A Note on Hybrid Mortgages

Brent W. Ambrose*, Michael LaCour-Little**, and Zsuzsa R. Huszar***

* Gatton College of Business and Economics, University of Kentucky, Lexington, KY 40506-0053 or ambrose@uky.edu.

**Wells Fargo Home Mortgage and Washington University in St. Louis, Clayton, MO 63105 or michael.lacour-little@wellsfargo.com.

*** Gatton College of Business and Economics, University of Kentucky, Lexington, KY 40506-0053 or zsuzsa.huszar@uky.edu.

Abstract

We extend previous research on traditional 1-year adjustable-rate mortgages by analyzing the performance of 3/27 hybrid instruments. Under this contract innovation, which first appeared in the mid-1990s, note rates are fixed for three years after which they convert to a traditional 1-year adjustment schedule with periodic and lifetime caps. We find high rates of prepayment, particularly at time of initial rate adjustment, and relatively high rates of default, as would be consistent with the payment shock that often affects adjustable-rate loans. The U.S. mortgage market continues to evolve, with contract innovations to adapt to economic and business

conditions. Adjustable-rate mortgages (ARMs) first appeared after the financial de-regulation of 1980, prompted by

the inflationary environment of the late 1970s. Today, ARMs are a popular mortgage alternative for many

households, especially those requiring larger loan amounts to finance higher priced housing.1 The ARM share is

much higher in the jumbo market compared to the conforming loan segment, reaching as high as 72% of total

originations during 1994 and 2000, which are both years with relatively higher rate levels (Nothaft 2003). The

relative share of adjustable-rate mortgages appears to rise in periods of rising interest rates, as consumers are

attracted to the lower initial interest rates relative to fixed-rate mortgages (FRMs). For example, in 1994 during a

period of rising market interest rates, adjustable-rate mortgages accounted for 39% of all mortgage originations

(Mortgage Bankers Association 2004).

Another stylized fact about the ARM product is that it appears to be the preferred instrument for portfolio lenders,

since it avoids much of the interest rate risks associated with long-term FRMs. As a result, a much smaller fraction

of ARM volume is securitized. Ambrose and LaCour-Little (2001) report that, as of 1999, only about $138 billion in

ARM-backed MBS were outstanding, as compared to about $1,776 billion in FRM-backed MBS. Since ARM loans

are mainly held in the portfolios of depository institutions, they can be funded with relatively low cost liabilities,

eliminating asset-liability mismatch during their initial fixed rate period.

This cost advantage allows lenders to offer lower rates to borrowers who accept a limited horizon for payment

stability, either because they do not expect to live in the house for an extended period or because they are

1

comfortable with future interest rate risk. As an example of the rate differential, as of July 30, 2004, the spread

between the 30-year FRM rate and a 3/27 ARM rate was 132 basis points (HSH Associates 2004). As an example of

balance sheet patterns, as of year-end 2002, Washington Mutual (the nation's largest thrift) reported $111 billion in

single-family residential mortgages at a weighted average coupon of 5.97% (a rate suggestive of a high percentage

ARMs, though not specifically broken out by financial reporting). Concurrently, their balance sheet showed $106

billion in interest-bearing deposits at an average rate of 2.50%, implying a net interest margin of 347 basis points on

that mortgage portfolio.

While adjustable-rate mortgages have received significant theoretical and empirical research interest,2 relatively

little attention has focused on one of the latest product innovations: the hybrid mortgage contract that contains

features of both traditional fixed- and adjustable-rate instruments.3 This study addresses that research gap by

examining the performance on a set of 3/27 adjustable-rate mortgages originated during the mid-1990s across the

United States for a major lender's portfolio.

To briefly summarize the loan features, the 3/27 adjustable-rate mortgage is a common hybrid mortgage that

provides borrowers with a fixed-rate mortgage for three years following origination, after which it converts to a

traditional 1-year adjustable-rate mortgage, indexed to the 1-year Treasury note, for the remaining 27 years of

amortization. Interest rate adjustments are capped at 2% at first and subsequent adjustments, and there is a lifetime

cap of 5% over the initial rate. Other hybrid ARM structures include fixed periods of 5, 7, or 10 years, effectively

allowing the borrower a menu of terms over which she may fix initial loan payments. In contrast, under a traditional

1-year adjustable-rate mortgage, the borrower faces a potential payment shock after only 12 months as the contract

rate is adjusted to market.4 Of course, the 3/27 design also exposes the borrower to interest-rate risk, though at a

later date. After its introduction in the mid-1990s, the hybrid mortgages gained increasing acceptance, first in the

non-conforming, then the conforming and then, most recently, the government segment of the market.5 Interest rate

adjustments are capped at 2% at first and subsequent adjustments, and there is a lifetime cap of 5% over the initial

rate.

