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Benford’s Law and Property Appraisals for Private-label Mortgages

1. Introduction A mathematical property, which has become known as Benford’s Law, was discovered

independently by Newcomb (1881) and Benford (1938). Benford’s Law holds that, contrary to

intuition, the digits in large sets of positive-valued, naturally occurring numbers that range over

many orders of scale are not uniformly distributed: instead they often (not always) follow a

logarithmic distribution such that numbers beginning with smaller digits appear more frequently

than those beginning with larger ones. Because manipulated, unrelated, or created numbers

usually do not follow a Benford distribution, Benford’s Law has been used to identify suspicious

data in a variety of settings. Financial auditors, for example, routinely check data for compliance

with Benford’s Law (Kumar and Bhattacharya, 2007). Although Benford’s Law is not without

its critics (e.g. Diekmann and Jann, 2010), a growing body of empirical evidence suggests that

fraud may be a possibility when data deviates from the Benford distribution.

This paper reports the first application of Benford’s Law to property appraisals for

private-label mortgages. Using ABSNet Loan which covers the majority of the private mortgage

securitization industry, we examine whether the distribution of the first digit in house appraisal

values significantly deviate from Benford’s distribution. Our empirical results show significant

deviations from a Benford’s distribution in the period leading up to the financial crisis. Also,

property appraisal values for large originators’ mortgages do not conform to what Benford

predicts with the largest deviation for WaMu that had $9.9 million settlement for its appraisal

fraud with eAppraiseIT. We also identify loan characteristics that are closely related to

nonconformity of appraisal data with the Benford’s distribution. Above all, ‘exotic’ loans with

the features of negative amortization, balloon payments, and interest-only payments significantly

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deviate from the distribution of natural data. First-lien mortgages show conformity to Benford’s

distribution while second-lien loans do not. Regarding loan purpose, cash-out refinancing and

purchase loans conform while refinance, debt consolidation, home improvement, and

construction loans do not conform. Owner occupied and second home loans acceptably conform

while non-owner/investment loans only marginally conform. Adjustable rate mortgages (ARM)

had slightly larger deviations than fixed rate mortgages (FRM) although both types of mortgages

conform to Benford’s distribution. Deviations in house value appraisal are not restricted to the

four states of Arizona, California, Florida, and Nevada hit the hardest by the subprime mortgage

crisis. In most of the states where non-agency mortgages were popular, property appraisals show

significant deviations.

The remainder of the paper is organized as follows. Section 2 provides a brief review of

Benford’s Law. The third section contains a review of the literature related to Benford’s law.

Section 4 introduces the data and presents the empirical results regarding the conformity or

deviation in property appraisals from Benford’s distribution in terms of loan vintages, originators,

and major loan-level characteristics. Section 4 concludes.

2. Benford’ Law

According to Benford’s Law the expected occurrence or proportion of a given number (a)

as the first digit in a number set (P1a) can be calculated using equation (1).

P1a = log10 (a + 1) – log10 (a) (1)

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Further, the expected proportion of a given number (a) as the first digit and the number (b)

as the second digit (P1a2b) can be calculated using equation (2).

𝑃1𝑎2𝑏 = log10 (𝑎 +𝑏+1

10) − log10 (𝑎 +

𝑏

10) (2)

And equation (3), which sums equation (2) over all possible a values for a particular b

value yields an overall expected proportion for b as the second digit (proportions are shown in

the third column of Exhibit 1).

𝑃1𝑎2𝑏 = ∑ (𝑙𝑜𝑔10 (𝑎 +𝑏+1

10) − 𝑙𝑜𝑔10 (𝑎 +

𝑏

10))9

𝑎=1 (3)

The expected proportion of each number in the third, and all subsequent, digits can be

similarly derived. Exhibit 1 shows the proportion of each number in the first through fourth

digits as predicted by Bedford’s Law. Note that the proportions shown in Exhibit 1 are skewed

towards 1 for the first digit (because zero cannot be a first digit) and towards zero for subsequent

digits.

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Table 1

Expected Proportions Based on Benford’s Law

Number 1st digit 2nd digit 3rd digit 4th digit

0 .11968 .10178 .10018

1 .30103 .11389 .10138 .10014

2 .17609 .19882 .10097 .10010

3 .12494 .10433 .10057 .10006

4 .09691 .10031 .10018 .10002

5 .07918 .09668 .09979 .09998

6 .06695 .09337 .09940 .09994

7 .05799 .09035 .09902 .09990

8 .05115 .08757 .09864 .09986

9 .04576 .08500 .09827 .09982

Source: Nigrini (1996)

3. Literature Review

Newcomb’s mathematical discovery was ignored for nearly six decades until Bedford

rediscovered it, and for another six decades after its rediscovery published empirical applications

of Bedford’s Law were sparse. In recent years, however, empirical studies have mushroomed.

A variety of data has been shown to follow Benford’s Law including, among others, aggregated

data reported to American (Nigrini, 1996) and Italian (Mir, et al, 2014) taxing agencies, prices in

various stock markets (Ley, 1996) and eBay auctions (Giles 2007).

