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An Analysis of Prepayment & Claim Rates for FHA-Insured Multifamily Mortgages

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A research paper sharing econometric models estimating claim and prepayment rates for FHA insured multifamily mortgages.

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An Analysis of Prepayment & Claim Rates for FHA-Insured Multifamily Mortgages

  1. 1. An Analysis of Prepayment and Claim Rates for FHA-Insured Multifamily Mortgages by Lawrence G. Boyer, James R. Follain, Wenyi Huang, and Jan Ondrich June 2001 Boyer, Follain, and Huang are with Freddie Mac and Ondrich is with the Center for Policy Research, Syracuse University. Richard Piccirillo also played an important role in the development of the project before his move to GE Capital. Special thanks are due Drew Schneider of PricewaterhouseCoopers for his overall leadership of the PricewaterhouseCoopers projects regarding the assessment of multifamily mortgage insurance risk and his support for this particular paper. We are also grateful for the support of Barry Dennis and Isaac Megbolugbe of PricewaterhouseCoopers. Raimundas Razanauskas and John Larson also provided helpful assistance at various stages of the lengthy process underlying this paper. The assembly of the database greatly benefited from the input of several people within FHA and Ginnie Mae including Joe Malloy, Sue Donahue, Rick Calvert, and—Scott Werdal, Michael Najjum and S. l Daniel Raley. Charles Capone, Larry Goldberg, Tyler Yang also provided helpful comments. The conclusions and opinions expressed in this paper represent those of the authors and not necessarily; of; PricewaterhouseCoopers, Freddie Mac, or the U. S. Department of Housing and l Urban Development.
  2. 2. I. Introduction State of the art analysis of the riskiness of single—family mortgages (e. g., Deng, Quigley, and Van_Order 2000) is built upon sophisticated econometric models of the probability of mortgage prepayment and default. These models explain the probability of mortgage default and prepayment during each year of the mortgage as functions of economic variables such as the amount of homeowner equity and the level of interest rates. They permit the computation of default costs for a wide variety of economic scenarios regarding housing prices and interest rates over the life of the loan. These capabilities allow these models to play critical roles in the pricing of mortgage credit risk and the allocation of risk—based capital for portfolios of single—family mortgages. The development of similar econometric models for multifamily mortgages has been severely hindered by data limitations. These limitations include the lack of a long time series of experience with comparable multifamily mortgages, complex and frequently changing underwriting criteria, and data tarnished by episodes of fraudulent reporting. Because multifamily lending has not been subjected to the same rigorous statistical examination, the risks inherent in multifamily mortgage lending are often perceived to be higher than for single—farnily mortgages. This is probably one of the factors underlying the lower rate of securitization of multifamily mortgages. The importance of examining econometric models of multifamily mortgages has been underscored by two important public policy issues. The first pertains to FHA-insured multifamily mortgages. Credit reform legislation of the early 1990s requires insurance programs such as FHA to compute the expected income (expected losses less premium income) for each program.
  3. 3. Programs that are projected to generate net losses require an allocation from a credit budget available to HUD. The current approach relies on a single set of average annual conditional probabilities of mortgage default for each year in the life of an FHA-insured mortgage. If, in fact, these probabilities vary widely among economic scenarios, then the current approach likely understates the true riskiness of FHA multifamily lending. The second policy issue is the assignment of risk—based capital for investments in multifamily mortgages by the Office of Federal Housing Enterprise Oversight (OFHEO) for the two government-sponsored enterprises (GSES), Fannie Mae and Freddie Mac. Although a substantial amount of excellent research was conducted to produce the model underlying OFHEO’s 1999 proposed risk-based rules for multifamily mortgages, both GSES expressed concerns about the data underlying the estimation of OFHEO’s econometric model. All parties seem to agree that more research on this topic with different and better data would be welcome. This study estimates conditional prepayment and claim rates (prepayment and claim hazards) for multifamily moztgages' using data from the largest FHA multifamily mortgage-insurance program from 1965 to 1995, the market-rate 221(d)(4) programz Although the data set lacks some important information such as prepayment penalties and initial debt-coverage (DCR) and loan—to-value (LTV) ratios, the data set has many desirable features. Most importantly, the data cover a wide- ranging time period, which no other multifamily mortgage database can match. As to the omission of initial DCRs and LTVS, most FHA loans seem to be heavily concentrated around initial DCRs of 1.05 to 1.10 and LTVs are 90 to 95 percent. The model specification employed in this study is similar to the hazard models typically estimated for single-family mortgage models. Duration dependence is represented by a third- order polynomial to impose a minimum of structure on the form of the baseline. The impact of
  4. 4. interest—rate movements is captured by piecewise linear specifications (splines) in which the responses of the prepayment and claim hazards depend upon the remaining maturity of the loan and the extent to which the contract interest rate exceeds the market interest rate. A variety of other variables are included to measure differences among regions and particular periods. A critical aspect of our model is the specification of the economic determinants of claims. Our approach uses the key components of the cash flow available to the borrower—~rental income and vacancies. Increases in the indices of rental price since loan origination are assumed to reduce the likelihood of a claim, while higher vacancy results in lower cash flow and therefore higher claim rates. However, a flexible specification of these variables is examined to take account of the nonlinearities predicted by option pricing theory. One specification includes an estimate of the probability that the cash flow available to the property is sufficient to make the mortgage payment on the property; this approach follows earlier work by Goldberg (1998) and Capone and Goldberg (1998). A model with a more flexible specification of the effects of rental income and vacancies is also estimated and the results of the two approaches are compared. The measures of rental income and vacancies vary over the life of the mortgage and are typically based upon government-provided indices available for the metropolitan area in which the property is located. The models are estimated by multinomial and binomial logit. The multinomial logit estimation can be characterized as a competing-risk hazard model that estimates coefficients efficiently relative to single—risk binary logits. The potential for bias in the multinomial logit estimation suggests the importance of comparing results. A number of interesting insights emerge from the analysis. First, the estimated responses of both claims and prepayments to economic variables support some of the key predictions of the option-pricing model. Specifically, prepayments are highly sensitive to interest-rate declines; in
  5. 5. addition, higher rates of growth in rental income are associated with higher rates of prepayments, all else equal. The responses of claims to increases (decreases) in rents (vacancies) have the expected signs and are highly nonlinear. Second, the results from applying the estimates of the multinomial logit model to an important policy question are quite similar to those obtained with the single-risk (binary logit) model. Third, application of the model to the computation of federal credit subsidies for the FHA 221(d)(4) program confirms that the approach currently being used to compute the credit subsidy understates the true riskiness of FHA multifamily lending. The remainder of the paper is divided into seven sections. The next section summarizes some of the existing literature in which multifamily mortgage prepayment and claim hazards have been estimated. The third section discusses the process and sources used to assemble the FHA database. The key characteristics of the FHA database are discussed in the fourth section. The fifth and sixth sections explain the econometric model and the results of the estimation. The seventh section demonstrates the calculation of the federal credit subsidies for the FHA program using the model estimates. The final section summarizes the main conclusions of the paper. II. Literature Survey Interest in the performance of multifamily mortgage loans has increased substantially in the 1990s for a variety of reasons} First, several multifamily programs generated substantial losses in the late 1980s and early 1990s. The most famous example is probably the FHA co- insurance programs of the early and mid-1980s. Another example is the pool of multifamily loans developed by Freddie Mac; indeed, the performance was so poor among Freddie Mac loans that it—Freddie Mac withdrew them from the market for several years in order to conduct a major evaluation and overhaul of its multifamily business.
