Portfolio value estimates using mass appraisals and portfolio confidence intervals.

Moss, Steven E. ; Williams, Susan R. ; Dyer, John N. 等

INTRODUCTION

The purpose of this paper is to present a methodology for estimating a confidence interval for the aggregate portfolio value of real estate assets underlying a commercial mortgage-backed security, a portfolio of real properties securing mortgages, or an equity real estate portfolio. Large portfolios of bank held mortgages commonly consist of residential properties. Banks are interested in the market value of the real estate against which they hold mortgages to determine loan to value (LTV) ratios. The market value of the portfolio and outstanding mortgage balances are used to determine the banks LTV ratio.

A commercial mortgage-backed security (CMBS) is a pool of real estate loans that have been securitized, allowing many individuals to invest relatively small amounts in the pool of loans. Any type of real estate (such as residential, fast food restaurants, and branch banks) may be included in the CMBS, and characteristics of individual loans within the pool may vary widely. For example, loans may be based on fixed or variable interest rates, specify various maturity dates, require amortized or balloon payments, and may be currently performing or non-performing. The loans within the pool are typically split into classes, or tranches, representing varying levels of risk. Securities are issued against each tranche, with returns and repayment precedence adjusted to reflect the corresponding levels of risk.

CMBS popularity began in the in the early 1990's when the Resolution Trust Corporation (RTC) used CMBS as a vehicle to liquidate distressed portfolios (Green, 1999). Since that time, CMBS have become one of the largest public real estate investment channels (Hess and Liang, 2001). The CMBS market continues to grow rapidly, with commercial banks playing a major role in its expansion. A recent FDIC report states that banks' holdings of mortgage-backed securities grew by record amounts ($51.6 billion and $53.3 billion) in the third and fourth quarters of 2001 (C&I Loans ... 2001). Commercial real estate loans also continue to grow at a rapid rate. In 1998 commercial banks held real estate mortgages totaling $467.5 billion, and issued more than 41% of new real estate debt (Green, 1999). In the fourth quarter 2001, the FDIC reports that commercial real estate loans rose by $13.8 billion, an increase of 2.8% over the previous quarter.

Packaging loans into a CMBS generates origination and securitization fees for the originator. Top CMBS originators include Morgan Stanley, Credit Suisse First Boston, Goldman Sachs, J.P Morgan Chase and Banc of America Securities (Morgan Stanley, 2001). Larger pools of loans can provide higher degrees of diversification but increase warehousing costs and risks for the originator of the CMBS. Larger pools of loans take longer to package. To minimize risk the originator has an interest in issuing securities against the pool of loans quickly (Green, 1999).

Although timely, accurate information is important for determining risk, the CMBS market does not always have adequate information regarding the underlying assets. Loan to value ratios are important in establishing the ratings and tranches for a CMBS (Hudson-Wilson and Pappadopoulos, 1999). LTV ratios also are also important to investors. Nineteen percent of outstanding CMBS have LTV ratios in excess of 90%. Many investors, however, including insurance companies prefer to invest in CMBS with LTV ratios in the 65-70% range (Levy, 2001).

Appraised value (or estimated appraised value) is a key factor in determining LTV ratios. Credit agencies (such as Moody's, Fitch, and S&P) that earn fees for rating CMBS and establishing loss determination rely on appraisal-based information (Riddiough, 2000). However, cost and time significantly limit the number of appraisals that can be performed for any one CMBS portfolio (some portfolios have more than 2,000 individual properties). To overcome this problem, a multiple regression model is often used to predict the values of properties that are not individually appraised (mass appraisal technique), enabling a point estimate for the entire portfolio to be determined. Establishing a confidence interval for the aggregate portfolio value is more difficult, and is the focus of the remainder of this paper. In estimating a confidence interval for the value of a CMBS real estate portfolio, factors such as time, cost, ease of repeated implementation, and margin of error (confidence interval half width) are critical to CMBS originators.

DATA

The data used to demonstrate the procedure developed in this study are residential real estate assets. The data was obtained from PricewaterhouseCoopers (PWC) real estate research division. The data consists of a sample of comparable residential sales and a subject portfolio to be valued. Property specific variables include building square footage, lot size, ocean view (yes/no), age, and location. This type of portfolio is most commonly valued to determine LTV ratios for a residential CMBS or a bank's outstanding mortgage loan portfolio. The procedure is easily applied to commercial real estate portfolios and equity portfolios.