As mentioned in the introduction, financial auditors routinely check data for compliance

with Benford’s Law. For example, McGinty (2014) relates the results an audit of a national call

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center. Several hundred call center operators were authorized to issue refunds up to $50

(anything larger required the permission of a supervisor), and each operator had processed more

than 10,000 refunds over several years. Auditors decided to check whether the first digit of each

operator’s refunds was consistent with Benford’s Law. For most operators no discrepancy was

discovered, but for a small group there was a large spike in the 4 category indicating that lots of

refunds just below the $50 threshold were being issued. Further investigation revealed that these

operators had issued thousands of dollars in fraudulent refunds to themselves, family and friends.

Deviations from a Benford distribution are not necessarily a result of fraud. McGinty

(2014a) describes another case in which reasonable explanations for incongruities were

discovered. Auditors ran a Benford test on three types of a client’s expense accounts. Two

ended up exactly as predicted by Benford’s Law. For the third, auto and truck expenses, 9s were

overrepresented, and 1s were underrepresented. Further investigation, however, indicated the

discrepancies were not fraudulent as many employees were simply following company policy

which allowed them to expense gas purchases and to combine expenses as long as the combined

amount didn’t exceed $100. The price of a tank of gas had effectively eliminated 1s from the

equation, and combining expenses increased the frequency of 9s.

Because Benford’s Law works best with large data sets, many researchers using

Benford’s Law to analyze the private sector use data from an entire industry or groups of

companies rather than focusing on a particular company; a procedure followed in the present

study. Some of these studies report small irregularities or data that conformed to the Bedford

distribution. Alali and Romero (2013a) conducted tests on a variety of accounts from financial

statements of American banks that failed between October, 2000 and February 2012. First, they

compared the distribution of the first digit in the accounts to Benford’s theoretical distribution.

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They also computed what Nigrini (1996) coined the distortion factor model which equals the

difference in the mean of the observed first two digits compared to the expected mean according

to Benford’s Law. They report no significant anomalies. Özer and Babacan (2013) examine the

first digit in annual off-balance sheet disclosures of Turkish Banks over the period 1990-2010

and report significant deviations between the distribution of the reported numbers and Benford’s

theoretical distribution for only one year: 1999. Gava and Vitiello (2014) compared the

distribution asset accounts first digits for fourteen Brazilian companies over the time period 1986

through 2009 to the Benford distribution. The study period contains periods of high and low

inflation, and they found that the data from the low-inflation period fit better to Benford’s law

than data from the high-inflation period, and suggest that high inflation increases the possibility

of fraud.

Other researchers report suspicious data. Johnson (2009) used Benford’s Law to analyze

the first digit of quarterly net income and earnings per share data for twenty-four randomly

selected publicly-traded companies for fiscal years 1999 through 2004 to identify firm

characteristics that may be associated with earnings management. Johnson identified several firm

characteristics where earnings management appeared possible because the earnings distributions

were inconsistent with Benford’s Law, including (1) companies with low capitalization (below

$45 billion), (2) companies with higher levels of inside trading (3% and higher), and (3)

companies that have been publicly traded for less than 25 years.

Hsieh and Lin (2013) analyzed the second digit of quarterly net income reported by 8,817

firms in the U.S. marine industry between the 1st quarter of 1980 and the 1st quarter of 2009.

Finding significantly more zeros in the second digit than would be expected in a Benford

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distribution, they conclude that managers in the industry engage in managing earnings through

rounding earnings numbers to achieve key reference points.

Several researchers have used Benford’s Law to scrutinize government entities.

Michalski and Stoltz (2013) analyzed data from 1989 through 2007 using Benford’s Law and

conclude that some countries strategically provide manipulated financial data to economic agents.

They observed non-Benford distributions for the first digits of data issued by groups of countries

that: are more vulnerable to high capital outflows, have fixed exchange rate regimes, have the

highest levels of net indebtedness and those that were running current account deficits. In

addition, they report rejection of the Benford distribution for the first digits of the balance of

payments statistics for euro-adopting countries after these countries joined the euro zone.

Johnson and Weggenmann (2013) subjected the first digits in a small set of American state

government data to Benford’s Law. The accounts for each of the fifty states examined were: (1)

total general revenues of the primary government, (2) total fund balance of the general fund, and

(3) total fund balance of governmental funds; all of which are often used as benchmarks in

financial analysis. Most authorities (e.g., Durtschi, 2004) agree that Benford’s Law is most

effective when applied to large data sets, but in Johnson and Weggenmann study only three

(unidentified) years of data were collected, yielding 150 data points for each state/balance. The

authors report distributions in conformity with Benford’s Law for the first two accounts, but

nonconformity for the total fund balance of governmental funds. de Freitas Costa, et.al. (2012)

analyzed 134,281 contracts issued by 20 management units in two Brazilian states and

discovered significant deviations in the distribution of the 1st and 2nd digits from the distribution

predicted by Benford’s Law. The first digit of the contract data contained an excess amount of

the numbers 7 and 8 while 9 and 6 were rare occurrences which the authors assert denoted a

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tendency to avoid conducting the bidding process. Analysis of the 2nd digit revealed a

significant excess of the numbers 0 and 5 which indicated the use of rounding in determining the

value of contracts.

