Renter Mobility and Multifamily Mortgage Default Risk

Journal of Housing Economics 10, 21–40 (2001)

doi:10.1006/jhec.2001.0277, available online at http://www.idealibrary.com on

Renter Mobility and Multifamily Mortgage Default Risk1

Charles A. Capone, Jr.

Congressional Budget Office, Ford House Office Building, Rm 489, Washington, DC 20515E-mail: [email protected]

and

Lawrence Goldberg

Institute for Defense Analyses 1801 N. Beauregard Street, Alexandria, Virginia 22301E-mail: [email protected]

Received January 18, 2000

Renter mobility is a major concern for the performance of multifamily mortgages. Ifenough new renters are not found to replace those that move, vacancy rates can quicklyescalate to where cash flows are negative and property mortgages are in jeopardy. In thisstudy we examine how differences in renter mobility patterns by property type can affectmortgage credit risk within submarkets of an MSA. We expand the default model developedby Goldberg and Capone (2000) to use unique distributions of rental unit turnover andvacancy durations for large and small multifamily rental properties. Monte Carlo simula-tions then show how credit risk on multifamily mortgages is affected if owners of smallproperties are able to keep tenants longer than owners of larger properties can. Themodel can be used to explore other potential intra-MSA differences in property marketdynamics. q 2001 Academic Press

I. INTRODUCTION

The real estate literature has rich traditions of analyzing the relationship be-tween vacancy rates and rental prices and the relationship between vacancy ratesand underlying tenant mobility rates.2 These two strands of literature are connected

1The views expressed in this paper are those of the authors and should not be interpreted as thoseof the Congressional Budget Office or the Institute for Defense Analyses.

2Blank and Winnick (1953) are generally credited with first empirical analysis of the relationshipbetween vacancy rates and rental proces. Another seminal study was performed by Rosen and Smith(1983), and a good summary of the literature can be found in Sivitanides (1997). Belsky and Goodman(1996) provide an analysis of why some empirical studies fail to find the hypothesized negativerelationship between vacancy rates and rental prices. Seminal work in the area of rental mobilityand turnover rates was conducted by Rydell (1979). More recent work was performed by Belsky(1992a), (1992b).

21

1051-1377/01 $35.00Copyright q 2001 by Academic Press

All rights of reproduction in any form reserved.

22 CAPONE AND GOLDBERG

through the idea of a natural vacancy rate that signifies market equilibrium. Morerecently, researchers have begun to explore a dynamic relationship between avacancy–rental-price nexus and the credit risk of multifamily property mort-gages.3 In this study, we draw together the housing mobility, vacancy, and credit-risk literatures by analyzing the impacts of variations in household mobilityrates on mortgage credit risk. Simulations are used to show potential credit riskdifferences between small and large multifamily properties, based on differencesin turnover rates found in the Property Owners and Managers Survey (POMS).

Multifamily mortgages are unique among mortgage types in that they areexposed to risk from relatively short-term leases and high rates of mobility amongrenters–tenants. High rates of renter turnover mean that multifamily propertiesmust constantly court new customers to survive. Data compiled by Belsky (1992a)from the 1980 Census show that renter mobility rates range from 20 to 55%across metropolitan statistical areas (MSAs). In a study of mobility rates in the1970s using the Michigan Panel Study of Income Dynamics data, Ioannides(1987) found a national average rate of 37% for renters versus just 11% forhomeowners. Downs (1983, p. 26), analyzing 1980 American Housing Survey(AHS) data, reports that 37% of renters were recent movers (last 12 months),while only 8% of homeowners were recent movers. Recent AHS numbers (1997)show identical mobility rates as those recorded in 1980.4 Work by Israeli andNelson (1992) on 1985 and 1987 AHS data indicate a one-year mobility rate of42% for renter households. The upshot of all of these studies is that renterhouseholds have significant mobility rates, which implies that managing turnoverrates is an important task in apartment risk management.

Many researchers have studied what a “natural” vacancy rate might be, in anygiven market. This natural rate reflects vacancy durations landlords are willingto accept, given turnover rates, in order to maximize profits through rental priceincreases. The point of this literature is that deviations in real prices of rentalunits are a function of deviations in vacancy rates around natural rates. Anessential contribution of Rosen and Smith (1983) was to show that rents andvacancies do have a negative correlation, but that the relationship is only seenin cross-section data when researchers control for differences in MSA levelnatural vacancy rates. Hendershott and Haurin (1988) provide a nice summaryof the theory of natural vacancy rates and rental price adjustments.

Researchers have long noted that vacancy rates alone are an insufficient statisticfor understanding the dynamics of rental market equilibrium (see Rydell, 1978).This is because turnover rates underlying those vacancy rates vary considerablyacross MSAs. Changes in vacancy rates within an MSA are best understood bylooking at changes in vacancy durations. Durations can have both supply- and

3See Goldberg and Capone (1998), (2000).4See summary tables 1A-1 and 2-11. These can be found at http:/www.census.gov/hhes/www/

housing/ahs/ahs97/.

MOBILITY AND MORTGAGE CREDIT RISK 23

demand-side influences—from new construction becoming available for rent, tochanges in rent-ask prices, to changes in underlying household demographics.

If different property types have unique vacancy and/or rental price growthpotential, then they will have different credit risk profiles. For example, Downs(1983, p. 35) hypothesizes that holders of small rental properties tend to beturnover-minimizers. He cites other research that suggests landlords reduce rental-price increases below market levels to keep current residents. When taken to-gether, these points suggest that smaller properties should experience both lowerturnover rates and lower rental price increases. Guasch and Marshall (1985)confirm, with AHS data (1973–1979), that small properties (less than 5 units)have lower turnover rates (vacancy frequencies) and lower vacancy durations,implying lower vacancy rates.5 The increased risk born by large-property ownersis in increasing durations of vacancies when asking prices (rents) are increased.

