What motivates a subprime borrower to default?

Journal of Banking & Finance 33 (2009) 681–693

Contents lists available at ScienceDirect

Journal of Banking & Finance journal homepage: www.elsevier.com/locate/jbf

What motivates a subprime borrower to default? Toby Daglish * Victoria University of Wellington, School of Economics and Finance, Level 3, Rutherford House, 23 Lambton Quay, Wellington, New Zealand

a r t i c l e

i n f o

Article history: Received 10 April 2008 Accepted 29 November 2008 Available online 16 December 2008 JEL classification: G12 G21 G33 Keywords: Mortgages Real options Default risk

a b s t r a c t This paper uses a real options approach to analyse the exercise of the default option embedded in mortgages. In particular, it examines a subprime household who borrows at a premium, but hopes to refinance at prime rates if their house appreciates. We show how these optimal default decisions can be used to calculate probabilities of default – an important input for risk management and pricing purposes. Numerical examples are provided, calibrated to US data. In a low interest rate environment, the credit-upgrade potential may discourage subprime borrowers from defaulting. However, default probabilities are highly sensitive to changes in interest rates and house prices. This provides a rational explanation for the prevalence of adjustable rate mortgages among subprime borrowers, and the subsequent large numbers of defaults, when interest rates rose and house prices declined. Ó 2008 Elsevier B.V. All rights reserved.

1. Introduction A great deal of press has been made of the subprime market crisis both in the US and internationally. Although many of the vectors by which contagion has spread through financial markets are to be found in the structured product market, or through the trading of credit derivatives between banks, at a grass roots level, the crisis has been caused by the default of households on their home loans. This paper focuses on the household’s decision, asking the question: what motivates a household to default on its mortgage? To address this question, we focus on a real options style analysis of the household’s decision. Such a model presumes that a household chooses to default or prepay their mortgage in such a way as to maximise their wealth.1 Given recent experiences with the securitisation of subprime mortgages through the use of collateralised debt obligations (CDOs) this analysis is particularly pertinent. One of the main advantages of the formation of CDOs is the ability to create some securities whose payoffs are relatively secure, even if the individual underlying securities have relatively high credit risk. This will only work well when the underlying securities’ defaults are weakly correlated. As has been discovered in practice, mortgage defaults may not be so independent as one might hope. In particular, as interest rates change, whether due to changes in market rates, or * Tel.: +64 4 463 5451; fax: +64 4 463 5014. E-mail address: [email protected]. 1 Or, since the loan is a liability to the household, to minimise the value of the loan. 0378-4266/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jbankfin.2008.11.012

the movement of hybrid adjustable rate mortgages from their fixed leg to their floating leg, a large number of households may simultaneously find themselves in a position where exercise of their default option is optimal. Correlation of default may be largely caused by the optimal use of the mortgage’s embedded options, rather than idiosyncratic factors affecting individual households. Thus the considerations in this paper have very real impacts on risk management for investors in mortgage portfolios. Thinking about the default decision using options is not entirely a new concept. Early work on pricing mortgages presumed that they can be valued as a financial derivative, whose underlying state variables are the interest rate and house price. Kau et al. (1987, 1993) outline the valuation of mortgage backed securities where the underlying securities are fixed rate and adjustable rate mortgages, respectively (or FRMs and ARMs as they are often abbreviated). In both cases, the household chooses to default or prepay their mortgage in a wealth-optimising fashion. Of course, not all decisions to default or refinance are motivated by the level of house prices and interest rates. Many households, for example, choose to prepay their mortgages as a result of selling the house, which may be motivated by other economic or personal decisions. As a result, in considering the real options problem faced by a household, one should consider that a household may also exercise their options suboptimally. Dunn and McConnell (1982) initially used this approach to model prepayment, and Kau et al. (1992) show how it can be blended with optimal option use in the valuation of fixed rate mortgages. In our analysis, we approach the problem entirely from the household’s perspective, to be

682

T. Daglish / Journal of Banking & Finance 33 (2009) 681–693

contrasted with the conventional pricing problem where we would be thinking about the value from the viewpoint of an investor in a mortgage backed security. This allows us to further explore the household’s response to costs associated with default, whether financial or social. More importantly, it allows us to consider the effect of a subprime credit spread on the household’s decision to default. Some readers may question the validity of a real options type analysis when modelling the behaviour of households, many of whom may be considered to lack financial sophistication. However, recently, the availability of micro level data for individual mortgages has enabled econometricians to test whether households exercise their options optimally or not. Stanton (1995) examines the decision by FRM borrowers to refinance their loans, while Deng et al. (2000) and Calhoun and Deng (2002), examine FRM and ARM loans (respectively) providing evidence that empirically, households default and refinancing decisions seem to be strongly influenced by the moneyness of the relative options: households most often default when their house’s value lies well below the present value of their future payments, and they refinance when the market value of their future payments is higher than the principal outstanding. As noted in Deng et al. (2000), although the real options problem faced by households is a complicated one, the conditions under which they should exercise their options can often be quite easy to determine, if the household can observe market prices for their mortgage.2 One critical difference between prime and subprime borrowers is the interest rate faced by the two types of borrowers. In general, subprime borrowers face higher rates, to compensate the lender for the credit risk involved in their loan. However, many subprime borrowers during a period of relatively rapid house price appreciation, such as during the early 2000s, would reason that the higher interest rate is an acceptable price to pay, since if house prices rise, they will accumulate sufficient equity in their property to refinance into a prime loan. Pennington-Cross and Chomsisengphet (2007) note that for many subprime borrowers, their home loan can also be a useful source of (relatively) low interest financing enabling them to consolidate their higher interest rate personal debt. This paper uses a real options analysis based on the pricing methodologies mentioned above, to trace the optimal exercise frontier for mortgage default. This enables us to explain which combination of house prices and interest rates would trigger a household to want to default and, for situations where the household should not default immediately, the probability that they will default in the future. We compare and contrast traditional, amortising, mortgages, with the recently popular interest-only mortgages. Our findings demonstrate a strong conditionality of default: the level of house prices which will trigger default depends critically on the prevailing interest rate. This conditionality also depends on the structure of the loan, both in terms of the interest rate structure (ARM versus FRM) but also upon whether the loan is amortising or interest-only. We show that under some circumstances, interest-only loans may be more credit-worthy than amortising loans. We demonstrate the robustness of our results to the potential for suboptimal default or refinancing, as discussed above. We also consider the possibility that a household faces some stigma cost to defaulting, whether financial or nonfinancial, and show that this does not qualitatively change our findings. The outline of the paper is as follows: Section 2 develops a model for house price and interest rate movements. Section 3 describes

2 An analogy can be made to regular financial option exercise. If one can observe the market price of a call or put option, exercise is optimal when the exercise value equals the option’s value.

the options available to a household and how the optimal exercise of these can be determined to find the states in which the household will default (and the probability of reaching one of these states). Section 4 presents empirical estimates of the housing model in Section 2. Section 5 uses these estimates and the analysis from Section 3 to present numerical examples of household default behaviour. Lastly, Section 6 concludes. 2. A model for housing Critical to understanding the relation between the structuring of residential mortgages and the optimal default behaviour of mortgagors is an understanding of the relation between interest rates and residential property prices. To examine that relation we assume a continuous time model for house prices (H) and interest rates, as suggested by Kau et al. (1993):

dH ¼ ðr þ nH ÞHdt þ rH HdW H ;

ð1Þ

where dW H is a Brownian motion and nH and rH are constants, reflecting the risk premium for house price growth and the volatility of house prices. r is the instantaneous interest rate faced by households borrowing in the prime market. We assume that housing provides a continuous stream of rent, proportional to the house’s value (qHdt) which is either consumed directly by the house’s owner or earned as revenue by leasing the house. In the risk-neutral world, the process for the house’s value is:

dH ¼ ðr  qÞHdt þ rH HdW H :

ð2Þ

We assume a Cox–Ingersoll–Ross (CIR) process for the short interest rate:

pffiffiffi dr ¼ ða  brÞdt þ rr r dW r ;