4. Data and empirical findings

4.1. ABSNet Loan Data 1 ABSNet Loan is one of the most popular loan-level data sources competing with

LoanPerformance from First American CoreLogic, Lender Processing Servies (LPS or formerly

known as McDash Analytics), and BBx data from BlackBox Logic LLC. ABSNet Loan

normalizes and provides loan-level information based on non-agency RMBS performance data

from various trustees and servicers. ABSNet data contains loan-level underwriting characteristics

at the time of origination, and monthly performance and payment information for 22 million

mortgages securitized by private institutions. Vintages for mortgages and HELOCs in ABSNet

date back to 1950 for loan origination and to 1988 for deal closing, However we focus on the

period from 2002 and 2007 when the data has reasonable coverage for the entire private

mortgage securitization industry. As shown in Tables 1 and 2, ABSNet covers 18 million loans

and 5,564 deals between 2002 and 2007. We do not consider the post-crisis period after 2007

when the private-label mortgage market was practically frozen.

Table 1. Loan vintages covered in ABSNet Loan

Loan origination year

Freq Pct Loan origination year

Freq Pct

1950 177 0 1984 11701 0.06

1951 1 0 1985 8134 0.04

1954 8 0 1986 9869 0.05

1955 2 0 1987 15262 0.07

1957 1 0 1988 18942 0.09

1958 4 0 1989 24689 0.12

1959 1 0 1990 15829 0.08

1 See http://www.lewtan.com/products/absnetloan.html.

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1960 4 0 1991 25986 0.12

1961 4 0 1992 48175 0.23

1962 6 0 1993 69817 0.33

1963 11 0 1994 58503 0.28

1964 6 0 1995 40602 0.19

1965 28 0 1996 80094 0.38

1966 31 0 1997 181093 0.86

1967 39 0 1998 440716 2.09

1968 115 0 1999 513264 2.43

1969 293 0 2000 529794 2.51

1970 201 0 2001 878389 4.17

1971 510 0 2002 1454214 6.9

1972 1055 0.01 2003 2381305 11.29

1973 1129 0.01 2004 3711168 17.6

1974 1031 0 2005 4924732 23.36

1975 1321 0.01 2006 4375701 20.75

1976 6136 0.03 2007 1213662 5.76

1977 3411 0.02 2008 3500 0.02

1978 3725 0.02 2009 387 0

1979 4006 0.02 2010 459 0

1980 3020 0.01 2011 1122 0.01

1981 2516 0.01 2012 4164 0.02

1982 1853 0.01 2013 5157 0.02

1983 5316 0.03

Table 2. Deal vintages covered in ABSNet Loan

Deal closing year

Freq Pct Deal closing year

Freq Pct

1988 6 0.09 2002 505 7.41

1989 13 0.19 2003 785 11.52

1990 6 0.09 2004 990 14.53

1991 27 0.4 2005 1254 18.41

1992 26 0.38 2006 1226 18

1993 77 1.13 2007 804 11.8

1994 68 1 2008 26 0.38

1995 30 0.44 2009 2 0.03

1996 69 1.01 2010 1 0.01

1997 72 1.06 2011 2 0.03

1998 166 2.44 2012 6 0.09

1999 157 2.3 2013 11 0.16

2000 151 2.22 2014 4 0.06

2001 329 4.83

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To examine whether the distribution of the first digit in property appraisal values, we

focus on OriginalAppraisedValue in ABSNet loan that is defined as “the appraised value of the

property at the time of underwriting.” Table 3 shows two sets of summary statistics for

OriginalAppraisedValue for the entire period and for the period of interest between 2002 and

2007. To avoid the possibility that our results are driven by the difference in distribution between

mortgages and HELOC, and we strictly focus on mortgages for our analysis. ABSNet documents

that OriginalAppraisedValue is calculated using LTV and original loan balance if the appraisal

prices are not provided from trustees and servicers. Among 20 million mortgages whose

appraisal information is available, we focus on 17.8 million loans that averages $353,960 and

ranges between $195 and $230 million.

Table 3. Summary statistics for original appraisal values in ABSNet Loan

N Obs N Miss Mean Min Max

the entire period 20,781,988 644,136 $339,195 $1 $230,000,000

2002 - 2007 17,839,015 221,767 $353,960 $195 $230,000,000

4.2. Deviations by loan vintage In this section, we examine whether deviations in property appraisal values from

Benfords’ Law are related to when mortgages were originated. We first calculate the actual

distributions of the first digit in house appraisal prices year by year from 2002 to 2007 for which

ABSNet Loan has a decent coverage of industry. As shown in Table 4, the number of

originations for loans privately securitized dramatically increases in the period leading up to the

financial crisis hitting 4.86 million in 2005. Interestingly, the portion of loans whose appraisal

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information is missing generally increases from 0.62% in 2003 to 3.81% in 2007, which implies

the possibility of deterioration in appraisal process.