II. CREDIT RISK MODEL

To link renter mobility differences to mortgage credit risk, we use the multifam-ily mortgage credit risk of Goldberg and Capone (2000).6 The centerpiece ofthat model is a variable that captures the joint probability of negative equity andnegative cash flow for each loan, during each observation period (annual). Equityis measured by loan-to-value ratios (LTV) and cash flow by debt coverage ratios(DCR). The joint probability variable is also a function of average LTV and DCRwithin an MSA, other mortgage terms, and assumptions about distributions ofproject-level vacancy rates and rental price growth, each around their MSAaverages, during each observation period. Project-level vacancy rates are assumedbinomial random variates, and rental price growth is assumed to arise from alognormal diffusion process.7 After some data transformations, Goldberg andCapone use a bivariate normal distribution to estimate the joint probability ofnegative equity and negative cash flow for each mortgaged property.

Here, we more closely examine the underlying distributions of project-levelvacancy and rental-price-growth rates to see how differences in these distributionslead to differences in credit risk across property types. In particular, we generatestochastic vacancy rates from distributions of turnover and duration. Rental price

5Guasch and Marshall (1985) look for reasons why this is true and believe the difference lies ina negative correlation between actual unit size and turnover rates. Large rental properties tend tohave smaller units than do smaller rental properties. However, in their regression analysis, theseresearchers compare 5–50 unit properties (“large”) with 1–2 unit properties (“small”). The latterwill be single-family homes and duplexes, which attract larger households with higher moving costs.

6Their work explains the default model found in the Notice of Proposed Rulemaking issued bythe Office of Federal Housing Enterprise Oversight, 64 Fed. Reg., 18192f, April 13, 1999.

7The binomial is approximated in the Goldberg-Capone study with a continuous, lognormal distribu-tion.


growth rates have a separate frequency distribution and are assigned to propertiesaccording to property vacancy rates. Using this structural framework, we decom-pose changes in mortgage credit risk into changes in underlying factors. Oncewe allow for unique distributions of the underlying factors, we can generateunique credit-risk profiles by property type within an MSA or across MSAs.This is done by using the numerical results of stochastic factor realizations tocalculate values for the joint-probability variable used in the Goldberg-Caponemortgage default model.

The particular research question we explore stems from the Danter (1997)hypothesis that large properties are more risky than small properties. Accordingto Danter, this is because with high renter mobility (turnover) rates, large rentalproperties must find large numbers of new renters each year—or each month.This requires that large properties have very large target market segments todraw from. Smaller properties, on the other hand, need smaller numbers of newrenters per year, even if they have the same mobility–turnover rates as do largerproperties. Thus, they can effectively survive in smaller target market populations.So, for any given rate of tenant turnover, larger properties may have highervacancy durations, which then lead to higher vacancy rates.

While the Danter hypothesis is appealing as a topic of study, data on vacancyduration by property size are not available. However, POMS, undertaken by theCensus Bureau in 1995, shows that turnover rates do vary by property size. WhilePOMS is just one, national snapshot, it provides some evidence that smallermultifamily properties may have lower average turnover rates—confirming theDowns (1983) hypothesis and leading to lower vacancy rates. The contrast be-tween large and small property turnover rates is seen in Table I, which is derivedfrom POMS summary table 23.8 We confine ourselves to exploring the implica-tions of the relationships found there.

POMS classifies multifamily properties into four groups: 5–9, 10–19, 20–49,and 50 or more units. There is virtually no difference between turnover rates inthe first two groups and little between the first two and the third. Thus wecombine the first two groups into a 5–10 unit group and also show a 5–49 unitgrouping to contrast with the 501 units group. We then classify “small” multifam-ily properties as under 50 units and “large” as more than 50 units. The last threecolumns of Table I show average per-percentile-unit turnover rates within eachof the rate categories denoted by the rows (none, less than 5 %, etc.). These helpto form the basis for our simulation model, and a graph of the relationships foundthere is provided in Fig. 1.

8This table is available from the Bureau of the Census at http://www.census.gov/hhes/www/housing/poms/multifam/mfmgmt/mftab23.html.


TABLE IFrequency Distribution of Turnover Rates from Property Owners and Managers Survey

Turnover rate per turnoverUnits in property rate category percentilea

Turnover ratecategories Total 5–19 20–49 5–49 501 Total 5–49 501

None 0.0558 0.1808 0.0429 0.1237 0.0154 0.0558 0.1237 0.0154Less than 5% 0.2399 0.2969 0.3386 0.3142 0.1957 0.0480 0.0628 0.03915 to 9% 0.1773 0.1519 0.1729 0.1606 0.1873 0.0355 0.0321 0.037510 to 19% 0.1766 0.1461 0.1815 0.1607 0.1860 0.0177 0.0161 0.018620 to 49% 0.2209 0.1476 0.1824 0.1620 0.2559 0.0074 0.0054 0.008550% or more 0.1295 0.0767 0.0818 0.0788 0.1597 0.0043 0.0026 0.0027Meansb 0.2014 0.1402 0.1620 0.1492 0.2324

Source. Author calculations from POMS summary Table 23. This table is available from the Bureauof the Census at http://www.census.gov/hhes/www/housing/poms/multifam/mfmgmt/mftab23.html.

a Turnover rate by category divided by size (width/range) of category. Last category (50% or more)is assumed to be 30 units long, which creates a maximum turnover rate of 79%.

b To calculate means, turnover rates within rate categories were assumed be the midpoint of therange. For the highest range class (50% or more) the turnover rate is assumed to be 60%.

III. Simulation Model

Our simulation model has five components, each of which will be describedbelow. These are:

• Generating random turnover rates, by property type;• Generating random vacancy durations within an MSA;• Calculating vacancy rates from turnover and durations;

FIG. 1. Frequency distribution of annual renter turnover rates by unit percentile. Source. Table I.


• Generating rental price growth rates within an MSA;• And calculating mortgage performance measures.

A. Turnover Rate Distributions

The distributions of annual turnover rates used here follow the patterns seenin the POMS data (see Fig. 1). To accomplish this, turnovers within each propertytype are generated as gamma random variates. For small properties, the shapeparameter equals one, which makes the distribution resemble an exponentialpattern, with the modal range being zero turnovers each year. For large properties,the shape parameter is set to two, which allows for a unimodal distribution witha lengthy (right) tail. Scale parameters for the gamma random variates are alsoset to allow for the relationship shown in the POMS data, where the meanturnover rate for large properties is twice that of small properties. The generationof these gamma random variates is explained further in Appendix A.

