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1
The Improved Net Rate Analysis
A discussion paper presented at Massey School Seminar of Economics and Finance, 30
October 2013.
Song Shi
School of Economics and Finance, Massey University, Palmerston North, New Zealand
Email: [email protected]; Tel: 0064 6 3569099-84070; Fax: 0064 6 3505660; Address: SST Building 4.29, School of Economics and Finance, Private Bag 11222, Massey University, Palmerston North 4442, New Zealand
2
The Improved Net Rate Analysis
Abstract
This paper proposes an improved net rate analysis using the assessed land values in real
estate appraisals. Compared to the traditional sales comparison approach, the method has
greatly simplified the comparison process and increased the selection of comparable
properties from the neighbourhood level to the suburb or even city level. This is very
attractive in thin markets where comparable properties are limited. Simulations based on the
theoretical and empirical data suggest that the method benefits much from the law of
compensation of errors. The method performs well at the widely accepted 10% margin of
valuation errors, at least for this New Zealand dataset. Guidance on the optimum selection of
comparable properties, the size and directions of the appraisal errors are also discussed in the
paper.
Keywords: net rate analysis, sales comparison approach, property valuations, assessed values,
thin markets
3
1. Introduction
The sales comparison approach has been seen as a key traditional appraisal theory for
many years. The method involves collecting sales of comparable properties at the time of
valuation, and placing them on a sales adjustment grid to derive the estimated market value
for the subject property. The methodology is simple and is taught in many appraisal
textbooks (see e.g. Betts and Ely (2008) and others). The successfulness of the sales
comparison method largely depends on the selection and number of comparable sales.
Several statistical techniques exist in the literature on the optimal selection and weighting
of comparable properties. For example, Vandell (1991) suggested the use of the minimum
variance for selecting and weighting comparable sales, while Gau, Lai and Wang (1992, 1994)
presented the use of coefficient of variation as the selection criteria. However, as pointed out
by Epley (1997) all these methods are
“based on the presumption that a sufficient number of closed sales of comparable
properties always exist in a finite time period such that a statistically reliable sample can be
found…The theory is good, but not measurable or applicable (p.175-176)”.
More recently, Lai, Vandell, Wang and Welke (2008) proposed the use of replication
method as an alternative to the traditional grid (and/or regression) method in estimating
property values. The proposed replication method shall reduce the degree of subjectivity in
the grid method and require a small sample size when compared to the regression model. For
the replication method to be successful, it requires the number of comparable properties to be
more than the number of property attributes. Again, this makes the method difficult to apply
with a small sample size of comparable properties being less than five.
In selecting comparable properties, appraisers often confine their selection of sales within
the same neighbourhood or suburb in order to minimise the location difference in the
4
appraisals. This could worsen the problem in a downward market when the number of
periodical property transitions is small. The purpose of this paper is to propose a technique
called “the improved net rate analysis” in property valuations. The method involves using the
assessed property land values to separate improvement values from sales for comparisons,
based on a net rate of per square meter of dwelling floor area (called “the net rate”).
Nowadays properties are typically reassessed on a regular basis for taxation purposes. When
combined with transaction data, the rating valuations can be used for improving the
traditional grid analysis. The results show that the proposed net rate analysis can potentially
increase the selection of comparable properties to the city level rather than at a
neighbourhood or suburb level, without compromising the appraisal accuracy.
The remainder of this study is organised as follows: Section 2 describes the net rate
methodology. Section 3 presents the simulation framework. Section 4 describes the empirical
data utilised. Section 5 reports the empirical results. Section 6 provides conclusions.
5
2. Methodology
2.1 The traditional sales comparison approach
For the traditional sales comparison method, the ith property’s sale price at time period
t1can be written as:
(1)
Where represents the ith property’s sale price at time period t1, is itsland
value and is its structure value at time t1.
Based on equation (1), the sale price for the jth property at time t2can be written as:
( ( ) ) ( ( ) ) ∑ (2)
Where ( ) represents the vector of location adjustments in percentage between
the ith and jth property, and ( ) represents the vector of structure adjustments in
percentage between the two properties. is the time impact for property sale prices
between the time period of t1and t2.