ð3Þ

where dW r is a second Brownian motion, and a, b and rr are constants. The correlation between dW r and dW H we define as q. The CIR model for interest rates allows interest rates to vary over time, but features mean reversion: in the long term, interest rates will return to some (fixed) steady state. Interest rates can be correlated with house prices, reflecting demand side effects (the cost of financing for purchasers) and supply side effects (the cost of constructing new housing capital). The assumption that house prices follow a geometric Brownian motion is a simple one, and is a similar assumption to that commonly used in financial option pricing. However, taken in conjunction with a stochastic interest rates process, it provides plenty of traction for analysing option exercise by households. When dealing with ARM loans, it is worth noting that the T year zero coupon rate in this model is given by:

rðTÞ ¼

2a r logðAðTÞÞ þ DðTÞ; T r2r T

ð4Þ

where we define: ðbþcÞT

2ce 2 ðb þ cÞðecT  1Þ þ 2c   2 ecT  1 DðTÞ ¼ ðb þ cÞðecT  1Þ þ 2c qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c ¼ b2 þ 2r2r : AðTÞ ¼

If the actual underlying house price for a borrower was observable at regular intervals, the parameters of (1) and (3) could be estimated using Generalised Method of Moments (GMM) estimation, based upon the work of Chan et al. (1992). Note that this presumes that the risk-neutral process for interest rates is identical to the physical process. An alternative approach, which would obviate this assumption, would be to work with yield curve information

T. Daglish / Journal of Banking & Finance 33 (2009) 681–693

to estimate the interest rate parameters. However, this would not enable us to estimate the correlation between house prices and interest rates.3 Since individual house prices are not available, in estimation, one must conventionally resort to the use of a house price index as a proxy for H. However, since this index represents an average across many house prices, it will understate the volatility of an individual house, and overstate the correlation between house prices and interest rates. If information is available as to the variability of individual house prices, relative to this index’s growth, we can correct the parameter estimates, by assuming that individual house price returns are composed of a common component and idiosyncratic noise:

rH ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2H þ r~ 2H ;

 H is the estimate of house price volatility obtained from the where r ~ H is the estimate of idiosyncratic house house price index, and r price return volatility. Idiosyncratic returns, if correlated with interest rate changes, would necessarily no longer be idiosyncratic. Hence correlation of a house’s price with interest rates must come through the common component. We can thus estimate the correlation parameter as:

q¼

r H q : rH

Further details of this breakdown of house price volatility is found in Calhoun (1996). 3. Real options analysis of mortgage default A residential mortgage can be thought of as being composed of three assets. The most important of these is the set of payments promised by the household. This takes the form of a bond, with either fixed coupon payments (for a FRM) or floating payments (for an ARM). The bond may either entail a payment of principal at its termination (in the case of an interest-only mortgage) or may have the principal paid over the life of the mortgage (in the case of an amortising mortgage). In the latter case, the periodic payments made will be larger to account for the principal being paid. The second component, with which we are especially concerned, is a put option written upon the house purchased with the mortgage. The mortgage is secured by this property, which is subject to seizure by the bank in the event of the household failing to make payments on its debt. Seen from the household’s perspective, the household has the option to cease making payments on the mortgage, in exchange for the house, in effect giving the household the right to sell their house (to the bank) for the value of the mortgage. Note that this option is peculiar to non-recourse lending states in the United States. In recourse lending states, and many European nations, a household who defaults on their loan will be accountable for debt in excess of the house’s sale value. In this situation, default is not a profitable strategy, except in situations where either the household’s assets are sufficiently low as to be unable to make up the shortfall, or the legal costs entailed in the bank pursuing the household preclude prosecution. The third component is a call option. The household is entitled to pay back their mortgage at any time by paying the principal outstanding. This gives the household the right to repurchase the bond they have sold to the bank. The strike price of this option will be constant for an interest-only option (since principal remains the same over the life of the mortgage) however, for amortising mort3 Specific details of the GMM estimation used here are available, on request, from the author.

683

gages, the strike will decline over time. For the case of an adjustable rate mortgage, the strike price will depend upon the path taken by interest rates over the preceding years of the mortgage: if interest rates have been high, relatively less principal will have been repaid. The call option is generally exercised in two situations. The first situation is one in which the house is sold prior to the maturity of the mortgage. In this case, the household must pay off any remaining principal on their loan. The second situation is one in which the household refinances their mortgage, by acquiring a new mortgage (presumably at a lower rate than their existing one) and paying off the existing mortgage. Refinancing is only viable if the household has positive equity in the house. Again, this option may not be so relevant in non-US markets. In the United Kingdom, for example, longer term fixed mortgages normally have a ‘‘mortgage redemption penalty” for early prepayment, which discourages exercise of this option. Although it is convenient to break up the mortgage into these three components, it is important to realise that they are inter-related. The strike price of the put option depends expressly on the market value of the bond, just as the strike price of the call option depends on the interest rate structure of the bond. Further, the options themselves are not independent. Exercise of one option necessarily eliminates the other. Mathematically, we can denote the value of the mortgage (as a liability) to the household as M, the principal outstanding on the loan as P and the house’s value as H. The borrower’s objective is to minimise M. Provided the homeowner has positive equity (i.e. H > P) the call option requires that the mortgage can never be a greater liability than the principal outstanding on the loan.

M6P

if H > P:

ð5Þ

Similarly, if we denote the stigma cost of default (as a portion of the loan’s principal) as S we can see that the put option also creates a bound on the mortgage’s value:

M 6 H þ SP:

ð6Þ

If the mortgage’s value is sufficiently low then neither default nor exercise will be optimal, and the household’s optimal behaviour will be to continue to service the loan. The stigma effect is an important one. It captures two effects. First, a household who choose to default on their mortgage will face (at least in the short term) credit rationing. Secondly, it is a stylised fact that most households attach a value to their own home over and above its market value. In such a situation, a household will not default completely ruthlessly: they may well choose not to surrender their house until financial pressure becomes quite strong. We would like to further allow for a situation where the homeowner may ‘‘involuntarily” sell the house, similar to Kau et al. (1992) and Dunn and McConnell (1982). Such a situation could develop as a result of a number of factors: the household could face cash flow difficulties, leading to their being unable to meet mortgage payments, they might be moving house for personal or economic reasons, or a divorce could force sale of the property. The householder now has a choice between selling the house and paying off the remainder of the mortgage (in which case the mortgage’s value jumps to its principal value) or defaulting on the loan, and letting the bank seize the house (in which case the mortgage value jumps to the house value, plus any default costs borne by the householder). The household’s decision will be based on the cheaper of the two exit strategies. Note that even in the case of a rental property, portfolio rebalancing or divorce settlement could trigger sale of the property. Since unemployment could be an important cause of house sales, and unemployment in turn could be correlated with interest rates, we assume that this intensity could, in general, be a function of interest rates.

684

T. Daglish / Journal of Banking & Finance 33 (2009) 681–693

Mathematically, in such a situation, the household must choose whether to sell the house and pay off their loan, in which case the mortgage’s value becomes P, or to default, in which case the mortgage’s value becomes H þ SP. As outlined in Kau et al. (1993), in the risk-neutral world, short term interest rates and house prices obey (2) and (3). Hence we can infer that, over the course of its life, the value of the mortgage to the household will be governed by the partial differential equation (PDE):

06

1 2 1 r rMrr þ r2H H2 MHH þ ða  brÞMr þ qrH rr rHMrH þ ðr 2 r 2  qÞHMH þ M t  ðr þ dðH; PÞÞM þ Cðt; dðH0 ; P0 ÞÞ þ kðrÞ  minðH þ SP  M; P  MÞ;

C ARM;io;t ¼ ðrARM;t þ dðH0 ; P0 ÞÞP0

PARM;io;t ¼ P0 ;

however, the coupon at any time is now based on the current level of the ARM rate, rather than the time zero level of mortgage rates. The ARM rate (after the initial fixed period) is determined each year as:

rARM;t ¼ maxðminðr t ð1Þ; r ARM;t1 þ cap1 ; cap2 Þ; r ARM;t1  cap1 Þ: ð7Þ

subject to the two exercise conditions (5) and (6). Here Cðt; dÞ is the (continuous) coupon payment of the mortgage, dðH; PÞ is the spread faced by the household over prime mortgage rates, and kðrÞ is the intensity of house sales. In the event of a house sale, the household must (as discussed above) decide whether to sell the house themselves or allow the bank to foreclose. Our household’s parameter d will affect their coupon payment C, since this will be calculated based upon their mortgage’s current interest rate. A household with positive d (a subprime borrower) will face a higher coupon rate, but will discount at a higher rate to compensate. For prime borrowers, we fix d at zero. Subprime borrowers will initially have a positive spread. However, if their loan-to value ratio decreases sufficiently (either by paying off principal on their loan or due to an appreciation in their house’s price) they will be able to access prime credit markets. Mathematically, for subprime borrowers, we consider d to be zero if the loan-to-value ratio is less than 0.8, and to take some positive value if the loanto-value ratio is greater than 0.8. In the case where d ¼ 0, since the household will now discount at a lower rate than their loan’s rate, calling the mortgage will be optimal. In effect, the household will refinance into a prime loan. This ability to move between the two markets is an important motivation for subprime borrowers, particularly in a rapidly growing housing market. One can think of this as the converse of the effect noted by Caplin et al. (1997), Peristiani et al. (1997) and Bennett et al. (2000), who note that households whose credit rating has declined (whether as a result of house price declines or unemployment) will often forego refinancing that would be optimal for other prime borrowers. Mortgages can be described by two attributes: first, the fashion in which interest rates are set, and second, the way in which the loan amortises. In this paper, we focus on FRM loans, where the rate is fixed for the entire life of the loan, and 1/1 ARM loans, where the rate resets every year. We also examine fully amortising loans and interest only loans, where no principal is repaid.4 Fixed rate loans are relatively straightforward: coupon rates (C) are fixed. In the first (interest-only) case, principal remains fixed over the life of the loan. In the second (amortising) case, principal will decay over the life of the loan, however the coupon will be slightly higher. If we denote the initial principal of the loan as P 0 , these can be expressed as:

C FRM;io ¼ ðr þ dðH0 ; P0 ÞÞP0 

where r þ dðH0 ; P0 Þ is the fixed interest rate of the loan (negotiated at the inception of the loan). For the case of an ARM for an interest-only loan, the coupon and principal are similar to that of an interest-only FRM:

ðr þdðH0 ;P0 ÞÞT

0 ÞÞP 0 e C FRM;amort ¼ ðrþdðHe0ðr;PþdðH 0 ;P0 ÞÞT 1

PFRM;io;t ¼ P0 C

FRM;amort PFRM;amort;t ¼ P0 eðrþdðH0 ;P0 ÞÞt þ rþdðH ð1  eðrþdðH0 ;P0 ÞÞt Þ; 0 ;P 0 Þ

4 Other possible structures can also be modelled within this framework. A 5/1 ARM, for example, has a fixed rate for the first five years, and subsequently resets its interest rate every year.

ð8Þ

Here r t ð1Þ represents the one year interest rate (see (4)), cap1 represents the yearly cap on interest rate changes and cap2 represents the lifetime cap on the ARM’s interest rate. When t is not an integer, r ARM;t ¼ r ARM;t . Note that the cap on changes in the ARM rate from year to year renders the mortgage’s value path dependent: knowing the current level of the short rate is not enough information to determine the ARM’s current interest rate. Amortising ARM mortgages introduce an added level of complexity, since the principal of the loan, in these cases, is also a path dependent function of interest rates. Specifically, if rARM;t is the ARM rate set at date t, for any s 2 ðt; t þ 1Þ

ðrARM;t þ dðH0 ; P0 ÞÞPt eðrARM;t þdðH0 ;P0 ÞÞðTtÞ eðrARM;t þdðH0 ;P0 ÞÞðTtÞ  1 PARM;amort;s ¼ P ARM;amort;t eðrARM;t þdðH0 ;P0 ÞÞðstÞ   C ARM;amort;t 1  eðrARM;t þdðH0 ;P0 ÞÞðstÞ : þ r ARM;t þ dðH0 ; P 0 Þ

C ARM;amort;s ¼

Kau et al. (1993) demonstrate that this path dependence on principal can be broken by normalising the mortgage’s value and the house’s price by P, in effect considering a value to loan ratio for the housing input and computing the mortgage’s value as a fraction of principal outstanding. We solve the problem of path dependence of ARM rates by constructing separate lattices for each possible level of the mortgage’s current ARM rate. At the end of each year, we transition between the lattices according to the current level of the short rate. Appendix A provides more details of the handling of the path dependence.5 Once the value of the mortgage has been determined, the optimal exercise frontier for the default option can be traced by finding the points on our finite difference grid where M ¼ H þ SP. 3.1. Calculating probability of default The evaluation of optimal exercise frontiers is a very useful output from such a model for a household, or for a lender seeking to evaluate the specific states under which default is likely to occur for the mortgages he writes. However, a more general picture of default behaviour can be obtained by calculating the actual probability that a household facing a particular combination of house value and interest rates will default at some time in the future. Such a probability can be relatively easily calculated in conjunction with solving the household’s optimal default/refinancing decisions. As discussed in Kau et al. (1994), if we denote pðH; r; tÞ as the probability of default over the remaining life of the mortgage, the probability of default will be characterised by the partial differential equation:

5 Further details of the finite difference scheme for solving (7) are available on request from the author.

T. Daglish / Journal of Banking & Finance 33 (2009) 681–693 Table 1 Parameter estimates for the interest rate and house price processes, annualised. Corrected estimates adjust the parameters for idiosyncratic variability in housing returns. Parameter

GMM Estimate

Corrected estimate

a

0.0067 0.1519 0.0263 0.0246 0.0101 0.5927

0.0897 0.0668

b

rr nH

rH q

0¼

1 2 1 r rprr þ r2H H2 pHH þ ða  brÞpr þ qrH rr rHprH 2 r 2 þ ðr þ nH ÞHpH þ pt þ kðrÞFðH; PÞ:

ð9Þ

Note that this equation is identical to (7) except for three aspects. The coefficient of the first partial derivative with respect to H (pH for probabilities, and MH for mortgage values) is no longer the risk-free rate minus the house’s rental stream, but is now the house’s expected return. Secondly, the discounting term (ðr þ dÞp) and the coupon term (C) are no longer present. Lastly, the jump term is now a function of house price and principal:

FðH; PÞ ¼



p

if H þ SP > P

1  p if H þ SP < P

:

This last equation suggests that if the household decides to move house, they will either sell the house and pay the mortgage (in which case the default probability jumps to zero) or default on the loan (in which case the default probability jumps to one). The boundary conditions for (9) are determined from the solution of the optimal exercise problem. The household’s default probability will equal one in states where default is optimal, and will equal zero in states where refinancing is optimal. For ARM loans, transition between the different lattices proceeds as in the pricing problem (see Appendix A). Amortising loans must be adjusted for the fact that house price relative to principal will be increasing, even absent any change in house price. However, since we do not scale probabilities by P, no rescaling of the solution is necessary at each time step.6 4. Parameter estimates In order to obtain estimates of the parameters of (1) and (3), we use quarterly data from Freddie Mac’s Primary Mortgage Market Survey (PMMS) ARM rate series, in conjunction with the Office of Federal Housing Enterprise Oversight (OFHEO) House Price Index. We use quarterly data spanning from 1984 quarter 1 to 2007 quarter 2.7 Our resulting parameter estimates are given in Table 1. In correcting the GMM estimates for idiosyncratic risk, we made use of OFHEO’s volatility estimates (as described in Calhoun (1996)). We obtained short term estimates of the idiosyncratic volatility for the nine regional house price indices, and averaged these to ob~ H ) of 0.0891. The tain a single figure for idiosyncratic volatility (r variability among the regions was not substantial, the highest being the Pacific region, with 0.0979, and the lowest being East South Central, with 0.0756.8

6

See step 1 in Appendix A. The PMMS series is available on a monthly basis, and was therefore converted to a quarterly series by averaging. 8 There is some evidence of mean reversion among the idiosyncratic component of house prices. Using OFHEO’s volatility statistics, a volatility can also be calculated for longer horizons. Considering a 30 year horizon, the annual volatility, averaged across the regions is 0.0672. In this case, regional estimates vary from 0.0538 (Mountain) to 0.0888 (Middle Atlantic). 7