B. Duration Time Distributions

Durations of vacancy (percentage of a year) are also generated as gammarandom variates. The modal duration is fixed to be half of the mean, whichallows the shape and scale parameters to be derived solely from a user-input meanduration value.9 In the present version of the simulation model, no distinctions aremade by property type; each receives the same parameters for random numbergeneration. Durations are measured in percentages of one year. See Appendix Afor more detail.

C. Calculating Vacancy Rates

Properties within each group are assigned random turnover and durationrates which are then multiplied together to calculate property-specific vacancyrates:

nijt 5 tijt*dit (1)

where: nijt is the vacancy rate for property i, property type j, year t; tijt is theturnover rate, property i, property type j, year t, and tijt , G(bj,cj); and dit is thevacancy duration for property i, year t, and dit , G(bd,cd).

First year vacancy rates are adjusted to reflect that full occupancy at mortgageorigination generally means 6% vacant units. So, for year one, vacancy ratesfrom Eq. (1), vij1, are adjusted upward by (0.06?(dit/2)), the component of the

9There is no particular theoretical rationale for this relationship. But, for our base case scenario,with mean durations of 2 months, it suggests a modal duration of 1 month.


annual vacancy rate due to vacancies existing at the start of the year. The durationof these initial vacancies, dit , are divided by 2 because they are not new vacancies.The resulting, adjusted vacancy rate formula is:

nijt 5 nijt 1 0.06 ? 1dijt

2 2, dijt 5 H1 if t 5 10 otherwise.

(2)

In addition, to create serial correlation in vacancy series, we use a stock-adjustment process whereby the actual vacancy rate for each project is a weightedaverage of the current random variable and the lagged random variable:

Vijt 5 0.80 ? nijt 1 0.20 ? Vij,t21. (3)

Stock adjustment models are typically used to analyze long-run supply re-sponses to demand shocks in real estate markets.10 Our rationale for using it hereis to temper the stochastic processes underlying Vijt by indicating that propertieswith above- or below-normal vacancy rates may have management, location, orreputation effects that persist, to some degree, over time. We do not have anempirical basis for choosing a 20% weight for the lagged value and the simulationmodel can easily be changed to accompany other values.

D. Rental Price Growth Rate Distributions

Annual rental price growth rates by property (i), Rit , are assumed normallydistributed around an MSA mean growth rate, R1, with a standard error of 2 ?R0.11 Following the work of Sivitanides (1997), we allow the mean rental pricegrowth in time t to vary as mean property-type vacancy rates (Vjt) vary froma natural rate, here assumed to be 6.5%, the long-term national average ratefound by Goldberg and Capone (1998) in their dataset.12 Our adjustment coeffi-cient is 0.50, which is the modal value across MSAs studied by Sivitanides(1997).13

10See, for example, Eppli and Shilling (1995).11Goldberg and Capone (1999) use a standard error of 0.075, which is an intra-MSA, but intertempo-

ral average provided by the Bureau of Labor Statistics (BLS). Here, we provide somewhat moreflexibility by allowing the standard deviation to vary with the mean. For historical mean rental pricegrowth rates of around 4%, our formula here will provide a standard error slightly higher than thatreported by BLS.

12Other authors, such as Gabriel and Nothaft (1988), find wide variation in natural vacancy ratesacross MSAs, and others (see Sivitanides, 1997) find variations within MSAs over time.

13Sivitanides (1997, p. 205) reports 4 (of 19) values around 0.50. Because inflation is lower todaythan it was during the time period of Sivitanides’ study (1980–1988), we are hesitant to use anadjustment as large as the mean of his study, which we calculate to be 0.67 (median 5 0.63). Eventhough Sivitanides uses real rental prices, downward adjustments in real prices are likely to be afunction of inflation, as are downward adjustments in real wages. Like Sivitanides, Gabriel andNothaft (1988, Table 1) also find large elasticities of real rental price with respect to vacancydeviations (0.92) in a study of 16 MSAs, 1981–1985.


Rjt 5 R0 1 0.5 ? (0.065 2 Vjt) (4)

R0 can be interpreted as providing an expected MSA trend inflation rate forrents.

Property-specific rental price growth rates, Rijt , N(Rjt ,(2 ? R0)2), are drawnrandomly, sorted, and matched to properties in an ordinal fashion, according tovalues of Vijt. This is done separately within each property type, j. Thus, propertieswith the lowest vacancy rates are given the highest rental price growth rateswithin their type class and vice versa. This matching is performed for eachobservation year, and then property-specific rental price indices are calculatedfrom the growth rates.

E. Calculating Mortgage Performance Statistics

Vacancy and rental-price-growth rates are simulated for 1000 properties ofeach property type. The results are used to generate values of the financialindicator variables, DCRijt and LTVijt , using algorithms provided in Goldbergand Capone (2000). With these financial indicators, we compute the joint probabil-ity of negative cash flow and negative equity, JPjt , as the joint frequency ofDCRijt , 1.00 and LTVijt . 1.00, for each property type, j, in each year, t.Annual default probabilities, by property type, are calculated using JPjt , andother covariates described in Goldberg and Capone (2000). These covariates aremortgage age and the value of depreciation write-offs in property sale (valueto a new investor).14 Annual prepayment probabilities are generated with theexponential function used by Goldberg and Capone (2000).15 Annual, conditionalprepayment rates mimic those of 15-year balloon mortgages with 10-year yieldmaintenance terms, under stable interest rates.

One final adjustment to our stochastic vacancy rates, Vijt, is necessary to usethem in the Goldberg–Capone (2000) model. That model uses statistical equationswhose coefficients were estimated using Bureau of Census vacancy rates. Theserates combine single- and multifamily properties. Single-family rental vacanciestend to be lower than those on multifamily properties, while the vacancies wegenerate here are for multifamily only. A simple adjustment equation is con-structed by regressing Census 51 unit property vacancy rates on Census (total)rental vacancy rates:16

V cijt 5 exp{20.1096 1 0.919 ? ln(Vijt ? 100)}/100. (5)

14Age enters in quadratic form, and there is an additional effect for JPjt in the balloon maturity year.15The functional form is, annual conditional prepayment rate 5 0.005e0.29957t

16This equation is estimated using national, annual data from 1970 to 2000. This data is available intables found at: www.census.gov/hhes/www.housing/hvs/q400tab1.html and http://www.census.gov./hhes/www/housing/hvs/historic/histtab5.html. The annual vacancy rates for all properties used hereare a simple average of the quarterly rates reported by the Census.