For the vector of location variables, this includes both the land and neighbourhood
considerations. For land, this mainly includes the consideration of the land shape, size,
contour, access and view. For neighbourhood, this mainly includes the distance to school,
hospital, CBD and neighbourhood aggregated incomes, employments, etc. For the vector of
structure variables, this includes the main dwellings, out buildings and other site
improvements. For the main dwellings, comparisons will take place among floor area,
exterior cladding materials, quality of construction, number of bathrooms and modernisation,
etc.
6
The concept and process of traditional comparing approach are based on the above
elements adjustment technique. Real estates are regarded as heterogeneous products. Often,
there are many property attributes need to be considered. For the above method to be
workable in practice, appraisers often choose similar properties sold in the same locality in
order to minimise the element adjustment process for the location difference. When there are
limited sales, the traditional comparable approach will not be working effectively and the
appraisal results may be subjective.
2.2 The net rate analysis
The technique of the net rate analysis involves the following steps
1) Estimate the land value of the each comparable property at sales first.
2) Calculate the building value of each comparable property by deducting the estimated
land values from their respective sale prices.
3) Calculate the building value of the subject property. Comparison adjustments are
based on the structure (building) difference between the subject property and each
comparable property.
4) Calculate the market value of subject property by adding the estimated building value
in step 3) to its market land value.
The above net rate analysis procedure can be written into following equations:
(3)
( ( ) ) ∑ (4)
Where
7
is the estimated building value of the ith comparable property at time t1;
isthe sale price of the ith comparable property at time t1;
is the estimated market land value of the ith comparable property at sales;
represents the estimated market value for the jth(subject) property at time t2;
isthe subject property’s market land value at time t2
( ) represents the building structure adjustments in percentage between the
subject property and the ith comparable property.
represents the time impact on the building value during the time period from t1to t2.
denotes for the market value.
The above method posts a tremendous burden of estimates in practice. Not only because
there are limited comparable land sales, but also the land values must be estimated as at the
date of sale (Jefferies, 1991).
2.3 The improved net rate analysis
To simplify the above estimation procedure, we propose to replace the market land values
with their assessed land values (rating land valuations) respectively in the above equations (3)
and (4), and we have:
( ( ) ) ∑ (5)
Where represents the estimated market value of subject property using assessed
land values. represents for the subject property’s assessed land value at time
t2. is the derived building value for the ith comparable property using the
assessed land value at time t1. A denotes for the assessed value.
8
Compared to Equation (4), Equation (5) has greatly simplified the comparison process as
market land values are no longer to be estimated. Instead, property assessed land values are
freely to obtain and ready to use. However, there are problems when using assessed land
values to proxy property’s market land values. First, there are random measurement errors in
assessed values. Second, assessed values may not be consistently estimated. Although
systematic errors are discouraged and audited by various statistical tests at the time of
assessment, both horizontal and vertical inequities have been found in empirical studies(Allen
& Dare, 2002; Cornia & Slade, 2005; Goolsby, 1997). The problem of random errors in
assessed values can be addressed by including more comparable sales in the net rate analysis.
On the other hand, empirical studies on vertical inequities in tax assessment generally show
the problem of inconsistency is small (see, e.g., Clapp (1990), Sirmans, Diskin & Friday
(1995) and Cornia & Slade (2005)). For land, the problems of random and systematic errors
in assessed land values are likely to be even smaller. This is because that land tends to be
homogeneous in nature and is assessed as no improvements on it1.