Table 4. Actual distribution of the first digit in property appraisal values

Originated in

First digit 2002 2003 2004 2005 2006 2007 1 34.43% 34.59% 33.99% 30.63% 28.16% 25.72% 2 15.84% 17.57% 20.10% 20.90% 21.62% 19.46% 3 8.67% 10.04% 12.77% 14.05% 14.52% 13.03% 4 8.66% 7.83% 8.57% 9.38% 9.69% 9.15% 5 7.92% 7.41% 6.48% 7.39% 7.65% 8.85% 6 7.25% 6.74% 5.40% 5.84% 6.48% 8.84% 7 6.41% 5.88% 4.61% 4.63% 4.82% 6.46% 8 5.94% 5.46% 4.37% 3.97% 3.92% 4.88% 9 4.88% 4.48% 3.71% 3.22% 3.12% 3.61%

N Obs 1,429,814 2,366,622 3,689,843 4,864,008 4,321,249 1,167,479

N missing 24,400 14,683 21,325 60,724 54,452 46,183

missing rate 1.68% 0.62% 0.57% 1.23% 1.24% 3.81%

Table 5 presents how much the distribution of each number in the first digit of actual

appraised values deviates from the expected distribution based on Benford’s Law. We calculate

the amount of deviations using mean absolute deviation (MAD), which is the average of absolute

values of the difference between actual and expected portions for each significant digit.

Following Drake and Nigiri (2000), we break down MAD value into four different ranges to

determine the goodness-of-fit.

MAD: 0.000±0.004 (close conformity)

MAD: 0.004±0.008 (acceptable conformity)

MAD: 0.008±0.012 (marginally acceptable conformity)

MAD: greater than 0.012 (nonconformity)

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As shown in Table 3-2, the distribution of the first digit in loan appraisal values

marginally conforms to Benford’s Law only for 2003 and 2005 with MAD values of 1.1% and

1.19%. The year with the largest MAD is 2004 for which the first digit is populated with 1 or 2

more than Benford’s Law by 3.89% and 2.49%. In 2002 and 2003, the actual distribution is

skewed more to 1 sacrificing the portion of 3, while in more recent years in 2006 and 2007, the

portion of 1 is smaller than natural and digit 2 or 6 appear more frequently than Benford predicts.

Table 5. Deviations in the first significant digit from Benford’s Law

Originated in

First digit 2002 2003 2004 2005 2006 2007 1 4.33% 4.49% 3.89% 0.53% -1.94% -4.38% 2 -1.77% -0.04% 2.49% 3.29% 4.01% 1.85% 3 -3.82% -2.45% 0.28% 1.56% 2.03% 0.54% 4 -1.03% -1.86% -1.12% -0.31% 0.00% -0.54% 5 0.00% -0.51% -1.44% -0.53% -0.27% 0.93% 6 0.55% 0.04% -1.30% -0.86% -0.22% 2.15% 7 0.61% 0.08% -1.19% -1.17% -0.98% 0.66% 8 0.83% 0.35% -0.75% -1.15% -1.20% -0.24% 9 0.30% -0.10% -0.87% -1.36% -1.46% -0.97%

MAD 1.47% 1.10% 1.48% 1.19% 1.34% 1.36%

3.3 Deviations by originators Mortgage properties are appraised in the process of underwriting for which originators

have major controls. Therefore, we break down our sample by originators to examine whether

appraisal deviations are more severe for a particular group of originators. Table 6 presents how

mortgages closed by different group of originators have the difference in the portion of each

number in the first digit of property appraisal values between actual and Benford’s distributions.

The distributions are listed by size in terms of their market share. Originators are defined to be

large if they are ranked above 20th in terms of the number of originations they have made.

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Originators ranked below 20th in terms of their business size are separately categorized to be

“small originators.” Mortgages whose originator identity information is not available is

categorized to be “unidentified originators.” For all three groups, actual distributions of the first

digit in appraised prices acceptably conform to Benford’s Law, however MAD is higher for large

originators by 0.19% than for small originators, which implies the possibility of poorer appraisal

process among large originators.

Table 6. Difference in proportion of the first digit between actual and expected distributions

First digit Large originators small originators Unidentified originators

1 2.08% 0.15% -0.46%

2 1.06% 1.63% 1.05%

3 -0.31% 0.51% 0.44%

4 -0.92% -0.18% 0.06%

5 -0.63% -0.08% 0.25%

6 -0.18% -0.09% 0.21%

7 -0.36% -0.49% -0.28%

8 -0.21% -0.52% -0.38%

9 -0.53% -0.93% -0.88%

N obs 12,068,607 8,713,381 5,994,009

N missing 338,884 305,252 269,665

Missing rate 2.73% 3.38% 4.31%

MAD 0.70% 0.51% 0.45%

If we break down the aggregate sample for large originators into individual institutions,

the hypothesis of poorer appraisals among large originators seem to be more likely.