TABLE IISimulation Parameter Values and Results

Scenarios

Parameters 0 1 2 3 4 5

DCR0 1.20LTV0 0.80R0 .03Mortgage coupon rate .08MSA turnover rate .35 .35 .50 .50 .35 .35

(year 1) (year 1)Maximum turnover at cycle — — — — .525 .35

troughMSA vacancy duration rate, 2 3 2 1 2 2

months (year 1) (year 1)Maximum duration at cycle — — — — 3 4

trough, monthsSimulation results (in percent)

Mean vacancy, small 4.50 6.95 6.55 3.33 6.35 6.15properties, all years (max 5 10.17) (max 5 9.34)

Mean vacancy, large, properties, 6.80 10.27 9.88 4.87 9.43 9.07all years (max 5 14.98) (max 5 13.71)

15-year cumulative default rate, 3.31 6.56 5.94 2.62 8.40 6.93small properties

15-year cumulative default rate, 4.09 11.49 10.46 2.97 15.54 13.05large properties

IV. COMPARATIVE RESULTS

To explore relationships between tenant mobility, vacancy rates, and creditrisk, we run the simulation model under six different scenarios. Scenario parame-ters and result summaries are provided in Table II. More detailed result statisticsare in Table III.

Scenario 0 is our base case, with an MSA-wide tenant turnover rate of 35%and a mean vacancy duration of 2 months. This is a fairly benign scenario, froma standpoint of credit risk.17 Given the distinctions in submarkets—small versuslarge properties—the MSA average vacancy rate (across time) is 6%. The re-sulting cumulative default rate for large properties is 22% higher than that forsmall properties, but the absolute difference is a minor 61 basis points (3.04versus 2.57%).

Figure 2 shows turnover- and vacancy-rate distributions, by property type, for

17Belsky (1992b) finds a national average turnover rate of 41% and duration of 12.7 weeks. Atypical MSA in the 1980 Census data used by Belsky with a turnover rate of 35% had a durationof about 10 weeks.


TABLE IIISimulation Results

Age Vjt RPI DCRjt LTVjt JPjt CDR CPR CUMD CUMP

a. Base case, Scenario 0—Small properties, 5–49 units1 0.051 1.013 1.282 0.945 0.048 0.000 0.007 0.000 0.0072 0.047 1.049 1.335 0.904 0.052 0.000 0.009 0.000 0.0163 0.046 1.086 1.386 0.869 0.041 0.001 0.012 0.001 0.0284 0.045 1.123 1.435 0.832 0.039 0.002 0.017 0.003 0.0445 0.048 1.160 1.476 0.804 0.035 0.003 0.022 0.005 0.0656 0.047 1.199 1.528 0.775 0.030 0.004 0.030 0.009 0.0937 0.046 1.242 1.584 0.740 0.029 0.005 0.041 0.013 0.1308 0.044 1.287 1.646 0.701 0.022 0.005 0.055 0.017 0.1779 0.044 1.328 1.697 0.671 0.024 0.004 0.074 0.021 0.237

10 0.045 1.376 1.756 0.642 0.023 0.003 0.100 0.023 0.31111 0.046 1.425 1.819 0.614 0.029 0.002 0.135 0.025 0.40112 0.045 1.475 1.883 0.580 0.025 0.001 0.182 0.025 0.50513 0.045 1.531 1.956 0.548 0.016 0.001 0.246 0.026 0.62114 0.047 1.587 2.021 0.518 0.010 0.000 0.331 0.026 0.73815 0.044 1.641 2.097 0.486 0.010 0.000 0.447 0.026 0.844

b. Base case, Scenario 0—Large properties, 50 or more units1 0.068 1.013 1.251 0.978 0.074 0.000 0.007 0.000 0.0072 0.072 1.051 1.297 0.955 0.094 0.000 0.009 0.001 0.0163 0.069 1.088 1.344 0.905 0.075 0.001 0.012 0.002 0.0284 0.065 1.123 1.393 0.863 0.063 0.002 0.017 0.003 0.0445 0.067 1.162 1.437 0.832 0.053 0.003 0.022 0.006 0.0656 0.068 1.203 1.485 0.800 0.064 0.005 0.030 0.011 0.0937 0.066 1.249 1.546 0.762 0.052 0.006 0.041 0.016 0.1308 0.070 1.293 1.592 0.734 0.048 0.006 0.055 0.021 0.1779 0.067 1.336 1.654 0.697 0.038 0.005 0.074 0.025 0.236

10 0.071 1.382 1.703 0.674 0.042 0.004 0.100 0.028 0.31011 0.066 1.430 1.773 0.632 0.034 0.002 0.135 0.029 0.39912 0.070 1.481 1.828 0.676 0.036 0.001 0.182 0.030 0.50313 0.069 1.530 1.887 0.574 0.034 0.001 0.246 0.030 0.61814 0.068 1.581 1.952 0.539 0.019 0.000 0.331 0.030 0.73515 0.070 1.639 2.019 0.510 0.019 0.000 0.447 0.030 0.840

Note. Parameters in Table II.

the base case scenario. Stark differences in turnover rates are muted in the vacancyrates because vacancy durations are generated from the same stochastic processfor each property type. However, as we increase average turnover and/or durationsat the MSA level in additional scenarios, vacancies on large properties rise tolevels that depress rental price growth and trigger substantially higher default rates.