To address the concern of appraisal errors by using the assessed land values instead of
using the estimated market land value in equation (5), we measure the possible appraisal
errors as follows:
( ( ) ) (6)
Since
(
,
( ( ) ) (7)
1 At least this is the case in New Zealand (see the Rating Valuations Act 1998 for the definition of land value).
9
Assuming the assessed land valuations are consistent and market value of lands are
proportional to their assessed land values. The appraisal errors in equation (7) can be further
re-arranged as the change of (The proof can be found in the appendix):
((β-1))/((γ*β(1+δ))/(α-(1+δ))+1),
(8)
Where
α=
, the ratio of the jth (subject) property’s assessed land value to the ith
(comparable) property’s assessed land value
β =
, the ratio of the ith (comparable) property’s assessed land value to its
market land value at time of sale
γ= , the ratio of the ith (comparable) property’s sale price to its assessed
land value at time of sale
δ= ( ) , building structure adjustments in percentage between the subject property
and comparable property
3. Simulation
The above equation (8) shows that the appraisal errors using the assessed land values in
estimating the market value of subject property are depending on four factors. First, the ratio
of assessed land values between the subject and each comparable property. Second, the ratio
of assessed land values to market land values. Third, the ratio of each comparable property’s
sale price to its assessed land value. Fourth, the level of structure differences between the
10
subject and each comparable property. It is interesting to see under what combinations, the
appraisal error under the improved net rate analysis is acceptable.
One advantage of using the improved net rate analysis is that it will increase the sample
size of comparable properties. Since land components do not require adjustments in the
valuation process, the whole suburb or even the whole city’s sales can be potentially used as
comparable properties purely based on the building structure difference. As a result, the
method will be much useful for property appraisals in thin markets. Moreover, the valuation
accuracy will also benefit much from the law of compensation for errors by including more
sales for comparisons. A theoretical simulation procedure to test the overall performance of
the improved net rate analysis method by varying sample sizes of comparable properties is
arranged as follows:
1) Let changes from 0.2 to 5 with a step of 0.1
2) is drawn from a log-normal distribution
3) γ is drawn from a log-normal distribution
4) δ is drawn randomly between -0.30 and 0.30
5) Calculate the appraisal errors in equation (9) conditioned on
6) Calculate the average result of step 5) for 5 and10 times, respectively
7) Calculate the absolute value of step 6)
8) Repeat 1,000 times
9) Calculate the average result of step 7)
In the above simulation, β is set between 0.2 and 5. It is expected that assessed land
values will be no more than 5 or no less than 0.20 times of their market values. For the
11
simulation results to be useful, we change β gradually from 0.2 to 5. For , the log-normal
distribution will give a most likely range of 0.55 to 1.82 (one standard deviation). For , the
log-normal distribution will give a most likely range of 1.67 to 3.03 (one standard
deviation).For δ, it is set between -30% and 30%. This is because for properties to be
comparable, building structure difference is unlikely beyond the above range. Assumptions
for , and δ are empirically supportive, at least for this New Zealand dataset. Step 6) will
reveal the benefit of law of compensation of errors under different sample sizes. The results
of Step 9) will show the average appraisal errors under the method of improved net rate
analysis.
For checking the results from the above theoretical simulation, we also carry out
empirical tests using actual transaction data. The simulation is arranged as follows:
1) Let changes from 0.3 to 5 with a step of 0.1
2) Randomly choose a property to be valued from the entire empirical dataset
3) Randomly choose 5 and 10 comparable sales from the same dataset
4) Calculate α and γ for each comparable sale, conditioned on
5) δ is drawn randomly between -0.30 and 0.30
6) Calculate the appraisal errors in equation (9)
7) Calculate the average result of step 6)
8) Calculate the absolute value of step 7)
9) Repeat 1,000 times
10) Calculate the average result of step 8)
4. Data
12
The data contains 1,171 single family sales in Palmerston North City, New Zealand
between March 2011 and February 2012. For each sale, it contains information of the total
sale price, sale date, assessed property value, assessed land value, floor area, land area and
other building variables including age and condition of buildings. General reassessments for
taxation purposes are carried out regularly on a 3-year basis. For this particular dataset,
assessed values were last carried out in September 2009. The summarised statistics of sales
data are presented in Table 1.
<Insert Table 1>
The Palmerston North city is a provincial city with an estimated population of 80,000 in
2012. There are currently about 30,000 owner occupied dwellings2. The average number of
property transactions is about 100 per month (see Table 1), which is about 0.3% of total
housing stock. The low level market activity could cause the problem for the use of
traditional sales comparison approach for estimating property values due to the lack of recent
comparable sales within the vicinity of the subject property to be valued. The city is inland
with a predominately flat land. Property’s land values are likely to be consistently assessed;
as such it provides a good exemplar for testing the proposed net rate analysis.
