220 Week 2 A /For WIZARD KIM

J Real Estate Finan Econ (2014) 48:561–588 DOI 10.1007/s11146-013-9449-5

First Mortgages, Second Mortgages, and Their Default

James B. Kau · Donald C. Keenan · Constantine Lyubimov

Published online: 24 November 2013 © Springer Science+Business Media New York 2013

Abstract Using 35,437 pairs of first and second mortgages matched from within a much larger set of subprime mortgages, this paper tracks and describes the tendency for either one of the mortgages to enter default, as well as the tendency for either the one or the other mortgage to ever return to being current, all this in a possibly repeated manner. Thus, the entire, interconnected default history of pairs of first and second mortgages is explored, as well as compared to theoretical predictions.

Keywords Piggyback mortgages · Default · Loan modifications

Introduction

There have been a number of papers in the mortgage literature looking at the effect of individual second-lien loans on the default behavior of primary residential mort- gages (Gerardi et al. 2009; Sherlund 2008; Demyanyk and Van Hemert 2011; Elul et al. 2010; Jagtiani and Lang 2010; Eriksen et al. 2011), along with a smaller number

The views expressed here are those of the authors and do not indicate opinions of other members of the research staff of FNMA.

J. B. Kau Department of Insurance, Legal Studies and Real Estate, Terry College of Business, University of Georgia, Athens, GA, USA

D. C. Keenan Department of Economics and Management, Université de Cergy-Pontoise & THEMA, Cergy-Pontoise Cedex, France

C. Lyubimov (�) Federal National Mortgage Association, Washington DC, USA e-mail: Konstantin [email protected]

562 J. B. Kau et al.

of papers looking at the default behavior of second loans in the presence of firsts (Agarwal et al. 2006a, b; Jagtiani and Lang 2010). To the best of our knowledge, however, this is the first paper to consider pairs of such loans simultaneously, and so the timing of default for either the first or second loan as affected by the status of the other loan. Furthermore, our pairs of loans continue to be observed beyond the initial default, and so, an accounting is made of whether the other loan also eventu- ally defaults and whether either loan ever returns to being current, possibly only to become delinquent again at a later date, and so on, in a recurrent fashion.

This is achieved by employing a multistate competing hazard framework, much like a Markov chain, where the states are the various combinations of being current or in default for the two mortgages, yielding four such primary states.1 The various transitions between these states are modeled and estimated, including both the for- ward directions, where one or the other mortgage becomes delinquent, as well as the backward directions, where one or the other loan returns to being current. See Fig. 1 for the definition of the states and the transitions between them. In addition, we account for unobserved heterogeneity among the mortgages and across the transi- tions, thus creating a dependency among all the various transitions, which must then be estimated in a simultaneous fashion.2

Piggyback Loans

The pairs of mortgages are determined by matching the times of origination, as well as the combined loan-to-value (CLTV) ratios, borrower’s FICO (Fair Isaac Corpo- ration) score, and zip code of both first and second loans found in a large pool of securitized GMAC mortgages originated between 1999 and 2007 and observed from January 2002 until June 2011.3,4 This matching yielded 35,437 loan pairs, of which 30,314 primary loans are ARMs (adjustable rate mortgages) and 5,123 primary loans are FRMs (fixed rate mortgages.) Observation of the loan pair’s status is made

1 Technically, what we have is a semi-Markov process, since the transition probabilities are allowed to depend on time spent in the state. The baseline hazard is completely free, being estimated using a sequence of dummy variables. 2 While these papers have not allowed for recurrence, a separate literature employing multiple states has followed the process from default through foreclosure (Ambrose and Capone 1998; Capozza and Thomson 2006; Pennington-Cross 2010; Chan et al. 2011). Since we suppress this foreclosure process, this literature is in some sense complementary to ours. One reason we avoid this further distinction as to the fate of mortgages, beyond the need to keep our state model within tractable dimensions, is that there have, as yet, been relatively few actual foreclosures in our data. 3 GMAC is the acronym of the General Motors Acceptance Corporation, now rebranded as Ally Financial Inc. 4 The matching task is not a trivial one; in the words of LaCour-Little (2007): “While an important area for future research, the data requirements to jointly analyze the performance of first and junior loans are quite daunting.”

First Mortgages, Second Mortgages, and Their Default 563

C

7

8 6

2 10

1

9

5

4 3

B

DA

Fig. 1 The scheme of transitions and states determined by possible statuses of the first and the second loans, without prepayment. A both loans are not in arrears, B the second lien in arrears, C the first lien in arrears, D both loans are in arrears

monthly and default is also indicated on a 30-day delinquency basis. Table 1 gives a summary of typical characteristics of the entire set of mortgages, whereas Table 2 breaks down the loans by year of origination and type, as well as listing average val- ues for some of these loans’ more important characteristics. Note that our matching procedure assures that these second loans are so-called piggybacks, originated at the same time as the primary loan. The usual explanation for such loans is that they per- mit the primary loan to be of 80 % or less LTV (loan-to-value) ratio, even for a person who wants to make less than a 20 % overall down payment, and so avoids the need for mortgage insurance on the primary loan. It might occur to most economists that the resulting benefit for the primary loan would need to be offset by the higher rates on the second loan, given an efficient market for insurance, but it should be observed that it is only the 80 % or less LTV ratio loans that are traditionally securitizable, and so for which a deep secondary market exists. The piggyback arrangement is then a convenient device for extracting, from a non-conventional loan, that part which can be expected, because of its greater market liquidity, to have particularly favorable terms not available through the equivalent larger loan.5,6

Previous Literature

As indicated, there is by now a substantial literature on the default behavior of pri- mary loans as they are affected by the presence of second-lien loans. We mention

5 It has also been suggested (Calhoun 2005) that originating piggybacks in place of higher LTV single loans helped banks avoid certain capital requirements, another explanation for better terms being offered on the pair of loans than on an equivalent single loan. 6 While we cannot entirely preclude the possibility of additional so-called “silent seconds”, which are unobserved second loans occurring at a later date, typically home-equity loans for a purpose other than funding the house itself, this seems especially unlikely in our sample, given that there is already an explicit second loan at origination.

564 J. B. Kau et al.

Table 1 Summary statistics for select variables

Variable No. obs. Mean St. dev. Min Max

Adjustable-rate first mortgages

Rt1 30314 7.90 1.41 3.4 13.88

Rt2 30314 11.61 1.69 6.49 16.99

LTV1 30314 80.91 2.64 54 90

LTV2 30314 18.61 3.14 4 43

Term1 30314 360 0.49 300 360

Bal1 30314 162.7 95.8 22 880

Bal2 30314 38.0 24.6 6 250

Marg1 30314 6.13 1.53 1.8 12.7

origCLTV 30314 1.00 0.02 0.73 1.01

No. modif. 3357

Fixed-rate first mortgages

Rt1 5123 8.44 1.61 4.85 13.88

Rt2 5123 11.48 2.01 6.70 16.99

LTV1 5123 80.90 3.11 22 90

LTV2 5123 18.33 3.41 4 33

Term1 5123 327.9 68.25 120 360

Bal1 5123 123.4 72.4 21 840

Bal2 5123 27.9 17.8 8.25 197.8

origCLTV 5123 0.99 0.02 0.40 1.01

No. modif. 495

Rt the rate at origination; Term the contract term, months; Bal the balance of the loan at origination, thousands of dollars; Marg the contract margin; origCLTV the combined loan-to-value ratio at origination; No. modif. number of modified first liens

Gerardi et al. (2009), Sherlund (2008), Demyanyk and Van Hemert (2011) and Elul et al. (2010) as outstanding examples. Except for Eriksen et al. (2011) and Jagtiani and Lang (2010), however, these papers lack further information on the the second loan beyond origination, except possibly as is reflected in the combined loan-to-value ratio. Eriksen et al. (2011) does have full information on second loans, as well as the firsts, for a smaller set of 3,078 FRM mortgages (taken from the same data set as the current one), but they do not fully exploit that data, in the sense that they look only at the effect of seconds on firsts, rather than treating them simultaneously.7 The same limitation exists in the nonetheless exceptional work of Jagtiani and Lang (2010), who match home equity loans with primary loans, but concentrate on such issues as

7 The data set of Eriksen et al. (2011) is a bit small to engage in the sort of analysis done here, and so in most of their analysis the matched FRMs are combined with other primary FRM loans who have no known second match, thus making these latter loans subject to “silent seconds.” The latter is a problem typically encountered in most empirical mortgage analysis, though as noted, the problem is minimal here.