In Scenario 1 we increase average durations from 2 to 3 months. Averagevacancies rise from 4.45 and 6.98% to 6.58 and 10.28% for small and largeproperties, respectively. Cumulative default rates nearly double for small proper-ties (from 2.57 to 4.67) and more than double for large properties (from 3.04 to


TABLE III—Continued


c. Scenario 1—Small properties, 5–49 units1 0.083 1.007 1.218 0.998 0.109 0.000 0.007 0.000 0.0072 0.076 1.026 1.255 0.886 0.109 0.001 0.009 0.001 0.0163 0.069 1.046 1.293 0.951 0.100 0.001 0.012 0.002 0.0284 0.068 1.068 1.323 0.925 0.110 0.003 0.017 0.005 0.0445 0.069 1.089 1.348 0.910 0.105 0.005 0.022 0.009 0.0656 0.067 1.111 1.379 0.877 0.113 0.007 0.030 0.016 0.0937 0.069 1.133 1.402 0.842 0.108 0.009 0.041 0.023 0.1298 0.066 1.157 1.437 0.821 0.100 0.009 0.055 0.031 0.1769 0.072 1.179 1.451 0.787 0.108 0.008 0.074 0.037 0.235

10 0.070 1.203 1.485 0.773 0.108 0.007 0.100 0.042 0.30811 0.070 1.225 1.512 0.749 0.103 0.004 0.135 0.045 0.39512 0.072 1.251 1.542 0.730 0.110 0.002 0.182 0.046 0.49713 0.069 1.276 1.579 0.691 0.095 0.001 0.246 0.047 0.60914 0.068 1.299 1.605 0.660 0.073 0.000 0.331 0.047 0.72315 0.066 1.320 1.640 0.629 0.055 0.000 0.447 0.047 0.826

d. Scenario 1—Large properties, 50 or more units1 0.105 1.005 1.179 0.930 0.159 0.000 0.007 0.000 0.0072 0.111 1.023 1.200 1.002 0.198 0.001 0.009 0.001 0.0163 0.103 1.038 1.224 1.032 0.187 0.002 0.012 0.004 0.0284 0.102 1.057 1.247 0.990 0.173 0.004 0.017 0.008 0.0445 0.101 1.080 1.276 0.948 0.158 0.007 0.022 0.015 0.0656 0.101 1.102 1.303 0.886 0.174 0.011 0.030 0.025 0.0937 0.100 1.123 1.328 0.911 0.174 0.014 0.041 0.038 0.1298 0.102 1.148 1.354 0.877 0.154 0.013 0.055 0.049 0.1759 0.101 1.173 1.386 0.851 0.151 0.012 0.074 0.058 0.232

10 0.100 1.200 1.420 0.826 0.150 0.009 0.100 0.065 0.30311 0.099 1.223 1.450 0.764 0.152 0.006 0.135 0.068 0.38812 0.103 1.245 1.467 0.776 0.152 0.003 0.182 0.070 0.48713 0.105 1.267 1.492 0.774 0.139 0.001 0.246 0.071 0.59614 0.106 1.289 1.512 0.880 0.129 0.000 0.331 0.071 0.70615 0.103 1.317 1.553 0.725 0.106 0.000 0.447 0.071 0.806

7.09). The much larger increase in defaults for large properties is because of thelarger (though still unchanged in this scenario) turnover rate.18

Scenario 2 increases MSA turnover rates from 35 to 50% which has about thesame effect as increasing duration from 2 to 3 months (Scenario 1). Scenario 3maintains the higher 50% turnover rate of Scenario 2, but decreases averagedurations to just 1 month. This reduces defaults on all properties to under 2.5%

18Higher vacancy rates have a nonlinear effect on default probabilities, first through the jointprobability of negative cash flow and negative equity and second through the nonlinear nature ofthe logistic probability function. Each percentage increase in average LTV or decrease in averageDCR has an increasing effect on the joint probability variable.




e. Scenario 2—Small properties, 5–49 units1 0.070 1.010 1.244 0.973 0.082 0.000 0.007 0.000 0.0072 0.066 1.029 1.278 0.960 0.092 0.000 0.009 0.001 0.0163 0.064 1.054 1.314 0.912 0.089 0.001 0.012 0.002 0.0284 0.061 1.078 1.346 0.893 0.087 0.002 0.017 0.004 0.0445 0.062 1.101 1.374 0.872 0.091 0.004 0.022 0.008 0.0656 0.064 1.123 1.398 0.857 0.089 0.006 0.030 0.013 0.0937 0.069 1.148 1.421 0.845 0.107 0.009 0.041 0.021 0.1308 0.070 1.171 1.447 0.824 0.105 0.009 0.055 0.029 0.1769 0.068 1.197 1.484 0.805 0.103 0.008 0.074 0.035 0.235

10 0.071 1.222 1.506 0.724 0.087 0.006 0.100 0.039 0.30811 0.065 1.249 1.552 0.724 0.086 0.004 0.135 0.042 0.39612 0.066 1.271 1.577 0.705 0.080 0.002 0.182 0.043 0.49913 0.066 1.298 1.611 0.675 0.079 0.001 0.246 0.043 0.61114 0.065 1.323 1.647 0.648 0.063 0.000 0.331 0.043 0.72615 0.061 1.349 1.683 0.608 0.048 0.000 0.447 0.043 0.829

f. Scenario 2—Large properties, 50 or more units1 0.097 1.010 1.197 1.034 0.129 0.000 0.007 0.000 0.0072 0.101 1.029 1.222 1.032 0.174 0.001 0.009 0.001 0.0163 0.091 1.050 1.259 0.979 0.154 0.002 0.012 0.003 0.0284 0.097 1.075 1.278 0.981 0.155 0.004 0.017 0.007 0.0445 0.097 1.097 1.303 0.948 0.157 0.007 0.022 0.013 0.0656 0.099 1.121 1.327 0.869 0.143 0.009 0.030 0.022 0.0937 0.101 1.145 1.354 0.901 0.169 0.014 0.041 0.034 0.1298 0.096 1.168 1.387 0.863 0.134 0.011 0.055 0.043 0.1759 0.097 1.190 1.413 0.891 0.135 0.010 0.074 0.052 0.233

10 0.098 1.217 1.442 0.774 0.126 0.007 0.100 0.057 0.30411 0.099 1.248 1.481 0.781 0.139 0.005 0.135 0.060 0.39112 0.095 1.274 1.519 0.735 0.116 0.002 0.182 0.062 0.49113 0.101 1.298 1.533 0.726 0.107 0.001 0.246 0.062 0.60114 0.096 1.327 1.576 0.678 0.084 0.000 0.331 0.062 0.71215 0.101 1.357 1.606 0.653 0.078 0.000 0.447 0.062 0.813

showing the near irrelevance of high turnover rates in tight markets. Note thatan average (mean) duration of 1 month implies a modal duration of just 2 weeks.