5. Results
5.1 Theoretical simulation
Table 2a and 2b show the point estimates of appraisal errors of equation (8) for a set of
values α and δ. Table 2a shows the estimated appraisal errors when α=2.0 and δ=0.15.
Several observations are in order. First, when , i.e. the assessed land value is less than
2 Statistics New Zealand 2006 census data shows that there are 27,849 owner occupied dwellings in
Palmerston North City.
13
its market land value, the appraisals are negatively biased. When , the appraisals are
positively biased. When , the appraisal errors will be equal to zero. Second, the higher
values of γ, the lower appraisal errors will be. This make senses as a high value of γ will
indicate a less proportional weight of land values in property’s sale prices, i.e. land values
become less important in analysing total property sales. Therefore, it can tolerate more
estimated errors. Third, when assessed land values are close to their market values, the
appraisal errors are small. For between 0.75 and 1.25, the appraisals are within 10% margin
of errors.
<Insert Table 2a>
For comparisons, Table 2b shows the estimated appraisal errors of equation (8) when
α=0.5 and δ=0.15. It is worth to note that the signs of estimated errors are opposite to the
results of Table 2a. When , the appraisal errors are positive. When , the appraisal
errors are negative. When , the estimated appraisal errors are equal to zero. The results
reveal the benefit to include more comparable sales in the improved net rate analysis. The
negative errors and positive errors could cancel out each other in the grid adjustment process.
The results of table 2a and 2b are also empirically useful in the optimum selection of
comparable properties. Depending on the pre-determined margin of errors, appraisers can
check comparable sales first even before put them into the grid adjustment system. What is
more, appraisers can even look for “opposite” comparable properties in order to minimise the
final appraisal errors when using the improved net rate analysis.
<Insert Table 2b>
14
Table 3 presents the point estimates of required minimal values of γ for a set of
combinations of α and δ. For example, the table shows that whenα=0.5, δ=0.30 and =0.75,
the required minimal value of γ is 3.00 at 10% margin of appraisal errors. Since α, δ and
can be easily pre-estimated by appraisers, the results of table 3 provide some useful
guidelines in selection of comparable properties when using the improved net rate analysis.
<Insert Table 3>
Apart from the above point estimates, the average appraisal errors under different sample
sizes of comparable properties are also tested through the simulation. The results in Table 4a
and 4b show that the average appraisal errors could be effectively reduced by including more
comparable sales in the improved net rate analysis. For example, when =0.5, using 5
comparable sales will produce an average appraisal error9.6%. In contrast, the error will be
reduced to 6.9% when using 10 comparable sales. At the5% margin of errors, the acceptable
range for will be between 0.60 and 2.40 when using 10 comparable sales. At the 10%
margin of errors, the range for will be extended to 0.40 - 5.00. The findings show that the
improved net rate method could be applicable in a wide range of . The benefit of
compensation for errors will be gradually reduced when is close to unity.Figure 1 shows the
benefit of compensation for errors between 5 and 10 comparable sales.
<Insert Table 4a and 4b>
15
<Insert Figure 1>
5.2 Simulation using empirical data
Table 5a and 5b show the results of simulation using empirical data. There is virtually no
difference between the average errors estimated using 5 comparable sales and 10 comparable
sales. At the 5% margin of errors, the range for is between 0.75 and 1.50 for using both 5
and 10 comparable properties. Whilst at the 10% margin of errors, the acceptable range for
is between 0.5 and 2.70. Compared to the results from the theoretical test, simulations using
empirical data show a narrow range of values at a given margin of errors, but in general
support the findings in the theoretical test. The difference might be due to the variable’s
distribution assumptions in the theoretical test.
<Insert Table 5a and 5b>
<Insert Figure 2>
16
6. Conclusions
In this paper we proposed an improved net rate analysis by using assessed land values.