  6. 6. Second, multifamily rental housing plays an important role in the ability of the federal govemment to deliver affordable housing because it is generally less expensive than single- family and owner-occupied housing. More specifically, such housing is also critical to the ability of the GSEs to meet the specific affordable housing goals recently established by HUD under a géongressional mandate. In order to assess accurately the cost of federally operated and insured multifamily rental housing programs and the impact of such mandates on the GSEs, it is important to better understand the performances of multifamily mortgage portfolios. Third, the secondary market for multifamily mortgages has lagged behind the secondary single-family mortgage market. One explanation often put forth to explain the lag is the lack of information about multifamily loan performance. Little has been written about the performance of multifamily mortgages relativecoinpared to the voluminous literature on the performance of single-family mortgages.4 The discussion below focuses upon several recent contributions to the topic and highlights the major differences between these papers and our own. Follain, Ondrich, and Sinha (1997) examine prepayment behavior for a portfolio of Freddie Mac multifamily mortgages originated throughout the 1970s and 1980s. The theoretical basis of their study is an option-pricing model in which prepayments depend upon the present value of future mortgage payments discounted at the current market rate of interest. A semiparametric proportional hazard specification with unobserved heterogeneity is employed to estimate the prepayment hazard. The estimated response of the prepayment hazard to a change in the market rate of interest is significant with the expected sign; nonetheless, the magnitude of the response is substantially less than that predicted by the ruthless option-pricing model. Elmer and Haidofer (1997) use prepayment data on securities issued by the Resolution Trust Corporation (RTC) from 1991 to 1992. Both adjustable and fixed rate mortgage pools are
  7. 7. examined. The difference between the average contract rate of the pool and the current market rate proves to be important in explaining variations in multifamily prepayment rates during this period for both adjustable and fixed rate mortgages. All else equal, balloon and adjustable rate mortgages tend to prepa-y—be prepaid more rapidly than others and larger loans tend to prepay—@ mflmore slowly. As in Follain, Ondrich, and Sinhajfl), seasonal effects do not appear to be important. Dillon and Belanger (1997) of Fitch Investors Service use RTC data to examine trends in commercial mortgage default rates and loss severity for the period 1991-1995. Multifamily apartments are just one of the real estate types included in this sample; others are warehouses, industrial property, office property, nursing homes, retail centers, and lodging. Apartments tum out to have a annual default rate of about 4 percent per year, which is second lowest among the various real estate uses (warehouses have the lowest). The authors present several important findings. First, loans with fixed interest rates tend to have lower default rates. Second, loan size is not strongly related to the default rate. Third, balloon mortgages have a substantially higher default rate than fully amortizing loans. The initial DCR is strongly correlated with defaults. (The authors do not test the role of the initial loan-to-value ratio because they feel that these data are flawed. ) The average loss severity in the sample is 37 percent of the initial loan balance and loss severity has a large variance. For example, 25 percent of the loans have a loss severity rate greater than 64 percent and 50 percent have a loss severity rate less than 27 percent. No major patterns that explain this variation are identified. Abraham and Theobald (1997) use multifamily data from Freddie Mac's portfolio for the years 1984 to 1994; thus, they exclude some of the data from earlier years used by Follain, Ondrich, and Sinha (1997) and include an additional five years of data. Their data contain a larger number of loans with yield—maintenance provisions, five-year prepayment lockouts, and
  8. 8. periods of large declines in the interest rate. The additional years of data include years in which credit quality problems among multifamily loans were more pronounced and many multifamily borrowers were unable to refinance due to inadequate equity. Abraham and Theobald pay special attention to this problem in their analysis. They conclude that the prepayment rates of eligible multifamily borrowers (beyond the lookout period with adequate equity) are more sensitive to interest rate declines than the prepayment rates of single-family borrowers. Conditional prepayment rates among eligible borrowers who are deeply in the money are double those for single-family borrowers. Nonetheless, the interest—rate sensitivity for these borrowers is still less than What might be predicted by a ruthless option-pricing model; moreover, the model underestimates prepayment rates during the refinancing boom of 1989. mcher _e1_aL (1999) test the hypothesis that the central measure of default risk is the LTV ratio predicted by option-pricing models. They argue that the key variables in the option model are endogenous to the loan origination and property sale process. Because of this endogeneity no empirical relation may be observed between default and the LTV. Examining the default experience of more than 9,000 multifamily mortgage loans seeuritizeel secured by the Resolution Trust Corporation (RTC) and the Federal Deposit Insurance Corporation (FDIC), the authors find that the DCR is more important in explaining default than t*l; _e_LTVltio. Fu, LaCour—Little, and Vandell (1999) focus on multifamily prepayment behavior and prepayment—penalty structure. They examine a nationwide disaggregated sample of 2,573 multifamily loan payment histories obtained from a large multifamily mortgage originator and servicer. Their findings confirm that most of the option-theoretic factors, such as the ratio of coupon rate to current market rate, interest—rate expectations, and interest volatility are important in explaining prepayment behavior. Region of the country, loan size, and the fractional remaining
  9. 9. loan balance are also found to be significant. The most important finding, however, is that the nature and terms of the prepayment penalty significantly affect the pattern of prepayment on multifamily mortgages. In particular, yield maintenance and lockouts are the most effective constraints, while fixed step—down penalties are less effective. The paper by Goldberg (1998) is the closest in spirit to the present study. Goldberg focuses upon FHA-insured multifamily mortgages, specifically; the 22l(d)(4) (OMI) and the 223(f) (HRI) programs. The periods of study are similar as well; we use 1965 to 1995, while Goldberg uses 1970 to 1993. The present study differs from the study by Goldberg in two several important respects. First, we model both claim and prepayment risks while Goldberg looks at only the claim risk. Second, the estimation strategy followed by Goldberg differs from the estimation strategy of the present study in a fundamental way. We take a reduced form approach in which the covariates that affect claim and prepayment risk are entered into the hazards directly without being modified in a way that reflects the economic structure of making the claim and prepayment decisions. Goldberg, on the other hand, takes a structural approach in which he attempts to explicitly model the multivariate random processes that he believes determine default. Goldberg derives an expression for the probability of the event that leads to default, namely an LTV ratio that exceeds one at the same time that the DCR is less than one. Because this event is defined by conditions on two variables, Goldberg calls this event the double-trigger. The calculated double—trigger probability is then treated as a regressor in the default hazard. To obtain variation in the double—trigger probability across mortgages at each point in calendar time, Goldberg includes residential rent and the rental vacancy rate at the metropolitan level. Their inclusion has a cost. The pure metropolitan—level series are unavailable for some MSAs in any year and for all MSAS at the beginning of the sample period. To overcome this problem, predicted values of vacancies and rental information from two different sources are
  10. 10. combined to fill in some of the unavailable data. Still, 3,300 of the 8,700 available loans are deleted from the sample because of missing values. Because we take a reduced form approach, we can avoid introducing measurement error by not using these series and other problem variables, such as the initial LRV and DCR, as well as information regarding the type of project-based rental assistance available. In sum, Goldberg estimates a structural model that takes into account more explanatory variables that are conceptually important but may also be difficult to work with in practice. We estimate a reduced form model and use more observations to estimate the hazards. Our potential weakness is omitted variables. Capone and Goldberg (1998) conduct an analysis similar to L Goldberg (l998). They use multifamily cash purchases acquired by Fannie Mae and Freddie Mac between 1983 and 1995. The sample includes 7,564 loans and focuses upon loans in 28 large metropolitan areas for which government data on rents and vacancies are available. Prepayment is not considered. As in Goldberg, Capone and Goldberg emphasize the role of tax policy as an explanation of the high default rates experienced in the late 1980s and early 1990s. They include an interesting discussion of the sometimes suspect underwriting criteria used in the early 1980s and present a new approach to update DCRs and LTVs. Their simulations examine the impact of the Tax Reform Act of l986,—whiel>. ;_@: y indicate that conditional default rates increased by 50 basis points due to the tax reform. More generally, the simulations identify tax reform, worsening economic conditions, and lax underwriting criteria as the reasons for the substantial increase in claim rates during the period of study. These are exactly the kinds of questions and simulations that must be conducted to accurately price mortgage insurance. Future and ongoing research will also determine the