In Scenarios 4 and 5 we take loan portfolios through 10-year housing cycles,with year 1 being an initial peak, year 6 the trough, and year 11 the full recoverypoint. In scenario 4, both (MSA average) turnover rates and durations increaseto 150% of their original level at the market trough and then return again to theirstarting values. In Scenario 5, only average duration increases, but it doubles inlength at the trough. Scenario 4 represents a situation where there is both over-building and mobility out of an area, like situations in oil patch cities during thelate 1980s. This provides the worst of our six scenarios, with default rates of




g. Scenario 3—Small properties, 5–49 units1 0.035 1.018 1.316 0.914 0.019 0.000 0.007 0.000 0.0072 0.034 1.059 1.371 0.873 0.021 0.000 0.009 0.000 0.0163 0.034 1.103 1.431 0.834 0.020 0.001 0.012 0.001 0.0284 0.035 1.151 1.490 0.796 0.016 0.001 0.017 0.002 0.0445 0.031 1.204 1.565 0.753 0.013 0.002 0.022 0.004 0.0656 0.033 1.257 1.630 0.718 0.013 0.003 0.030 0.007 0.0937 0.032 1.308 1.699 0.685 0.014 0.004 0.041 0.011 0.1308 0.033 1.358 1.762 0.652 0.006 0.004 0.055 0.015 0.1779 0.032 1.418 1.843 0.617 0.011 0.004 0.074 0.018 0.237

10 0.033 1.477 1.916 0.586 0.011 0.003 0.100 0.020 0.31211 0.032 1.541 2.001 0.551 0.006 0.002 0.135 0.022 0.40212 0.033 1.607 2.086 0.521 0.007 0.001 0.182 0.022 0.50713 0.034 1.680 2.178 0.490 0.005 0.000 0.246 0.022 0.62214 0.032 1.750 2.275 0.458 0.003 0.000 0.331 0.023 0.74015 0.030 1.826 2.380 0.426 0.000 0.000 0.447 0.023 0.846

h. Scenario 3—Large properties, 50 or more units1 0.052 1.020 1.288 0.939 0.040 0.000 0.007 0.000 0.0072 0.054 1.063 1.344 0.908 0.067 0.000 0.009 0.001 0.0163 0.053 1.113 1.405 0.859 0.047 0.001 0.012 0.001 0.0284 0.048 1.159 1.473 0.811 0.022 0.001 0.017 0.003 0.0445 0.047 1.205 1.532 0.775 0.023 0.002 0.022 0.005 0.0656 0.048 1.256 1.594 0.738 0.018 0.003 0.030 0.008 0.0937 0.051 1.311 1.658 0.705 0.022 0.004 0.041 0.012 0.1308 0.049 1.366 1.734 0.667 0.019 0.005 0.055 0.016 0.1779 0.048 1.426 1.808 0.630 0.015 0.004 0.074 0.019 0.237

10 0.049 1.486 1.884 0.597 0.009 0.003 0.100 0.022 0.31111 0.052 1.551 1.961 0.566 0.016 0.002 0.135 0.023 0.40112 0.049 1.617 2.049 0.531 0.008 0.001 0.182 0.024 0.50613 0.050 1.685 2.135 0.501 0.011 0.000 0.246 0.024 0.62214 0.048 1.755 2.229 0.468 0.004 0.000 0.331 0.024 0.73915 0.050 1.830 2.316 0.437 0.002 0.000 0.447 0.024 0.845

6% for small properties and 10% for large properties. Scenario 5, while not assevere, represents significant overbuilding (4 month vacancy durations), but withno out-migration-induced increase in turnover rates. Under these conditions, largeproperty default rates are 7.71% while those on small properties are 5.57%.

V. SUMMARY, CONCLUSIONS, AND POSSIBLE FUTURE RESEARCH

In this paper we present a simple model of the interaction between structuralcomponents of rental markets and the derivative credit risk of multifamily mort-gages. We start with MSA-level turnover and vacancy-duration rates, match them




i. Scenario 4—Small properties, 5–49 units1 0.052 1.015 1.282 0.948 0.055 0.000 0.007 0.000 0.0072 0.059 1.043 1.307 0.903 0.070 0.000 0.009 0.001 0.0163 0.062 1.064 1.326 0.914 0.081 0.001 0.012 0.002 0.0284 0.076 1.078 1.322 0.931 0.101 0.003 0.017 0.004 0.0445 0.086 1.088 1.317 0.891 0.137 0.006 0.022 0.010 0.0656 0.094 1.086 1.299 0.867 0.163 0.011 0.030 0.019 0.0937 0.089 1.089 1.313 0.922 0.163 0.013 0.041 0.031 0.1298 0.079 1.099 1.341 0.866 0.153 0.013 0.055 0.042 0.1759 0.068 1.124 1.392 0.839 0.143 0.011 0.074 0.051 0.233

10 0.060 1.152 1.442 0.794 0.120 0.007 0.100 0.056 0.30511 0.048 1.189 1.513 0.740 0.102 0.004 0.135 0.059 0.39112 0.046 1.228 1.567 0.703 0.088 0.002 0.182 0.060 0.49113 0.047 1.271 1.621 0.677 0.070 0.001 0.246 0.060 0.60214 0.047 1.313 1.676 0.634 0.055 0.000 0.331 0.060 0.71415 0.048 1.359 1.730 0.597 0.039 0.000 0.447 0.060 0.815

j. Scenario 4—Large properties, 50 or more units1 0.070 1.016 1.249 0.979 0.078 0.000 0.007 0.000 0.0072 0.074 1.049 1.290 0.963 0.117 0.001 0.009 0.001 0.0163 0.094 1.076 1.285 0.965 0.139 0.002 0.012 0.002 0.0284 0.114 1.094 1.272 0.946 0.171 0.004 0.017 0.007 0.0445 0.134 1.098 1.241 1.068 0.219 0.011 0.022 0.017 0.0656 0.149 1.099 1.215 1.341 0.253 0.021 0.030 0.037 0.0937 0.136 1.106 1.246 1.035 0.237 0.023 0.041 0.057 0.1288 0.119 1.117 1.288 0.953 0.232 0.024 0.055 0.077 0.1739 0.106 1.135 1.331 0.884 0.195 0.017 0.074 0.089 0.229

10 0.088 1.164 1.401 0.824 0.152 0.009 0.100 0.095 0.29711 0.074 1.201 1.473 0.765 0.123 0.005 0.135 0.098 0.37912 0.070 1.241 1.530 0.723 0.106 0.002 0.182 0.099 0.47413 0.072 1.281 1.576 0.694 0.088 0.001 0.246 0.100 0.57914 0.073 1.322 1.621 0.655 0.068 0.000 0.331 0.100 0.68515 0.069 1.367 1.688 0.609 0.057 0.000 0.447 0.100 0.781

to unique submarket dynamics, and then use Monte Carlo techniques to see howsubmarket differences result in differences in the credit risk profile of real estateportfolios based on those submarkets.