Under the assumption that land values are uniformly assessed, the improved net rate method
has greatly simplified the traditional grid adjustment method. The main advantage of using
the improved net rate method is that the method can accommodate more sales for
comparisons, thus increasing the sample size of comparable properties. The method provides
a very attractive solution when estimating property values in thin markets, where there are
limited comparable sales per period. In practice, the whole suburb’s sales or even the whole
city’s sales can be potentially used as comparable properties in the analysis.
One weakness of using the assessed land values in the net rate analysis is that when the
assessed land values and the market land values are not equal to each other, appraisal errors
are inevitable in the improved net rate method. The results show that the size and direction of
inherent errors in the improved net rate method are determined by a set of factors and its
overall effect could be offset through a careful selection of comparable properties. Therefore,
the findings in this study provide some useful guidelines in optimal selecting and weighting
comparable properties for the use of net rate analysis in practice. Furthermore, simulation
results show that the method could tolerate a wide range of assessment errors and market
conditions when assessed land values and market land values are not equal to each other, due
to the law of compensation for errors.
17
Appendix: The proof of equation (8)
( ( ) )
( ( ) ) ∑
Since
, t1≈t2 or ∑ ≈0, we have
( ( ) )
( ( ) )
Simplify the above equation by dividing both the numerator and denominator by , we
have:
( ( ) )
( ( ) )
Since lands are uniformly assessed, let
( )
We have:
19
References
Allen, M. T., & Dare, W. H. (2002). Identifying determinants of horizontal property tax inequity:
evidence from Florida. Journal of Real Estate Research, 24(2), 153-164.
Betts, R., & Ely, S. (2008). Basic Real Estate Appraisal: Principles & Procedures (7th ed.): Thomson
South-Western.
Clapp, J. M. (1990). A new test for equitable real estate tax assessment. The Journal of Real Estate
Finance and Economics, 3(3), 233-249.
Cornia, G. C., & Slade, B. A. (2005). Property taxation of multifamily housing: an empirical analysis of
vertical and horizontal equity. Journal of Real Estate Research, 27(1), 17-46.
Epley, D. R. (1997). A Note on the Optimal Selection and Weighting of Comparable Properties.
[Article]. Journal of Real Estate Research, 14(1/2), 175.
Gau, G. W., Lai, T.-Y., & Wang, K. (1992). Optimal Comparable Selection and Weighting in Real
Property Valuation: An Extension. [Article]. Journal of the American Real Estate & Urban
Economics Association, 20(1), 107-123.
Gau, G. W., Lai, T.-Y., & Wang, K. (1994). A Further Discussion of Optimal Comparable Selection and
Weighting, and A Response to Green. [Article]. Journal of the American Real Estate & Urban
Economics Association, 22(4), 655-663.
Goolsby, W. C. (1997). Assessment error in the valuation of owner-occupied housing. Journal of Real
Estate Research, 13(1), 33.
Jefferies, R. L. (1991). Urban valuation in New Zealand (Vol. 1). Wellington: New Zealand Institute of
Valuers Inc.
Lai, T.-Y., Vandell, K., Wang, K., & Welke, G. (2008). Estimating Property Values by Replication: An
Alternative to the Traditional Grid and Regression Methods. [Article]. Journal of Real Estate
Research, 30(4), 441-460.
20
Sirmans, G. S., Diskin, B. A., & Friday, H. S. (1995). Vertical inequity in the taxation of real property.
National Tax Journal, 48(1), 71-84.
Vandell, K. D. (1991). Optimal Comparable Selection and Weighting in Real Property Valuation.
[Article]. Journal of the American Real Estate & Urban Economics Association, 19(2), 213-239.