First Mortgages, Second Mortgages, and Their Default 565

Table 2 Sample by year of origination

Origination No. of Perc. of No. of Perc. of CLT V Rt1 Rt2

year ARM ARM FRM FRM

1999 234 0.8 44 0.9 0.78 10.98 14.36

2000 954 3.1 116 2.3 0.84 11.38 14.47

2001 1640 5.4 838 16.4 0.87 10.06 12.97

2002 3463 11.4 793 15.5 0.89 9.28 12.27

2003 4426 14.6 1239 24.2 0.89 8.28 10.86

2004 2867 9.5 747 14.6 0.89 7.39 9.40

2005 5540 18.3 363 7.1 0.98 7.17 8.29

2006 10467 34.5 856 16.7 1.07 7.75 8.72

2007 723 2.4 127 2.5 1.13 7.39 8.76

Total 30314 100 5123 100

The third and the fifth column represent the share of originations in a given year to the total number of, respectively, adjustable- and fixed-rate first mortgages in our sample. The sixth column (“CLT V ”) displays the mean current combined LTV over time for that origination cohort (both ARM and FRM first liens), the seventh and the eighth columns (“Rt1” and “Rt2”, respectively) display the average current rates on the first and the second lien over time for that origination cohort

who continues to maintain their second loan while nonetheless defaulting on the first, rather than providing a comprehensive estimation of all default activity among the loan pairs over time.

Agarwal et al. 2006a, b face the opposite problem to most of those articles men- tioned above, in the sense that they have full information on the second-lien loans but little information on the first loan, other than as reflected in combined loan-to-value ratios. Their analysis is restricted to lines of credit (Agarwal et al. 2006b) or to home equity loans together with lines of credit (Agarwal et al. 2006a).

LaCour-Little et al. (2011) engage in matching of piggyback loans, but keep the analysis at the state or zip code level, rather than the individual loan level. Finally, while it did not engage in a similar empirical analysis, since it was written before the recent events which now provide us with so much information on default behavior, we would remiss if we did not mention Calhoun (2005), whose prescient analysis of piggyback loans portended many of the difficulties which have more recently came to pass.

We note, finally, that our approach, with its multiple states and the risk of default, is reminiscent of the vast literature on rating transitions of corporate debt (see, for instance, Lando (2004) for a partial review.) One important difference, though, is that this rating transition literature is necessarily concerned with the market’s view of the imminence of default, whereas we are concerned with actually occurring default, and not market perceptions. The occurrence of actual corporate default is, of course, a much rarer event, particularly in absolute numbers, than is default on residential mortgages.

566 J. B. Kau et al.

The Empirical Framework

Default

An overall theme of this paper is that there is not just a first loan that is influenced by a second, nor just a second that is influenced by a first, there is a pair of loans that the borrower considers together at all times, whether one or the other is in default, until such time as there is final foreclosure on the house. Since this is how the borrower is presumed to think, this is how we must approach the problem: we have tried to take this view seriously in developing our estimation model.

As already indicated, Fig. 1 illustrates the overall setup of our state transition scheme. State A is the initial state of both loans being current, state B is the second being in default with the first current, state C is the opposite, and state D is both loans being in default.

Pride of place among the forward transitions is given to transition 1, where begin- ning from both loans being current, only the second loan goes into default, whereas transition 2 is where, instead, only the first loan goes into default. In between is transition 3, where both loans go into default simultaneously.

Unlike much analysis, we do not, however, stop with these competing risks from the initial state A, but instead follow the pairs of loans throughout their lives. Tran- sition 4 is the transition from only the second in default to both being in default, whereas transition 5 is the corresponding transition from only the first being in default to both being in default. Note that one could have treated transition 3, both simultaneously defaulting, as transition 1 immediately followed by transition 4, or alternatively, as transition 2 immediately followed by transition 5, but besides the question of which way to treat it, this simultaneous decision to default seemed a distinct and significant enough choice to warrant its own transition.8,9

Not only have we included all the possible forward transitions toward default, but we have also included the corresponding backward transitions restoring loans to currency. After some preliminary investigation, it was decided in the backwards direction to treat the pair of paths 6 and 10 as obeying the same transition law, as well as treating the pair of paths 7 and 9 in the same manner. Given the large number of possible transitions, further elaborated below, and the limited amount of data, it was necessary that some consolidation occur, and the backward directions seemed the most promising candidates, given that they are of less importance to us and usually come with less observations. Note that comparing the two paths in each of above pairs, the same mortgage is returning to currency, it is just a matter of whether the other mortgage is in default or not.

While some transitions are obviously more common than others, none are vac- uous: all possibilities occur with some frequency in our data. Furthermore, it is

8 In part, the distinction is warranted because while the logic of why one would default on, say a first and not a second has been called into question, no one questions that one might default on the two loans together. 9 We treat movements from B to C or vice versa as a return to A followed immediately by the other leg of the trip.

First Mortgages, Second Mortgages, and Their Default 567

7 5

4

1

86

9

11

2 10

3 E

A’’

A’

C’

C’’

B

D

Fig. 2 The scheme of transitions and states determined by possible statuses of the first and the second loans, prepayment of the first lien included. A′ both loans are not in arrears and the second loan has not been prepaid, A′′ the first loan is not in arrears and the second loan has been prepaid, B the second lien in arrears, C′ the first lien in arrears and the second loan has not been prepaid, C′′ the first lien in arrears and the second loan has been prepaid, D both loans are in arrears

possible, and sometimes happens, that one or the other of a loan pair may enter into default, then one or the other may return to being current, and then, once again, a default reoccurs for one or the other loan. Indeed, our scheme permits any history of recurrent default behavior to be accounted for among the loan pairs.10

Prepayment

It must now be admitted that we have not been entirely forthcoming as to the com- plexity of the situation. In order to stress what we are primarily interested in, default, we have avoided mention, till now, of another possibility, prepayment. We have not in fact ignored prepayment, though we have treated it in a rather more cursory fashion than default. The first point to note is that, though we continue to follow a loan pair if only the second prepays, if the first prepays we cease observing the pair. We thus have an additional state E representing the first loan having prepaid, which constitutes the only absorbing state of the model. See Fig. 2 for an illustration.

What we have referred to as state A is then formally two states, A′ and A′′, where A′ is both loans fully current and A′′ is the first loan fully current but the second prepaid. The same distinction exists for state C (and, if you wish, for state E, though not for B, nor D), so C′ is the first loan in default with the second current, whereas C′′ is the first loan in default with the second prepaid. The reason we feel entitled to refer to either A′ or A′′ as state A is that we assume that transition 2 is unaffected by which state, A′ or A′′, the pair is in, though, of course, for transitions 1 and 3 it does make a difference, in the somewhat trivial sense that a pair in state A′′ cannot actually transition to state B or D, since a prepaid second loan can obviously never

10 Note that the unobserved heterogeneity assigned to an individual for a particular transition may vary with the recurrence.

568 J. B. Kau et al.

go into default.11 The same obvious logic applies to other states and transitions, both forward and backward. The consequences are further illustrated in Fig. 2. Note, also, that in the spirit of limiting the complications arising from the opportunity to prepay, we have treated all transitions to state E as obeying the same law, that of transition 11, no matter the state of origin. There are then 9 different transitions that need to be estimated.

