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.
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Notes
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.
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.
GMAC is the acronym of the General Motors Acceptance Corporation, now rebranded as Ally Financial Inc.
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.”
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.
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.
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.
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.
We treat movements from B to C or vice versa as a return to A followed immediately by the other leg of the trip.
Note that the unobserved heterogeneity assigned to an individual for a particular transition may vary with the recurrence.
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.
See Clapp and Deng (2006) for a discussion of the use of such models in the context of mortgages.
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).
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.
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.
The covariate modif is a dynamic indicator variable activated when the loan is modified and will be discussed further below.
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.
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.
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.
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.
We are ignoring the possibility of prepayment, for convenience.
Interestingly, it makes no difference to the logic whether the holder of the first and second loans are the same or different entities.
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.
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.
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.
We also did a run where, rather than using a quadratic, we used the bump function \(e^{-\frac {z}{2}^{2}}\), familiar as the Gaussian density function, where we chose z = (L T V 1 − 1) / L T V 2 = L 1 / L 2 − 1 / L T V 2. The thought was that the quadratic has extreme consequences far from L T V 1 = 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.
As discussed further below, modif and balcomb are insignificant in transition 1 and origCLTV is insignificant 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.
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.
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.
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.
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 current, whereas the second might also go delinquent. All these possibilities do sometimes occur and are so recorded within our scheme.
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.
As noted before, though, a true value maximizer would at most substitute between defaulting only on the second loan and defaulting on both.
The closest analogy to our hypothesis of the behavior of a borrower with respect to L T V 1 and L T V 2would be the behavior of a consumer of two goods, x 2 and x 1, with respect to prices p 1 and p 2when the goods are normal: for simplicity one may consider the case where preferences are homothetic. An increase in CLTV, given L 2 / L 1, would be like increasing p 1 + p 2given p 2 / p 1, and so moving along the ray p 2 / p 1, resulting in less x 1 and x 2, 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 L 2 / L 1given CLTV would be like increasing p 2 / p 1given p 1 + p 2, which would result in less x 2 and more x 1, or by our analogy, a higher probability of defaulting on the second loan but a lower probability for the first.
We note that origCLTV which was insignificant in Table 3 is now significantly positive, like the other two transitions.
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.
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.
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.
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Kau, J.B., Keenan, D.C. & Lyubimov, C. First Mortgages, Second Mortgages, and Their Default. J Real Estate Finan Econ 48, 561–588 (2014). https://doi.org/10.1007/s11146-013-9449-5
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DOI: https://doi.org/10.1007/s11146-013-9449-5