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Table 7. Difference in proportion of the first digit between actual and expected distributions

First digit Countrywide RFC Wells Fargo

Option One

New Century

First Franklin IndyMac WaMu Ameriquest Long Beach Fremont

1 -0.98% 8.57% -3.67% 6.55% 3.22% 7.92% -9.23% -6.40% 12.32% 2.01% -1.11%

2 3.28% -1.15% -4.68% 2.68% 4.03% 3.50% 1.05% -7.54% 3.17% 1.99% 6.37%

3 1.39% -2.55% -3.45% -0.51% 2.13% -0.97% 3.53% -5.24% -1.88% 0.82% 3.93%

4 0.27% -2.74% -0.41% -2.77% -0.67% -2.70% 3.34% 0.23% -4.45% -0.62% 1.12%

5 -0.04% -2.06% 2.84% -3.00% -2.16% -2.71% 2.60% 4.80% -4.51% -1.05% -1.06%

6 -0.30% -0.84% 4.16% -1.77% -2.11% -1.97% 1.59% 6.19% -3.46% -0.93% -2.08%

7 -1.05% -0.21% 2.93% -0.84% -1.94% -1.56% -0.17% 4.36% -1.77% -0.95% -2.53%

8 -1.15% 0.51% 1.81% -0.03% -1.31% -0.85% -1.00% 2.73% 0.15% -0.63% -2.41%

9 -1.43% 0.46% 0.49% -0.31% -1.19% -0.66% -1.70% 0.86% 0.42% -0.65% -2.26%

N Obs 2,613,012 2,404,130 872,110 749,190 643,231 558,741 462,769 453,532 443,755 435,476 375,878

N missing 86,172 33 513 9,072 183 2 504 125 1 52 3

missing rate 3.19% 0.00% 0.06% 1.20% 0.03% 0.00% 0.11% 0.03% 0.00% 0.01% 0.00%

MAD 1.10% 2.12% 2.72% 2.05% 2.08% 2.54% 2.69% 4.26% 3.57% 1.07% 2.54%

Conformity Marginally acceptable No No No No No No No No

Marginally acceptable No

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Table 7. (cont’d) Difference in proportion of the first digit between actual and expected distributions

First digit BOA WEYERHAEUSER Argent Impac Chase

Manhattan AHM Centex GMAC First

Horizon

1 -2.33% -4.94% 6.94% -0.55% 2.91% -5.41% 7.18% -4.95% -3.24%

2 -6.00% 5.59% 5.52% 5.13% -3.23% 3.89% -3.04% -2.61% 4.06%

3 -4.76% 5.98% 0.95% 3.78% -3.37% 3.79% -4.16% -2.54% 2.29%

4 0.12% 2.18% -2.25% 0.91% -2.37% 1.51% -3.50% 0.30% -0.68%

5 3.70% -0.22% -3.67% -1.16% -0.03% 0.59% -1.62% 3.99% -0.59%

6 4.05% -1.55% -3.73% -1.89% 1.91% -0.06% 0.40% 3.70% 0.35%

7 2.84% -2.34% -2.15% -2.30% 1.85% -0.98% 1.47% 1.85% -0.19%

8 1.83% -2.36% -0.91% -2.02% 1.65% -1.48% 2.01% 0.67% -0.71%

9 0.57% -2.34% -0.70% -1.91% 0.68% -1.86% 1.26% -0.40% -1.31%

N Obs 323,127 331,696 332,917 286,715 286,610 226,962 161,547 49,102 58,107

N missing 21,783 3,576 342 135 119 22,773 3,238 103,956 86,347

missing rate 6.32% 1.07% 0.10% 0.05% 0.04% 9.12% 1.96% 67.92% 59.77%

MAD 2.91% 3.05% 2.98% 2.18% 2.00% 2.17% 2.74% 2.33% 1.49%

Conformity No No No No No No No No No

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As shown in Table 7, Among the group of large originators, only mortgages closed by

Countrywide and Long Beach show marginally acceptable conformity to Benford’s Law with

MADs of 1.1% and 1.07%. If deviations from Benford’s distribution is meaningfully related to

manipulation of data, it would not be surprising to see the largest MAD of 4.26% for WaMu that

had $9.9 million settlement for its appraisal fraud with eAppraiseIT.

The nonconformity in terms of MAD prevalent among the largest players including

WaMu, Ameriquest, WEYERHAEUSER, Argent, BOA, etc. doesn’t seem to be related to a

particular business model or loan processing channels. Also, as shown in Figure 1, this non-

conformity doesn’t seem to be driven by any selective missing for appraised values with a

particular number as the first digit.