The Statistical Technique

The statistical framework is essentially the same as the other mixed proportional haz- ard models that have already been widely employed for mortgages facing competing risks, given unobserved heterogeneity.12,13 The main difference is that here a loan does not necessarily terminate or cease being observed after its first transition, as in the standard competing risk models of default and prepayment, and, indeed, here there is the possibility of repeated returns to the same state, limited in principle only by the finite life of the loan. Note that it is assumed that the hazard from a particular state depends only on the the most recent duration in that state, though of course the covariates affecting the baseline may evolve in either mortgage or calendar time.

No distributional assumptions were made as to the frailty distribution, which is approximated by masspoints.14 The advantage of the discrete masspoint method is that it can arbitrarily well approximate any actual distribution and need not result in the biases inherent in the choice of a specific functional form for the frailty distribu- tion, as is inevitably required when adopting a continuous frailty distribution. (See, for instance, the discussion in Han and Hausman 1990).15 In order to assure computa- tional feasibility, though, we did limit ourselves to four masspoints. As noted earlier, though, the assigned frailty term of an individual may vary by the source state, the tar- get state, and the particular recurrence. Prior experience with competing risk models (see, for instance, Deng et al. 2000) showed that using only two masspoints seemed adequate to the task of treating unobserved heterogeneity among mortgage holders.

Contractual Features Affecting the Transition Hazards

Note that the setup and estimation technique permits covariates to vary at will among the various transitions, but that we typically keep them the same, except when inves- tigating some particular feature of default. This is with the notable exception of

11 That is, if one is in, say, state A′, then one can technically only move to C′, but not to C′′ and if one is in state A′′, one can move to C′′, but not C′. This is, however, of little importance for these transitions, given that we have assumed the rules of the transitions are the same, though for further possible transitions, we do need to keep track of which state the pair is actually in. 12 See Clapp et al. (2006) for a discussion of the use of such models in the context of mortgages. 13 Identification of our model is achieved by results going back to at least (Sueyoshi 1992). See Brinch (2009) for a more recent discussion of such identification results. 14 See discussions in Wienke (2011) or Bijwaard (2011) for the importance of treating unobserved heterogeneity in the context of duration models. 15 Thanks to Simen Gaure and Knut Røed for graciously sharing their code. This software has also been used, for example, in the estimation of models of employment transitions; see, e.g. Gaure et al. (2008).

First Mortgages, Second Mortgages, and Their Default 569

transition 11, prepayment, which is modeled with a rather different set of covariates than are the default transitions.

The key contractual variables of the mortgages are in general dynamic, being at their current values, and include the ones most widely recognized in the mort- gage literature: i.e. the contract rate, the loan size, and the loan-to-value ratio. Being dynamic and current,16 these features are as applicable to a variable rate mortgage as a fixed one. Note, though, partly to conserve on variables, we have invoked elements of the combined loan hypothesis (see further discussion below), having aggregated such things as the loan sizes, in balcomb, and the contract rates, in ratecomb (see below for the exact definitions). We have, however, in the most basic model (Table 7) kept distinct what is traditionally considered the most important of these contractual variables, the two loan-to-value ratios. One non-dynamic, non-current contractual variable we do include among the covariates, though, is the original combined loan- to-value ratio, origCLTV, whose effect is sometimes thought to reflect self-selection of different borrower types, not fully captured by, say, their FICO scores. We note that these FICO scores have, indeed, also been included as another static covariate, fico.17 Other static covariates include lowdoc, indicating whether it is a low doc- umentation loan, together with a dummy variable distinguishing an ARM from an FRM, arm.18

Preliminary Data Analysis

In the lower triangle of Table 3, we present, near the lower right hand corner of each cell, the number of mortgages ever making the transition from the source state of that column to the target state of that row, and then conversely, near the upper left hand corner, the number of mortgages ever making the transition from the source state of that row to the target state in that column. Corresponding transitional prob- abilities are displayed in Table 4. The diagonal elements of Table 3 represent loans where, from the state of both being current, the second prepays (for states other than A and C this is not possible, so no number is indicated.) In the upper triangle of the same Table, we list in parentheses only the number of loans that are making the transition for the first time. While some transitions are obviously more frequent than others, most are well populated, giving one confidence that the various rules of tran- sition can be estimated, despite the general need in hazard analysis that there be a

16 Case-Shiller HPA index series were used to derive the current loan-to-value ratio for properties located in 20 largest MSA’s; for the rest of the sample, FHFA state-level series were used. 17 Loans, particularly, adjustable rate mortgages have many additional features, such as margins, teasers, caps and floors, but these can be regarded as adequately reflected in the current state of the dynamic features of the loans which we do account for, e.g. the current contract rate, though it must be admitted that in a truly rational model they might exercise an additional influence on the future terms of the loan anticipated by the borrower, and, as with our motivation for including the original combined loan-to-value ratio, they constitute potential, though increasingly obscure, margins on which borrowers might self-select. 18 The covariate modif is a dynamic indicator variable activated when the loan is modified and will be discussed further below.

570 J. B. Kau et al.

Table 3 Transitions by source and destination

State A State B State C State D

State A 805 (4439) (6912) (10444)

(3450) (4290) (3255)

State B 4987 (930) (3443)

7779 (859) (2394)

State C 6288 987 99 (6118)

10966 1037 (3139)

State D 4425 3075 4457

13684 4395 8248

Prepay (E) 7780 0 0 277

The lower left triangle of the transition matrix displays total numbers: the total number of transitions from the source row state is displayed in the upper left corner of a cell, whereas the total number of transitions from the source column state is displayed in the lower right corner of a cell. The upper right triangle of the transition matrix contains the counts of first-time transitions: transitions from the source column state are displayed in parenthesis in the lower left corner, whereas transitions from the source row state are displayed in the upper right corner of respective cells. The top left corner of the first cell on the main diagonal contains the number of the second loans prepaid from state A, the same spot in the third cell on the main diagonal contains the number of the second loans prepaid from state C; the bottom row displays the number of the first loans prepaid from the respective column state

substantial number of observations before accurate estimation of the effect of covariates can be achieved.

In Table 5 we list typical values of some of the characteristics of the loans at the time of a transition from state A to either state B, state C, or state D, respectively. We note that the average loan-to-value ratios are not as high as one might imagine, indicating that an overreliance on the principle that the borrower must be acting to

Table 4 Transition probability matrix

Destination Source state

state A A′ B C C′ D E

A 0.952 0.214 0.157 0.018 0

A′ 0.001 0.951 0.036 0 B 0.009 0.553 0.013 0

C 0.013 0.610 0.018 0

C′ 0 0.029 0.003 0.962 0 D 0.016 0.187 0.206 0.950 0

E 0.009 0.026 0.046 0.024 0.002 0.001 1

The empirical transition probability from the source column state to the destination row state averaged over all durations is displayed in a cell

First Mortgages, Second Mortgages, and Their Default 571

Table 5 Summary statistics for select variables at the time of select transitions

Variable No. obs. Mean St. dev. Min Max

Transition 1

Ht/H0 7779 1.031 0.133 0.441 1.358

Rt1t /Rt10 7779 1.037 0.182 0.235 1.972

LTV1 7779 0.79 0.141 0.217 1.833

currCLTV 7779 0.962 0.181 0.334 2.287

DurSource 7779 9.66 12.1 1.00 104

Transition 2

Ht /H0 10966 1.003 0.142 0.426 1.357

Rt1t /Rt10 10966 1.064 0.187 0.343 2.16

LTV1 10966 0.816 0.153 0.369 1.88

currCLTV 10966 0.988 0.194 0.459 2.338

DurSource 10966 12.47 12.97 1.00 105

Transition 3

Ht /H0 13684 0.987 0.144 0.439 1.338

Rt1t /Rt10 13684 1.058 0.171 0.164 2.076

LTV1 13684 0.827 0.161 0.225 1.901

currCLTV 13684 1.018 0.206 0.341 2.328

DurSource 13684 13.32 13.03 1.00 112

Ht /H0 the ratio of derived house value at the time of transition to the house price at origination, Rt1t /Rt10 the ratio of contract rate on the first loan at the time of transition to that rate at origination, DurSource number of months that the borrower spent in the state from which a given transition occurred

minimize the market cost of the loan (discussed further below) is liable to run into difficulties.19