As with traditional FRM and ARM, hybrid ARMs contain the usual explicit and implicit options for the borrower to

terminate the mortgage through either prepayment or default. The theoretical and empirical literature on mortgage

prepayment and default (both for fixed-rate as well as adjustable-rate instruments) is now well developed so we do

not belabor it here. Rather, we simply extend the empirical literature by addressing hybrid ARM performance in

general and the effect of the adjustment period in particular.

Data

Our data consists of 2,192 hybrid mortgages originated during 1995 and 1996 by a large national financial

institution for portfolio, with performance observed through June 2000. Hence, the data contains loans that are

seasoned between 3.5 and 5.5 years, a time period sufficient to observe the first re-set event for all loans but just

2

barely sufficient to encompass peak default and prepayment periods (generally thought to occur in years 3-5 of loan

life). A longer observation period would be desirable, of course, so our results should be viewed as an early

assessment of termination risk for the hybrid mortgage category. Since these are all 3/27 in contract design, we

observe the 3-year adjustment period for all loans in the sample, assuming they survive up to that point. After

adjustment, note rates are indexed to the 1-year Treasury rate with a 250 basis point margin. Of these mortgages,

1,685 (76.8%) had prepaid by June 2000, 181 (8.2%) had defaulted (defined as the first instance of a 90-day

delinquency, regardless of ultimate outcome)6 and the remaining 326 (14.9%) were still active (censored) as of the

end of the observation period.

Conventional wisdom holds that since ARMs are subject to payment shock on adjustment, they will tend to have

higher default rates compared to FRMs. In support of this view, for example, the Mortgage Bankers Association

National Delinquency survey reports that as of Q1-2004, 0.44% of prime FRMs were in foreclosure and 0.73% of

prime ARMs were in foreclosure. As a result, lenders tend to more conservatively underwrite ARMs compared to

FRMs, typically by imposing lower maximum loan-to-value ratios and/or higher minimum credit scores.

Underwriting for hybrid ARMs, however, appears to be relatively less conservative, at least when compared to the

traditional 1-year ARMs we previously studied. In this data, the average loan-to-value ratio is 74.9% compared to

73.4% as reported by Ambrose and LaCour-Little (2001) for traditional 1-year adjustable-rate mortgages. Moreover,

approximately 9% of the hybrid mortgages in this data set had LTV ratios greater than or equal to 95%; in contrast,

fewer than 1% of the 1-year adjustable-rate mortgages analyzed by Ambrose and LaCour-Little (2001) had LTV

ratios that high (see Table 1). These difference may simply reflect a secular trend toward higher LTV lending over

the decade of the 1990s. Another difference between the 3/27 mortgages studied here and the traditional 1-year

adjustable mortgages studied by Ambrose and LaCour-Little is the significant difference in defaults. The dataset

used by Ambrose and LaCour-Little (2001) contained very few defaults; as a result, their default model had very

little explanatory power. In contrast, we find that just over 8% of our sample defaulted during the observation period

providing a much richer dataset for studying this phenomenon. Moreover, the data represents a geographically

diverse cross-section of hybrid mortgages. We note significant concentrations in the West (23%), consistent with the

higher house prices and loan balances in that region. The average (median) loan amount is $259,878 ($224,350), and

the amount ranges from $22,700 to $2,050,000. Since the conforming loan limit was $203,150 in 1995 and $207,000

in 1996, the majority of these loans were nonconforming at origination. The average initial fixed rate is 7.09%, with

0.30 origination points. Mean borrower credit score (FICO) at origination is 730 and ranges between 536 and 821.7

Over the course of the study period, the general level of interest rates declined during 1997 and 1998, reaching a

new low in October 1998 and then moving consistently higher during 1999 and 2000. As a result, we have a good

cross section of rate environments at which adjustments occurred. Figure 1 shows the average contract rate at

origination (by monthly cohort), the 30-year FRM rate and the adjustment rate (the 1-year Treasury rate plus the 250

basis point margin) over the period from 1995 through 2000. The figure indicates that a loan originated in midyear

3

1995 would have reset into a very low environment in 1998; in contrast, a loan originated in late 1996 would have

encountered a rate structure roughly 200 basis points higher in 1999.