  11. 11. superior approach. Finally, a greater investment in data collection would improve both approaches. Ill. Assembling the FHA Data Set The “ideal” data set for pricing mortgage insurance would include th_eLRV ratio and the DCR, the contract interest rate, prepayment penalties, and property characteristics. Although much of this information is collected during the FHA application process, only some of this information is in electronic format. Therein lies the root of the problems. The problems are not related to the type and volume of information theoretically available. If all of the forms associated with FHA data were properly filled out and available in computerized form, the FHA data would represent an exceptional source of information on multifamily loan performance. Unfortunately, this is not the present state of affairs. There are three general categories of problems that limit the usefiilness of FHA data for econometric analysis: 0 Non-computerized information: A significant amount of information on FHA—insured multifamily mortgages exists on paper forms stored in the HUD headquarters and field offices. - HUD database infrastructure: Computerized data are stored in multiple databases at HUD. Each of these databases is designed and created to track specific information for different purposes. Obtaining information often involves relating several databases. - Missing and miscoded information: Data in HUD systems are not always complete and may contain miscoded information. The combination of these three factors results in high data-assembly costs for econometric estimation of claim and prepayment hazard rates. Most of the loan-record data is—_a_r_e_ obtained from H'UD’s F47 database, which is used to track loans that enter and exit FHA’s multifamily insurance programs. ltThe database contains 10
  12. 12. mortgage information collected at endorsement and related property information. The data are collected at endorsement by the field offices and are periodically sent electronically to the HUD headquarters office in Washington, D. C. IV. Data Description We examine one of the largest FHA multifamily mortgage-insurance programs, the market; ~rate 221(d)(4) New Constructions and Substantial Rehabilitation program (OMI). The numbers refer to the section of the housing act under which the program is authorized. The three- letter designations are codes that distinguish the many subprograms. The OMI program insures mortgages issued at market ratesgs on new and substantially rehabilitated properties, although subsidies are sometimes associated with the properties or tenants. For example, some OMI- insured loans are associated with properties that receive Section 8 assistance. However, in our study; we use only loans without Section 8 assistance in order to fully identify the market—based “economic” reasons for borrowers to prepay or to default. The OMI loans used in this study have amortization start dates from 1965 to 1995. There are 2,747 loans in the OMI program—with 450 claims and 703 prepayments. The loan size in this study ranges from $43,800 to $100 million, with a mean of $3.74 million. Approximately 54 percent of the properties are located in central cities. An average property has about 13 units; the biggest property in the estimation sample has 172 units»-in~tetal. About 23 percent of the prepeitieslci are originated between 1981 and 1986. More than 19 percent of the properties participate in the Tandem program. Insights about claim and prepayment hazard rates are ofien best provided by life tables. The life tables are presented in Figures 1 and 2. The claim rates and the prepayment rates are ll
  13. 13. separately calculated for each year of mortgage life and span all books of business and endorsement years. Tl1e claim rate rises steadily in the beginning years of loan life and reach_e_s a maximum of 2.3 percent in year 5. It drops to the vicinity of 1 percent and stays at this level until year 20. There is a surge of claims in year 22 and the claim rate reaches about 1.6 percent. The last claim among the sample observations occurs at year 23. The prepayment rates rise§ steadily until year 20, in which year the prepayment rate reaches about 5.0 percent. Then sharply after year 20 and reach_e_s_ i= ates—ef—10 percent in year 21 and over 17 percent in year 24. This is a sign of another covariate besides time or policy life influencing behavior. One possibility is the “G4” loan policy by which the participating loans can “cash out” after year 21. The loan data contain the term, status, and origination and termination years of the loan, the contract interest rate, the number of units, the initial mortgage amount, the MSA and state in which the property is located, and whether the property is included in the central city of the MSA. As noted in the previous section, there is information about other variables in the data, but their frequency of missing values led us to drop them from in our analysis. The initial LTV ratio and the DCR are the most important of these. Such an omission might be a source of concern for typical conventional multifamily mortgage programs because the initial values of DCR and LTV vary widely and are likely to have a large impact upon ultimate default performance. However, conversations with people familiar with the history of the FHA program suggest that the actual range of these Values was quite small. Typical initial DCRs were around 1.10 and initial LTVs were near 90 percent. As a consequence, we view the lack of such information as a weakness in the data, but not a fatal one and not even a serious source of concern. 12
  14. 14. The 1oan—level information is supplemented by three economic variables of the MSAS in which the properties are located. The MSA-level information includes annual information about the growth rate of households, the residential rental component of the CPI, and the vacancy rate. For properties not located in an MSA, or for locations where the MSA—level information is not available, regional or national indices are used instead. Definitions of all explanatory variables are provided in Table 1 and summary statistics are presented in Table 2. Modern research concerning prepayment and default on financial obligations usually takes an option-pricing perspective. Borrowers have two separate options: a prepayment (call) option and a default (put) option. The prepayment option is a call option because the borrower has the option to pay off the unpaid balance prior to maturity; the borrower essentially buys the original mortgage for the amount of the unpaid balance (exercise price). The most important economic variable that influences the exercise of the call option is the gap between the book Value of the mortgage (outstanding unpaid balance) and the market Value of the mortgage (the future mortgage payment flow evaluated at the current market rate). The prepayment rate is expected to be low when the book value of the loan exceeds its market value; however, the prepayment rate will use when the market value of the mortgage exceeds its book value. The value of the call option is calculated as the difference between the book value and the market value of the mortgage, expressed as a percentage of the market value. Transaction costs are assumed to be 1 percent of the unpaid balance. Ideally, the ca1l—option value should incorporate information about prepayment penalties and lockouts, as well as yield—maintenance agreements; however, these data are unavailable in the FHA database. The effect of the value of the call option enters the prepayment and claim risks as a quadratic polynomial spline with a knot at zero. (For a theoretical discussion of splines, see Poirier 1974, or Greene 1999. An empirical application is given in Garber and Poirier 1974.) For 13
  15. 15. both prepayment and claim risks, this means that, for positive values of the call, the effect on the risk is that of a quadratic. For negative values of the call, the effect on the risk is a second quadratic unrelated to the first. Both quadratics meet at zero, the knot-point. Finally, the two quadratics for the prepayment risk are allowed to have different coefficients than the two quadratics for the claim risk. We expect the slope of both quadratics in the prepayment risk to be positive, but w_e have no expectations regarding the quadratic slopes in the claim risk. The default option is a put option because the borrower sells (puts) the property to the lender for the value of the property (exercise price). The default decision therefore depends on the amount of negative equity associated with the property. Because estimates of the market value of the property are unavailable, we do not have a direct measure of negative equity. Furthermore, unlike the case of the single-family market, price indices for multifamily housing by market area are unavailable for most observations in the sample. As a result, proxies for the current asset~price of the property are included. The first proxy is the cumulative rental price appreciation rate since the origination of the mortgage. Presumably, higher rates of rental appreciation are associated with higher values of the property. The rental appreciation rate is generated by the residential rental component of the US. Consumer Price Index (CPI). MSA indices of rental appreciation are preferred, but they are unavailable for some of the time periods and market areas included in the sample. In this case, the regional or national index is used and adjusted to replace the MSA level information. The second proxy is the vacancy rate at the MSA level. A higher vacancy rate results in lower net cash flow for a property and thus implies higher probability of default. The third proxy is the unexpected growth in households since the origination of the mortgage. All else equal, higher rates of unanticipated growth in households lead to higher demand for housing and higher rental 14