Rationality and Value Maximization

Value Maximization without Transaction Costs

We make the distinction between being rational, which in economics means acting in a goal-seeking manner, and so responding appropriately to incentives, and the much more narrow assumption, often employed in the finance-oriented mortgage literature,

19 There is of course also the inevitable problem that, even looking only at the averages, one would still expect our constructed loan-to-value ratios to underestimate the “actual” loan-to-value ratios of those houses going into default, since default will presumably be especially chosen among houses experiencing exceptionally high falls in their value compared to those in the region represented by the house price index. It is not, however, even clear that the notion of a particular house’s “actual” price is operationally defined; even if one had, say, recovery values after foreclosure, these would undoubtedly overstate the original fall in house value. As discussed below, such difficulties suggest that one concentrate on such qualitative issues as the direction of change in the default hazard when covariates change.

572 J. B. Kau et al.

where the persons’ goals are explicit and impersonal, that is they act to minimize the market value of the mortgage to the lender, meaning that they are then maximizing the value to themselves.20 It is this strict assumption that allows the precise predictions of the options approach to mortgages (see Kau and Keenan 1995). We consider briefly what would be the consequences of such strict behavior in the present context. We also assume, with considerably greater confidence, that the lender also behaves in a corresponding manner, and so acts to maximize the market value of the mortgage.21

The relevant comparison is between a loan pair and a single combined loan with the same contractual terms, so that if there is a primary loan in amount L1 and a sec- ondary one in amount L2, then the combined loan would be of size L = L1 + L2, the combined loan-to-value ratio would be CLT V = L/H = (L1 + L2)/H = LT V1 + LT V2, where H is the current house price, and the combined contract rate would be a = (L1/L)a1 +(L2/L)a2, where, of course, a1 is the current contract rate on the primary loan and a2 the rate on the second.

22 It should never be disadvanta- geous to have the two component mortgages, since one always has the right to treat them as one, but the question is whether one would ever actually take advantage of this possibility to distinguish them: if not, the combined loan hypothesis would be said to hold. As we are about to argue, in the idealized setting of no recourse, but where the lender always retains the option to costlessly foreclose, the combined loan hypothesis should hold.

Indeed, the only failure of the combined loan hypothesis would occur if the bor- rower were to ever default on one mortgage and not the other.23 However, one sees that this would only occur if by not defaulting on the other loan the borrower could forestall foreclosure, since, if not, he might as well default on both. In the case of defaulting on the first-lien loan, there would surely be some value left in the house, leading the primary lender to foreclose, and so, in fact, the borrower might as well default on the second as well. The interesting case occurs when we start with the borrower defaulting on the second.

The mortgage literature is by now accustomed to the idea of a value to the future right to default, so that one doesn’t exercise the option to default as soon as it comes “into the money” (see Kau et al. 1992), but rather one waits. It thus seems possi- ble that the house might have fallen enough in value that there is no equity for the

20 The source of value maximization is, of course, the classical separation principle that when the only factors that affect your utility are goods transacted in perfectly competitive markets, then all production and investments should be done with a view to value maximization, no matter your utility, given only that you prefer more goods to less. It may be questioned, however, whether one’s home and its financing can be always be satisfactorily viewed within this strict framework, and whether, in regard to all transactions, one indeed only faces the single constraint of your limited overall market wealth. In particular, this framework implies the ability to borrow and lend in perfect capital markets across all states of nature, and so avoids issues of moral hazard and adverse selection, as to the borrower. 21 We may regard the lenders, in the ideal case, as facing infinitesimal transaction costs, so that they only foreclose when there is value in the house to be had. 22 In order that everything be exactly expressible in terms of the loan-to-value ratio, one needs that the house price process be homogenous of degree one, as is the commonly used lognormal (geometric Brownian motion) process. 23 We are ignoring the possibility of prepayment, for convenience.

First Mortgages, Second Mortgages, and Their Default 573

second-lien holder to collect (H < L1), yet the house price has not fallen enough that you would yet default on the primary mortgage. Thus, one might default on the second, without fear of foreclosure, yet wish to maintain your payments on the first loan. However, one needs to recall why the future value of default for the primary mortgage typically exists; it is because of the upside that can be gained when the house rises in value, given that the downside of further house price losses can be avoided by this right to future default. However, in the current case, there is no upside potential for the borrower on his primary loan. If ever the house value again rises, the benefit will immediately be seized by the second loan holder, who will foreclose at that time, and so, given this, the borrower might as well default on the first as soon as they default on the second, thus assuring the combined loan hypothesis.24

Beyond Rationality and Value Maximization: Strategic Behavior

Notice that in the above logic we have not only assumed that the borrower and lender are value maximizers, but also that the borrower knows the lender to be a value max- imizer and acts on this knowledge. Following the traditions of game theory, it is this awareness by the borrower of the lender’s rationality that leads to what we refer to as “strategic default”, and not, as this term is sometimes utilized in the literature, the mere fact that the borrower acts in a goal-seeking manner, which we would find rather less remarkable and not at all strategic.25

Value Maximization with Transaction Costs

Of course, in reality, there are considerable costs to a lender in foreclosing, and so, extensive opportunities for the combined loan hypothesis not to hold, even with value-maximizing behavior by all parties. We note in passing, though, that there surely must be some upper limit on such transaction costs, else a second-lien holder would never foreclose, and so, again acting strategically, rational borrowers would always immediately default on these loans, rendering impossible the existence of a market for such second loans, which, however, we know not to be so in actuality.26

An interesting phenomenon that arises, given moderate transaction costs to the lender, and so, the possibility of the second defaulting but not the first, is that strategic default will then result in non-monotonic behavior with regard to default of the first. That is, as the house price falls, the borrower will begin defaulting on the second loan, whereas for the first loan, when the house price is only a little below L =

24 Interestingly, it makes no difference to the logic whether the holder of the first and second loans are the same or different entities. 25 Another assumption being invoked, in the terminology of game theory, is that this is a situation of complete information, which is to say the players are mutually aware of the situation both face, in this case the main doubtful condition being that the borrower is aware of the transaction costs faced by the lender. 26 Several authors (e.g. Jagtiani and Lang 2010) have emphasized the difficulty in foreclosing on seconds, since one needs to effectively take over the first loan as well. Be that as it may, in an ideal setting this would present no particular barrier, since one could presumably acquire the loan for its market value, and though, in reality, there may be transaction costs to this, they would appear not to be qualitatively different from other transaction costs arising when considering foreclosure.