To put these rates into perspective, consider three hypothetical mortgages originated in January 1995, January 1996

and December 1996. At the first adjustment date (month 36), the January 1995 mortgage would have experienced a

5.6% payment reduction as the contract rate fell from 8.4% to 7.7%, while the January 1996 mortgage would have

experienced a payment reduction of only 0.9% since the contract rates for this cohort only declined from 7.1% at

origination to 7% in month 36. In contrast, the December 1996 mortgage would have experienced a payment

increase of 14.4% at the first adjustment as the contract rate increased from 6.9% at origination to 8.3% in month 36.

Then, at the second adjustment date (month 48), the January 1995 mortgage would have experienced another

payment reduction of 6.4% as the contract rate declined from 7.7% in month 36 to 7% in month 48, while the

January 1996 mortgage would have experienced a payment increase of 15.4% as the contract rate increased from 7%

to 8.6%. The December 1996 mortgage would have experienced a 4% payment drop as the contract rate declined

from 8.3% to 8.1%. As this simple exercise demonstrates, depending on when the hybrid mortgage was originated,

the borrower faced significantly different payment shocks at the 36-month adjustment date.

Empirical Approach

Following standard practice, we model prepayment and default as competing risks. This approach recognizes the

mutually exclusive nature of the prepayment and default options. Recent studies such as Deng, Quigley and Van

Order (2000), Ambrose and LaCour-Little (2001), Ambrose and Sanders (2003) and Calhoun and Deng (2002) use

this framework.

We first recognize that during our observation period, a borrower either prepays, defaults, or remains current

through the end of the time-period of study (censored). We define Tj (j = 1, 2, 3) as the latent duration for each loan

to end by prepaying, defaulting or being censored, and the observed duration, τ, is the minimum of the Tj.

Conditional on a set of explanatory variables, xj, that include personal risk characteristics and market conditions, the

probability density function (pdf) and cumulative density function (cdf) for Tj are

( ) ( ) ( )( )jjjjjjjjj xrIxThxTf |exp|| −= (1.)

( ) ( )( )jjjjjj xrIxTF |exp1| −−= (2.)

where Ij is the integrated hazard for outcome j:

( ) ( )∫=jT

jjjjj dsxshxTI0

|| (3.)

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and hj is the hazard function.

The joint distribution of the duration and outcome is

( ) ( ) (( xIxhxjf jj |exp||, 0 ))τττ −= (4.)

where x=(x1, x2, x3) and I0=Σ Ij is the aggregated integrated hazard. Thus, the conditional probability of an outcome

is

( ) ( )( )∑

=

= 3

1|

|,|Pr

jj

jj

xh

xhxj

τ

ττ . (5.)

In order to simplify estimation, we specify a separate exponential hazard function for each mortgage outcome

( ) ( )jjjjj xxh βτ ′= exp| . (6.)

and estimate (5) in a multinomial logit framework.

Following Gross and Souleles (2002), we separate xj into components representing borrower risk characteristics and

economic conditions having the following linear form:

iititittjj adjusteconriskagex 43210 ββββτββ ++++=′ (7.)

where τt represents a series of dummy variables corresponding to calendar quarter of origination, ageit is a fourth

order polynomial of time since origination, riskit represents a set of characteristics that reflect the lender’s

underwriting criteria (including borrower credit score (FICO), loan-to-value ratio (LTV) and payment-to-income

(INCOME)), econit is a set of variables capturing changes in economic conditions (including current yield curve,

mortgage interest rates and property values) and adjusti is a dummy variable denoting the 3-month window

surrounding the anniversary marking the conversion of the loan from a fixed-rate to an adjustable-rate mortgage.

The calendar quarter dummy variables allow the propensity to default or prepay to shift over time. The age function

allows for nonparametric variation in the prepayment and default hazard.

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Following Ambrose and LaCour-Little (2001), we capture the dynamics of the prepayment option value by

calculating the relative position of the market interest rate to the contract rate. Thus, we include the variable spread

over fixed rates (SPD_FRM) defined as

)()()(

)(_tr

trtrtFRMSPD

f

fc −= , (8.)

where rc is the current contract interest rate at t and rf is the current 30-year conventional fixed-term mortgage

interest rate at t, as measured by the FreddieMac 30-year primary mortgage market survey (PMMS) rate. The spread

over fixed rates represents the relative interest rate savings associated with switching from the current hybrid 3/27 to

a fixed-rate mortgage.8 Positive values of this variable indicate that the contract interest rate is greater than the

market rate and, accordingly, the prepayment option is “in-the-money;” negative values indicate that the prepayment

option is “out-of-the-money.” The current period t yield curve (TERMSTRU) is estimated as the difference between

the 10-year and 1-year Treasury bond rates. This variable provides an indication of market expectations about future

interest rates.