Figure 1.MAD and missing rate for house appraisal values by originators

Interestingly, for small originators ranked below 20th and among unidentified originators,

MAD is significantly small implying acceptable conformity to Benford’s law. This doesn’t seem

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

3.00%

3.50%

4.00%

4.50%

0.00% 10.00% 20.00% 30.00% 40.00% 50.00% 60.00% 70.00% 80.00%

MA

D

Missing rate

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to be simply driven by large number of observations for the group of loans whose originators are

small or unidentified because Countrywide and Long Beach loans achieve conformity with even

smaller number of observations.

3.4 Deviations by loan characteristics In this section, we examine the relation between different loan characteristics and the

amount of deviations in appraised values from Benford’s Law. The most interesting finding is

riskier loans are more likely to be associated with poor appraisals at least in terms of several

major mortgage risk characteristics including lien type, negative amortization, balloon payment,

interest-only, and occupancy types. We do not find evidence that more deviations are related to

particular property locations, loan documentation types, or purpose types.

Table 8. Deviations by lien type

Lien type

First digit 1st 2nd

1 0.49% 6.05%

2 0.74% 5.09%

3 -0.22% 1.32%

4 -0.53% -1.40%

5 -0.03% -2.65%

6 0.32% -2.78%

7 -0.06% -2.49%

8 -0.11% -1.80%

9 -0.60% -1.34%

N Obs 16,325,746 2,863,022

N Missing 197,950 168,719

Missing rate 1.20% 5.57%

MAD 0.34% 2.77%

Table 8 shows how deviations in the first digit distribution of property appraised values

vary depending on lien position. A mortgage is said to be in a first lien position on the

collateralized property if the first priority is given to the lender over all other claims in case of

default. If a borrower chooses to use a second mortgage to manage her loan-to-value ratio or to

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minimize the amount of private mortgage insurance premium, her second mortgage is said to be

in the second lien.

Mortgages in the second lien positions are exposed to more default risk because those

borrowers tend to have less equity in their houses and are less willing to make a scheduled

repayment under adverse situations. As presented in Table 5, the second lien loans have more

missing values for their property appraised values and the distribution of the first digit in

appraised values does not conform to Benford’s Law with MAD of 2.77% while first lien loans

show conformity to Benford’s distribution.

Table 9. Distribution by Negative amortization

Negative Amortization

First digit No Yes

1 2.27% -14.40%

2 1.35% 0.26%

3 -0.30% 5.03%

4 -1.02% 5.41%

5 -0.70% 4.31%

6 -0.30% 2.53%

7 -0.43% 0.12%

8 -0.28% -1.19%

9 -0.61% -2.08%

N Obs 19,270,659 1,167,539

N Missing 371,532 6,872

Missing rate 1.89% 0.59%

MAD 0.81% 3.92%

Mortgages are said to be ‘exotic’ when they include one or more features to help

borrowers qualify for mortgages they couldn’t have successfully obtained otherwise. An

example of an exotic is onethat allows negative amortization by giving the borrower an option to

add deferred interests to the principal balance. Since negative amortization is engineered

particularly for borrowers with short-term disability to make monthly payments, it is reasonable

think that mortgages with negative amortization are exposed to more default risk. Table 6

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presents the amount of deviation in appraissed values from Benford’s Law depending on whether

negative amortization is allowed. As shown in Table 6, riskier loans in terms of negative

amortization tend to have more deviation in the first digit of appraised value, with MAD of

3.92%, while loans with no negative amortization conform to Benford’s Law.

Balloon payment is the second exotic feature whose effects on appraisals are studied in

this section. Mortgages are said to require balloon payments if a lump-sum bulk payment for the

remaining outstanding balance is scheduled before the end of the amortization term. As shown

in Table 10, balloon payment is more associated with larger deviation in the first digit

distribution of appraised values with MAD of 2.54%.

Table 10. Deviations by Balloon payment type

Balloon payment

First digit No Yes

1 1.56% -1.63%

2 0.76% 6.63%

3 -0.36% 3.98%

4 -0.75% 0.81%

5 -0.34% -1.01%

6 0.04% -1.90%

7 -0.21% -2.44%

8 -0.15% -2.28%

9 -0.55% -2.17%

N Obs 18,893,408 1,888,580

N Missing 599,100 45,036

Missing rate 3.07% 2.33%

MAD 0.52% 2.54%

The last exotic feature we examine is the provision of interest-only payment. Table 11

shows the difference in the amount of deviations between non-IO and IO loans. As expected

from the results for other exotic features, IO loans do not conform to Benford’s Law with MAD

of 2.57% while non-IO loans show close conformity.

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Table 11. Deviations by interest-only type

Interest Only

First digit No Yes

1 2.94% -7.33%

2 0.84% 3.45%

3 -0.69% 3.64%

4 -1.12% 2.01%

5 -0.78% 1.62%

6 -0.32% 0.86%

7 -0.34% -0.70%

8 -0.10% -1.51%

9 -0.43% -2.03%

N Obs 16950671 3337174

N Missing 600904 24650

Missing rate 3.42% 0.73%

MAD 0.84% 2.57%

Mortgages can be categorized to three different groups based on their occupancy type. A

property is said to be owner-occupied if it is used for primary residence. A house can also be

used as a second home. If a house is purchased for an investment, the property is occupied by

non-owner, which shows weaker conformity than other occupancy types.