685

Our estimate of the rental yield (q) is obtained from the 2005 US Housing Survey. We take the median monthly rent ($694) divided by the median price of an owner-occupied house ($165 344), to obtain a monthly rental rate. Multiplying by 12, we obtain an annual rental income of 5.04%. Note that, after controlling for mean reversion in interest rates, there is a negative correlation between unexpected innovations in interest rates and house prices: the effect of changes in mortgage rates on demand for housing. It is interesting to note nH ’s negative sign: house price capital gains are generally lower than the interest rates by around 2.5%. However, when rent is taken into account, housing earns a return in excess of mortgage rates of around 2.5% per annum on average. Regarding specific borrowing rates, we presume that prime FRM borrowers pay 6.4% interest, consistent with PMMS rates as at 26 October 2006. When dealing with ARM loans, we set cap1 ¼ 0:02 and cap2 ¼ 0:09 for prime borrowers, and cap1 ¼ 0:02 and cap2 ¼ 0:12 for subprime borrowers. Lastly, we consider a possible relationship between suboptimal default and interest rates (kðrÞ). From Pennington-Cross and Chomsisengphet (2007), we estimate that for a 1% increase in unemployment, there is a 0.72558% increase in mortgage terminations for subprime borrowers. Next we regress the Bureau of Labour Statistics’ seasonal unemployment figure on the ARM rate series to find that a 1% rise in ARM rates correlates to a 0.1816% increase in unemployment. The net result is a sensitivity of termination to interest rates of 0.13176. For an average level of suboptimal termination intensity  k the default intensity takes the form:

  a kðrÞ ¼ k þ 0:13176 r  ¼ k  0:005812 þ 0:13176r; b

ð10Þ

since a=b is the long run steady state level of mortgage rates. 5. Optimal default We now present the results of our analysis. We begin by discussing the case of prime mortgages, where the borrower does not face a spread. We use this to illustrate the optimal behaviour of different types of borrower, and draw some general conclusions regarding the determinants of the optimal default policy. We then turn our attention to subprime borrowers, using this analysis to explain their differences and similarities when compared to prime borrowers. Next, we present some analysis of default probabilities (as per Section 3.1). These provide an overview of the default risk involved in the different types of mortgage. Throughout these subsections, we assume that households behave ruthlessly. Finally, in Section 5.4 we consider the behavioural aspect of the mortgage decision, checking that our findings are robust to stigma costs and premature exercise. 5.1. Prime borrowers In the following analyses, we will often wish to show graphs where one axis is the ‘‘interest rate”. There are in fact two important interest rates for determining the household’s best plan of action. One rate is the prevailing short rate (r) which determines, at each time, the yield curve’s shape. However, for an ARM, because of its bundled caps, a second rate also affects the household’s decision: their own current ARM rate (r ARM ). For ease of display, in the graphs portrayed in this section, we only show situations where the household’s ARM rate is equal to the one year rate corresponding to the current short rate (see (4)).9 Since the ARMs con-

9 For interested readers, files containing the complete sets of numbers for each mortgage type are available on request.

T. Daglish / Journal of Banking & Finance 33 (2009) 681–693

A

1.6

t=0 t=10 t=20 t=25 t=29.9

Optimal Exercise Frontier (LTV)

1.5

1.4

1.3

1.2

1.1

1 0.02

B

1.6

t=0 t=5 t=10

1.5 Optimal Exercise Frontier (ILTV)

686

1.4

1.3

1.2

1.1

0.03

0.04

0.05 0.06 Interest Rate (r)

0.07

0.08

1 0.02

0.09

0.03

0.04

0.05 0.06 Interest Rate (r)

0.07

0.08

0.09

Fig. 1. Optimal default threshold for a 1/1 ARM. The axis r represents the short rate of interest. The one year ARM rate is assumed to be consistent with this short rate (see (4)). Graph (A) shows the loan-to-value ratio which will trigger default. Graph (B) shows this with loan-to-value calculated using initial principal, and assumes principal erosion occurs as if interest rates stayed at the long term rate of 4.41% throughout the mortgage’s life.

10

Although not, as we will see, for a subprime borrower.

Of course, this result is somewhat misleading, in that while the loan-to-value ratio that would trigger default for an amortising ARM mortgage may be lower than that which triggers default for an interest-only ARM, as time progresses, the amortising mortgage’s principal will decline, while the interest-only loan will (by definition) still have the same principal outstanding as it did on the day of its creation. In panel B of Fig. 1 we try to account for this effect by calculating an approximate initial-loan-to-value ratio which would trigger default. This is (in this case) rather a crude approximation. For an ARM, the actual principal outstanding depends not merely on the initial loan size and the current level of the ARM’s interest rate, but also on the sequence of ARM rates in preceding years. We have, in these calculations, assumed that the preceding reset dates of the ARM have all seen the ARM’s rate set at the long term interest rate of ab. Although, in most cases, this will not be correct, it does provide us with a representative rate of decline for the mortgage’s principal. Panel B of Fig. 1 makes it clear that this qualification of panel A’s story is important. While the increase in principal payments in later years may lead to the household defaulting at lower loan-to-value ratios, the outstanding 3

t=0 t=25 t=28 t=29 t=29.9

2.8 Optimal Exercise Frontier (LTV)

sidered allow for a 2% movement in their rates each year, any difference between the two rates is unlikely to persist for very long, nor to be very large. Examining the historical PMMS interest rate series, we note that there were only 13 cases where a borrower would have encountered these caps. The maximal amount by which a borrower’s ARM rate might have differed from the market on account of a ratcheting cap would have been 0.61%. In no cases would the difference have persisted after the next reset date. For a prime borrower,10 the call option becomes unimportant if the loan in question is an ARM. Since the coupon of the mortgage’s bond component resets each year, the value of the loan never strays far from par. As such, one might be tempted to suspect that interest rates have very little effect on the exercise of the put option. However, if we examine Fig. 1 we see that this is far from the case. Panel A of Fig. 1 shows that for early maturities, the investor’s optimal loan-to-value ratio at which to default is very strongly negatively affected by interest rates. The reason for this behaviour is the coupon payment required to maintain the mortgage. By foregoing default on a mortgage, the household must continue to make their monthly payments. As interest rates decline to zero, the household can continue the mortgage costlessly, and early exercise of the put option is never optimal. However, as interest rates rise, waiting becomes a more expensive decision. Note that this effect is entirely generated through optimal use of the options embedded in the mortgage: it is not a function of credit constraint on the part of the household. A ruthless household who can easily afford their mortgage’s monthly payments will still default on their mortgage if interest rates rise. As time progresses, the household will default at lower and lower loan-to-value ratios. This is caused by the amortisation of the loan. As time to maturity declines, the payments made each month are composed of greater and greater amounts of principal. As such, the cost of keeping the option alive increases, as a fraction of the loan’s principal. Fig. 2 shows a corresponding graph for an interest-only ARM. Now the household’s default threshold is almost totally insensitive to time. At the end of the mortgage’s life, the household will make a snap decision whether to default or not, but prior to that, the loan-to-value ratio which triggers default, while very sensitive to interest rates, does not change much with time.

2.6 2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.02

0.03

0.04

0.05 0.06 Interest Rate (r)

0.07

0.08

0.09

Fig. 2. Optimal default threshold for a 1/1 interest-only ARM. The axis r represents the short rate of interest. The one year ARM rate is assumed to be consistent with this short rate (see (4)).

687

T. Daglish / Journal of Banking & Finance 33 (2009) 681–693

1.4

Optimal Exercise Frontier (LTV)

1.35 1.3

B

t=0 t=10 t=20 t=25 t=29.9

1.25 1.2 1.15 1.1 1.05 1 0.02

1.4 1.35

Optimal Exercise Frontier (ILTV)

A

t=0 t=5 t=10

1.3 1.25 1.2 1.15 1.1 1.05

0.03

0.04

0.05 0.06 Interest Rate (r)

0.07

0.08

0.09

1 0.02

0.03

0.04

0.05 0.06 Interest Rate (r)

0.07

0.08

0.09

Fig. 3. Optimal default threshold for a 6.4% fixed interest rate, amortising mortgage, as a function of the short interest rate for various times in the mortgage’s life. Graph (A) shows the loan-to-value ratio which will trigger default. Graph (B) shows the loan-to-value ratio, evaluated using the initial principal of the loan.