Recognizing that the 3/27 hybrid converts to a 1-year adjustable-rate mortgage in month 36, we include a variable

that captures the payment shock (PMT_SHOCK) encountered by the borrower at each interest-rate adjustment date.

Specifically, we calculate the payment shock as follows:

1__)(_ −=

pmtoldpmtnewtSHOCKPMT

(9.)

where new_pmt is the new payment based on the remaining balance and the 1-year Treasury rate in period t plus the

250 basis point margin, and old_pmt is the prior year’s payment. For example, for a mortgage still current between

months 48 and 60, new_pmt is the new payment based on the 1-year Treasury rate at month 48 (plus the margin) and

the old_pmt is the payment during months 36 to 47. Thus, payment shock is the percentage increase (or decrease) in

the monthly payment relative to the payment in the previous year. We expect that the prepayment hazard should

increase during periods of positive payment shock as borrowers attempt to control future interest rate shocks by

converting to fixed-rate mortgages. We also anticipate that positive payment shocks should correspond to an

increase in the default hazard, the conventional wisdom about adjustable-rate mortgages.

We capture the interaction of borrower equity and default-option value with prepayment probabilities by including

the probability that an individual borrower has negative equity in the property. The probability of negative equity is

calculated for each borrower based on the state-level house price growth and the variance of these growth rates

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around the state-level mean appreciation rate.9 Since house prices over the sample period were generally increasing,

the probability of negative equity is small. As a result, we create a one-zero indicator variable (HIGHPROB) to

capture months where the probability of negative equity exceeded 5%. In order to assess the variation in the

probability of negative equity over time, Figure 2 shows the average probability of negative equity for each month

since loan origination, segmented by quarterly origination cohort. For example, the line marked 95Q1 represents the

average probability of negative equity experienced by mortgages originated during the first quarter of 1995. Figure 2

clearly indicates that loans originated during 1995 experienced higher probabilities of negative equity. In addition,

we also incorporate the current period t loan-to-value ratio (LTV). LTV captures both the scheduled principal

amortization as well as changes in the underlying property as reflected in the state level OFEHO repeat sales index.

The adjustment period dummy variable is one of the key variables of interest. Our hypothesis is that the hazards of

prepayment and default should shift during the rate adjustment window. We also interact the current interest rate

spread with the adjustment window indicator variable in order to isolate the impact of interest rates at adjustment.

Results

Figures 3 and 4 report the baseline hazard rates and cumulative prepayment and default probabilities, respectively.

We estimate the baseline hazards using the following specification:

ittjj agex 10 βτββ +=′ . (10.)

This specification allows us to estimate the baseline hazard function without controlling for borrower characteristics,

time-varying economic conditions or specific features of the 3/27 hybrid. As evident in Figure 3, the prepayment

hazard displays the typical non-linear pattern of mortgage seasoning. Furthermore, the cumulative probabilities of

prepayment and default (Figure 4) indicate that the probability of a mortgage surviving beyond 5 years is less than

1%.

Table 2 reports results of the full competing risk model specification. In the prepayment function, we find

statistically significant effects for loan age (positive), spread over fixed rates (positive), credit score (positive), loan-

to-value ratio (positive), loan balance at origination (positive), adjustment date (positive), the ratio of the new

payment relative to the old payment at each adjustment date (positive) and the interest rate spread interacted with the

adjustment date period (negative). The estimates for the interest rate term structure and high probability of negative

equity are not statistically significant. For the default function, we find statistically significant coefficients for the

spread over fixed rates (positive) and credit score (negative). These results are consistent with prior expectations.

For example, borrowers with higher credit quality scores at origination have a higher probability of prepayment and

a lower probability of default.

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Also, consistent with the findings of Ambrose and LaCour-Little (2001), we find strong evidence that financial

factors drive prepayment. Following Ambrose and LaCour-Little, the spread over fixed rates is a proxy for the value

of the call option. The positive coefficient indicates that the probability of prepayment is significantly higher when

market interest rates (in this case, the 30-year fixed-rate mortgage) are below the contract rate (i.e., the prepayment

option is “in-the-money”). To put this result into perspective, a 1-basis-point increase in the spread over fixed rates

variable (from 0.0 to 0.01) corresponds to approximately a 7-basis-point difference between the fixed-rate mortgage

market rate and the contract rate when the contract rate is 7%. The significance of a change in this spread on

prepayment is apparent through simulations. For example, if by month 12, the FRM interest rate falls 100 basis