Table 12. Deviations by occupancy type

Occupancy type

First digit Non-owner Owner-occupied Second Home

1 3.92% 0.91% 0.01%

2 0.64% 1.30% 1.79%

3 -1.52% 0.25% -0.58%

4 -2.06% -0.40% -1.30%

5 -1.10% -0.29% -0.21%

6 0.05% -0.14% 0.38%

7 0.14% -0.47% 0.29%

8 0.25% -0.41% 0.18%

9 -0.32% -0.75% -0.55%

N Obs 1,831,377 16,877,757 621,999

N Missing 19,398 242,756 5,417

Missing rate 1.05% 1.42% 0.86%

MAD 1.11% 0.55% 0.59%

21

Borrowers can use either fixed rate mortgages (FRM) or adjustable rate mortgages (ARM)

As shown in Table 13, FRMs show acceptable conformity to Benford’s distribution with MAD

of 0.49% while ARMs only marginally conform with MAD of 0.87%.

Table 13. Deviations by interest rate type

Interest rate type

First digit FRM ARM

1 1.86% 0.73%

2 0.04% 2.44%

3 -0.76% 0.76%

4 -0.81% -0.42%

5 -0.26% -0.53%

6 0.23% -0.47%

7 0.01% -0.80%

8 0.06% -0.71%

9 -0.37% -1.00%

N Obs 9930452 10851536

N Missing 546495 97641

Missing rate 5.22% 0.89%

MAD 0.49% 0.87%

Table 14 presents the distribution of the first digit in the property appraisal prices for 20

states where private-label mortgages were particularly popular during the period leading up to

the crisis. Regardless of the property location, MADs are all higher than 1.2%, or the threshold

of non-conformity. Ohio shows the highest MAD of 5.66% while VA shows the least deviation

with MAD of 1.48%. NV, CA, AZ, and FL that suffered the most from the subprime crisis do

not seem to be associated with the amount of deviations.

22

Table 14. Deviations by property states

First digit CA FL TX IL NY AZ GA VA MI

1 -15.70% 7.83% 17.06% 6.05% -13.20% 8.20% 20.60% -4.25% 15.88% 2 -2.17% 8.21% -5.31% 6.57% -4.26% 11.52% 0.14% 0.43% -3.85% 3 5.17% -0.86% -6.81% -0.49% 5.64% -1.11% -4.65% 2.38% -6.73% 4 5.99% -3.64% -5.21% -3.05% 6.94% -4.07% -4.93% 1.95% -5.60% 5 5.12% -3.61% -3.09% -2.79% 3.86% -3.94% -3.93% 1.27% -3.22% 6 3.22% -2.81% -0.96% -2.00% 2.38% -3.44% -2.87% 0.64% -0.71% 7 0.77% -2.21% 0.41% -1.71% 0.38% -2.96% -2.19% -0.42% 0.87% 8 -0.68% -1.54% 1.81% -1.26% -0.51% -2.36% -1.34% -0.78% 1.92% 9 -1.72% -1.38% 2.10% -1.31% -1.23% -1.85% -0.83% -1.22% 1.43%

N Obs 5,002,396 1,901,792 1,114,388 795,898 783,259 724,914 637,270 593,493 593,548 N Missing 140,554 38,897 30,660 20,849 18,492 20,530 19,793 21,049 19,245 Missing rate 2.73% 2.00% 2.68% 2.55% 2.31% 2.75% 3.01% 3.43% 3.14%

MAD 4.50% 3.57% 4.75% 2.80% 4.27% 4.38% 4.61% 1.48% 4.47%

First digit MD NJ WA OH CO PA MA NV NC MN MO

1 -6.83% -9.32% 0.48% 14.47% 6.13% 7.69% -11.33% -6.56% 16.86% 8.36% 14.40%

2 6.52% 4.86% 11.47% -7.58% 12.21% -2.52% 5.67% 15.06% -4.05% 13.79% -6.72%

3 5.93% 7.62% 2.25% -8.52% -0.80% -4.54% 9.70% 7.48% -6.45% -3.00% -7.94%

4 0.87% 3.52% -2.01% -6.13% -3.26% -3.38% 3.31% -0.62% -5.45% -4.38% -5.51%

5 -0.30% 0.31% -2.49% -3.21% -3.24% -1.23% 0.77% -2.51% -3.29% -3.83% -2.37%

6 -0.64% -0.91% -2.43% -0.03% -2.93% 0.48% -1.09% -3.08% -1.12% -3.18% 0.67%

7 -1.57% -1.83% -2.60% 2.60% -2.93% 1.06% -2.14% -3.42% 0.39% -2.96% 2.15%

8 -1.82% -2.01% -2.37% 4.68% -2.66% 1.53% -2.31% -3.26% 1.58% -2.48% 3.04%

9 -2.14% -2.22% -2.29% 3.72% -2.53% 0.91% -2.57% -3.10% 1.53% -2.31% 2.27%

N Obs 576,836 575,909 525,652 514,907 503,789 499,255 420,328 423,628 394,860 311,317 296,892