  16. 16. prices. The specific variable is the difference between the annual growth rate in households and a trend estimated for the full sample period. It is measured at the metropolitan level. The three proxies enter the risks as piecewise-linear splines. This means that the effect of each proxy on a risk is linear for small changes in the value of the proxy but is potentially nonlinear for larger changes. A graph of the proxy effect on a risk is a series of line segments joined together at knot-points. Suppose first that the proxy variable, 2 , only takes on positive values (this is true of the vacancy rate proxy). Then the left-most segment, called Segment 1, has slope B, and begins at z = O and ends at z = a, . The next segment, Segment 2, has slope B, and begins at z = a, , where it is conjoined to Segment 1, and ends at z = a, . In general, Segment s, having slope BS, begins at a, _,, where it is conjoined to Segment s—-l, and ends at a, . An S-piece spline will have S such segments; the slope of each segment is to be estimated and is the coefficient of an associated regression variable.6 The Segment 1 variable, zl , takes on a minimum value (:20) of zero and a maximum value of a, . In general, the Segments variable, 2, , takes on a minimum value of zero and a maximum value of a, —a, _,. The values of the segment variables are determined by the value of the proxy as follows. If z > aS_, , then zs = x—aS_, and z, = as —aS_, for all .9 less than S, from which it is clear that the equation z= Zz. (1) holds. If it is not true thatz > aS_, , define m to be the largest value of s such that z > am . Then, if Equation (1) is to hold in all such cases as well, we let 2, :0 for all s greater than m+l, Z "H, and 2, = as — ah, for all s less than m+1. = Z~'am’ 15
  17. 17. Now suppose that the proxy variable can take on negative as well as positive values (this is true of both the cumulative rental price appreciation rate and unexpected growth in households proxies). Although it is possible to have a knot at zero, for both proxies we will have a segment passing through zero. For the cumulative rental price appreciation rate, the segment passing through zero is the left—most segment. When this proxy is positive, the values of the 25 are those defined in the last paragraph. For Equation (1) to hold when this proxy takes on a negative value, we set 21 = z and 2: = O for all s greater than 1. For the unexpected growth in households, there is a single negative knot, so that the segment passing through zero is not the lelt-most segment. Because S =1 for this proxy (where S is the number of segments over the positive reals), we may call its left-most segment; Segment 2. Thus, the proxy has 2 segments with a single knot at the negative value a, . When the unexpected growth in households is greater than or equal to a, , 2, = z and 22 =0.When z<a, ,we set z, =11, and 22 = z—a, . Although knot values can be estimated, they wei= e—gi@_ taken to be the quartile values for the two 4-piece splines (vacancy and rental price appreciation rates) and the median for the 2- piece piecewise-linear spline. The number of segments for each spline is chosen by trial and error in an effort to achieve the most reasonable fit. In this study, two distinct and competing approaches are taken to model the default option. The first employs the three spline functions discussed above to capture the nonlinearity of default behavior towards the economic conditions. We call this approach the more “flexible” approach because it does not impose assumptions about the probability distribution of the economic variables. The altemative method is what we call the “structural” approach and is the one developed by Goldberg and Capone (1998). The default option is captured by a single variable that measures the probability that the DCR is less than unity, i. e., cash flow from the 16
  18. 18. property is insufficient to meet the debt-service payments. The probability measure is a fiinction of the actual rental price growth rate and the change in the vacancy rate since the mortgage was originated for the MSA in which the property is located. This probability replaces the spline fimctions of the rental price growth rate and the vacancy rate. Other variables are included in the specification of the claim and prepayment risks. Both have a third—order polynomial in duration of the mortgage, which should be general enough to capture any major non—monotonicity in the hazard baselines. Regional variation in the prepayment and claim hazards is captured in four regional indicators for the Northeast, South, Midwest and West. The omitted category is the South region. An indicator for central city location of the property is included to measure whether financial returns to multifamily lending mayediffers aer= ess—between suburbs and the central city, where most of the census tracts deemed to be under—served are located. The size of the property is included as a quadratic function in the number of housing units within a property. Property size is a proxy for the fixed cost component of transaction costs associated with prepayment and default. An indicator for whether an insured mortgage was part of the Tandem program of the late 1970s and early 1980s is also included- these loans had below-market interest rates at 7.5 percent. The specification also includes an indicator for the period 1981—1986.7 These were tumultuous years in the market for multifamily housing. Changes in tax laws in the early 1980s increased tax shelter benefits to multifamily owners. Upheaval in financial institutions, especially savings and loan associations, and variations in the quality of underwriting for multifamily mortgages also characterized the early 1980s. In order to identify the “pure” effect of the economic factors that affect a borrower’s decision to prepay or default, we exclude all loans that receive Section 8 rental assistance from our estimation. Some Section 8 projects provide above—ma. rket rental payments to landlords; this 17
  19. 19. property is insufficient to meet the debt-service payments. The probability measure is a function of the actual rental price growth rate and the change in the vacancy rate since the mortgage was originated for the MSA in which the property is located. This probability replaces the spline functions of the rental price growth rate and the vacancy rate. Other variables are included in the specification of the claim and prepayment risks. Both have a third—order polynomial in duration of the mortgage, which should be general enough to capture any major non-rnonotonicity in the hazard baselines. Regional variation in the prepayment and claim hazards is captured in four regional indicators for the Northeast, South, Midwest and West. The omitted category is the South region. An indicator for central city location of the property is included to measure whether financial returns to multifamily lending n= i&y—differs aeress—between suburbs and the central city, where most of the census tracts deemed to be under—served are located. The size of the property is included as a quadratic function in the number of housing units within a property. Property size is a proxy for the fixed cost component of transaction costs associated with prepayment and default. An indicator for whether an insured mortgage was part of the Tandem program of the late 1970s and early 1980s is also included- these loans had below—market interest rates at 7.5 percent. The specification also includes an indicator for the period 1981-1986.7 These were tumultuous years in the market for multifamily housing. Changes in tax laws in the early 1980s increased tax shelter benefits to multifamily owners. Upheaval in financial institutions, especially savings and loan associations, and variations in the quality of underwriting for multifamily mortgages also characterized the early 1980s. In order to identify the “pure” effect of the economic factors that affect a borrower’s decision to prepay or default, we exclude all loans that receive Section 8 rental assistance from our estimation. Some Section 8 projects provide above—market rental payments to landlords; this 17
  20. 20. differential from market rents may serve as a financial incentive to continue with the mortgage even if pure financial considerations suggest prepayment or default. Furthermore, FHA and HUD intervene occasionally in a troubled project by providing additional subsidies to cover operating costs. These loan-management set-aside programs (LMSAS) may affect the borrower’s decision to prepay or default. Rather than model this intervention decision or treat all properties with LMSA as a claim, we choose to exclude such mortgages from the estimation. V. Econometric Methodology The discrete~tirne econometric models that we estimate will be used to examine the determinants of the competing risks of prepayment and default for (FHA—market rate) multifamily mortgages in the OMI insurance program. Conditional on not having prepaid or having a claim filed before year t, the mortgagor must decide whether to prepay, default (represented by the claim risk), or continue with the mortgage in year t. Associated with each option is a utility, respectively, U5’ , U P, and U, °. The mortgagor chooses the option with the highest utility value. The utilities are not directly observable by the econometrician, who views them each as the sum of a detenninistic component; and a random component. The deterministic components are linear in the components of a mortgagor—specifrc vector, x,. The coefficient vectors for x, ; vary with the decisions to prepay, default, or continue, but are common across mortgagors. The ordinality of utility requires that the deterministic utility of one decision type serve as the utility baseline, which is accomplished by setting the coefficient vector for the baseline utility equal to zero. In the analysis for this study, the baseline decision will be continuing with the mortgage. The coefficient (for the prepayment and default choices) for a given component of x, gives the effect of a unit increase in that component on the deterministic utility expressed as a 18