In general portfolio sizes can range from as few as 50 to more than 2,000 properties per portfolio. The asset types can include raw land, fast food parcels, other business parcels (grouped by type of business), and residential properties. The geographic locations studied are all within the U.S. Each portfolio studied is homogenous by geographic location and property type. Total portfolio values studied range from $10 million to over $1.5 billion per portfolio. Portfolios studied included CMBS portfolios and portfolios of real estate securing mortgages held by banks.

METHODOLOGY

The objective of this research is to develop a methodology that can be used to quickly, economically, and accurately to predict the portfolio value for a large group of real estate assets. For residential portfolios comparable sales data is collected. The comparable sales data is then used to estimate a multiple regression equation to estimate sales price. The multiple regression equation is then used to appraise each property in the portfolio to be valued.

The procedure for non-residential portfolios is to draw a simple random sample from the portfolio to be valued and perform a complete fee-based appraisal for each of the selected properties. The recommended sample size is determined by multiplying the estimated number of independent variables to be included in the regression model by 20, then adding 10 to 20 extra observations to adjust for any appraisals that are not comparable with the rest of the portfolio (Hair, 1995).

A multiple regression model is estimated with appraised value as the dependent variable. Independent variables include property specific variables that are different for each portfolio valued. Each portfolio contains one type of real estate, such as residential, fast food restaurant, or branch bank. A unique multiple regression equation is estimated for each portfolio valued. Some multiple regression models require correcting for non-linearity with log transformations.

Residuals of the model are checked for outliers, non-constant variance, and normality. Any outliers are reviewed for systematic differences such as extreme lot size in relationship to building size. If an outlier is found to have characteristics that are inconsistent with the remainder of the portfolio it is set aside. The entire portfolio is then scanned for properties with characteristics similar to outlier properties. These properties are removed from the portfolio, and if they have not already been appraised, are scheduled for an actual appraisal. The multiple regression equation is then re-estimated (excluding the outlier properties that have been removed). The modified regression model is used to establish the value of the remainder of the portfolio.

A point estimate for the entire portfolio is obtained by summing all of the actual appraised values in the sample and outlier groups, together with the predicted values obtained for the remainder of the properties from the multiple regression equation. While the point estimate is useful, it is important to establish a confidence interval for the overall portfolio value. Confidence intervals for individual observations or means are well documented and can be produced by various statistical software packages. These procedures all assume that the objective is a confidence interval for a single prediction or the mean prediction for a very large group of observations all with the same independent variable vector (Xh vector). In this analysis, the confidence interval desired is for the aggregate value of a new group of observations each with a different Xh vector.

For simple linear regression, the confidence interval for an individual prediction is given by formula 1 (Neter, Wasserman and Kutner, 1990). The confidence interval for mean response from the sample used to estimate the model is shown in formula 2.

[Y.sub.h] +- t * [[MSE * (1 + 1/n + [([X.sub.h] - [X.sub.bar]).sup.2]/Sum[([X.sub.i] - -X.sub.bar]).sup.2])].sup..5] Formula (1)

[Y.sub.h] +- t * [[MSE * (1/n + [([X.sub.h] - [X.sub.bar]).sup.2]/Sum[([X.sub.i] - [X.sub.bar]).sup.2])].sup..5] Formula (2)

The mean response for m new observations, all with the same [X.sub.h] vector, in a simple linear regression model is shown in formula 3 (Neter, Wasserman & Kutner, 1990).

[[Y.sub.h] +- t * [[MSE * (1/m + 1/n + [([X.sub.h] - [X.sub.bar]).sup.2]/Sum[([X.sub.i] - [X.sub.bar]).sup.2])].sup..5]] Formula (3)

The corresponding formulas for multiple regression are shown in formulas 4, 5 and 6 (Neter, Wasserman & Kutner, 1990).

[Y.sub.h] +- t * [[MSE * (1 + [X'.sub.h][(X'X).sup.-1][X.sub.h])].sup..5] Formula (4)

[Y.sub.h] +- t * [[MSE * ([X'.sub.h][(X'X).sup.-1][X.sub.h])].sup..5] Formula (5)

[Y.sub.h] +- t * [[MSE * (1/m + [X'.sub.h][(X'X).sup.-1][X.sub.h])].sup..5] Formula (6)

For simple linear regression if the group size m is equal to 4, the resulting confidence interval for the mean response of the group is 4 times the confidence limits found in formula 3 (Neter, Wasserman & Kutner, 1990). Although not directly stated in (Neter, Wasserman & Kutner, 1990), it follows that for multiple regression the confidence limits for a group of m new observations is m times the limits found by formula 6. Note, that in both formulas 3 and 6 the [X.sub.h] vector is the same for all m cases estimated.