loan’s decline in principal dominates this effect considerably. A household with an amortising loan will, as their equity in the house grows, become considerably more credit-worthy than a household with an interest-only loan, as can be seen by comparing panel B of Fig. 1 to Fig. 2. This analysis might tempt one to reach the conclusion that lending on interest-only terms is a far riskier proposition than lending with amortisation. It is unambiguous that during the later phases of the mortgage’s life, this is true. However, during the very early days of the mortgage, this is not true. A household who has an amortising loan will face a larger monthly payment than one whose loan is interest-only. Eventually, these extra payments will erode the principal of the loan. In the short run, however, a household whose property has precipitously declined in value will see these payments as wasted: if they default next month, the extra principal they have paid off will be of no benefit to them, it will only help offset the lender’s loss. To see this difference, examine the levels of the optimal default threshold for t ¼ 0 (when loanto-value and initial-loan-to-value are identical) in Figs. 1 and 2: the amortising loan will default at lower loan-to-value levels. Consider next the case of a FRM borrower, whose mortgage amortises over time. The payments made by the borrower are fixed, and do not vary as interest rates change. However, the value of those payments does change. If interest rates rise, the remaining payments on the mortgage become less valuable, while if interest rates fall, the payments become correspondingly more valuable. However, their value can never exceed the outstanding principal of the loan, since then the household would choose to refinance the loan (exercising their call). Fig. 3 plots the household’s optimal default exercise frontier. As was the case in Fig. 1, the figure is composed of two panels. In panel A, we display the loan-to-value ratio at which the household should default, for various maturities, as a function of the short interest rate. We note that for low interest rates, the household will default if the loan-to-value ratio exceeds one by 6% or more. In this interest rate range, the mortgage is not economical to maintain: if the homeowner had any equity in the property, he should refinance the loan, setting the loan’s value equal to the principal outstanding. If the house is worth less than the loan’s principal (less the time value of the put) default is optimal, since the value of the bond component of the mortgage is above par. In contrast, as interest rates rise, the bond component of the mortgage becomes less valuable than its face value. As such, the

call option is out of the money (although it still has value) and the net value of the bond and call becomes less (in absolute terms) than the outstanding principal. That being so, the household will not default unless the loan-to-value ratio is considerably greater than one. In layman’s terms, the household is continuing to service the mortgage because they have an attractive rate of interest locked in, which they would forfeit by defaulting. Note that as time progresses, the remaining life of the mortgage decreases, and the duration of the bond also declines. Hence, the sensitivity of the bond’s value to interest rates decreases, and this effect diminishes. In the limit, just before maturity, the bond’s value is totally insensitive to interest rates, and is simply equal to the outstanding principal.11 Panel B of Fig. 3 displays the initial-loan-to-value ratio which would trigger default. We can again see the benefit for credit-worthiness of an amortising loan. However, the sensitivity of the household’s decision to the level of interest rates still declines as time progresses. It is only borrowers during the early phases of their loan whose default decisions will be materially affected by interest rates. 5.2. Subprime borrowers As noted previously, the critical difference between a subprime mortgage and a prime one is the spread the borrower must pay to compensate the lender for the loan’s increased credit risk. The borrower will also, generally, have little or no (perhaps even negative) equity in the mortgage at inception. However, if house prices appreciate and/or the borrower pays off a sufficient portion of the mortgage, he may find himself able to enter the regular (prime) mortgage market. Clearly, if house prices appreciate rapidly, and the household is able to relatively quickly enter the prime market, this may prove an effective means for a low wealth household to enter the housing market. Note that this credit-motivated refinancing means that any type of mortgage may see the call option being used to the household’s advantage. In the case of a prime mortgage, the ability to pay off the mortgage early is only valuable for fixed rate mortgages, since adjustable rate mortgages will generally see the value of the bond

11 This effect is further reinforced by the increased principal payments of the loan (as noted for the amortising ARM case).

T. Daglish / Journal of Banking & Finance 33 (2009) 681–693

component of the mortgage hover around par. This has an important effect for default risk, since defaulting on the mortgage will require the household to abandon not only the house, but also the call option embedded in the mortgage in order to write off their debt. Increasing the value of this option will move the optimal LTV ratio for default higher than that of a prime borrower. On the other hand, the subprime borrower is paying a higher level of interest on his loan, for any given level of prime rate. As we saw in the previous section, when examining prime ARMs, this will make the subprime borrower more inclined to default if their put option moves into the money. The subprime borrower’s decision to default, relative to a prime borrower, is thus influenced by two competing effects: a desire to maintain their good standing in the hope of a future credit-upgrade, but also an incentive to default caused by their higher interest payments. Depending on which effect dominates, we will see higher or lower loan-to-value ratios triggering default when compared to a prime mortgage. The relative size of the two effects varies across different mortgage types and also varies as market interest rates change. Consider a subprime borrower, whose loan is an amortising ARM loan. Fig. 4 shows his optimal exercise frontier. Contrast this to Fig. 1, which shows a prime amortising ARM borrower’s optimal default frontier. In panel B of both figures, we consider initial loanto-value, calculated as discussed in the previous subsection. Examining first the behaviour of the two loans if a high rate of interest prevails, we see that a ruthless subprime borrower has a lower initial LTV ratio which will cause him to exercise his put option relative to a ruthless prime borrower. However, contrast this result with the state of affairs if interest rates are low. For a market ARM rate of 2%, at inception of the loan, even though the subprime borrower pays a 3% premium (i.e. paying interest at 5%) he would only default at a loan-to-value ratio of 139%, while a more modest level of 133% would trigger default by a prime borrower in similar circumstances. This difference in behaviour is caused by the high value of the call option under these circumstances. As noted at the beginning of Section 3, the value of the options contained in a mortgage are strongly inter-related. If one option is exercised, the others must necessarily be abandoned. Hence, when interest rates are high, and house prices low, the call option declines in value, as the put option becomes more likely to be exercised. However, in the situation where interest rates are low, the put is far less likely to be exercised (with little mainte-

A

1.5

t=0 t=10 t=20 t=29.9

Optimal Exercise Frontier (LTV)

1.45 1.4 1.35 1.3 1.25 1.2 1.15 1.1 1.05 1 0.02

nance cost, delay becomes a more viable strategy). With this effect diminished, the value of the call becomes quite apparent. One should not, however, infer that a subprime borrower is a lower credit risk than his prime counterpart. A prime borrower will, at inception, have a loan-to-value ratio considerably less than one, while a subprime borrower’s loan-to-value ratio will be closer to (and in some cases larger than) one. Further, since the prime borrower faces a lower interest rate than the subprime borrower, his principal will amortise at a more rapid rate, further lowering his LTV relative to the subprime borrower’s. To make a fair comparison of the actual credit risk involved in lending to prime and subprime borrowers, one must compute actual default probabilities. This analysis is undertaken in Section 5.3. A FRM, as we have seen in Section 5.1, has a fixed maintenance cost. Fig. 5 shows the optimal default strategy for a subprime borrower with a fixed rate mortgage. This should be contrasted to the results for a prime borrower in Fig. 3. Here, we see that, with the exception of very low interest rates, the subprime borrower will default at lower initial-loan-to-value levels. Comparing Figs. 4 and 5, one can see the appeal, from a lender’s perspective, of encouraging a subprime borrower to choose an ARM over a FRM. Providing interest rates are low, as was the case during the 2002–2004 period, an ARM borrower, because of their (relatively) low cost of servicing the mortgage, and relatively valuable call option to refinance as a prime borrower, would be unlikely to default, absent a substantial rise in interest rates, coupled with a depreciation of their house price. Note also that, from the subprime lender’s perspective, the exercise of the call option to refinance as a prime loan would remove the mortgage from the subprime market: the subprime lender is only concerned with default in the early part of the mortgage’s life. In contrast, encouraging a client to take a FRM would create a loan where even a small fall in house price would trigger default. The logical extension of this reasoning is to pose the question: what type of subprime mortgage would, for low interest rates, offer the highest loan-to-value ratio that would trigger default? The answer, as one might suspect from the analysis in Section 5.1, is an interest-only ARM (see Fig. 6). Note that, when compared to Fig. 2, the subprime borrower has a lower LTV ratio which triggers default when compared to his prime counterpart. Given that we noted higher optimal exercise frontiers for subprime borrowers at low interest rates for amortising ARMs, this may come as some-

B

1.5

t=0 t=5 t=10

1.45 Optimal Exercise Frontier (ILTV)

688

1.4 1.35 1.3 1.25 1.2 1.15 1.1 1.05

0.03

0.04

0.05 0.06 Interest Rate (r)

0.07

0.08

0.09

1 0.02

0.03

0.04

0.05 0.06 Interest Rate (r)

0.07

0.08

0.09

Fig. 4. Optimal default threshold for a subprime borrower with 3% spread whose house is financed using a 1/1 ARM. The axis r represents the short prime rate of interest. The one year prime ARM rate is assumed to be consistent with this short rate (see (4)). The subprime short rate and one year rate are each 3% higher than their prime counterparts. Graph (A) shows the loan-to-value ratio which will trigger default. Graph (B) shows this with loan-to-value calculated using initial principal, and assumes principal erosion occurs as if prime interest rates stayed at the long term rate of 4.41% throughout the mortgage’s life.