  21. 21. difference from the effect of a unit increase in that component on the deterministic utility of the baseline decision. McFadden (1973) shows that if the random utility components for the three choices are standard extreme—value§ independently distributed across 1 and the three choices r, then the probability, P", that the mortgagor makes choice n at time t in year 2.‘ has the l‘ multinomial lo git form: P, " = exp(x, ’B, ,)/ Zexp(x; p,), n= P, D, C, (21 where the summation index r ranges across choices, P, D, and C. Multinomial logit estimation is by maximum likelihood (ML) using the LIMDEP 7.0 econometrics package (Greene 1995}. If ML estimation of the multinomial logit model just described is to produce consistent coefficient estimates, the application must satisfy the assumption of independence of irrelevant alternatives (IIA) assumption (see McFadden 1978). This assumption states that the log-odds of any two choices are independent of the presence of a third choice in the choice set. If this assumption is empirically valid (true of the data), ML estimation of the multinomial logit yields (asymptotically) efficient estimates of the utility coefficients. If this assumption is not empirically valid, the estimates will be inconsistent. An asymptotic test of the empirical validity of the HA assumption is described in Hausman and McFadden (1984). Rather than use the I-Iausman—McFadden test, we present, along with the multinomial logit results, the results of two binomial logit estimations (prepay versus continue with mortgage and default versus continue with mortgage) that have only two choices in the choice set. The first of these estimations drops claim from the choice set and drops from the data set all mortgage years in which a claim is made; the second estimation drops prepayment from the choice set and drops from the data all mortgage years in which a prepayment is made. If the HA assumption is valid, multinomial logit coefficient estimates will be asymptotically equivalent to those of the 19
  22. 22. two binomial logits. If not, coefficient estimates across multinomial and binomial logits will diverge for at least one of the two risks. In this case, it is possible that one of the two binomial logits will still yield consistent estimates. In this sense, it is safer to use binomial logit results i= atl= ier—than multinomial logit results. VI. Estimation Results The multinomial logit estimation results are presented in Tables 3 and 4. The results for the more “flexible” specification with the 4-piece splines for the cumulative rental price appreciation rate and the vacancy rate are in Table 3, while the “structura ” specification results with the variable used by Capone and Goldberg (1998) are in Table 4. The Capone-Goldberg Variable measures the probability that the cash flow for a property is negative at each year in the life of the mortgage. Likelihood-ratio tests for both models reject the null hypothesis that all of the coefficients are zero at the 1 percent level. Deciding whether the flexible or structural specification is prefei‘able is a difficult econometric problem. We analyze the results of two simple model selection criteria. Akaike (1974) develops an information criterion, now known as Akaike’s Information Criterion (AIC), for the problem of choosing among a set of models with different numbers of parameters. Akaike’s procedure is to maximize the likelihood separately for each model j and elieesing choose the model for which AIC}. =lnLj —kJ. is largest, where lnLJ. is the log-likelihood of model j and k 1. is the number of parameters in model j . The AIC for the flexible specification is —4997.544, while the AIC for the structural specification is -5043.107. Thus, at least by AIC, the flexible model is preferred. 20
  23. 23. Schwarz (1978), noting that maximum likelihood estimators can be obtained as large- sample limits of Bayes estimators for arbitrary nowhere vanishing priors, examines the asymptotic behavior of Bayes estimators under a special class of priors. Based on his analysis, he suggests a criterion, now called the Schwarz Criterion (SC), whereby the log-likelihood of model j is penalized by half the product of k}. and sample size. Thus, SC}. = In Lj —%kj. ln N. For any reasonable sample size, then, the SC approach will choose the “smaller” model more often than the AIC approach. In the present application N = 2747, implying an SC of -5151.419 for the flexible specification and —5l55.554 for the structural specification. The flexible specification is preferred even when the SC approach is used. Finally, to check the sensitivity of the results to the imposition of the independence of irrelevant assumption required by the multinomial logit estimation, we perform two binary logits (prepay-continue and default-continue) using the flexible specification. The parameter estimates from the binary logits do not change much from the multinomial lo git. Call Option Variables. As expected, the coefficients of the two variables that measure the value of the call option when it is out_-of_—the money (i. e., the market rate exceeds the contract rate) are insignificant in the prepayment model. The coefficients of the variable depicting a positive interest rate spread is significant and positive, as expected; while the coefficient of the quadratic term is insignificant. Thus, evidence of a strong nonlinear effect is not found. Figure 3 is presented to show the nonlinearity of the prepayment rate corresponding to the “in-the-moneyness” of the call-option value. The prepayment rates are plotted for a loan corresponding to different call—option values ranging from -10 percent to 50 percent. We plot the loan at durations of 5, 10, 15, and 20 years. When the option is “out of the money, ” the prepayment rate is below 5 percent for the four durations. As the call—option value increases, the 21
  24. 24. prepayment rate picks up steadily. For the same prepayment incentive, the 20-year; ~old loan has the highest prepayment probability. However, the prepayment behavior is not as “rutliless” as option-pricing theory would predict: when the call option is “deep in the money” at 50 percent, the prepayment is still only 44 percent for the 20-year; -old loan. Housing Price Proxies. Increases in the amount of rent inflation since origination of the mortgage significantly reduce the claim rate. All four segments of the rental price appreciation splines are negative and significant at the 5 percent level. For the multinomial logit estimation, the likelihood—ratio statistic for the joint significance of the rental price splines in both prepayment and claim risks is 124.458 (chi-squared with 8 degrees of freedom under the null). The signs of the coefficients for rental price appreciation in the prepayment equations are positive as hypothesized. That is, higher rates of rental appreciation increase prepayments; on the other hand, prepayment can be hindered for properties with negative cash flow. Properties located in areas with relatively higher vacancy rates are expected to have a higher propensity to default, all else equal. A 4-piece spline for the vacancy rates tests this hypothesis. The four coefficients are all positive and significant at the 5 percent level in the claim risk. In the multinomial logit estimation, the likelihood—ratio test statistic for the joint significance of the eight coefficients in both prepayment and claim risks is 143.258 (chi-squared with eight degrees of freedom under the null). Finally, growth in the number of households above the trend growth rate for an MSA tends to reduce claim rates, all else equal. Both coefficients of the spline are significantly negative for the claim risk at the 5 percent level in the multinomial logit estimation; the coefficient for Segment 2 is significantly negative at the 1 percent level for the binomial logit for claim. In the multinomial logit estimation, the likelihood-ratio test statistic for the four 22
  25. 25. coefficients in both prepayment and claim risks is 65.47 (chi—squared with four degrees of freedom under the null). Mortgage and Property Characteristics. The size of the mortgage is measured by the units in a property. A quadratic function of the units is used in the estimation equation. In both models, higher prepayment and claim rates are associated with properties that have more units. Mortgages in the Tandem program do not demonstrate significantly different performance in the claim equation, although the prepayment rate is significantly higher (at the 1 percent level) for Tandem loans. Lastly, the performance of the dummy for originations in the 1981-1986 period is strong. As hypothesized, claims are significantly higher during this period. Indeed, experimentation with dummy variables for each of these years revealed the same strong pattern between claims and books of business 5'; in the early l980s. Prepayments among these books of business are also higher than for others. Further investigation of mortgages from these books of business is a high priority for future research. Regions. We include indicators for regions of Northeast, Midwest and West regions in the estimation. The omitted region is the South region. The Northeast has the highest claim rate, while prepayment rates for the included regions are significantly higher (at the 5 percent level) than in the South. Duration Dependence. For both claim and prepayment equations, duration dependence is incorporated as a third degree polynomial in current duration. Each of the terms in the polynomial is significant for the claim risk and insignificant for prepayment. We plot a baseline claim and prepayment hazard in Figure 4. The baseline scenario is set for an average property with 13 apartment units, located in the central city of the Northeast region. The loan is not originated between 1981 and 1986, and the property does not participate in the Tandem program. 23