21
Table 1: summarised statistics of dwelling sales for Palmerston North City, March 2011 to February 2012
Total sale price ($)
Assessed total values ($)
Assessed land values ($)
Age of dwelling (year) Floor area (M2) Land area (M2)
Ratio of sale price to assessed land value (γ)
Mean 302,010 303,695 137,480 47 154 798 2.28
Median 272,000 270,000 118,000 50 134 689 2.19
Maximum 1,101,500 1,375,000 780,000 >100 500 9589 5.71
Minimum 100,000 113,000 55,000 1 67 220 1.01
Std. Dev. 112,591 115,302 62,757 27 58 624 0.67
Skewness 1.65 2.08 3.00 0.13 1.20 8.49 0.81
Kurtosis 7.74 11.95 20.52 2.11 4.73 91.56 4.01
Observations 1,171 1,171 1,171 1,171 1,171 1,171 1,171
22
Table 2a: Estimated appraisal errors, when and
Notes: -- line denotes for 10% margin of errors
0.250 0.500 0.750 1.000 1.250 1.500 1.750 2.000 2.250 2.500 2.750 3.000 3.250 3.500 3.750 4.000 4.250 4.500 4.750 5.000
0.250 1.275 1.333 1.388 1.439 1.486
0.500 0.425 0.496 0.557 0.612 0.660 0.703 0.742 0.778 0.810 0.839 0.865 0.890 0.913
0.750 0.198 0.270 0.330 0.381 0.424 0.462 0.495 0.524 0.549 0.572 0.593 0.612 0.629 0.644 0.659
1.000 0.000 0.093 0.165 0.223 0.270 0.309 0.342 0.371 0.395 0.417 0.436 0.453 0.468 0.481 0.494 0.505 0.515
1.250 0.000 0.080 0.141 0.189 0.228 0.260 0.287 0.310 0.329 0.346 0.361 0.375 0.386 0.397 0.406 0.415 0.423
1.500 -0.099 0.000 0.071 0.124 0.165 0.198 0.225 0.247 0.266 0.282 0.296 0.309 0.319 0.329 0.338 0.345 0.352 0.359
1.750 -0.090 0.000 0.063 0.110 0.146 0.174 0.198 0.217 0.233 0.247 0.259 0.269 0.278 0.287 0.294 0.300 0.306 0.312
2.000 -0.213 -0.083 0.000 0.057 0.099 0.131 0.156 0.176 0.193 0.207 0.219 0.230 0.239 0.247 0.254 0.260 0.266 0.271 0.275
2.250 -0.198 -0.076 0.000 0.052 0.090 0.119 0.141 0.159 0.174 0.187 0.197 0.207 0.215 0.221 0.228 0.233 0.238 0.243 0.247
2.500 -0.186 -0.071 0.000 0.048 0.082 0.108 0.129 0.145 0.159 0.170 0.179 0.188 0.195 0.201 0.206 0.211 0.216 0.220 0.223
2.750 -0.175 -0.066 0.000 0.044 0.076 0.100 0.118 0.133 0.146 0.156 0.164 0.172 0.178 0.184 0.189 0.193 0.197 0.201 0.204
3.000 -0.165 -0.062 0.000 0.041 0.071 0.093 0.110 0.123 0.135 0.144 0.152 0.159 0.164 0.170 0.174 0.178 0.182 0.185 0.188
3.250 -0.156 -0.058 0.000 0.038 0.066 0.086 0.102 0.115 0.125 0.134 0.141 0.147 0.153 0.157 0.161 0.165 0.168 0.171 0.174
3.500 -0.148 -0.055 0.000 0.036 0.062 0.081 0.096 0.107 0.117 0.125 0.132 0.137 0.142 0.147 0.150 0.154 0.157 0.160 0.162
3.750 -0.141 -0.052 0.000 0.034 0.058 0.076 0.090 0.101 0.110 0.117 0.123 0.129 0.133 0.137 0.141 0.144 0.147 0.149 0.152