If the m new observations are based upon m different [X.sub.h] vectors, the multiple regression confidence intervals must be individually obtained and aggregated rather than multiplying formula 6 times m (see formula 7). Notice that if all m [X.sub.h] vectors are identical and the portfolio estimate includes sample and non-sample observations, the result obtained using formula 7 will be identical to the result using formula 6.

[Sum.sub.h=1.sup.m] {[Y.sub.h] +- t * [[MSE * (1/m + [X'.sub.h][(X'X).sup.-1][X.sub.h])].sup..5]} Formula (7)

Source: Authors extrapolation of formulas 1-6

It should be noted that the resulting confidence interval using formula 7 is narrower than summing the m individual confidence intervals obtained by formula 4. The narrower confidence interval using formula 7 is due to the clustering effect of estimating mean portfolio value. It would not be reasonable to assume that 300 properties all simultaneously would be at the low or high end of their respective individual confidence intervals.

A confidence interval obtained using formula 7, which is estimated by summing over the observations in the portfolio that require predicted appraised values, is wider than a confidence interval obtained using formula 6, which is estimated in most statistical programs and then summed over all observations in the portfolio. This is partially due to the value of m used in formula 6. Statistical software programs default to a value for m that is extremely large. In portfolio sizes studied, m can be as small as ten. In addition, when a random sample is used to estimate the regression equation and the sampled properties are left in the portfolio, the fact that there is no variation in value for these observations (actual appraisals are obtained for these properties) distorts the results. Summing formula 6 over all the observations in the portfolio results in a large overestimation of m and an underestimation of the confidence interval half width needed for this application. Formula 7 corrects for this by summing over only those observations in the portfolio that require estimation.

Using the mean vector of the m new observations and formula 6 to obtain an estimate of the mean response for m new observations is also flawed. This approach fails to capture the variability in the m new observations' [X.sub.h] vectors. The confidence interval using the mean vector, repeated m times, will generally be narrower than the confidence interval using formula 7 and the m individual [X.sub.h] vectors.

Scheffe and Bonferroni intervals were also considered. Both Scheffe and Bonferroni intervals are interpreted as limits such that we are (1-alpha) % confident that all m observations will be within the limits. If the Bonferroni or Scheffe limits are summed across m observations the resulting interval is much wider than required for a portfolio value. For the portfolio value it is not important if individual properties deviate outside of given limits. The aggregate value of all properties in the portfolio is of interest.

RESULTS

Tables 1 through 3 show the SPSS regression results for one selected CMBS portfolio. The multiple regression equation is used to predict appraised value of each real estate asset in the portfolio.

The MSE from Table 2 and the vector of un-standardized regression coefficients from Table 3 are transferred from SPSS to Excel to facilitate implementation of formula 7. The contribution of each property to the portfolio confidence interval is calculated in Excel using formula 7, as shown in Table 4.

The resulting confidence intervals using formulas 4, 5, and 7 for individual, mean, and portfolio confidence intervals are shown in Table 5.

The results in Table 5 show that individual confidence limits are much wider than means or portfolio limits. Confidence limits generated using formula 6 (means) and confidence limits generated within Excel using formula 7 (portfolio) have a half width difference of $183,594 in the sample portfolio shown. Recall that the degree of error introduced by using the confidence intervals for means from formula 6 is dependent on the value of m. Formula 6 uses a very large number for the number of properties, m, which in this example is actually 386. For portfolios consisting of a smaller number of properties, the error is larger, resulting in a larger underestimation of the confidence interval half width. For CMBS portfolios, it is common to have a relatively small number of properties, each with a high value. For portfolios with these characteristics, the error can be substantial.

CONCLUSIONS

For portfolio valuations it is critically important not to underestimate the risk of the portfolio as measured by the LTV ratio. Many CMBS originators and banks utilize mass appraisal techniques (multiple regression) to predict the portfolio value and LTV ratio. Risk for these portfolios is determined in part by the confidence interval half width. It is, therefore, also critical not to underestimate the portfolio confidence interval. At the same time, it is also key not to overestimate risk. An overestimation of the half width of the confidence interval can result in lower loan amounts or higher risk ratings.

Estimating a confidence interval for the aggregate value a portfolio of assets via multiple regression is an important business application left undocumented in the literature. Most practitioners are likely defaulting to using the means confidence intervals produced by software programs.

As has been demonstrated in this paper, using confidence intervals for means generated by formula 6 underestimates the half width of the portfolio confidence intervals and therefore underestimates the potential LTV ratio risk. The amount of the error is dependent on the individual property values and the number of properties in the portfolio. It is recommended that multiple regression results and formula 7 be used for estimating portfolio value and the associated portfolio confidence interval.