574 J. B. Kau et al.

Panel A

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L1+L2L1

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Fig. 3 The region of default on the second lien. R1,2 - the region in which a borrower would prefer to default on both second and first liens, R2 - the region in which a borrower would prefer to default on the second lien but not on the first lien

L1 + L2, the borrower will think that the second-lien holder will surely foreclose, given that there will remain enough equity to justify the effort, and so the borrower might as well default on this first loan as well. On the other hand, at further lower house prices, there will not be enough equity for the second-lien holder to bother foreclosing, now or in the likely future, and so the borrower will rationally elect to continue the first loan. Finally, at low enough house prices, the borrower will again surely default on both mortgages. See Fig. 3 for an illustration. Note, though, that it might be difficult to actually observe this interesting nonlinearity in the actions of the borrower with regard to the first loan, providing house prices change in a gradual fashion, since the borrower is then likely never to make it to the final or even the intermediate situation without having first entered the earliest situation, where he would have already defaulted on both loans.27

Empirical Results Concerning Strategic Default

Guided by the reasoning of Fig. 3, particularly as seen in Panel B, we chose a rep- resentation of the role of the LT Vi’s involving two covariates: CLT V − 1 and (LT V1 − 1)2. The first, referred to as currCLTV-1 in the estimations, just captures the overall effect of the combined loan-to-value ratio, where, given the specificity of Fig. 3, we decided to express matters in terms of whether there was any equity left in the house or not. The second, called stratdef in the estimations, captures the notion of potentially strategic default behavior, already explained in the context of Fig. 3, occurring when the borrower is again near zero equity, but now when just counting the first loan. Thus, if one were to graph the effect of CLTV on the log hazard of,

27 On the other hand, payments are made monthly and one only rationally defaults at that time, and there is then always the possibility that, even with continuous house price movements, that the house price will have sufficiently fallen in that time such that one arrives in one of the later regions. Also, even though the value of ideal default is typically found to be low relative to the size of the loan, so one would normally be expected to default on the second loan long before its value had completely disappeared, certain borrowers seem to face substantial transaction costs of their own in defaulting, perhaps reputational costs, and for such individuals, all value in the second might indeed have vanished by the time they default.

First Mortgages, Second Mortgages, and Their Default 575

Table 6 Mixed proportional hazard estimates of the first specification (strategic default)

Variable Transition 1 Transition 2 Transition 3

Estim. Std. err. Estim. Std. err. Estim. Std. err.

currCLTV - 1 1.1080 0.0770 1.6648 0.0566 2.2152 0.0540

stratdef −1.7581 0.2207 −4.0434 0.1728 0.1802 0.0100 balcomb −0.0044 0.0129 0.1055 0.0096 0.1940 0.0077 ratecomb 0.2219 0.0096 0.1281 0.0080 0.1660 0.0087

modif 0.0159 0.0424 −0.6001 0.0351 −0.7211 0.0421 arm 0.3631 0.0304 0.4444 0.0256 0.5114 0.0282

fico −0.7163 0.0322 −0.7924 0.0266 −0.6311 0.0275 lowdoc 0.1274 0.0322 0.3460 0.0251 0.4739 0.0240

origCLTV 1.9642 0.5611 −0.3337 0.4546 2.4590 0.5738 No. of unique pairs 35,437

No. of month∗pairs 1,189,542 Log-likelihood −301,432

See Fig. 1 for the definition of transitions

say transition 3, it would be expected to appear linear and of positive slope insofar as the first covariate is concerned, while the second covariate would also be expected to have a positive coefficient, so that the effect of this covariate on the trend of the other would depress the probability of both loans together defaulting when near LT V1 = 1, which is to say near CLT V = 1 + LT V2 = 1 + L2/L1. The same logic would apply to transition 2. On the other hand, transition 1 should, instead, bump up there; that is, the coefficient of its (LT V1 − 1)2 term should be of opposite sign, which is to say negative, if there are to be signs of strategic default. We remind the reader, though, that we are here using a very demanding notion of what is meant by strategic default.

In Table 6, we report hazard runs for transitions 1, 2 and 3, where we included the above quadratic specification in a search for signs of strategic behavior.28 The results are supportive of strategic default, though not unambiguously so. The covari- ate currCLTV-1 appears in all three transitions with a positive sign, as expected, and transition 1 indeed shows signs of the posited strategic default effect, since stratdef does indeed appear with a significantly negative sign. Furthermore, the same vari- able, stratdef, appears in transition 3 with an opposite positive sign, as hypothesized for strategic default. However, this same covariate appears with a significantly nega- tive sign in transition 2, which is not what would be expected were strategic behavior

28 We also did a run where, rather than using a quadratic, we used the bump function e− z 2

2 , familiar as the

Gaussian density function, where we chose z = (LT V1 − 1)/LT V 2 = L1/L2 − 1/LT V2. The thought was that the quadratic has extreme consequences far from LT V1 = 1, and so it might be better to have a more localized functional form. However this alternative choice of functional form yielded roughly the same results as the quadratic, so having performed this test of robustness, we decided to report only the more usual quadratic form.

576 J. B. Kau et al.

occurring, so if strategic default is indeed present, it is just seen in transition 1 relative to 3, and not transition 1 relative to 2.29,30

Rationality without Value Maximization

This strategic reasoning we have been attributing to borrowers may be rather too much of an imposition; indeed, it is not obvious that borrowers even always act as value maximizers. As a prime example, in the absence of large transaction costs on foreclosing on the first loan but small ones on the second, it is very difficult to recon- cile value maximization with the fact that borrowers are known to sometimes default on the first loan but not the second. Now, we have chosen a finer definition of default then sometimes used, 30 day delinquency, rather than, say, 90 days. This is motivated in part by the fact that we have gone to a great deal of trouble to permit recurrence and so are not that eager to artificially suppress it by coarsening the data.31 It is our belief that becoming 30 days delinquent (a legal definition of being in default) is a serious issue, as is then the reporting of such an event, and so if it is recorded that a person went 30 days delinquent and then 60 days later returned to being current, then this is important, probably accurate, information that should not be ignored by acting as if nothing ever happened. Similarly, we believe that there is an important distinc- tion between the first and second loan going into default together and this happening, say, 60 days apart.32 In any case, one cannot remove the fact that people sometimes default on the first and not the second by simply obscuring the information about their behavior: even if one uses a 90-day, or indeed a 180-day, definition of default, you will still find numerous instances in our data where a person has gone delinquent on the first loan but maintained the second throughout. See Fig. 4.33

Various explanations, implicitly trying to maintain the value maximization hypoth- esis, but in the presence of additional constraints, have been offered as to why lenders are sometimes seen to default on firsts but not on seconds. One explanation, is that, when no recourse is permitted, all debt on the first loan disappears upon foreclosure

29 As discussed further below, modif and balcomb are insignificant in transition 1 and origCLTV is insignif- icant for transition 2. Except for these insignificant cases, the covariates other than currCLTV-1 and stratdef are consistent across transitions and of the expected signs. 30 The remaining transitions of the alternative hazard specifications are qualitatively similar to those recorded for the one specification below that is completely reported, and so, because of space limitations, have not been reported as well. 31 It is more natural to use a stricter definition of default in most other competing risk analyses of default, where one ceases to view the loan after it is declared in default, and so there is no way afterwards to distinguish between temporary and more permanent defaults. 32 The desire to equate two loans defaulting 60 days apart with two loans truly defaulting together probably stems, again, from the fact that most analyses stop observing the loans soon after the loan of most concern to them goes into default; however in the current analysis, the fact that the other loan will later default is fully accounted for at the appropriate time. 33 Obviously, the percent of mortgages with only the first in default does drop significantly the longer one waits to report this fact, since in the intervening time, the first may prepay or return to being cur- rent, whereas the second might also go delinquent. All these possibilities do sometimes occur and are so recorded within our scheme.

First Mortgages, Second Mortgages, and Their Default 577

40%

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Fig. 4 Time in the state where only the first lien is in default. C - the state of default on the first lien only, D - the state of default on both the first and the second liens

whether sufficient value exists in the property or not, but that this is not true of all sec- ond liens, which may continue to exist after foreclosure as ordinary debt. However, while this may be true of home equity loans, it is not obvious that the same principle applies to piggybacks, which have been solely used to finance the house purchase, and so are very much like the primary loan. The legal issue seems a bit opaque; in our data, very few second loans of any kind have actually led to foreclosure. On the other hand, borrowers act on their perceptions, not on later realized events, and so even the perhaps unfounded fear that they would be pursued after foreclosure on their second loans might be enough to explain their observed behavior.34

A second hypothesis that has been advanced (Jagtiani and Lang 2010) to explain default on the first only is that first loans are almost always securitized, that being most of the point of there being a first and second loan, whereas second loans are sometimes kept by the originator. Thus, the hypothesis is that, granting moral hazard, the direct lender has a far greater interest in the second loan and so exercises influence over the borrower to maintain the second, even after the borrower has defaulted on the first loan. Again, however, while this argument might sometimes apply, it would seemingly be inapplicable to the second loans in our sample, which have in fact all been securitized, just as were the primary loans.