points below the hybrid contract rate at origination (from a 7% hybrid to a 6% FRM), then the probability of

prepayment increases by 130 basis points (from 1.71% to 3.02%).10

As discussed earlier, the variables of most importance for this study are those that characterize the period

surrounding the switch from initial fixed-rate to fully adjustable-rate (ADJUST, AFTER_ADJUST, PMT_SHOCK

and SPDFRM_ADJUST). Since the adjustment period imposes significant interest rate risk on the borrower, we

anticipate that the prepayment and default hazards should increase during this period. Overall results are consistent

with this expectation. Figure 5 shows that the prepayment and default hazard rates increase significantly following

the first adjustment. Figure 6 shows cumulative default and prepayment rates, with a clear inflection point around

month 36. We note that with termination hazard rates this high, the probability of a hybrid mortgage surviving past

month 60 is less than 1%.

We include two dummy variables to estimate the impact of the initial adjustment shock and the subsequent ARM

period. The significant coefficient for the adjustment window dummy variable (ADJUST) indicates that the hazard

of prepayment increases significantly during the three-month window surrounding the origination anniversary when

the hybrid converts from a fixed-rate to an adjustable-rate mortgage. The marginal effect indicates that the

probability of prepayment during this 3-month window is 78% higher than the corresponding prepayment hazard in

other months.11 This significant spike is clearly evident in Figure 5, which shows the monthly prepayment and

default hazards. With the exception of this spike at the anniversary date, the prepayment hazard displays a

characteristic non-linear pattern of rising slowly to month 36 and then declining, mirroring the seasoning observed

in pool-level prepayment statistics.

We also include a dummy variable controlling for the period following adjustment (AFTER_ADJUST). This

specification provides an estimate of the shift in the hazard rate from being a FRM to an ARM. The insignificant

coefficients for this variable indicate that the ARM portion of the mortgage does not have greater default or

prepayment risk, on average, than the FRM portion of the mortgage. However, the highly significant coefficient for

PMT_SHOCK in the prepayment model (and marginally significant in the default model) indicates that termination

risk is clearly associated with adjustments to the mortgage payment. Recall, we include the PMT_SHOCK variable

to control for the relative change in mortgage payment at each interest rate adjustment. The significant positive

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coefficient in the prepayment model indicates that borrowers are more like to prepay their mortgage after

experiencing a positive shock to their payment. This result confirms that borrowers are sensitive to changes in

interest rates suggesting that a positive increase in the mortgage payment provides borrowers with the necessary

incentive to seek out refinancing alternatives. This pattern supports the results reported by Ambrose and LaCour-

Little (2001) for teased ARMs. The results for teased ARMs indicated that the hazard of prepayment increased

significantly during the window surrounding the adjustment anniversary; this effect was particularly strong during

the first anniversary when the teaser expired. Our results here indicate that the 3/27 hybrid mortgage displays a

similar pattern. That is, after the initial “teased” period wears off—or in this case, when the fixed-rate period

expires—the prepayment and default hazards increase substantially. Given this performance similarity, one might

characterize 3/27 ARMs as simply standard 1-year ARMs with a 3-year teaser period.

To isolate the impact of interest rates during the conversion window (months 35 to 37), we interact the adjustment

window dummy variable with the interest rate spread variable (SPDFRM_ADJUST). The coefficient for

SPDFRM_ADJUST is significantly negative for the prepayment model. Recall, this variable measures the marginal

impact of a change in interest rates at the time of adjustment from the original contract interest rate. Thus, the

negative coefficient suggests that during the adjustment period window the impact of a decline in interest rates

(relative to the origination contract rate) is lower than during other periods. In contrast, if interest rates are higher

(SPD_FRM is negative), then the prepayment hazard increases, implying that borrowers recognize the risks

associated with ARMs during periods of rising interest rates.

Conclusions

In this brief paper, we extended the research on the performance of adjustable-rate mortgages to the hybrid ARM

category. This product innovation is increasingly popular, especially in high-cost markets and higher-rate

environments; however, we have shown that there seem to be some unique risk characteristics. Both prepayment and

default risk are high, with particular concentrations around the time period of rate adjustment, though the exact

magnitude of the spikes in termination probability will depend on the rate environment at adjustment date. Of

course, since our data comes from the servicing records of a single lender and covers at most 5.5 years of history, it

may be premature to generalize results. But as this product innovation enters the public securities arena through

additional hybrid ARM pool securitization, additional empirical research is warranted to corroborate these initial

findings on loan performance.

