heterogeneous pool then Competing Risks several plausible scenarios to experiences lesser attrition Consider for a moment the understand the range of cumula- (greater survival). The remaining regular process that follow tive losses. At a more sophisticat- slower prepayers, as a result, have in setting their allowance for loan ed level, financial economists more opportunities to default, and loss reserves. When projecting the have been more successful in pro- they produce a higher lifetime cumulative risk of loss for a port- jecting the future probability dis- default rate. folio of mortgages (the life-of-loan tributions for interest rates and To avoid the errors intro- loss), the expected speed of pre- house prices. Thus, performing duced by heterogeneity, analysis payment must be taken into Monte Carlo analysis based on of mortgages should be carried account. Traditional approaches to these distributions provides statis- out at a fairly detailed level. estimating loss reserves cannot tically valid estimates of the Analysis of large pools based on account for the variability of pre- entire probability distribution of average characteristics can pro- payment speed. Moreover, the cumulative losses. MBS investors

Figure 6 Cumulative Default Incidence Monthly Default Incidence 2.0% 0.03% 1.5% Heterogeneous Pool Heterogeneous Pool 0.02% 1.0% 0.01% 0.5% Homogeneous Pool Homogeneous Pool 0.0% 0.0% 0 24 48 72 96 120 024487296120 Age in Months Age in Months

58 The RMA Journal September 2005 The Competing Risks Framework for Mortgages: Modeling the Interaction of Prepayment and Default use this approach when valuing ed losses with the amount allocat- with significant mortgage portfo- MBS by combining a hazard ed sufficient to cover any loss up lios, some coordination of credit model for prepayment model with to some threshold with only a and management Monte Carlo simulation of inter- very small probability of occur- could pay . ❐ est rates. rence. Prepayment risk, on the Contact Arden Hall by e-mail at The competing risks concept other hand, is treated as a compo- [email protected]; contact Kyle also has implications for the nent of market risk and managed Lundstedt at [email protected]. process of capital allocation for at an overall balance sheet level mortgage portfolios. If the size and because there are often offsetting Notes timing of losses vary with the positions within the balance 1 In contrast, the two GSEs put only $188 billion in speed of prepayment, then, in sheet. If there is net market risk conforming mortgage risk on their balance sheets dur- principle, capital requirements at an overall level, it may either ing the same period. should reflect that. Currently this be accepted or offset by hedging 2 For a background of various approaches to model- is not the case, at least with regard in wholesale financial markets. ing consumer default, see Lundstedt, K., "Credit Models in Banking: Past, Present, and Future," a pres- to regulatory capital for financial The negative correlation entation given at the OCC Conference on Credit institutions. The Basel II regulato- between losses and prepayment Rating and Scoring Models, May 2004. ry capital requirements for mort- and default, as measured using the 3 See Davidson, A., et al., : Structuring gages are based on the same theo- competing risks framework, and Investment Analysis, John Wiley & Sons, and Hayre, L. (ed.), Salomon Smith Barney Guide to retical approach—a Merton-type should reduce the market risk of Mortgage-Backed and Asset-Backed Securities, John model based on a single risk fac- mortgages. In principle, overall Wiley & Sons. tor—as is used for other could be broken into 4 See Hilligeist, S., et al., "Assessing the Probability of ," Review of Accounting Studies, March loans and for commercial loans. two components: one that is nega- 2004, 9:1. The Basel regulatory frame- tively correlated with prepayment 5 See Alexander, W., et al., "Some Loans Are More work therefore ignores the impact risk and another that is uncorrelat- Equal Than Others: Third Party Originations and of prepayment on cumulative ed. Netting the correlated compo- Default in the Subprime Mortgage Industry," Real Estate Economics, 30:4; Calem, P. and LaCour-Little, losses. Moreover, the single-factor nent with prepayment risk will M., "Risk-based Capital Requirements for Mortgage approach attempts to reduce the reduce its contribution to overall Loans," Journal of Banking & Finance, 28:3; and Office of Federal Housing Enforcement Oversight, impacts of varying risk drivers— market risk. The remaining uncor- "Risk-based Capital Rule," www.ofheo.gov. interest rates, house prices, related component of credit risk Alexander, et al. examine subprime mortgage per- formance; Calem and LaCour-Little offer the founda- employment, etc.—to one factor. then would be smaller than overall tion for the current Basel II regulations for mortgages; Though banks and thrifts must credit risk and produce smaller and OFHEO explains the application of hazard models to mortgages in the context of capital regulation for calculate regulatory capital accord- expected and unexpected losses. Fannie Mae and Freddie Mac. ing to the Basel guidelines, they As a final point, the inter- 6 Allison, P., Survival Analysis Using the SAS System: often perform their own calcula- twined nature of prepayment and A Practical Guide, SAS Institute, Inc., Cary, N.C., 2005. tions of for use default for mortgages may have 7 Hall, A. and Brown, S., "Modeling Mortgage Default in specific portfolio management consequences for governance in and Prepayment as Competing Risks," a presentation and pricing decisions. Given the financial institutions. Credit risk at the OCC Conference on and Scoring Models, May 2004. results shown herein, economic management and market risk 8 See Deng, Y., et al., "Mortgage Terminations, capital estimates clearly should management have usually been Heterogeneity and the Exercise of Mortgage Options," account for the competing risks of separate functions. For many Econometrica, vol. 68, 1998; and Hall, A., "Controlling for Burnout in Estimating Mortgage default and prepayment. financial products this is com- Prepayment Models," Journal of Housing Economics, As another implication, the pletely appropriate, and even for vol. 9, 2000. competing risks framework may mortgages, many of the functions 9 As the story goes, three economists went out hunt- offer mortgage investors some of credit risk management and ing and came across a large deer. The first econome- trician fired but missed, by a meter to the left. The sec- additional flexibility in the way market risk management are ond econometrician fired but also missed, by a meter they manage risks. Financial insti- unrelated. However, for mort- to the right. The third economist didn't fire, but shout- ed in triumph, "We got it! We got it!" tutions manage credit risk by allo- gages, prepayment and default are cating capital to absorb unexpect- competing risks. For institutions

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