  26. 26. The call—option value is set equal to zero and unexpected household growth rate is 0.08 percent. The claim hazard is relatively stable for the first 20 years in the vicinity of 1 percent to 3 percent. After year 20, the claim hazard increases steadily and exceeds 9 percent in year 30. The prepayment hazard increases slowly and peaks in year 25 at around 2.3 percent. Vll. Application: Computing the Cost of FHA Insured Multifamily Loans The estimation results are used to address a particular policy issue: what is the likely default cost associated with FHA’s unassisted Section 221(d)(4) mortgage insurance program? We evaluate the various claim and prepayment equations for three different scenarios for rental growth and the vacancy rate. Although our computations do not allow an exact comparison to current policy, the analysis offers several insights about the assessment of these costs using the current approach. In addition, the analysis demonstrates two more general points: i) the asymmetric response of claims to changes in rent and vacancies; and ii) the results based upon the multinomial approach for this policy application are virtually identical to those obtained using binary iogit estimation. Under current policy, attention is focused upon the credit—subsidy rates associated with each of FHA’s mortgage insurance programs.8 A positive subsidy program, such as the Section 221(d)(4) program, requires an armual appropriation from Congress to cover the expected losses that are not covered by insurance premiums. FHA assumes that the future performance (claim rates, loss rates, etc. ) of this program will be the same as its historical average performance. Our analysis will be similar to that performed by FHA, except that the conditional claim and prepayment rates will be based on econometric models. Our numbers are not directly comparable to existing policy because our analysis pertains to the loans with no Section 8 assistance, whereas 24
  27. 27. the actual credit subsidy assigned to this program (7.12 percent) takes account of properties that receive Section 8 assistance in the form of Loan Management Set Aside (LMSA) assistance. We forecast the cash flows by using the estimated models to compute the present value of claim costs for an average loan that experiences a typical or average path of rental growth (3 percent per year) and average vacancy experience (6.80 percent). The application examines the loss associated with an average loan. The property contains 13 units and is located in {hag central city e>l~, t'-in the Northeast region. The loan is not originated in the 1981 to 1986 period, and the loan is not part of the Tandem program. Interest rates do not change and, therefore, all the call-option values are set to zero. The recovery rate is assumed to be 55 percent (Abt Associates Lg_c_. J999, p. 7) and occurs three years after the claim. Loans are assumed to be fixed rate 40-year mortgages with an interest rate of 7 percent. We assume a discount rate equal to 6 percent.9 All of the costs are expressed as a percent of the original loan amount and are contained in the first column of Table 6. Application of the mean scenario to the multinomial logit model with “flexible” specification estimates implies a positive credit—subsidy rate of 2.8 percent. This means that for every $100 of insurance, FHA must provide $2.88&j80 of additional capital to keep the insurance fund actuarially sound. Although this is less than half the number used for current policy, we are unable to critique the current policy (7.12 percent) as being too high Without more information and study of the cost of the LMSA program. Such an analysis is beyond the scope of this study. Computation of the credit—subsidy rate using the same approach with the other two models provides insights about their applicability. The estimate based upon the binary logit model produces almost the exact same estimate (2.85 percent), which suggests that assuming the independence of irrelevant alternatives assumption required by the multinomial logit is 25
  28. 28. reasonable for this application. However, the estimate based upon the multinomial logit model with the DCR—probability measure is much different (-0.53 percent). '° Taken together with the Schwarz Qeriterion outcome, we interpret the difference between the “flexible” model results and the “structural” model results as supportive of the flexible functional form approach because it appears to capture the nonlinearity of the default function better than the “structura ” approach. Strong confirmation of the option-pricing model also emerges by conducting the same exercise for two different scenarios—one above the mean scenario and one below the mean scenario. The good or “up” scenario is the one in which the annual rent growth is set equal to 5 percent and the vacancy rate equal to those experienced by the 75”‘ percentile property, 5.4 percent. The bad or “down” scenario corresponds to 1 percent annual rent growth and tlge vacancy rate experienced by the 25"‘ percentile property, 8.77 percent. For the mean scenario, the annual rent growth is 3 percent and the vacancy rate is 6.8 percent. The option-pricing model predicts that the increase in defaults costs (and hence the credit—subsidy rate) for the down scenario relative to the mean path will be larger than the decrease in default costs for the up scenario relative to the mean path. In fact, the estimates strongly support such an outcome. The estimate for the down scenario based upon the multinomial logit model with the “flexible” specification is 18.90 percent, which is 15 percent higher than the cost along the mean path. The reduction in the credit—subsidy rate is less than one-half of this (from 2.88 to -5.23 percent). Obviously, simulating the cost of FHA insurance along the mean path understates the average cost of FHA insurance among a wide variety of paths around the mean path. For example, if we assign 50 percent to the mean path and 25 percent to the other two scenarios, the credit-subsidy rate would be about 2 percentage points higher. Finally, we use the “flexible” multinomial logit model to compute the cost of the FHA program using an alternative method for reserving against claims. This alternative approach is 26
  29. 29. known as the stress-test approach and is the one used by OFHEO in its analysis of the GSE lending programs. This is done by evaluating the econometric models for a stressful economic scenario. In our application, we use a ten-year path in which rent growth is 1 percent and the vacancy rate corresponds to the 25"‘ percentile experience, md8.77 . The estimate of the credit—subsidy rate in this case is 12.98 percent, which is over four times higher than an estimate based upon current FHA policy. Although this result is specific to the scenarios we employ, the finding is likely to be robust over most reasonable definitions of a stress-test scenario using our model. This holds because our model, like the option approach, places a severe penalty on scenarios in which the put option is in the money and most likely to be exercised. Vlll. Summary and Conclusions The purpose of this paper is to provide additional information about the performance of FHA~insu. red multifamily mortgages. We are specifically interested in the prepayment and claim rates associated with these mortgages because such information is vital to the accurate pricing and assessment of FHA-mortgage insurance. Although the computerized subset of the theoretically available infonnation on FHA multifamily mortgages is deficient on several counts, we believe these data offer one of the best available opportunities to study multifamily loan performance over the past 30 years. Estimates of a competing-risk hazard model of multifamily claim and prepayment are provided. Anaiysis of the coefficient estimates suggests a number of conclusions regarding the specification of the hazard models for prepayment and claims. First, the estimates confirm the interest-rate sensitivity of mortgage prepayments and the nonlinearity predicted by the option- 27
  30. 30. pricing model for mortgages in the money. Furthermore, prepayments are positively related to measures of the economic strength of the property—rental income and vacancies; higher rates of rent growth and lower levels of vacancies reduce prepayments, all else equal. Second, claims are positively and nonlinearly related to the growth in rents and vacancies since origination. We view this as confirmation of the importance of local market conditions on multifamily mortgage performance. Furthermore, the effects of these variables upon claims are highly nonlinear, as the option-pricing model predicts. Interest-rate movements appear to have much smaller impacts upon claims. Several conclusions also emerge when the models are used to compute the federal credit- subsidy rate for the 221 (d)(4) FHA multifamily insurance program. First, the competing-risk estimate of the subsidy rate using this model is nearly identical to the estimate obtained using the single-risk model. This suggests that the added assumptions required by the “efficient” multinomial logit estimation are not violated in the data. Second, the results obtained using a more flexible functional form appear to dominate those based upon a single probability specification used by Goldberg (1998). The credit—subsidy rates obtained using the flexible functional form are substantially higher for both the mean path and the stress-test path than the estimates obtained using the single probability specification. It is possible that the flexible approach picks up the nonlinearities in the default equation better than the single probability specification. Third, the analysis highlights the likelihood that the current approach used by FHA to assess its credit-subsidy rate iilée-l-y-understates the true cost of the pro gram. 28