689

T. Daglish / Journal of Banking & Finance 33 (2009) 681–693

1.4

t=0 t=10 t=20 t=25 t=29.9

Optimal Exercise Frontier (LTV)

1.35 1.3 1.25 1.2 1.15 1.1 1.05 1 0.02

B

1.4

t=0 t=5 t=10

1.35 Optimal Exercise Frontier (ILTV)

A

1.3 1.25 1.2 1.15 1.1 1.05

0.03

0.04

0.05 0.06 Interest Rate (r)

0.07

0.08

0.09

1 0.02

0.03

0.04

0.05 0.06 Interest Rate (r)

0.07

0.08

0.09

Fig. 5. Optimal default threshold for a subprime borrower with 3% spread whose house is financed using a fixed rate mortgage at 9.4%. The axis r represents the short prime rate of interest. Graph (A) shows the loan-to-value ratio which will trigger default. Graph (B) shows this with loan-to-value calculated using initial principal.

thing of a surprise. To see why this is the case, consider the household’s ability to use their call for refinancing: under an amortising loan, even without an appreciation in the house’s value, the household will, eventually, have a 20% equity in the property, since their principal declines over time. With an interest-only loan, the household can reach this level only through a 25% house price appreciation. A household with an amortising loan is likely to be able to refinance sooner than an interest-only loan. The call option effect is thus less for interest-only loans than for amortising loans. 5.3. Default probability results We now present results for the probabilities of default for new mortgages. This follows the analysis in Section 3.1. Fig. 7 illustrates the case of a fixed rate amortising mortgage, as investigated by Kau et al. (1994). The graphs illustrate default probabilities for a prime and a subprime borrower, at the inception of their mortgages, as a function of loan-to-value ratio and interest rates. The graphs can basically be divided into regions, based on

2

t=0 t=25 t=28 t=29 t=29.9

Optimal Exercise Frontier (LTV)

1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 0.02

0.03

0.04

0.05 0.06 Interest Rate (r)

0.07

0.08

0.09

Fig. 6. Optimal default threshold for a subprime borrower with 3% spread whose house is financed using a 1/1 interest-only ARM. The axis r represents the short prime rate of interest. The one year prime ARM rate is assumed to be consistent with this short rate (see (4)). The subprime short rate and one year rate are each 3% higher than their prime counterparts.

whether r is high or low and whether the default option is in the money or not. Consider first the case where interest rates are low. If the loan-to-value ratio is low enough, the mortgage should be refinanced, and the probability of default is basically zero. As the loan-to-value ratio increases, the household moves to a situation where default is optimal, since there is very little value to maintaining a mortgage paying an above market rate. If interest rates are high, in contrast, the mortgage has considerable value. Hence for high interest rates, even for quite high loan-to-value ratios, the default probabilities are modest. Examining the situation where interest rates are too high to motivate refinancing, and loan-to-value ratio is too low to motivate default, we see that default probabilities first increase in interest rates and then decrease. This is caused by two competing effects: first, as interest rates rise, refinancing becomes less likely, effectively increasing the probability of default. Second, as interest rates increase, the expected growth rate of house prices increases, reducing the probability of default. This hump shaped default probability for FRM loans is also noted by Kau et al. (1994). For FRM loans, the behaviour of prime and subprime default probabilities is qualitatively similar. More pronounced differences in default probabilities can be seen when examining ARM loans. Fig. 8 presents results for interest-only and amortising ARM loans. Considering first the case of a prime interest-only ARM, we note that there are relatively high probabilities of default, even for a borrower who has considerable equity in the house initially. This is because for this type of loan, there is almost no probability of refinancing.12 Hence this loan will only be paid off if the house’s value remains above the optimal exercise frontier for the entire thirty year life of the mortgage. In contrast, a subprime interest-only ARM borrower may choose to exercise their prepayment option if the loan-to-value ratio declines sufficiently, and so for low loan-to-value ratios, their default probability becomes zero. For both prime and subprime cases, as the default option moves into the money, we see a U-shaped effect. On the one hand, low interest rates make a further decline in house prices more likely, but on the other, a high interest rate makes exercise of the default option optimal for lower levels of loan-to-value (see Fig. 6).

12 The only circumstance under which refinancing might be optimal would be if the interest rate declined precipitously, resulting in the household’s rate set at the year’s start being substantially above the new market rate.

690

T. Daglish / Journal of Banking & Finance 33 (2009) 681–693

Prime Amortising FRM

A

Subprime Amortising FRM

B 1

0.8

0.8

0.6

0.6

π

π

1

0.4

0.4

0.2

0.2

0 1.1

0 1.1 1 0.9 0.8 0.02

0.7

LTV

0.06

0.04

0.08

1

0.1

0.9 0.8 0.7

LTV

r

0.06

0.04

0.02

0.08

0.1

r

Fig. 7. FRM default probabilities at inception of mortgage. LTV refers to loan-to-value ratio, r is current prime interest rate, and p is the probability of default for the mortgage given this initial-loan-to-value ratio, and short rate. The loan’s rate is 6.4% for the prime case and 9.4% for the subprime case. The subprime borrower borrows at a 3% spread compared to the prime borrower.

Prime IO ARM

A

Subprime IO ARM

B 1

0.8

0.8

0.6

0.6 π

π

1

0.4

0.4

0.2

0.2

0 1.1

0 1.1 1 0.9 0.8 0.7

LTV

0.02

0.06

0.04

0.08

0.9 0.8 0.7

LTV

r

Prime Amortising ARM

C

1

0.1

0.02

0.06

0.04

0.08

0.1

r

Subprime Amortising ARM

D 1

0.8

0.8

0.6

0.6

π

π

1

0.4

0.4

0.2

0.2

0 1.1

0 1.1 1 0.9 0.8 LTV

0.7

0.02

0.06

0.04

0.08

0.1

r

1 0.9 0.8 LTV

0.7

0.02

0.06

0.04

0.08

0.1

r

Fig. 8. ARM default probabilities at inception of mortgage. LTV refers to loan-to-value ratio, r is current prime interest rate, and p is the probability of default for the mortgage given this initial-loan-to-value ratio, and short rate. The one year prime ARM rate is assumed to be consistent with this short rate (see (4)). The subprime short rate and one year rate are each 3% higher than their prime counterparts.

This U-shaped boundary disappears for amortising loans (see panels (C) and (D) of Fig. 8). Here a third effect contributes to make

default unlikely in low interest rate environments: with lower interest rates, the loan amortises more rapidly. Hence the loan-

691

T. Daglish / Journal of Banking & Finance 33 (2009) 681–693 Table 2 Default probabilities for new mortgages as a function of class of investor and loan-tovalue ratio. Prime borrower is assumed to have 20% equity (LTV 0.8, which corresponds to H=P ¼ 1:25), while subprime borrower is considered as having 9% equity (LTV 0.91, which corresponds to H=P ¼ 1:1), no equity (LTV 1.0), or negative equity (LTV 1.05, which corresponds to H=P ¼ 0:95). r is assumed to be 6%, and the one year ARM rate is assumed to be consistent with Eq. (4). Loan type

Amortising FRM Interest-only FRM Amortising ARM Interest-only ARM

Prime

Subprime

0.80

0.91

1.00

1.05

0.0000 0.0131 0.0203 0.0778

0.0916 0.1093 0.0902 0.1038

0.3338 0.3362 0.2550 0.2361

0.5834 0.5397 0.4219 0.3501

Table 3 Zero and five year default thresholds. Fix. Am. is an amortising FRM. Fix. I.O. is an interest-only FRM. 1/1 I.O. is an interest-only floating loan while 1/1 Am. is an amortising 1/1 ARM. r represents the short prime rate of interest. The one year prime ARM rate is assumed to be consistent with this short rate (see (4)). The subprime short rate and one year rate are each 3% higher than their prime counterparts. Inception

Five years

r

to-initial-value ratio which would trigger default is declining more rapidly in low interest rate environments. This makes default less likely (for low interest rates) for an amortising ARM than for an interest-only ARM. Again, the default probabilities for the subprime ARM is lower than its prime counterpart (assuming identical LTV ratios) because of the refinancing option, but this effect is less pronounced than for the interest-only case. Perhaps the most striking overall feature of these graphs is the high sensitivity of default probability to interest rates, particularly as the option to default moves into the money (loan-to-value ratio greater than one). Further, the sensitivity is very different between ARM and FRM loans, and when comparing interest-only to amortising ARMs.13 Lastly, we can use these graphs to compare the probabilities of default for some realistic types of borrower. In Table 2, we compare the probability of default for a prime borrower (assumed to have a 20% deposit) against a subprime borrower with varying amounts of equity. Clearly, given their lower initial equity, Subprime borrowers have a considerably higher probability of default. Comparing across mortgage types, however, confirms the intuition of Section 5.2: when constructing a low equity subprime mortgage, ARM loans give considerably lower default probabilities than FRM loans. Further, choosing an interest-only loan reduces the probability of default for borrowers with zero or negative equity (although it does increase the default probability slightly for those with positive equity). This contrasts with prime borrowers, who are less likely to default when borrowing using amortising loans. 5.4. Behavioural considerations As noted in Section 3, there are two important considerations for the exercise of the embedded options in a mortgage that are not present in the analysis of financial options: stigma costs and suboptimal exercise. We present two tables (Tables 3 and 4) presenting the optimal exercise frontier at inception, 5 years into the mortgage, 10 years into the mortgage and 20 years into the mortgage. For each year, we examine six scenarios for a subprime borrower’s default decision.14 The first scenario is that presented in Section 5.2: ruthless default, with no suboptimal exercise. We refer to this as the ‘‘base case”. Next we examine a case where a household faces a 5% stigma cost, but no suboptimal default. The third case ignores stigma cost, but gives the household a 10% intensity for suboptimal exercise, so that the household moves roughly every 10 years. Last we consider three cases, by combining a 5% stigma cost with first a 10% suboptimal default intensity, a 20% default intensity, and finally a case

13 Results for interest-only FRM loans are available on request from the author. The results are qualitatively similar to those for amortising FRMs. 14 Corresponding results for a prime borrower are available, on request.