4.000 -0.319 -0.135 -0.049 0.000 0.032 0.055 0.072 0.085 0.095 0.103 0.110 0.116 0.121 0.125 0.129 0.132 0.135 0.138 0.140 0.143
4.250 -0.308 -0.129 -0.047 0.000 0.031 0.052 0.068 0.080 0.090 0.098 0.104 0.110 0.114 0.118 0.122 0.125 0.128 0.130 0.132 0.134
4.500 -0.297 -0.124 -0.045 0.000 0.029 0.049 0.064 0.076 0.085 0.092 0.099 0.104 0.108 0.112 0.115 0.118 0.121 0.123 0.125 0.127
4.750 -0.288 -0.119 -0.043 0.000 0.028 0.047 0.061 0.072 0.081 0.088 0.094 0.099 0.103 0.106 0.110 0.112 0.115 0.117 0.119 0.121
5.000 -0.279 -0.114 -0.041 0.000 0.026 0.045 0.058 0.069 0.077 0.084 0.089 0.094 0.098 0.101 0.104 0.107 0.109 0.111 0.113 0.115
β: the ratio of property's assessed land value to its market land valueγ:
th
e ra
tio
of
pro
per
ty's
sal
e p
rice
to
its
asse
ssed
lan
d v
alu
e
23
Table 2b: Estimated appraisal errors, when and
Notes: -- line denotes for 10% margin of errors
0.250 0.500 0.750 1.000 1.250 1.500 1.750 2.000 2.250 2.500 2.750 3.000 3.250 3.500 3.750 4.000 4.250 4.500 4.750 5.000
0.250 -3.900 -3.694 -3.534 -3.406 -3.302
0.500 -1.300 -1.262 -1.238 -1.221 -1.209 -1.200 -1.193 -1.187 -1.182 -1.178 -1.174 -1.171 -1.169
0.750 -0.505 -0.567 -0.605 -0.630 -0.647 -0.661 -0.671 -0.679 -0.686 -0.692 -0.696 -0.701 -0.704 -0.707 -0.710
1.000 0.000 -0.206 -0.302 -0.358 -0.394 -0.419 -0.438 -0.453 -0.464 -0.474 -0.481 -0.488 -0.494 -0.499 -0.503 -0.506 -0.510
1.250 0.000 -0.142 -0.216 -0.261 -0.292 -0.314 -0.331 -0.344 -0.355 -0.364 -0.371 -0.377 -0.382 -0.387 -0.391 -0.395 -0.398
1.500 0.252 0.000 -0.108 -0.168 -0.206 -0.232 -0.251 -0.266 -0.278 -0.287 -0.295 -0.302 -0.307 -0.312 -0.316 -0.320 -0.323 -0.326
1.750 0.189 0.000 -0.087 -0.137 -0.170 -0.193 -0.210 -0.223 -0.233 -0.241 -0.248 -0.254 -0.259 -0.264 -0.267 -0.271 -0.274 -0.276
2.000 0.650 0.151 0.000 -0.073 -0.116 -0.144 -0.165 -0.180 -0.191 -0.200 -0.208 -0.214 -0.220 -0.224 -0.228 -0.232 -0.235 -0.237 -0.240
2.250 0.505 0.126 0.000 -0.063 -0.101 -0.126 -0.144 -0.157 -0.168 -0.176 -0.183 -0.188 -0.193 -0.197 -0.201 -0.204 -0.207 -0.209 -0.212
2.500 0.413 0.108 0.000 -0.055 -0.089 -0.111 -0.127 -0.140 -0.149 -0.157 -0.163 -0.168 -0.173 -0.176 -0.180 -0.183 -0.185 -0.187 -0.189
2.750 0.349 0.094 0.000 -0.049 -0.079 -0.100 -0.115 -0.126 -0.134 -0.141 -0.147 -0.152 -0.156 -0.159 -0.163 -0.165 -0.168 -0.170 -0.171
3.000 0.302 0.084 0.000 -0.044 -0.072 -0.090 -0.104 -0.114 -0.122 -0.129 -0.134 -0.138 -0.142 -0.145 -0.148 -0.151 -0.153 -0.155 -0.157
3.250 0.267 0.075 0.000 -0.040 -0.066 -0.083 -0.095 -0.105 -0.112 -0.118 -0.123 -0.127 -0.131 -0.134 -0.136 -0.139 -0.141 -0.143 -0.144