A third, very attractive explanation for defaulting only on the first loan is that the borrower is strategically seeking a modification on that first loan, and does not default on the second because of fear that the plan will be disrupted by foreclosure on

34 The argument that borrowers continue secondary credit lines for use even after foreclosure obviously does not apply here either, since piggyback loans are not at all credit lines.

578 J. B. Kau et al.

the second loan. It is difficult to discern someone’s intentions, but we can make the following observations. Starting from December 2006, the first recorded date we have of a modification, 8.4 % of the borrowers initially defaulting on only the first loan received a modification within 3 months and 15.7 % received one within 6 months, and so, at least in retrospect, the anticipation of a modification is not unreasonable. Furthermore, the borrowers apparently did have something to fear from foreclosure of the second, since 3139 out of 3585, or 88 %, of the first-only defaults involved loans where the house value exceeded the value of the first loan, so that potential value remained in foreclosing on the second. It also suggests that pursuing a modification is a potentially viable strategy, rather than simply abandoning the property.

Rationality with Relative-Loan-Size and Combined Loan-to-Value Effects

The fact that borrowers may not be strict value maximizes is not at all, however, to say they are irrational; in economics, one is accustomed to the idea that individuals are goal-seeking even though one is not sure of the exact form of their goals (their utility function) nor perhaps of all the constraints they may face. Nonetheless, based just on the notion that they are rational, one can typically make hypotheses about how they will respond to changes in the constraints one does know about.

In our case, we expect the combination of (LT V1, LT V2) to be critical and that an increase in CLT V = LT V1 + LT V2 will roughly be in analogy to an income effect, and so, normally, encourage default. On the other hand, given the “income” level CLTV, we would expect a “substitution” or “relative loan size” effect to arise as LT V2/LT V1 = L2/L1 varies, so that, as the relative importance of the second loan increases, one becomes more likely to default on that loan and less likely to default on the first one. As for the possibility of defaulting on both loans, the substitution effects between the first and second loans act oppositely, so that we would expect to see only the income effect coming from the combined loan-to-value ratio. See Fig. 5 for an

Fig. 5 Rational default for a borrower with two liens. R1 - the region in which a borrower would prefer to default on the first lien but not on the second lien, R2 - the region in which a borrower would prefer to default on the second lien but not on the first lien, R1,2 - the region in which a borrower would prefer to default on both

First Mortgages, Second Mortgages, and Their Default 579

illustration. Note that one might expect this same general sort of behavior regardless of whether a person more resembles a value-maximizer or more resembles, say, a liquidity-constrained borrower, simply trying to manage his payments.35 We regard the predicted behavior as merely an indication of borrowers correctly responding to incentives.36

Empirical Results Concerning Rational Behavior

Empirical Results with No Structure on the Role of Loan-to-Value

In Table 7 we take an agnostic view as to the nature of the LT Vi variables as they are treated in the mind of the borrower, and merely report their separate effects on the three principal hazards, transitions 1 through 3. One sees that each LT Vi vari- able appears positively in every transition. Given our discussion, when considering the own effects, LT V2 in transition 1 and LT V1 in transition 2, the “substitution” and “income” effects should support one another for these transitions, and so yield positive signs, while one might expect the offsetting substitution effects in transition 3 to more or less cancel out, leaving only the income effect, and thus again positive signs for both LT V1 and LT V2 in transition 3. On the other hand, in the case of the cross effects for transitions 1 and 2, the substitution and income effects are at odds with one another, and so one can reach no firm conclusion in principle, though in fact the effects are seen to again be positive.

Thus, except for the same insignificance of balcomb and modif in transition 1 also seen in Table 3, the results are uniform across all three transitions, and so would seem to be consistent with a hypothesis that borrowers choose whether to default accord- ing to relevant covariates but then select the form of default more or less randomly. However, as was already seen to some extent in section 6.3, and as will be seen more dramatically in the next section, this is merely because we have not here specified the interactions between loan-to-value ratios in a revealing manner, and when this is done, more striking differences across transitions appear.37

35 As noted before, though, a true value maximizer would at most substitute between defaulting only on the second loan and defaulting on both. 36 The closest analogy to our hypothesis of the behavior of a borrower with respect to LT V1 and LT V2 would be the behavior of a consumer of two goods, x2 and x1 , with respect to prices p1 and p2 when the goods are normal: for simplicity one may consider the case where preferences are homothetic. An increase in CLTV, given L2/L1 , would be like increasing p1 +p2 given p2/p1, and so moving along the ray p2/p1 , resulting in less x1 and x2 , which is indeed like scaling up the various probabilities of default, where then, a form of default not previously chosen would continue to not be chosen. On the other hand, increasing L2/L1 given CLTV would be like increasing p2/p1 given p1 +p2, which would result in less x2 and more x1 , or by our analogy, a higher probability of defaulting on the second loan but a lower probability for the first. 37 We note that origCLTV which was insignificant in Table 3 is now significantly positive, like the other two transitions.

580 J. B. Kau et al.

Table 7 Mixed proportional hazard estimates of the second specification (separate loan-to-value ratios)

Variable Transition 1 Transition 2 Transition 3

Estim. Std. err. Estim. Std. err. Estim. Std. err.

LTV1 0.4514 0.1129 1.9699 0.0773 2.0355 0.0967

LTV2 3.2693 0.3439 0.4713 0.1897 2.7475 0.3205

balcomb −0.0014 0.0132 0.1193 0.0098 0.2115 0.0078 ratecomb 0.2404 0.0104 0.1348 0.0082 0.2004 0.0092

modif −0.0177 0.0425 −0.6073 0.0357 −0.8131 0.0428 arm 0.4158 0.0312 0.5032 0.0263 0.6093 0.0288

fico −0.7536 0.0340 −0.7753 0.0273 −0.5795 0.0283 lowdoc 0.0797 0.0331 0.3538 0.0258 0.4175 0.0246

origCLTV 1.2637 0.6183 1.4471 0.4825 2.1313 0.6275

No. of unique pairs 35,437

No. of month∗pairs 1,189,542 Log-likelihood −289,863

See Fig. 1 for the definition of transitions

Empirical Results Concerning Relative Loan Size and the Combined Loan-to-Value Ratio

In Table 8 we have imposed the structure of our “relative loan size” hypothesis on the form of the hazard estimated, in order to test the signs of the resulting terms. As may be seen, it is indeed the case that CLTV appears in each of transitions 1, 2, and 3 with an always significant, positive sign (the normality effect), while the relative loan-to-value ratio LT V2/LT V1 appears in 1 with a significantly positive sign and in 2 with a significantly negative one (the substitution effect). The sign of this effect in transition 3 is instead insignificant, just as one might anticipate. Together with the earlier results of Table 7, this seems a rather strong confirmation that, when choosing to default, borrowers react to combined loan-to-value ratios and relative loan-to-value ratios in a manner consistent with rational behavior.38

Alternatively, to appreciate the behavior of these three principal transitions in the time dimension, consider Fig. 6, where we graph the hazard functions of transitions 1, 2, and 3. Thus, in the first case, we have the probability over time that the second loan only defaults, conditional on having been in a state of full currency that many months prior; in the second case we have the corresponding probability that the first only loan defaults, and finally, in the last case, we have the probability of both loans

38 Following the famous observations of Becker (1962), one might instead attribute such regular behavior in the form of default, not to rationality, but to uniformly random choice. However, while this interpretation might be possible for just the response to LTV, we have a host of other covariates exhibiting quite regular behavior, which could never be similarly explained by purely random choice.

First Mortgages, Second Mortgages, and Their Default 581

Table 8 Mixed proportional hazard estimates of the third specification (loan size ratio)

Variable Estim. Std. err. Estim. Std. err. Estim. Std. err.