We thank Michael Fratantoni, Jim Follain, Andrea Heuson, Susan Wachter, Nancy Wallace and an anonymous referee for helpful comments and suggestions on earlier versions of this paper. The views expressed are the authors' alone and not necessarily those of Wells Fargo and Company or any of its affiliates.

9

References

Ambrose, B.W. and A. Sanders. 2003. Commercial Mortgage Backed Securities: Prepayment and Default, Journal of Real Estate Finance and Economics 26 (2/3): 175-192. Ambrose, B.W. and M. LaCour-Little. 2001. Prepayment Risk in Adjustable Rate Mortgages Subject to Initial Year Discounts: Some New Evidence. Real Estate Economics 29(2): 305-328. Brueckner, J.K. and J.R. Follain. 1988. The Rise and Fall of the ARM: An Econometric Analysis of Mortgage Choice. Review of Economics and Statistics 70: 92-102. Calhoun, C.A. and Y. Deng. 2002. A Dynamic Analysis of Fixed- and Adjustable-Rate Mortgage Terminations. Journal of Real Estate Finance and Economics 24(1/2): 9-33. Capone, C. and D.F. Cunningham. 1992. Estimating the Marginal Contribution of Adjustable-Rate Mortgage Selection to Termination Probabilities in a Nested Model. Journal of Real Estate Finance and Economics 5: 333-357. Cunningham, D.F. and C. Capone. 1990. The Relative Termination Experience of Adjustable to Fixed-Rate Mortgages. Journal of Finance 45(5): 1687-1703. Deng, Y. 1997. Mortgage Termination: An Empirical Hazard Model with Stochastic Term Structure. Journal of Real Estate Finance and Economics 14: 309-331. Deng, Y., J.M. Quigley and R. Van Order. 2000. Mortgage Terminations, Heterogeneity and the Exercise of Mortgage Options. Econometrica 68(2): 275-307. Dhillon, U., J.D. Shilling and C.F. Sirmans. 1987. Choosing Between Fixed and Adjustable Rate Mortgages. Journal of Money, Credit, and Banking 19: 260-267. Gross, D.B. and N.S. Souleles. 2002. An Empirical Analysis of Personal Bankruptcy and Delinquency. Review of Financial Studies 15(1): 319-347. Houston, J.F., J. Sa-Aadu and J.D. Shilling. 1991. Teaser Rates in Conventional Adjustable-Rate Mortgage (ARM) Markets. Journal of Real Estate Finance and Economics 4(1): 19-32. HSH Associates. 2004. Hybrid ARM Mortgage Statistics. http://www.hsh.com/hybrid-stats.html. Lea, M.J. and P.M. Zorn. 1986. Adjustable-Rate Mortgages, Economic Fluctuations, and Lender Portfolio Change. AREUEA Journal 14(3): 432-447. Linneman, P. and S. Wachter. 1989. The Impacts of Borrower Constraints on Homeownership. AREUEA Journal 17(4): 389-402. Mortgage Bankers Association. 2004. 1-to-4 Family Mortgage Originations. http://www.mortgagebankers.org/marketdata/data/03/1-4_originations.html. Nothaft, Frank. 2003. Will Interest Change Lead to Higher ARM Share? Special Commentary from the Office of the Chief Economist. http://www.freddiemac.com/news/finance/commentary/sp-comm_092203.html. Stanton, R. and N. Wallace. 1995. Arm Wrestling: Index Behavior and Prepayment of Adjustable Rate Mortgages. Real Estate Economics 23: 311-345. Stanton, R. and N. Wallace. 1999. The Anatomy of an ARM: Index Dynamics and Adjustable Rate Mortgage Valuation. Journal of Real Estate Finance and Economics 19(1): 49-67.

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Table 1: Descriptive statistics. The spread over fixed rates is the percentage by which the current note rate exceeds the current 30-year fixed mortgage rate. The term structure is defined as the difference between the 10-year and the 1-year Treasury rates. Payment-to-income is the monthly loan payment divided by borrower monthly income. Credit score is the primary borrower’s FICO score. Loan-to-value ratio is estimated current LTV based on loan amortization and estimate house price changes. Loan amount is the original loan amount at origination. Points (in percentage of loan amount) are prepaid finance charges paid at origination. Region 1 through Region 10 are regional dummy variables for U.S. census regions.