  31. 31. Endnotes 1. Throughout this paper the prepayment hazard is defined as the probability of prepaying a mortgage in year t (t less than scheduled maturity) conditional on the fact that it has survived (payments proceeded according to schedule) until year t. The claim hazard is defined similarly as the probability that a mortgage will result in an insurance claim in yeart conditional on survival until year t. I 2. The research incorporated in this paper is based upon a larger project between Price Waterhouse and the U. S. Department of Housing and Urban Development (HUD) to provide services helpful in the analysis of its various mortgage insurance programs. Some of the analyses presented in this paper were used with the most recent enhancements to the Government National Mortgage Association’s (Ginnie Mae) Policy and Financial Analysis Model (PFAM). Ginnie Mae and Price Waterhouse developed the PFAM principally to demonstrate compliance with the Credit Reform Act of 1990. Ginnie Mae satisfies the gAct’s requirements for monitoring liabilities associated with government- I backed credit programs with the PFAM, a cell-based simulation model that enables Ginnie Mae to assess its financial condition under a variety of economic and policy scenarios. The PFAM combines several econometric models with a financial model. The econometric models forecast the performance of loans that collateralize securities guaranteed by Ginnie Mae. The financial model simulates the effect of loan performance on the issuer/ servicers that issue Ginnie Mae-backed securities and service the underlying mortgages and the cash flows received by Ginnie Mae. This paper presents a portion of the multifamily econometric model, which is relevant to a variety of issues related to the pricing of mortgage insurance. 3. Follain and Szymanoski (1995) provide an overview of many of these issues. Crews, Dunsky, and Follain ( , Segal and Szymanoski (1997), Wachter et al. I (1996), provide information about the role of the GSEs in the primary and secondary markets for multifamily mortgages and the impact of the recently enacted mandates. 4. Kau and Keenan (l-9961995) and Vandell (+99-71995) offer recent surveys of the large I academic literature on the valuation of single-family mortgages. 5. An exception is the Ginnie Mae Tandem program. During the high interest rate period of the late 197015 and early 198033 below market-rate mortgages with FHA insurance were issued under the market; —rate program at 7.5 percent. The mortgages were sold on the secondary market, with Ginnie Mae making up the difference between the market and below market rates. We do not know which mortgages took part in this program, but we assume that all mortgages originated during this time with a 7.5 percent interest rate were Tandem program loans. 6. Note that the right—most segment is actually a ray starting at as_, , and therefore ax can be either set equal to infinity or defined to be the maximum value of the proxy. 29
  32. 32. 10. Experimentation with several versions of this approach yielded essentially the same results. The credit-subsidy rate is the present value of claims less premiums and recoveries on claims as a percentage of the loan disbursements. The credit-subsidy rate is used to calculate the total subsidy cost of the Feeler-alafederal goverii1nent’s direct I loan and loan guarantee programs as required by the Federal Credit Reform Act of 1990 and implemented through guidance found in Statement of Federal Financial Accounting Standards No. 2, Accounting for Direct Loans and Loan Guarantees, OMB Circulars A- 11 Preparation and Submission of Budget Estimates and A-34 Instructions on Budget Execution. Additional inputs include a 0.30 percent application fee, a 0.50 percent up-front premium, a 0.50 percent annual premium and a two-year construction period prior to the start of amortization. The difference is even more striking for the down scenario (see below in the text for an explanation)———18.90 percent for the flexible specification versus 10.70 percent for the structural specification. 30
  33. 33. Table 1. Variable Definitions Variable Description Prepayment (Call) Option Squared Positive Call The square (multiplied by 1,000) of the positive ratio of the difference between the Option Value market value of the mortgage and the book value of the mortgage (adjusted for 1 percent transaction costs) to the market value of the mortgage, 0 otherwise. Linear Positive Call The positive ratio (multiplied by 1,000) of the difference between the market value Option Value of the mortgage and the book value of the mortgage (adjusted for 1 percent transaction costs) to the market value of the mortgage, 0 otherwise. Squared Negative Call The square (multiplied by 1,000) of the negative ratio of the difference between the Option Value market value of the mortgage and the book value of the mortgage (adjusted for 1 percent transaction costs) to the market value of the mortgage, 0 otherwise. Linear Negative Call The negative ratio (multiplied by l,000) of the difference between the market value Option Value of the mortgage and the book value of the mortgage (adjusted for 1 percent transaction costs) to the market value of the mortgage, 0 otherwise. Housing Price Proxies Rental Price Growth Rate Rental Price Growth Rate The first segment of the cumulative rental growth rate in the residential rent Spline, Segment 1 component of the CPI for the market area of the loan, measured as a percentage. This is the segment below the 25 percentile of the cumulative rental price growth rate. Rental Price Growth Rate The second segment of the cumulative rental growth rate in the residential rent Spline, Segment 2 component of the CPI for the market area of the loan, measured as a percentage. This is the segment between the 25 and the 50 percentile of the cumulative rental price growth rate. Rental Price Growth Rate The third segment of the cumulative rental growth rate in the residential rent Spline, Segment 3 component of the CPI for the market area of the loan, measured as a percentage. This is the segment between the 50 and the 75 percentile of the cumulative rental price growth rate. Rental Price Growth Rate The fourth segment of the cumulative rental growth rate in the residential rent Spline, Segment 4 component of the CPI for the market area of the loan, measured as a percentage. This is the segment over the 75 percentile of the cumulative rental price growth l'2lt€. Vacancy Rate Vacancy Rate Spline, The first segment of the vacancy rate in the market area of the loan, measured as a Segment 1 percentage. This is the segment below the 25 percentile of the vacancy rate. Vacancy Rate Spline, The second segment of the vacancy rate in the market area of the loan, measured as a Segment 2 percentage. This is the segment between the 25 and the 50 percentile of the vacancy rate. Vacancy Rate Spline, The third segment of the vacancy rate in the market area of the loan, measured as a Segment 3 percentage. This is the segment between the 50 and the 75 percentile of the vacancy rate.
  34. 34. Variable Table 1. Continued Description Unexpected Growth in Households Unexpected Growth Households Spline, Segment 1 Unexpected Growth Households Spline, Segment 2 Structural Variable Probability that DCR is less than 1.0 The first segment of the difference (multiplied by 100) between the annual household growth rate and the historical trend of household growth at the MSA level, measured as a percentage. This is the segment below the 50 percentile of the unexpected household growth rate. The second segment of the difference (multiplied by 100) between the annual household growth rate and the historical lrend of household growth at the MSA level, measured as a percentage. This is the segment above the 50 percentile of the unexpected household growth rate. Predicted probability used by Capone and Goldberg (1998). Mortgage and Property Characteristics Units Units Squared Central City Tandem 1981 — 1986 Origination Regions Northeast Midwest West South Durations Current Duration Cubed Current Duration Squared Current Duration The number of apartment units in a property. The square of the number of apartment units in a property. A dummy variable equal to 1 if the property is located in a central city, 0 otherwise. A dummy variable equal to 1 if the mortgage originated between 1975 and 1981 and had a contract rate of 7.5 percent, 0 otherwise. A dummy variable equal to 1 if the mortgage originated between 1981 and 1986, 0 otherwise. A dummy variable equal to 1 if the property is located in the Northeast region, 0 otherwise. A dummy variable equal to 1 if the property is located in the Midwest region, 0 otherwise. A dummy variable equal to 1 if the property is located in the West region, 0 otherwise. A dummy variable equal to 1 if the property is located in the South region, 0 otherwise. Current duration of mortgage cubed, divided by 1000 (years cubed I 1000). Current duration of mortgage squared, divided by 10 (years squared / 10). Current duration of mortgage (years).