REFERENCES

C&I Loans Decline Further; MBSs, Commercial Real Estate Continue to Grow. (2001, 4th Quarter) The FDIC Quarterly Banking Profile, 3.

Green, G. (1999). CMBS Market Turmoil Opens Door for Commercial Banks, Commercial Investment Real Estate Journal, 18(2), 10-11.

Hair, J. F., Anderson, R. E., Tatham, R. & Black, W. C. (1995). Multivariate Data Analysis, (4th Ed.). Englewood Cliffs, NJ: Prentice Hall.

Hess, R. C. & Liang, Y. (2001). Trends in the U.S. CMBS Market, Real Estate Finance, 18(1), 9-23.

Hudson-Wilson, S. & Pappadopoulos, G. J. (1999). CMBS and the Real Estate Cycle, Journal of Portfolio Management, 25(2), 105-112.

Levy, J. B. (2001). The ground floor: Hot, hot, hot, Barron's, 81(31), 37-38.

Morgan Stanley. (2001, July 19). CSFB top CMBS league table, Commercial Mortgage Alert, 1-19.

Neter, J., Wasserman, W. & Kutner, M. (1990). Applied Linear Statistical Models, (3rd Ed.). Homewwod, IL: Irwin.

Riddiough, T. J. (2000). Forces changing real estate for at least a little while: Market structure and growth prospect for CMBS, Real Estate Finance, 17(1), 52-61.

Steven E. Moss, Georgia Southern University

Susan R. Williams, Georgia Southern University

John N. Dyer, Georgia Southern University

Steven Laposa, PricewaterhouseCoopers LLP

Table 1: [R.sup.2] Model Summary

 R R Square Adjusted R Square Std. Error of the Estimate
811 .658 .652 $24,032.48

Predictors: (Constant), Location dummy, Lot size, Sq Ft, Age

Table 2: ANOVA Table

 Sun of Squares df

Regression 252,627,861,593 4
Residual 131,106,153,522 227
Total 383,734,015,115

 Mean Square F Sig.

Regression 63,156,965,398 109.35 .000
Residual 577,560,148
Total

Predictors: (Constant), Location dummy, Lot size, Sq Ft, Age

Dependent Variable: Appraised Value

Table 3: Multiple Regression Coefficients

Coefficients Un-standardized Std. Error
 Coefficients B

Constant 110,882.83 8,555.55
Age -1,223.43 160.38
Lot Size 126,851.93 19,996.81
Sq Ft 45.53 5.35
Location Dummy 111,451.24 8,183.28

Coefficients Standardized t Sig.
 Coefficients Beta

Constant 12.96 .000
Age -.326 -7.628 .000
Lot Size .275 6.344 .000
Sq Ft .353 8.516 .000
Location Dummy .557 13.619 .000

Table 4: Calculation of Portfolio Confidence Interval

[X.sub.h] Age Lot SqFt LocD
 Size .
 1 10 .15 1367 0
 1 11 .10 1345 0
 1 11 .10 1345 0
 1 11 .13 1367 0

[X.sub.h] [X.sub.h.sup.T] *
 [([X.sup.T] * X).sup.-1]

 1 .026469 -.000445 .011
 1 .026680 -.000319 -.024
 1 .026680 -.000319 -.024
 1 .025475 -.000367 -.004

[X.sub.h] [X.sub.h.sup.T] *
 [([X.sup.T] * X).sup.-1]
 * [X.sub.h]

 1 -9.4E-6 -.012658 .010792
 1 -7.2E-6 -.011664 .011037
 1 -7.2E-6 -.011664 .011037
 1 -8.0E-6 -.011799 .009850

[X.sub.h] 1/m MSE MSE* sq.rt.
 (1/m+...)

 1 .002591 5.78E+08 7,729,517 2,780
 1 .002591 5.78E+08 7,870,610 2,805
 1 .002591 5.78E+08 7,870,610 2,805
 1 .002591 5.78E+08 7,185,136 2,681

[X.sub.h] t +- [Y.sub.h]

 1 1.97 5,478 179,912
 1 1.97 5,528 171,344
 1 1.97 5,528 171,344
 1 1.97 5,282 176,151

Continued to property number 386

Table 5: Confidence Intervals

 Lower Upper Half width

Individual (summed) $61,087,955 $98,097,648 $18,048,846
Means (summed) $76,986,041 $82,199,562 $2,606,761
Portfolio $76,802,447 $82,383,156 $2,790,355
Point Estimate (Yh) = $79,592,802
m = 386