0.02

r 0.04

0.05

0.06

0.08

0.02

0.04

0.05

0.06

0.08

Fix. Am. Fix. I.O. 1/1 Am. 1/1 I.O.

Base case 1.08 1.09 1.09 1.10 1.39 1.23 1.72 1.35

1.10 1.11 1.19 1.28

1.11 1.14 1.16 1.23

1.15 1.18 1.12 1.15

1.06 1.09 1.30 1.72

1.08 1.10 1.19 1.35

1.09 1.11 1.16 1.28

1.10 1.14 1.15 1.23

1.14 1.18 1.11 1.14

Fix. Am. Fix. I.O. 1/1 Am. 1/1 I.O.

Stigma 1.12 1.14 1.45 1.81

1.15 1.18 1.27 1.37

1.18 1.19 1.23 1.30

1.22 1.25 1.18 1.20

1.11 1.14 1.37 1.81

1.14 1.15 1.27 1.43

1.15 1.18 1.23 1.37

1.16 1.19 1.20 1.30

1.20 1.25 1.17 1.20

Fix. Am. Fix. I.O. 1/1 Am. 1/1 I.O.

10% Premature 1.08 1.09 1.09 1.10 1.37 1.22 1.69 1.32

exercise 1.10 1.11 1.11 1.12 1.18 1.15 1.27 1.20

1.15 1.16 1.11 1.12

1.06 1.09 1.28 1.69

1.08 1.10 1.18 1.32

1.09 1.11 1.15 1.27

1.10 1.12 1.14 1.20

1.14 1.16 1.10 1.12

Fix. Am. Fix. I.O. 1/1 Am. 1/1 I.O.

Stigma 1.12 1.14 1.43 1.79

and 10% premature 1.14 1.15 1.16 1.15 1.16 1.18 1.28 1.25 1.22 1.39 1.33 1.27

1.20 1.23 1.18 1.19

1.11 1.14 1.35 1.79

1.14 1.15 1.25 1.39

1.14 1.16 1.22 1.33

1.15 1.18 1.19 1.27

1.19 1.23 1.16 1.19

Fix. Am. Fix. I.O. 1/1 Am. 1/1 I.O.

Stigma 1.12 1.14 1.43 1.79

and 20% premature 1.14 1.15 1.16 1.15 1.16 1.18 1.27 1.23 1.20 1.37 1.32 1.25

1.20 1.22 1.16 1.18

1.11 1.14 1.33 1.79

1.12 1.15 1.23 1.37

1.14 1.16 1.20 1.32

1.15 1.18 1.19 1.25

1.19 1.22 1.15 1.18

Fix. Am. Fix. I.O. 1/1 Am. 1/1 I.O.

Stigma 1.12 1.14 1.43 1.79

and varying premature 1.14 1.15 1.16 1.20 1.15 1.16 1.18 1.23 1.28 1.25 1.22 1.17 1.39 1.33 1.27 1.19

1.11 1.14 1.35 1.79

1.14 1.15 1.25 1.39

1.14 1.16 1.22 1.33

1.15 1.18 1.19 1.27

1.19 1.23 1.16 1.19

cost 1.14 1.15 1.30 1.43

where there is a 5% stigma cost and intensity varies with interest rates, as described by Eq. 10, with a long run intensity k ¼ 0:1. The effect of a stigma cost is (not surprisingly) to move the price at which a household will default down, and therefore to increase the loan-to-value threshold that they will default at. This will make default less likely. The shift, if measured in the house price dimension, is essentially equal to the size of the stigma cost, with the only exception being at very low interest rates for an ARM loan. Here the price which triggers default is so low, the stigma effect becomes somewhat smaller.15 Introducing a ten percent intensity of suboptimal option exercise results in a very minor decrease in the loan-to-value ratio which triggers default. The possibility of suboptimal exercise in the future reduces the time value of both the options contained in the mortgage. Reducing the time value of the put option will encourage the household to be more trigger-happy in their decision to default, but this is also offset by the decline in the value of the call option embedded in the mortgage. Since the latter effect can be an important deterrent to default for subprime borrowers, it is not surprising to find that the impact of suboptimal default on the put option is quite weak.16 We note that premature exercise

15 Of course, since the tables record loan-to-value (the reciprocal of house price to loan value) these numbers become relatively larger, so this effect is somewhat harder to discern. 16 For prime borrowers, where the refinancing option is less valuable, the effect of a ten percent intensity of suboptimal exercise on the default decision is more substantive.

692

T. Daglish / Journal of Banking & Finance 33 (2009) 681–693

Table 4 10 and 20 year thresholds. Fix. Am. is an amortising FRM. Fix. I.O. is an interest-only FRM. 1/1 I.O. is an interest-only 1/1 ARM while 1/1 Am. is an amortising 1/1 ARM. r represents the short prime rate of interest. The one year prime ARM rate is assumed to be consistent with this short rate (see (4)). The subprime short rate and one year rate are each 3% higher than their prime counterparts. Ten years

Twenty years

r

r

0.02

0.04

0.05

0.06

0.08

0.02

0.04

0.05

0.06

0.08

Fix. Am. Fix. I.O. 1/1 Am. 1/1 I.O.

Base case 1.05 1.09 1.22 1.72

1.06 1.10 1.15 1.35

1.08 1.11 1.14 1.28

1.09 1.14 1.12 1.23

1.12 1.18 1.09 1.14

1.03 1.09 1.08 1.69

1.04 1.10 1.06 1.35

1.04 1.11 1.06 1.28

1.05 1.14 1.06 1.22

1.08 1.18 1.05 1.14

Fix. Am. Fix. I.O. 1/1 Am. 1/1 I.O.

Stigma cost 1.11 1.14 1.28 1.81

1.12 1.15 1.20 1.43

1.14 1.18 1.19 1.37

1.15 1.19 1.18 1.30

1.19 1.25 1.15 1.20

1.09 1.14 1.14 1.79

1.09 1.15 1.12 1.41

1.10 1.18 1.12 1.35

1.10 1.19 1.11 1.28

1.14 1.25 1.11 1.20

Fix. Am. Fix. I.O. 1/1 Am. 1/1 I.O.

10% Premature 1.05 1.09 1.20 1.69

exercise 1.06 1.10 1.14 1.32

1.08 1.11 1.12 1.27

1.09 1.12 1.11 1.20

1.11 1.16 1.09 1.12

1.03 1.09 1.08 1.69

1.04 1.10 1.06 1.32

1.04 1.11 1.06 1.25

1.05 1.12 1.05 1.20

1.06 1.16 1.05 1.12

Fix. Am. Fix. I.O. 1/1 Am. 1/1 I.O.

Stigma and 10% premature 1.11 1.12 1.14 1.15 1.27 1.20 1.79 1.39

1.12 1.16 1.19 1.33

1.14 1.18 1.18 1.27

1.18 1.23 1.15 1.19

1.09 1.14 1.14 1.79

1.09 1.15 1.12 1.39

1.10 1.16 1.11 1.33

1.10 1.18 1.11 1.27

1.12 1.23 1.11 1.19

Fix. Am. Fix. I.O. 1/1 Am. 1/1 I.O.

Stigma and 20% premature 1.11 1.12 1.14 1.15 1.25 1.19 1.79 1.37

1.12 1.16 1.18 1.32

1.14 1.18 1.16 1.25

1.18 1.22 1.15 1.18

1.09 1.14 1.12 1.79

1.09 1.15 1.12 1.37

1.10 1.16 1.11 1.32

1.10 1.18 1.11 1.25

1.12 1.22 1.11 1.18

Fix. Am. Fix. I.O. 1/1 Am. 1/1 I.O.