N Missing 18,495 16,115 17,232 13,380 15,589 14,935 13,395 9,619 13,141 6,426 8,459

Missing rate 3.11% 2.72% 3.17% 2.53% 3.00% 2.90% 3.09% 2.22% 3.22% 2.02% 2.77%

MAD 2.96% 3.62% 3.15% 5.66% 4.08% 2.59% 4.32% 5.01% 4.53% 4.92% 5.01%

23

There is also little evidence that loan documentation type is associated with poor

appraisal practices. As shown in Table 15, regardless of whether borrowers income, assets, and

employment status are fully, partially or never verified, the distributions of first digit in appraised

values do not conform to Benford’s Law.

Table 15. Deviations by loan documentation type

Documentation type

First digit Full Low No Unknown

1 5.55% -4.01% -4.55% 2.42%

2 0.47% 2.02% 0.52% 1.54%

3 -1.88% 2.23% 0.43% 0.28%

4 -2.15% 1.26% 0.80% -0.77%

5 -1.36% 0.85% 1.96% -1.06%

6 -0.55% 0.45% 2.26% -0.75%

7 -0.23% -0.50% 0.21% -0.69%

8 0.21% -0.90% -0.45% -0.37%

9 -0.07% -1.38% -1.16% -0.61%

N Obs 9,213,483 7,603,108 429,422 2,074,158

N Missing 137,342 75,184 8,594 355,151

Missing rate 1.47% 0.98% 1.96% 14.62%

MAD 1.39% 1.51% 1.37% 0.94%

Variation in loan purpose exhibited some diversity in terms of conformity to a Benford’s

distribution. Table 16 shows that appraisals prepared to support a mortgage for a house purchase,

cash-out refinancing, or refinancing are associated with acceptable conformity to Benford’s Law

with MAD values ranging from 0.75% to 1.17%, but appraisal prepared to support loans for debt

consolidation, home improvement or construction did not demonstrate any conformity.

24

Table 16. Deviations by loan purpose type

First digit Cash-out refi Debt consolidation

Home improvement Construction Purchase Refinance

1 0.34% 25.25% 8.42% 2.19% 1.46% 0.59%

2 2.59% -5.36% -2.91% -3.01% 1.59% -2.19%

3 0.93% -8.51% -3.51% -2.06% 0.31% -2.25%

4 -0.50% -7.22% -3.01% -1.27% -0.45% -0.82%

5 -0.80% -5.39% -1.63% 0.13% -0.32% 0.95%

6 -0.54% -3.03% -0.28% 0.91% -0.26% 1.65%

7 -0.67% -0.59% 0.61% 0.88% -0.68% 1.15%

8 -0.48% 1.78% 1.26% 1.48% -0.68% 0.84%

9 -0.85% 3.07% 1.05% 0.76% -0.97% 0.09%

N Obs 7,387,335 341,593 25,805 24,029 8,600,425 2,849,287

N Missing 110,243 32,145 6,366 811 86,122 26,827

Missing rate 1.47% 8.60% 19.79% 3.26% 0.99% 0.93%

MAD 0.85% 6.69% 2.52% 1.41% 0.75% 1.17%

5. Conclusion This paper reports the results of an investigation of whether real property appraisals for

private-label mortgages conform to Benford’s Law. Using data that covers the majority of

private mortgage securitization industry for the six year period leading up to the recent financial

crisis, we calculate the distribution of the first digit in house appraisal values and compare them

with a Benford’s distribution. Significant deviations were discovered. Property appraisal values

for large originators’ mortgages did not conform to what Benford predicts with the largest

deviation discovered for loans originated for WaMu which coincidently had a $9.9 million

settlement for its appraisal fraud with eAppraiseIT. We also identify loan characteristics that are

associatedwith nonconformity of appraised values with the Benford’s distribution. Exotic loan

features including payment terms that result in negative amortization, balloon payments, and

interest-only loans are significantly associated with more deviation from the distribution of

natural data. First-lien mortgages show conformity to Benford’s distribution while second-lien

loans do not. Regarding loan purpose, cash-out refinancing and purchase loans conform while

25

refinance, debt consolidation, home improvement, and construction loans do not conform.

Owner occupied and second home loans were found to be in acceptable conformance while non-

owner/investment loans only marginally conformed.

The results presented here suggest the possibility that appraised values for certain loan

types have been subject to manipulation in that the distribution of their first digit does not

conform to Benford’s Law. But, nonconformance does not guarantee that data has been

manipulated. Therefore, we consider this study preliminary in nature and plan next to search for

reasons that may explain the deviations reported here.

26

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