Transition 1 Transition 2 Transition 3

currCLTV 1.2960 0.0789 1.9728 0.0565 2.5257 0.0490

origCLTV 3.1631 0.6686 3.7347 0.5031 5.2033 0.6236

balcomb 0.0163 0.0138 0.1414 0.0100 0.2397 0.0078

ratecomb 0.2711 0.0110 0.1569 0.0085 0.2314 0.0092

fico −0.7499 0.0355 −0.7664 0.0278 −0.5791 0.0281 lowdoc 0.0905 0.0352 0.3842 0.0262 0.4607 0.0244

modif 0.0599 0.0463 −0.5574 0.0368 −0.7958 0.0420 arm 0.3846 0.0335 0.4924 0.0274 0.5685 0.0284

LTV2/LTV1 1.0988 0.2831 −1.9799 0.1646 −0.4339 0.2580

Transition 4 Transition 5 Transition 6/10

currCLTV 0.9312 0.0838 0.6662 0.0618 −1.4750 0.0712 origCLTV 1.8768 0.8836 2.6657 0.5572 −13.401 0.5955 balcomb 0.1426 0.0167 0.0592 0.0102 −0.2245 0.0127 ratecomb 0.0425 0.0136 0.0556 0.0101 0.1054 0.0102

fico −0.1190 0.0446 −0.0135 0.0319 −0.1770 0.0342 lowdoc 0.2369 0.0420 0.1672 0.0288 −0.7106 0.0350 modif −0.9323 0.0488 −0.2845 0.0424 0.5911 0.0384 arm 0.2110 0.0443 0.1624 0.0331 −0.2692 0.0323 LTV2/LTV1 11.402 0.2245

Transition 7/9 Transition 8 Transition 11

currCLTV −1.8320 0.0679 −2.6308 0.1112 −34.059 0.2340 origCLTV −13.025 0.5824 −17.899 0.7170 17.959 1.0203 balcomb −0.1983 0.0128 −0.2049 0.0245 0.4627 0.0204 ratecomb 0.0808 0.0098 0.0440 0.0233 −2.6220 0.0238 fico −0.0561 0.0340 0.0031 0.0432 −1.0773 0.0474 lowdoc −0.7212 0.0333 −0.9055 0.0475 1.3418 0.0513 modif 1.4535 0.0346 1.2678 0.1408 −2.1278 0.1203 arm −0.3150 0.0321 −0.4783 0.0466 1.5803 0.0440 LTV2/LTV1 9.1898 0.1242 12.738 0.1624

spreadfix 11.076 0.1550

spreadvar −1.7223 0.1023 No. of unique pairs 35,437

No. of month∗pairs 1,189,542 Log-likelihood −291,152

See Fig. 2 for the definition of transitions

582 J. B. Kau et al.

0.01

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Fig. 6 Estimated hazards for a representative mortgage pair. Actual covariate values for a selected loan pair (cf Table 10) were used in this display. The first lien is an ARM and the second an FRM, with their first payment dates being June 2006. The pair was observed until June 2011. The LTV1 was 80 %, while the LTV2 was 20 %, all at origination; the balance of the first lien at origination was $119,200, the balance of the second lien at origination was $29,800; the contract rates at origination were, respectively, 6.85 % and 9.24 %; the FICO score at origination was 728 for both loans, with full documentation having been submitted by the borrowers. Hazard values are calculated as the product of covariate values, parameter estimates for the covariates (presented in Table 8) and parameter estimates for baselines, not reported here

simultaneously defaulting from a state of full currency after that many months. A typical loan pair, further described in the legend to the Figure, was chosen, and the conditional probabilities depend on the actual circumstances encountered by these loans through their lives, starting from their origination, since the covariates need to be evaluated in order that probabilities be generated.

Returning to our main Table 8, then for the transitions other than 1, 2, and 3, there are, of course, CLTV effects as well, and in addition there are substitution terms for all the return transition estimations, each of which features one transition from a node where a choice has to be made as to which path to take. These loan-to-value results are of secondary importance to us in comparison to transitions 1, 2, and 3, but, nonetheless, they are not without interest. CLTV appears in a positively signifi- cant fashion in forward transitions 4 and 5, just as with the other, principal forward transitions, while our reasoning requires that no substitution effect be estimated for these two transitions, since starting from their associated states there is no choice of which forward direction to take.

The Empirical Effects of Non-Loan-to-Value Covariates

The behavior of the non-LTV variables in default exhibited in Table 8 appears even more regular than that of the LTV variables, though, perhaps for that very reason,

First Mortgages, Second Mortgages, and Their Default 583

their effects seem intrinsically less interesting. We would regard the purely contrac- tual non-LTV variables as being ratecomb, balcomb, arm, and modif. We comment on modif below, when treating backward transitions.39 In the forward direction, the effects of the remaining contractual variables are the same across transitions 1, 2 and 3, being significant and raising the probability of default, all except the already encountered insignificance of balcomb in transition 1. In some strict value maximiza- tion reasoning with a natural homogenous-of-degree-one house process, loan size is indeed predicted to be insignificant, since only the loan-to-value ratio matters, but the less purist consensus is that increases in loan size increase the probability of default, even given the loan-to-value ratio. In any case, it remains a bit peculiar that balcomb is insignificant in transition 1 but nowhere else. When we turn to the remain- ing covariates, origCLTV, lowdoc and fico, which involve sorting among borrowers, the behavior of these covariates is entirely consistent across transitions 1 through 3, where as one would imagine, the first two covariates raise the probability of default but fico lowers it. As for the remaining forward transitions 4 and 5, the same con- sistent patterns established in transitions 1 through 3 continue to hold for all their covariates as well, the only exception being the insignificance of fico in transition 5, which is rather surprising given the otherwise consistently negative effect of fico among transitions 1 through 4 and the clearness of its character.

With but three exceptions among thirty five, then, where in all the three cases the problem is only that the coefficient is insignificant, all the results for forward tran- sitions among the seven covariates discussed in this section are exactly as expected. They are less interesting than the current loan-to-value variables precisely because, in principle, and as we have seen, for the most part in practice, they do not qualitatively distinguish among the different forms of default.

Prepayment

Prepayment is not of central interest, but it is a potentially important option for the borrower, so we felt the need to account for it. The primary new covariates assigned to it are the two spread variables spreadfix and spreadvar, being the spread of the current contract rate over the current market rate on a typical newly originating fixed or adjustable rate mortgage, respectively, and so representing the alternatives of pre- paying into a new FRM or into a new ARM, respectively. The other covariates used to explain prepayment all already appear in the default transitions. The covariate spreadfix appears with a positive sign, as might be expected but, somewhat surpris- ingly, spreadvar appears instead with a negative sign. There are few further surprises among the other covariates, though for a number of them we have few preconceptions as to their consequences for prepayment, since they were mostly chosen to instead explain default.

It is to be expected that arm is positive, and the fact that currCLTV, the driving force of default, is negative is quite interesting, since it suggests the substitution

39 More than the others, one could argue that arm includes a selection effect among borrowers, in addition to its direct role, where these two possible effects would seem to work in the same direction, increasing the probability of default in comparison to an FRM.

584 J. B. Kau et al.

between prepayment and default much discussed in the literature treating default and prepayment as financial options (see Kau and Keenan 1995). It is also interesting that origCLTV has an opposite, significant sign to currCLTV, suggesting that, indeed, origCLTV represents an intrinsically different phenomenon than just the contractual rate, an effect presumably arising through sorting by the borrowers at origination. It is to be expected that balcomb has a positive significant sign, and while, at first, it seems peculiar that ratecomb should have a negative, significant sign, one must remember that this is the presence of the two spread variables, so that it is then far from obvious what its sign should be. Fico was designed to predict default, not prepayment, so the correct sign of fico is also not so evident, though, as with default, it turns out to enter negatively. It is perhaps not surprising that lowdoc turns out to be positive and significant, though again this could not be said to be an inevitable prediction. We would informally expect that a modified loan would be less likely to prepay than a corresponding one that had experienced no such arrangement, and this presumption is indeed borne out in the result for modif.