  35. 35. Table 2. Summary Statistics for Non—Section 8 OMI Data Means and Standard Deviations All Years First Year §N = 37,557} §N = 2,7471 Standard Standard Variable Mean Deviation Mean Deviation Prepayment (Call) Option Squared Positive Call Option Value 1.810 6.666 1.581 5992 Linear Positive Call Option Value 14.755 39.908 16.438 36.210 Squared Negative Call Option Value 24.257 49.387 7.653 60.466 Linear Negative Call Option Value -1 13.514 106.640 -39.601 78.019 Housing Price Proxies Rental Price Growth Rate” Rental Price Growth Rate, Segment 1 0.144 0.055 ——— ——- Rental Price Growth Rate, Segment 2 0.147 0.104 ——— ——— Rental Price Growth Rate, Segment 3 0.139 0.141 —~— ——- Rental Price Growth Rate, Segment 4 0.195 0.339 ——— -—- Vacancy Rate” Vacancy Rate, Segment 1 5.041 0.787 4.929 0.893 Vacancy Rate, Segment 2 0.792 0.623 0.634 0.603 Vacancy Rate, Segment 3 0.495 0.635 0.342 0.569 Vacancy Rate, Segment 4 0.367 1.046 0.247 0.841 Unexpected Growth in HouselzaIds° Unexpected Growth in Households, 0004 0.007 0.008 0.01 1 Segment 1 Unexpected Growth in Households, -0.003 0.006 -0.002 0.004 Segment 2 Structural Variable Probability that DCR is less than 1.0 0.286 0.148 0.513 0.029 Mortgage and Property Characteristics Units 12.532 9.652 13.239 10.119 Units Squared 250.210 693.601 277.634 778.334 Central City 0.519 0.500 0.542 0.498 Tandem 0.228 0.419 0.197 0.398 1981-1986 Origination 0.177 0.382 0.230 0.421 Regions Northeast 0.317 0.465 0.297 0.457 Midwest 0.076 0.265 0.083 0.276 West 0.259 0.438 0.249 0.432 Duration Dependence Current Duration Cubed 8.971 5.819 1.000 0.000 Current Duration Squared 1 1.434 12.73 1 0.100 0000 Current Duration 1.737 2.659 0.001 0.000 “Spline-segment endpoints are defined by the extremes and quartile values for this variable: min = a0 = -0.031; a, = 0.172; a, = 0.403; a, = 0.703; and max = a4 =3.255. bSp1ine-segment endpoints are defined by the extremes and quartile values for this variable: a0=0; min = 0.500; al = 5.4; (12 = 6.775; as = 8.225; and max = a4 = 23.6. “Spline-segment endpoints are defined by the extremes and median for this variable: min = a2 = -0.927; a, = —0.0009; and max = no = 0.086.
  36. 36. Table 3. Multinomial Logit Results: “Flexible Approach” (Relative to Utility of Continuing with Mortgage) Claim Eguation Prepayment Eguation Standard Standard Variable Coefficient Deviation Coefficient Deviation Constant -10.634 1.253 -7.846 0.614 Prepayment (Call) Option Squared Positive Call Option Value -0.082 0.022 -0.012 0.013 Linear Positive Call Option Value 0.015 0.004 0.013 0.003 Squared Negative Call Option Value -0.040 0.009 0.000 0.002 Linear Negative Call Option Value -0.010 0.003 0.000 0.001 Housing Price Proxies Rental Price Growth Rate Rental Price Growth Rate, Segment 1 -7.775 1.926 2.133 3.949 Rental Price Growth Rate, Segment 2 -8.328 1.191 3.916 1.638 Rental Price Growth Rate, Segment 3 -1.791 0.966 2.785 0.944 Rental Price Growth Rate, Segment 4 -1.312 0.538 1.576 0.268 Vacancy Rate Vacancy Rate, Segment 1 0.575 0.246 -0.035 0.085 Vacancy Rate, Segment 2 0.809 0.201 0.130 0.114 Vacancy Rate, Segment 3 0.279 0.124 -0.127 0.102 Vacancy Rate, Segment 4 0.079 0.031 0.003 0.048 Unexpected Gro wt]: in Households Unexpected Growth in Households, -13.255 10.775 -18.361 9.757 Segment 1 Unexpected Growth in Households, -48.232 6.453 9.276 7.348 Segment 2 Mortgage and Property Characteristics Units Units Squared Central City Tandem 1981-1986 Origination Regions Northeast Midwest West Duration Dependence Current Duration Cubed Current Duration Squared Current Duration Likelihood Function 0.027 0.000 0.285 -0.042 0.722 0.605 0. 149 -0.099 0.701 -0.364 0.721 0.010 0.000 0.107 0.179 0.128 0.218 0.140 0.157 0.130 0.111 0.276 0.012 0.000 -0.187 0.448 1.207 0.359 0.322 0.5 18 0.084 0.023 -0.154 -4945 .544 0.007 0.000 0.080 0.124 0.175 0.177 0.124 0.134 0.176 0.116 0.241
  37. 37. Table 4. Multinomial Logit Results: “Structural Approach” (Relative to Utility of Continuing with Mortgage) Claim Equation Prepayment Equation Standard Standard Variable Coefficient Deviation Coefficient Deviation Constant —l0.647 0.415 -7.172 0.584 Prepayment (Call) Option Squared Positive Call Option Value -0.100 0.022 -0.008 0.013 Linear Positive Call Option Value 0.019 0.004 0.01 1 0.003 Squared Negative Call Option Value -0.034 0.009 -0.002 0.003 Linear Negative Call Option Vaiue -0.008 0.003 -0.002 0.001 Housing Price Proxies Probability that DCR is less than 1.0 8.980 0.595 -2.348 0.744 Unexpected Growth in Households, -20.794 10.872 -19.2041 9.753 Segment 1 Unexpected Growth in Households, -45.754 6.281 8.401 7.284 Segment 2 Mortgage and Property Characteristics Units 0.016 0.009 0.011 0.007 Units Squared 0.000 0.000 0.000 0.000 Central City 0.375 0.106 -0.181 0.080 Tandem -0.264 0.171 0.369 0.120 1981-1986 Origination 0.501 0.122 0.980 0.164 Regions Northeast 0.206 0.211 0.572 0.164 Midwest -0.033 0.126 0.256 0.113 West -0.069 0.149 0.869 0.120 Duration Dependence Current Duration Cubed 0.433 0.078 0.249 0.108 Current Duration Squared -0.203 0.076 0.004 0.078 Current Duration 0.411 0.213 -0. 103 0.179 Likelihood Function -5005 .107
  38. 38. Table 5. Binary Logit Results: “Flexible Approach” Claim Equation Prepayment Eguation Standard Standard Variable Coefficient Deviation Coefficient Deviation Constant -10.615 1.254 -7.848 0.614 Prepayment (Call) Option Squared Positive Call Option Value -0.082 0.022 -0.012 0.013 Linear Positive Call Option Value 0.015 0.004 0.013 0.003 Squared Negative Call Option Value -0.039 0.009 0.000 0.002 Linear Negative Call Option Value -0.010 0.003 0.000 0.001 Housing Price Proxies Rental Price Growth Rate Rental Price Growth Rate, Segment 1 -8.003 1.930 2.279 3.909 Rental Price Growth Rate, Segment 2 -8.350 1.193 3.952 1.640 Rental Price Growth Rate, Segment 3 -1.886 0.968 2.777 0.945 Rental Price Growth Rate, Segment 4 -1.313 0.53 8 1.5 62 0.269 Vacancy Rate Vacancy Rate, Segment 1 0.571 0.246 -0.034 0.085 Vacancy Rate, Segment 2 0.812 0.201 0.129 0.1 14 Vacancy Rate, Segment 3 0.278 0.124 -0.130 0.102 Vacancy Rate, Segment 4 0.079 0.031 0.004 0.048 Unexpected Growth in Households Unexpected Growth in Households, -12.909 10.764 -17.945 9.783 Segment 1 Unexpected Growth in Households, -49.052 6.443 9.336 7.368 Segment 2 Mortgage and Property Characteristics Units 0.027 0.010 0.012 0.007 Units Squared 0.000 0.000 0.000 0.000 Central City 0.276 0.107 -0.186 0.081 Tandem -0.042 0.179 0.453 0.124 1981-1986 Origination 0.723 0.128 1.207 0.176 Regions Northeast 0.599 0.218 0.357 0.177 Midwest 0.150 0.140 0.321 0.125 West -0099 0.157 0.520 0.135 Duration Dependence Current Duration Cubed 0.710 0.130 0.074 0.175 Current Duration Squared -0.366 0.111 0.031 0.115 Current Duration 0.719 0.275 -0. 169 0.240 Likelihood Function -2003.695 -2933.843
  39. 39. Table 6. Credit Subsidy Rates Over 40 Years Mean Up Scenario Scenario Multinomial Logit (Flexible Specification) 2.88 -5 .23 Multinomial Logit (Structural Specification -0.53 -4.46 Using Probability Measure) Bivariate Logit (Flexible Specification) 2.85 -5.23 Down Scenario 18.90 10.70 19.01
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