# Validation of Basic Valuation Models: A Multi-Family Housing Example

## Article excerpt

There are a number of simple mathematical models used in the valuation of income-producing property. These include the various income multipliers and the overall rate (OAR) which are associated with the income capitalization approach.(1) Similar multipliers are used in the sales comparison approach to value.(2) Expenses often are estimated on a per dwelling unit or per net rentable square foot basis or as a percentage of gross income or effective gross income. All of these mathematical models are directly proportional, almost the simplest models we can imagine.(3) How good are these simple models? They are certainly intuitively appealing and appear to conform with the realities of the market, at least to a first order of approximation over the relevant range of most real estate transactions.

The objective of the study reported in this article was to test and compare the simple, proportional models used in income property valuation. The testing and comparison used a basic statistical technique (multiple linear regression) on a good-quality, relatively homogeneous sample of 30 multi-family transactions in suburban collar areas in the Atlanta, Georgia, metropolitan area for the years 1984 through 1988. Although not large by statistical standards, the sample was of high quality.

THE DATABASE

Each record (transaction) included the number and type of units (DU) for the apartment complex, the annual gross rental income (GR), other income (OI), gross income (GI), vacancy and collection loss (VC), effective gross income (EGI), expenses (EXP), net operating income (NOI) and units by type. In addition, the actual or contract selling price (SP), the year built (YR) and the net rentable area (SF) were included. From these, a number of measures were derived, including age at sale (AGE), gross income multiplier (GIM), gross rent multiplier (GRM), effective gross income multiplier (EGIM), net income multiplier (NIM) and expenses per square foot of net rentable area (ESF), rooms (ERM), units (EDU) and bedrooms (EBD). The selling price also was analyzed as a ratio of square footage of net rentable area (SPSF), rooms (SPRM), units (SPDU) and bedrooms (SPBD).

In this analysis, all figures for income, expenses and selling price were rounded and reported in thousands of constant dollars.(4) In cases where the apartment complex was too new to have an extensive income history, the projected figures for the transaction were used.

A summary of some descriptive statistics is reported in Table 1 (Table 1 omitted). These statistics include the mean, minimum, maximum and coefficient of variation for each measurement. For example, the smallest complex contained 64 units, while the largest contained 490 units. The mean for the variable "number of units" was 244.30. Annual gross rental income varied from \$493,300 to \$3,958,000 with a mean of \$1,819,500, and so forth.

VALUATION MODEL TESTS

The database was analyzed using ordinary least squares regression techniques as embodied in the BMDP statistical software developed at the University of California, Los Angeles. A series of multiple linear regressions were run initially, with selling price (SP) as the dependent variable against the number of units (DU), age at sale (AGE) and a time (market conditions) variable (T), along with one of the commonly used value indicators listed in Table 1. These value indicators included gross rent (GR), gross income (GI), effective gross income (EGI), net operating income (NOI), area (SF), rooms (RMS), bedrooms (BDRMS) and number of units (DU).

The first of the analyses regressed selling price (SP) on gross income (GI), units (DU), age (AGE) and time (T).(5) This regression was highly significant and had a coefficient of determination of 99.7%, indicating the regression based on the four variables explained 99.7% of all the variation in the data. The units and age variables each had student t values of less than 1. …