At Origination At Termination

Variable Definition Mean Std. Dev Min Max Mean Std. Dev Min Max

Spread Over

Fixed Rates )()()(

trtrtr

f

fc −

-0.0956 0.0546 -0.2782 0.1518 -0.0246 0.0869 -0.2844 0.3216

Term Structure 10 yr Treasury – 1yr Treasury 0.0112 0.0018 0.0080 0.0144 -0.0008 0.0218 -0.0618 0.0144

Payment-to- Income

Ratio of loan payment to monthly income 0.1788 0.0712 0.0104 0.8498

Credit Score Primary borrower FICO 730.0 47.6 536.0 821.0

Loan to Value LTV Ratio (time-varying) 0.7491 0.1596 0.1180 0.9580 0.6215 0.1756 0.0973 0.9945

Loan Amount Loan Amount at Origination $259,878 $184,585 $22,700 $2,050,000

Points Points paid at Origination 0.301 0.572 0.0000 4.076

Region 1 New England 6.9%

Region 2 Northwest 2.1%

Region 3 Upper-West 5.6%

Region 4 Mid-West 17.8%

Region 5 NY/NJ 8.1%

Region 6 West 22.8%

Region 7 Southwest 6.5%

Region 8 Plains 3.5%

Region 9 South 14.0%

Region 10 Mid-Atlantic 12.6%

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Table 2: Competing risks model of prepayment and default. Parameter estimates for the following competing risks model estimated via maximum likelihood in a multinomial logit framework

( ) ( )( )∑

=

= 3

1|

|,|Pr

jj

jj

xh

xhxj

τ

ττ

where we specify a separate exponential hazard function for each mortgage outcome

( ) ( )jjjjj xxh βτ ′= exp|

and separate xj into components representing borrower risk characteristics and economic conditions having the following linear form: iititittjj adjusteconriskagex 43210 ββββτββ ++++=′ .

AGE is the number of month since origination. AGE2, AGE3 and AGE4 are AGE squared, raised to the third and raised to the fourth power, respectively. ϑ represents quarter dummies (Q1 1995 through Q4 1996) that control for mortgage origination, where Q1 is the reference group. RISK represents the following variables designed to control for borrower risk characteristics: PAYMENT-TO-INCOME is loan payment relative to borrower monthly income; credit score (FICO) is the primary borrower’s FICO score; LTV is a loan –to-value ratio, monthly adjusted for the mortgage payments and housing price movements; ORIBAL is the loan amount at origination; HIGPROB is a dummy variable, equal to 1 when the probability of negative equity exceeds 5%. ECON represents the following variables designed to control for economic risk factors: TERMSTRU is the difference between the ten-year and the 1-year Treasury bond rates; SPD_FRM variable is the loan interest rate spread over the fixed-mortgage rate. ADJUST represents the set of variables that control for risks associated with the switch from fixed- to adjustable-rate mortgage: ADJUST is a dummy variable denoting the 3-month window surrounding the anniversary date of the conversion of the loan from a fixed-rate to an adjustable-rate mortgage; AFTER_ADJUST is a dummy, equal to 1, for months after the adjustment; SPDFRM_ADJUST is an interaction variable, which is the adjustment window dummy (ADJUST) multiplied with SPD_FRM; PMT_SHOCK is the spread of new mortgage payment over the old mortgage payment at the adjustment date.

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Prepayment Model Default Model

Parameter Coefficient Chi-Sq p-value

Parameter Coefficient Chi-Sq p-value

INTERCEPT -7.6898 238.03 <.0001 0.8974 0.55 0.457 AGE 0.2145 30.91 <.0001 0.0932 1.17 0.279

AGE2 -0.0066 6.88 0.009 -0.0013 0.04 0.835 AGE3 9.6E-05 2.37 0.124 -3.0E-05 0.04 0.851 AGE4 -6.5E-07 1.54 0.214 4.9E-07 0.13 0.715

QUARTER2 0.0052 0.00 0.965 0.4188 1.27 0.259 QUARTER3 0.1736 2.14 0.143 0.3680 0.91 0.340 QUARTER4 0.0622 0.24 0.625 0.4298 1.13 0.289 QUARTER5 0.1217 0.56 0.453 -0.2488 0.18 0.671 QUARTER6 0.0733 0.32 0.574 0.2871 0.47 0.491 QUARTER7 0.0791 0.41 0.521 0.1538 0.14 0.709 QUARTER8 0.1255 0.82 0.364 0.3732 0.73 0.394

