While the overall goodness-of-fit of equation (11) is very low, our interest is only in the marginal influence of interest rate changes, as reflected by the coefficient on In r

Time Dimension

Individual projects are most vulnerable to economic shocks in the early years after loan origination because DCR, is typically at its lowest level and LTV, is relatively high. However, a troubled project may not default immediately for a number of reasons. First, there may be large and valuable depreciation writeoffs that we are not capturing in PVTAX. Also, working-capital cushions required at loan origination provide a shield from default if there are temporary dips in occupancy rates. Furthermore, it may be optimal to "bleed the project" through deferred maintenance and other expenditures for a few years prior to default (Quercia, 1995). Such bleeding can be extended through nonpayment of the mortgage for between 12 and 24 months, the time it takes to foreclose on a defaulted property (Riddiough and Thompson, 1993). For all of these reasons, we expect default rates to rise slowly at first and then peak after a sizable period of time. Preliminary data analysis suggests that the peak default period is between 6 and 7 years from loan origination on new properties. To model this relationship, we use a quadratic function of time since origination.

Underwriting Changes

DCR and LTV are adjusted for loans underwritten in earlier time periods to make them consistent with current practice, which uses actual values of project rents and expenses, and a conservative (high) vacancy factor. Our approach is first to reduce NOI on early originations by 15 percent and then to add a shift variable (OLDRITE) to capture the lower level of due diligence at loan origination. The 15-percent adjustment factor is the midpoint of the range mentioned earlier. It captures a combination of historical rent trending (adding 1 year of inflation to rental income), assumptions of immediate and continued full occupancy, and the disregard for capital replacement reserves when computing expense ratios.

Data Sources

The mortgage sample for this study includes multifamily cash purchases acquired by Fannie Mae and Freddie Mac between 1983 and 1995. The original sample includes 14,211 loans. However, after deletion of observations with missing variable values and restricting loans to cities for which rental market data are available, the final estimation sample includes 7,564 loans, representing 52,222 loan-years.16 Default in this study refers to a forfeiture of property rights, so it includes foreclosure, third-party sale, note sale, and short sale events.

Data on apartment vacancy rates (from Bureau of the Census, Housing Vacancy Surveys) and rent growth (from the consumer price index (CPI) residential rent series) are for 28 MSAs for which government data are available back to 1983. The CPI residential rent series is used to compute the RPI, indices for each loan.17

The distribution of the sample (loan-years) by MSA is presented in exhibit 3, which also gives the average annual growth rate of rents and average annual vacancy rates throughout the observation period, 1983–95. Annual growth rates of rents over the entire period ranged from a low of 2.1 percent in Houston to a high of 4.6 percent in New York City. These cities were also the polar extremes in terms of average vacancy rates throughout this time period: 13.41 percent for Houston and 3.78 percent for New York.

Sources: Rental growth rates are from the Bureau of Labor Statistics CPI index, rental cost component. Vacancy rates are from the Bureau of the Census, H-111 series.

Annual averages for default rates and key variables by calendar (exposure) year are given in exhibit 4. As mentioned earlier, the default rate rises from 1983 to 1993, and then declines from 1993 to 1995. During this period, there are increases in vacancies and the average age of the portfolio increases from 1 to 7 years. There is also a sharp decline in depreciation writeoffs in 1987. Until 1988, the portfolio consists entirely of OLDRITE loans; then the mix starts improving and, by 1995, better underwritten loans account for about one-third of the portfolio. The regression analysis will measure the effects of risk factors and provide a framework for explaining the contribution of various factors to the trend in default rates.

Statistical Model Specification

Investor wealth maximizing choices can be modelled in a logistic regression. The logistic model estimates choice probabilities based on a linear wealth (indirect utility) function, which in our case is:

where

Wi.m.

i, m, y, t

= B12+ B, (1 /DCR;, m. y.) + B2LTV. m. y. + BPVTAX ̧y.t

++ẞ, OLDRITE + B2t+ B ̧t2 + H., m. y. t

= borrower,

y

t

(14)

y,

This equation is estimated with maximum likelihood techniques on a cumulative logistic probability function:

The one change made here for purposes of the statistical model is to use the inverse of DCR, as the parameter of interest. The reason for this is to provide a similar scale and interpretation of coefficients on DCR and LTV at the trigger points: probabilities of default increase as both 1/DCR and LTV increase.19

We assume that the relevant choice here is binary: to default or not to default. Because of the prevalence of yield maintenance clauses, or outright prepayment lockouts on sampled loans, we do not attempt to estimate a joint decision function for defaults and prepayments. The model can be thought of as a discrete hazard function, in which loans are tracked from acquisition to default or to a censoring point defined by payoff or the end of the sample period, 1995.

Estimation Results

All coefficient signs are correct and indicate statistically significant effects (see exhibit 5). The effects of DCR and LTV are additive, indicating that default risk increases sharply for loans with deteriorating underwriting ratios.20 Exhibit 5 also provides mean values and marginal probability estimates for each variable (evaluated at sample means). These show the slightly stronger effect of LTV over DCR. The baseline hazard curve (constants plus

b

The constant term is deleted.

⚫ Effect of a one-unit change in an explanatory variable on probability of default. Computed at mean values of all variables.

time and time-squared) is also strong, verifying the underlying time dimension of credit risk discussed above.

The 1987 tax-change shock to PVTAX implies a roughly 50-basis-point increase in annual default rates on multifamily mortgages (0.0004 marginal probability times the 13-point drop). OLDRITE measures changes in underwriting practices not captured by adjustments to DCR and LTV. The mean-values marginal probability of 30 basis points suggests that improvements in due diligence at loan underwriting have had a substantial effect on default risk. The effects of changes in tax laws and underwriting are fully discussed in the next section, where we simulate default risk under different scenarios.

Simulations and Issues for Today

We have built a statistical model that allow us to decompose economic and tax-law effects on the default risk of multifamily mortgage loans. This double-trigger model of multifamily loan default allows for updating of DCR and LTV over time and for measuring default rates directly from these variables. In the 1983-92 period, market default rates steadily increased. In our simulations, we track how these movements can be attributed to declines in tax benefits and rental market conditions and to the aging of poorly underwritten loans made in the mid-1980s. We show that defaults would have been greater had it not been for declines in interest rates that increased the value of project cash flows.