Stigma and varying premature 1.11 1.12 1.14 1.15 1.27 1.20 1.79 1.39

1.12 1.16 1.19 1.33

1.14 1.18 1.18 1.27

1.18 1.23 1.15 1.19

1.09 1.14 1.14 1.79

1.09 1.15 1.12 1.39

1.10 1.16 1.11 1.33

1.10 1.18 1.11 1.27

1.12 1.23 1.11 1.19

has a greater influence on ARM loans, since in this case, the call option is less valuable to begin with, and so the effect of the decline in the put option’s value clearly dominates. The effect of combining the two behavioural considerations is basically additive. It is easy to see, by comparing the results from the combination of stigma cost and a twenty percent intensity of suboptimal default to the base case, that even for a quite transitory population who routinely exercise their call and put options prematurely, the stigma effect dominates. Since the stigma effect effectively shifts our curves from the previous sections by the amount of stigma cost the household faces from default, it is clear that our results for ruthless default can (qualitatively) be extended to the case of stigma costs and premature exercise of the options. Households may be, across the board, less inclined to default than the ruthless case suggests, but their sensitivity to changes in interest rates and time to maturity will remain the same. Lastly, we observe that introducing interest rate sensitivity into the intensity function has almost no effect on the optimal exercise frontier. We conclude that a model with a constant intensity of premature exercise is probably (at least for US data) a good parsimonious solution for capturing suboptimal default/prepayment effects. 6. Conclusions We provide a tractable model for the analysis of the default decision on the part of subprime (and prime) borrowers in the mortgage market. Our model can easily provide default probabilities or values of individual mortgages, and so could prove a valuable tool for the valuation of mortgage based CDOs. We show

that subprime borrowers have a valuable additional use for the call option embedded in a mortgage, in order to refinance into a prime loan if their house price appreciates. The value of this option can, in a low interest rate environment, provide a significant deterrent to default, for ARM loans. Hence during a period of relatively low interest rates, it may be quite feasible to make ARM loans to subprime borrowers, with relatively little credit risk attached. However, under higher interest rate regimes, or for fixed rate borrowing, we note an increased tendency toward default on the part of subprime borrowers. Since their interest payments will in general be high relative to prime borrowers, the expense of maintaining a loan larger than the associated property’s value may trigger default more rapidly than for a conventional borrower. Perhaps our main finding could be simply stated as: subprime borrowers’ credit quality is highly sensitive to interest rate fluctuations. Our results are robust to behavioural considerations. While costs associated with default may render households less willing to default on their house, the effect is merely to shift the optimal exercise frontier in a parallel fashion. Further, if the household faces the possibility of suboptimally exercising their mortgage’s embedded options due to a need to liquidate the property (such as due to unemployment or moving house) there is little effect on the conditions which would trigger optimal default. None of our reasoning is dependent on some credit constraint on the part of the household. Thus, one might not view the decline of the subprime market as a terrible blow to low income borrowers who took out ARM loans during the 2002–2004 period, but rather see that period as a fleeting opportunity when it was possible for such households to enter the housing market. Their subsequent

T. Daglish / Journal of Banking & Finance 33 (2009) 681–693

exercise of the embedded put options in their mortgages was a perfectly rational exit strategy when interest rates rose and the housing market declined in value. Further, while it is popular to blame some of the recent spate of subprime defaults on the manner in which household incomes were investigated (or, more specifically, not investigated) by mortgage brokers during the subprime boom period, our results suggest that even had due diligence been taken, the outcome would still have been the same: even households who could have made payments on their loans would have chosen not to.

2.

3.

4. Acknowledgements The author thanks Jon Garfinkel, Jay Sa-Aadu, David Tripe, Calvin Schnure, Andreas Lehnert, Graeme Guthrie, Lyndon Moore, Glenn Boyle, an anonymous referee, participants at seminars at Victoria University of Wellington, Massey University, University of Canterbury and Auckland University of Technology, along with participants at the Financial Management Association, Northern Finance Association, Australasian Banking and Finance Conference and New Zealand Finance Colloquium Meetings. The material in this paper previously appeared as part of a joint working paper: ‘‘Default Risk in the US Mortgage Market”, with Jon Garfinkel and Jay Sa-Aadu. The author was at one point himself a subprime borrower, although he did not default on his loan. Appendix A. Path dependence of ARM rates Since an ARM loan’s interest rate may not vary by more than cap1 per year, the current rate paid by a borrower will, in general, depend on the previous history of rates. This introduces path dependence into the problem: the value of an ARM mortgage depends on the path of the short rate. We take care of this problem by considering a separate finite difference lattice for each level of r ARM . After solving each of the lattices for a given year, a new set of lattices are created whose terminal conditions are consistent with the solutions of the previous lattices, and the transition between r ARM governed by (8). We thus alternate between solving the PDE for one year using finite difference techniques and interpolating between the lattices for r ARM . The algorithm for valuation of an amortising ARM mortgage proceeds as follows: 1. Set up and solve the PDE (7) for a spectrum of different values of r ARM;ðT1Þ , starting at the mortgage’s maturity. At each time step, we interpolate between levels of H=P to account for the fact that

5. 6.

693

even if the house price remains constant, H=P will increase. We also scale the solution at time t þ Dt by P tþDt =P t , since we are solving for mortgage value per dollar of principal. We now know the value of the mortgage for any level of the three state variables: the short rate (rT1 ) arm rate (rARM;T1 ) and house value relative to loan value (HT1 =P T1 ). Now consider the mortgage between times T  2 and T  1, for a given starting level of ARM rate (r ARM;T2 ). For each level of terminal short rate (r T1 ) on a given lattice, we can determine rARM;T1 , using (8). Using our solutions for the mortgage’s value at time T  1 as a function of r ARM;T1 , we can interpolate to find the appropriate mortgage value as a function of r ARM;T2 and r T1 . Solve (using (7)) backwards to find the value of the mortgage at time T  2, as a function of r ARM;T2 , rT2 and HT2 =P T2 . Repeat this process to find the mortgage’s value at T  3, etc.

References Bennett, P., Peach, R., Peristiani, S., 2000. Implied mortgage refinancing thresholds. Real Estate Economics 28, 405–434. Calhoun, C.A., Deng, Y., 2002. A dynamic analysis of fixed- and adjustable-rate mortgage terminations. Journal of Real Estate Finance and Economics 24, 9–33. Calhoun, C.A., 1996. OFHEO House Price Indexes: HPI Technical Description, Working Paper, Office of Federal Housing Enterprise Oversight. Caplin, A., Freeman, C., Tracy, J., 1997. Collateral damage: Refinancing constraints and regional recessions. Journal of Money, Credit, and Banking 29, 496–516. Chan, K.C., Karolyi, G.A., Longstaff, F.A., Sanders, A.B., 1992. An empirical comparison of alternative models of the short-term interest rate. Journal of Finance 47, 1209–1227. Deng, Y., Quigley, J.M., van Order, R., 2000. Mortgage terminations, heterogeneity and the exercise of mortgage options. Econometrica 68, 275–307. Dunn, K.B., McConnell, J.J., 1982. Valuation of GNMA mortgage-backed securities. Journal of Finance 36, 599–616. Kau, J.B., Keenan, D.C., Kim, T., 1994. Default probabilities for mortgages. Journal of Urban Economics 38, 278–296. Kau, J.B., Keenan, D.C., Muller, W.J., Epperson, J.F., 1987. The valuation and securitization of commercial and multifamily mortgages. Journal of Banking and Finance 11, 525–546. Kau, J.B., Keenan, D.C., Muller, W.J., Epperson, J.F., 1992. A generalized valuation model for fixed-rate residential mortgages. Journal of Money, Credit and Banking 24, 279–299. Kau, J.B., Keenan, D.C., Muller, W.J., Epperson, J.F., 1993. Option theory and floatingrate securities with a comparison of adjustable- and fixed-rate mortgages. Journal of Business 66, 595–618. Pennington-Cross, A., Chomsisengphet, S., 2007. Subprime refinancing: Equity extraction and mortgage termination. Real Estate Economics 35, 233–363. Peristiani, S., Bennett, P., Monsen, G., Peach, R., Raiff, J., 1997. Credit, equity and mortgage refinancings. FRBNY Economic Policy Review 3, 83–103. Stanton, R., 1995. Rational prepayment and the valuation of mortgage backed securities. Review of Financial Studies 8, 677–708.