Transitions to Currency and the Role of Modifications

To find the anticipated signs of covariates in the backward direction, one generally just reverses the anticipated sign in the forward direction. The logic, of course, is sim- ply that whatever makes one more likely to default ought to make one less likely to come out of default. This reasoning seems especially appropriate for the contractually based covariates: ratecomb, balcomb, arm, and modif. The covariate modif seems particularly relevant for return to currency, and so, we have not discussed it till now. While its effect might at first seem obvious, one must recall that we have already included the other current contractual variables, so if, for instance, a modification lowers the contract rate, this is already captured in ratecomb. Having said this, there are very possibly other unobserved changes in the modification, along with an antic- ipation of now better terms in the future, as well as in the present, all of which might then not be adequately captured by a covariate such as ratecomb. Thus, all in all, one anticipates that modif will have a negative sign for the forward transitions, and so, a positive signs for the backward ones. Now it turns out that in the forward directions modif does have the expected negative sign for all transitions except 1, where some- what surprisingly its sign is positive but insignificant,40 On the other hand, in the backward direction, where modif is of greatest interest, the signs, reassuringly, are always positive and significant.

As one might anticipate, the other contractual variables don’t match predictions quite so satisfactorily in the backward direction as they did in the forward one. The covariate balcomb does well, being significantly negative in the three backward directions, and arm performs adequately as well, having the correct negative signs in all cases. The difficulty comes with ratecomb, which is, rather inexplicably pos- itive in transitions 6/10 and 7/9 while insignificant in transition 8. As for the less

40 It is very possible that one is seeing signs here of the strategic behavior, often attributed to borrowers who default only on the second loan, where after a modification, presumably on the first loan, borrowers then default on the second, confident that no foreclosure will then ensue.

First Mortgages, Second Mortgages, and Their Default 585

contractual variables, both lowdoc and origCLTV do indeed have a consistently neg- ative effect throughout, but fico behaves quite oddly, being negative in transition 6/10 and insignificant in the other two return transitions.

As for the current LTV variables, in the return direction CLTV acts as expected, always having a negative impact, but perhaps not completely surprisingly, the substi- tution argument fares less well in the backward direction than in the forward one. The variable LT V 2/LT V 1 appears negatively in transition 7/9, which conforms with the predicted patterns, but it also appears negatively in 6/10 which is against predictions, and it is in transition 8, instead, that it appears positively. The sign of transition 8 is formally indeterminate but, as suggested earlier, given the dominance of the first loan over the second, if there is to be a sign, a positive sign indeed makes better sense. All in all, given our limited interest in the backwards directions and their smaller number of observations, they have behaved quite adequately, contradicting our pre- dictions only on four among twenty seven occasions, though being insignificant three additional times.

Probability Distributions

In order to take advantage of the interconnected and repeated nature of the transitions that compose our model, we decided to calculate the probability that a chosen mort- gage pair will be in a given state at the end of 5 years. This consists of taking our hazards, forming the time-varying transition matrix and propagating it sixty times over the 5 years. This is then applied to the known state of the loans at origination, which is to say beginning from state A. In order to apply this to a given mortgage pair, as in composing Fig. 6, we had to follow the actual history encountered by this pair, since only in this manner could we evaluate the relevant covariates. The result is dis- played in Table 10 for the chosen mortgage pair, whose characteristics are explained in the legend of the Table, with this indeed being the same pair chosen for Fig. 6.

For comparison, in Table 9 we display the empirical final distribution of our actual mortgages, among those whose history is known for 5 years. A comparison of the two Tables, however, is only provided as a matter of interest, since there is certainly no precise law of large numbers at work here between the theoretical and empirical results; the latter covers different time periods depending on the date of origination of the individual mortgage, with all these mortgages being of quite heterogenous char- acters, as opposed to the single mortgage which went into the predicted probabilities constituting Table 10. While we find the current predictions of some note, and so

Table 9 Empirical five-year terminal probability distributions

Probability A A′ B C C′ D E

Empirical 0.460 0.030 0.028 0.050 0.016 0.403 0.010

The empirical probability of a given final state was calculated as the ratio of the number of mortgage pairs in that state to the total number of surviving pairs at the end of the fifth year of the contractual term

586 J. B. Kau et al.

Table 10 Predicted five-year terminal probability distributions

Probability A A′ B C C′ D E

Predicted 0.314 0.099 0.015 0.196 0.017 0.229 0.129

Actual covariate values for a selected loan pair were used to calculate final model-implied probability distribution. The first lien is an ARM while the second one is an FRM, originated in TX, with their first payment dates being June 2006. The LTV1 was 80 %, while the LTV2 was 20 %, all at origination; the balance of the first lien at origination was $119,200, the balance of the second lien at origination was $29,800; the contract rates at origination were, respectively, 6.85 % and 9.24 %; the FICO score at origination was 728 for both loans, with full documentation having been submitted by the borrowers. The observation period for this exhibit was from September 2006 until June 2011.The borrower first fell behind on the first lien in September 2010 followed by delinquency on the second lien the next month. The second lien is shown as liquidated in March 2011 whereas the first lien entered foreclosure in May 2011

have presented them, we should remark that we have far greater faith in our previous estimations of the effects of covariates in a given period, since in the current case any errors in estimation of the transition matrix have been allowed to multiply some sixty times, as well as this transition matrix having to depend on the baseline hazard estimations and not just the estimations of the effects of covariates.

Conclusion

We have modeled the transition of pairs of first and second loans to default, where the novelty is that we continue following the loans throughout their history, rather than stopping observation after an initial default. This then requires taking account of the fact that one or both loans may return to being current, and indeed, may subsequently enter into default once again. We thus have, in effect, a Markov chain between the various possible states of currency or default among each pair of mortgages.

Beyond the act of just estimating the laws of transition within this setup, we have tried to pursue theoretical issues of interest. Most notably, for value-maximizing bor- rowers, we considered the theoretical merits of strategically defaulting on only the second mortgage, where our notion of strategic default was quite strict, involving not just rational behavior but cognizance of the rational behavior of one’s opponent. It should be noted that the assumption of value-maximizing behavior is quite limiting as well. Be that as it may, the empirical evidence for such strategic behavior proved suggestive but hardly convincing, since while it was true that borrowers exception- ally increased their propensity to default on seconds only near where there ceased being any collateral in that second loan, they increased their propensity to default on just the first loan as well.

On a more empirical note, our investigation of the return of defaulted loans to cur- rency seems somewhat novel. We found that modifications played a significant and positive role in this, even beyond the immediate improvement in, say, the contract rate coming from a modification. As one would expect, when a covariate increased the propensity to default for the forward transitions, it in general decreased the propensity to return to currency for the backward ones.

First Mortgages, Second Mortgages, and Their Default 587

Throughout, our greatest interest was in the role of the individual loan-to-value ratios, particularly in relation to the choice in the manner of default, starting from the initial state of being fully current. Economists typically do not inquire deeply into why a person wants a good; they simply assume that they do, and then make predictions about how demand for the good changes as circumstances change; e.g. as its price goes up the consumer demands less. We engage in similar reasoning. We are not exactly sure why borrowers default on second loans but not first ones; we advanced the strategic default thesis but the results were decidedly mixed. A much greater mystery is why borrowers default on first loans and not seconds, but default they do. The best explanation we can advance is that they are seeking modification of the first loan but fear foreclosure of the second. Whatever their exact reasons may be, though, borrowers mitigate their decisions according to changing circumstances in exactly the manner one would predict. Looking at, say, the most provocative case - default of the first only - borrowers do more of this the greater the contract rates, the loan sizes, and the combined loan-to-value ratio, and perhaps, most reassuringly, they do less of it as the size of second loan increases relative to the first.

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