The model is simple because we do not attempt to get at underlying causesof changing turnover rates or durations. Our work does not attempt to use morecomplicated models of vacancy chains, where net new construction, migrationpatterns, and household demographic changes are used to generate turnover ratesand vacancy durations (see Emmi and Magnusson, 1994, 1995). Such modelsare suited to differentiating between major property-type classes (rental vs owner




k. Scenario 5—Small properties, 5–49 units1 0.050 1.015 1.284 0.940 0.045 0.000 0.007 0.000 0.0072 0.056 1.040 1.307 0.926 0.070 0.000 0.009 0.001 0.0163 0.065 1.067 1.327 0.929 0.072 0.001 0.012 0.002 0.0284 0.071 1.082 1.335 0.949 0.094 0.002 0.017 0.004 0.0445 0.080 1.092 1.332 0.947 0.117 0.005 0.022 0.009 0.0656 0.096 1.096 1.310 0.906 0.143 0.009 0.030 0.017 0.0937 0.085 1.108 1.343 0.889 0.158 0.013 0.041 0.028 0.1298 0.087 1.120 1.354 0.888 0.163 0.014 0.055 0.040 0.1769 0.069 1.145 1.414 0.814 0.122 0.009 0.074 0.048 0.234

10 0.058 1.172 1.470 0.773 0.095 0.006 0.100 0.052 0.30611 0.049 1.208 1.535 0.728 0.092 0.004 0.135 0.054 0.39212 0.044 1.254 1.604 0.685 0.075 0.002 0.182 0.055 0.49313 0.044 1.297 1.659 0.649 0.059 0.001 0.246 0.056 0.60414 0.045 1.341 1.712 0.613 0.041 0.000 0.331 0.056 0.71715 0.046 1.386 1.769 0.580 0.038 0.000 0.447 0.056 0.819

l. Scenario 5—Large properties, 50 or more units1 0.071 1.015 1.247 0.996 0.081 0.000 0.007 0.000 0.0072 0.075 1.048 1.287 0.988 0.112 0.001 0.009 0.001 0.0163 0.091 1.071 1.285 0.966 0.117 0.001 0.012 0.002 0.0284 0.102 1.094 1.292 0.957 0.153 0.004 0.017 0.006 0.0445 0.119 1.107 1.276 1.185 0.192 0.009 0.022 0.014 0.0656 0.132 1.115 1.262 1.053 0.205 0.015 0.030 0.028 0.0937 0.126 1.128 1.289 0.942 0.206 0.018 0.041 0.044 0.1298 0.116 1.140 1.321 1.004 0.185 0.017 0.055 0.058 0.1749 0.098 1.159 1.377 0.858 0.163 0.013 0.074 0.068 0.231

10 0.086 1.190 1.435 0.793 0.123 0.007 0.100 0.073 0.30111 0.072 1.232 1.513 0.741 0.111 0.004 0.135 0.076 0.38612 0.067 1.275 1.577 0.700 0.081 0.002 0.182 0.077 0.48413 0.068 1.320 1.631 0.662 0.065 0.001 0.246 0.077 0.59214 0.068 1.360 1.677 0.627 0.043 0.000 0.331 0.077 0.70115 0.068 1.406 1.736 0.588 0.035 0.000 0.447 0.077 0.800

Vjt , average vacancy rate, across properties ( j 5 property type, t 5 year) (this is the apartmentvacancy rate and not the adjusted Census-equivalent used to calculate default probabilities); RPI,rental price index, average across properties; DCRjt , debt coverage ratio, average across properties;LTVjt , loan-to-value ratio, average across properties; JPjt joint probability of negative equity andnegative cash flow; CUMP, cumulative prepayment rate; CDR, conditional default rate; CPR, condi-tional prepayment rate; CUMD, cumulative default rate.

occupied), and they may be helpful in discerning potential changes in MSA-average turnover and duration rates over time.

The credit risk model developed by Goldberg and Capone (2000) provides auseful vehicle for our task. Because it is built on frequency distributions ofvacancy rates and rental price growth, we can use our own assumptions about


FIG. 2. Turnover and vacancy rate distributions, base case (Scenario 0), Year 1. (a) Vacancyrates for small properties. (b) Turnover rates for small properties. (c) Vacancy rates for large properties.(d) Turnover rates for large properties.

the generation of those distributions, and about how they might change overtime, to create a more flexible credit-risk analysis tool. In particular, we extendthe Goldberg–Capone model by:

• Creating more flexible vacancy rate distributions through use of underlyingturnover- and duration-rate distributions;

• Creating credit-risk subsectors within MSAs;


0.04 0.08 0.12 0.16 0.20 0.24 0.28

0.2 0.6 1.0

020

4060

8010

00

2040

6080

100

120

FIG. 2—Continued

• Tying rental price growth to vacancy rates on individual properties and todeviations in MSA vacancy rates from a natural rate.

Our simulations show the sensitivity of default risk to changes in both turnoverand duration rates. For any given average turnover rate per apartment project,extensions in vacancy durations of just a few weeks can have a significant impacton default risk (annual and cumulative rates). Likewise, increases in turnover


rates within the range historically observed across MSAs requires compensatingreductions in duration to avoid substantial increases in default risk. Simulationsshown here also indicate that a doubling of durations from 2 to 4 months is notquite as severe as increasing both turnovers and durations by 50%, from the basecase assumptions of 35% and 2 months.