SPD_FRM 3.4788 94.44 <.0001 2.6894 6.49 0.011 TERMSTRU 0.9643 0.09 0.760 16.1832 2.00 0.157

PAYMENT-TO-INCOME 0.5208 2.22 0.137 -0.0493 0.00 0.964

FICO 0.0016 9.33 0.002 -0.0123 92.26 <.0001 LTV 0.4125 2.79 0.095 0.6456 0.71 0.400

ORIBAL 8.8E-07 46.41 <.0001 4.6E-07 1.21 0.272 HIGHPROB 0.0142 0.04 0.844 -0.1372 0.38 0.537

ADJUST 0.5737 42.00 <.0001 -0.0545 0.02 0.877 AFTER_ADJUST 0.1081 0.75 0.388 0.6783 2.51 0.113

PMT_SHOCK 4.5405 24.41 <.0001 4.5474 2.67 0.102 SPDFRM_ADJUST -1.7140 2.63 0.105 3.5758 0.72 0.395

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Figure 1: Monthly interest rates. Origination rate is the average mortgage rate at origination for the specific month during the period of January 1995 through December 1996. FRM rate is 30-year fixed-rate mortgage from the Primary Mortgage Market Survey (PMMS). Adjustment rate is the 1-year Treasury rate plus 250 basis points measured each month, 36 month after origination.

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Interest Rates Over Study Period: Origination Rate 1995-1996; Adjustment Rate 1998-1999;

FRM Rate 1995-2000

5.00

5.50

6.00

6.50

7.00

7.50

8.00

8.50

9.00

9.50

10.00

Jan-95 Jul-95 Jan-96 Jul-96 Jan-97 Jul-97 Jan-98 Jul-98 Jan-99 Jul-99 Jan-00 Jul-00

Date

%

Origination RateAdjustment RateFRM Rate

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Figure 2: Average probability of negative equity by month from mortgage origination, segmented by quarterly origination cohort.

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Figure 3: Baseline prepayment and default hazard rates.

Baseline hazard function shows the estimated probability of default and prepayment as a function of time since origination.

Baseline Hazards

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61

Month

%

prepaydefault

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Figure 4: Cumulative prepayment and default probabilities for the baseline model.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61

Month

%

prepaydefault

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Figure 5: Prepayment and default hazard rates for the competing risk model (Table 2).

Estimated prepayment and default rates, conditional on flat term structure, as function of time since loan origination.

Prepayment and Default Hazards(flat term structure)

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61

Months

prepaydefault

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Figure 6: Cumulative prepayment and default hazards for the competing risk model (Table 2).

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61

prepaydefault

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1 Linneman and Wachter (1989) note that adjustable-rate mortgages reduce the constraints on homeownership. 2 For important early research on adjustable-rate mortgages, see, for example, Lea and Zorn (1986), Dhillon, Shilling and Sirmans (1987), Brueckner and Follain (1988), Cunningham and Capone (1990), Houston, Sa-Aadu and Shilling (1991), Capone and Cunningham (1992) or Stanton and Wallace (1995, 1999). 3 See Ambrose and LaCour-Little (2001) for a more comprehensive literature review. 4 Ambrose and LaCour-Little (2001) address adjustment shock on traditional 1-year adjustable-rate mortgages with initial "teaser" rates. 5 For example, on July 29, 2004, Ginnie Mae announced that it would begin to guarantee hybrid ARM loans in pools backed by FHA 3/27, 5/25, 7/23 and 10/20 hybrid ARMs effective September 1, 2004. 6 Some fraction of the loans that reached 90-day delinquency (that, using our terminology, "defaulted") ultimately paid off in full, so the lender realized no net credit loss from the default. After a 90-day delinquency, however, loans were transferred to a special servicing department for foreclosure, workout or other resolution, and we do not have records from that period of time to examine ultimate outcome. 7 Fair Isaac and Company (FICO) is the industry leader in credit scores, with scores ranging from 300-900. A higher score indicates better credit. Scores above 700 are generally considered prime credits. 8 Obviously, borrowers could also refinance into a new 3/27 hybrid mortgage or a number of alternative mortgages (e.g., fully-adjusting ARMs or other hybrid products). 9 Using the variance of the OFHEO repeat transaction index (ε2), the probability of negative equity is calculated as

( ) ( )⎟⎟⎠

⎞⎜⎜⎝

−Φ=

2

logε

mktvalpvbalPROBNEQ⎛ log

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where pvbal is the present value of the remaining mortgage payments, mktval is the current market value of the property estimated using the repeat transaction index and Φ is the cumulative normal density function (see Deng 1997 and Deng, Quigley and Van Order 2000). 10 To isolate the effect of the spread over fixed-rate variable, the simulations were performed assuming a 75% LTV mortgage, a term structure value of 0.011, a FICO score of 730 and a payment-to-income ratio of 0.1788. 11 The marginal effect is calculated as eβ-1.

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