The small versus large property distinction is an important area of study forcredit markets. If they are truly different markets, they need to be evaluatedseparately. Credit markets favor large projects because of the high fixed cost ofunderwriting commercial loans. If small properties pose less credit risk, thenperhaps they could go through less rigorous underwriting examinations or beallowed more lenient financial ratios, reducing the cost of commercial propertyfinancing to where it is more affordable for small-property owners. Small multi-family properties are an important part of our nation’s housing, making up over37% of multifamily units and over 13% of total housing units in the nation.19

There are many possible extensions of this work that could be explored. Someof these are:

• Explore data sets that might provide MSA level turnover rate distributionslike the national distribution found in the POMS data;

• Use resampling techniques to analyze the credit risk of different sized portfo-lios of loans within a property-type market segment;

• Employ lumpy (discrete), rather than continuous turnover rate distributionsfor small properties;

• Explore the thesis that small-property owners maintain lower turnover withlower average rental price growth;

• Allow for different vacancy duration assumptions by property type (the fullDanter, 1997, hypothesis);

• Explore applications of the simulation model to other potential market dichot-omies.

APPENDIX A: DERIVING PARAMETERS FOR RANDOMNUMBER GENERATORS

Turnover Rates

Annual property turnover rates, tijt , are assumed to be gamma random variates,tijt , G(b,c), where i is a property index (i P (1, . . . , Nj), j is the property-typeindicator ( j P {s,l}), and t denotes the time period (t P {1, . . . ,15}). The twoparameters represent scale (b) and shape (c).

In order to run simulations with a fixed relationship between property type

19Statistics from 1993 AHS and POMS, respectively.


results (means) in all scenarios, we first fix the values of c. For small properties,cs 5 1, and for large properties, c1 5 2.20 These two-parameter values allow usto model the POMS results, where the modal frequency for small-property turn-over is zero, the distribution of turnovers for large properties is unimodal witha long tail, and the small-property mean is two-thirds the large-property mean.We then solve for scale parameters using these POMS relationships, from anygiven MSA-level average turnover rate.

Primary relationships from POMS:

mm 5 0.3717*ms 1 0.6283*m1 (property-type sample weights) (A.1)

ms 5 0.667* m1 (property-type sample means) (A.2)

where mm is the MSA mean turnover rate, ms is the small-property mean turnoverrate within MSA, and m1 is the large-property mean turnover rate within MSA.Substitute A.2 into A.1 and solve for m1 and then for ms:

m1 5 1.179*mm (A.3)

ms 5 0.787*mm. (A.4)

Property-type specific mean turnover rates are then computed directly from anMSA average. The scale parameters for the random number generation arefunctions of these means and the shape parameters, cs 5 1 and c1 5 2.

bs 5 ms /cs 5 0.787*mm (A.5)

b1 5 m1 /c1 5 0.5895*mm (A.6)

Vacancy Duration Distribution

Average vacancy durations per apartment project (percentage of year) are alsomodeled as gamma random variates, dijt , G(bd , cd). In this case we fix themode (md) to equal one-half the mean (md) and then can derive both the scaleand shape parameters from any assumed MSA mean duration.

md 5 bd ? cd (A.7)

md 5 bd (cd 2 1) [ md /2 (A.8)

Solving Eq. (A.7) for bd , and substituting into (A.8), yields cd 5 2. Thus, the

20Setting the shape parameters to equal whole numbers reduces the gamma distribution to the Erlang.


scale parameter bd 5 md/2. The mean duration is md 5 dt , where dt is used inthe text to denote the MSA-wide duration and is constant across property types( j ), though random numbers are generated separately for each property type, ineach year (t).

REFERENCES

Belsky, E. (1992a). “Rental Vacancy Rates and Rental Markets,” Housing Economics, February,9–12.

Belsky, E. (1992b). “Rental Vacancy Rates: A Policy Primer,” Housing Policy Debate 3, 793–813.

Belsky, E., and Goodman, J. L. (1996). “Explaining the Vacancy Rate-Rent Paradox of the 1980s,”J. Real Estate Res. 11, 309–323.

Blank, D. M., and Winnick, L. (1953). “The Structure of the Housing Market,” Quart. J. of Econ.67, (May), 181–203.

Danter, K. (1997). “The Many Levels of Multifamily,” Mortgage Banking, 57 July, 34–41.

Downs, A. (1983). Rental Housing in the 1980s. Washington, DC: The Brookings Institution.

Emmi, P. C., and Magnusson, L. (1994). “The Predictive Accuracy of Residential Chain Models,”Urban Stud. 31, 1117–1131.

Emmi, P. C., and Magnusson, L. (1995). “Opportunity and Mobility in Urban Housing Markets,”Progr. Planning 43, 1–88.

Eppli, M. J., and Shilling, J. D. (1995). “Speed of Adjustment in Commercial Real Estate Markets,”Southern Econ. J. 61, (4 April), 1127–45.

Gabriel, S., and Nothaft, F. (1988). “Rental Housing Markets and the Natural Vacancy Rate,” Amer.Real Estate Urban Econ. 16, 419–429.

Goldberg, L., and Capone, C. A. (1998). “Multifamily Mortgage Credit Risk: Lessons from RecentHistory,” Cityscape 4, 93–113.

Goldberg, L., and Capone, C. A. (2000). A Dynamic Double-Trigger Model of Multifamily MortgageDefault, Alexandria, VA: Institute for Defense Analyses, unpublished manuscript.

Hendershott, P., and Haurin, D. (1988). “Adjustments in the Real Estate Market,”Amer. Real EstateUrban Econ. 16, 343–353.

Ioannides, Y. M. (1987). “Residential Mobility and Housing Tenure Choice,” Reg. Sci. Urban Econ.17, 265–287.

Israeli, M., and Nelson, C. B. (1992). “Distribution and Expected Time of Residence for U.S.Households,” Risk Anal. 12, 65–72.

Rosen, K. T., and Smith, L. (1983). “The Price Adjustment Process for Rental Housing and theNatural Vacancy Rate,” Amer. Econ. Rev. 73, 779–786.

Rydell, P. (1978). Vacancy Duration and Housing Market Conditions. Santa Monica: Rand Corpora-tion, January, WN-10074-HUD.

Sivitanides, P. (1997). “The Rent Adjustment Process and the Structural Vacancy Rate in the Commer-cial Real Estate Market,” J. Real Estate Res. 13, 195–209.

Documents

Renter Mobility and Multifamily Mortgage Default Risk