3.500 0.239 0.069 0.000 -0.037 -0.060 -0.076 -0.088 -0.097 -0.104 -0.109 -0.114 -0.118 -0.121 -0.124 -0.126 -0.128 -0.130 -0.132 -0.134
3.750 0.216 0.063 0.000 -0.034 -0.056 -0.071 -0.082 -0.090 -0.096 -0.101 -0.106 -0.109 -0.113 -0.115 -0.117 -0.119 -0.121 -0.123 -0.124
4.000 0.975 0.197 0.058 0.000 -0.032 -0.052 -0.066 -0.076 -0.084 -0.090 -0.095 -0.099 -0.102 -0.105 -0.108 -0.110 -0.112 -0.113 -0.115 -0.116
4.250 0.852 0.181 0.054 0.000 -0.030 -0.049 -0.062 -0.071 -0.079 -0.084 -0.089 -0.093 -0.096 -0.099 -0.101 -0.103 -0.105 -0.107 -0.108 -0.109
4.500 0.757 0.168 0.050 0.000 -0.028 -0.046 -0.058 -0.067 -0.074 -0.079 -0.084 -0.087 -0.090 -0.093 -0.095 -0.097 -0.099 -0.100 -0.102 -0.103
4.750 0.681 0.156 0.047 0.000 -0.026 -0.043 -0.055 -0.063 -0.070 -0.075 -0.079 -0.083 -0.086 -0.088 -0.090 -0.092 -0.094 -0.095 -0.096 -0.098
5.000 0.619 0.146 0.044 0.000 -0.025 -0.041 -0.052 -0.060 -0.066 -0.071 -0.075 -0.078 -0.081 -0.083 -0.085 -0.087 -0.089 -0.090 -0.091 -0.093
β: the ratio of property's assessed land value to its market land valueγ:
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24
Table 3: The required minimal values of γ, at 10% margin of errors
0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00
α=0.5
δ=0.30 3.00 1.00 1.75 2.50 3.00 3.50 3.75 4.00 4.25 4.50 4.50 4.75 4.75 4.75 5.00 5.00 5.00 >5.00
δ=0.15 2.75 1.00 1.75 2.50 2.75 3.25 3.50 3.75 4.00 4.00 4.25 4.25 4.50 4.50 4.50 4.50 4.75 4.75
δ=0.00 2.50 1.00 1.50 2.00 2.50 2.75 3.00 3.25 3.50 3.50 3.75 3.75 4.00 4.00 4.00 4.00 4.25 4.25
δ=-0.15 4.75 2.00 1.00 1.25 1.75 2.00 2.50 2.50 2.75 3.00 3.00 3.00 3.25 3.25 3.25 3.25 3.50 3.50 3.50
δ=-0.30 3.25 1.50 1.00 1.00 1.25 1.50 1.75 1.75 2.00 2.00 2.00 2.25 2.25 2.25 2.25 2.25 2.50 2.50 2.50
α=1.0
δ=0.30 3.00 1.50 1.00 1.00 1.00 1.25 1.50 1.50 1.50 1.75 1.75 1.75 1.75 1.75 2.00 2.00 2.00 2.00 2.00
δ=0.15 4.50 2.00 1.50 1.00 1.00 0.75 0.75 0.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.25 1.25 1.25 1.25
δ=0.00 4.00 2.00 1.50 1.00 1.00 0.75 0.75 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.25 0.25 0.25 0.25 0.25
δ=-0.15 4.50 2.00 1.50 1.00 1.00 0.75 0.75 1.00 1.00 1.00 1.25 1.25 1.25 1.25 1.25 1.50 1.50 1.50 1.50 1.50
δ=-0.30 3.50 1.50 1.00 1.00 1.25 1.75 1.75 2.25 2.50 2.75 2.75 3.00 3.00 3.25 3.25 3.25 3.25 3.50 3.50
α=2.0
δ=0.30 4.50 1.50 1.00 1.00 1.50 2.00 2.50 2.75 3.00 3.25 3.50 3.75 3.75 4.00 4.00 4.00 4.25 4.25 4.25
δ=0.15 1.50 1.00 1.00 2.00 2.75 3.50 4.00 4.25 4.50 4.75 5.00 >5.00 >5.00 >5.00 >5.00 >5.00 >5.00 >5.00
δ=0.00 2.00 1.00 1.25 2.75 3.75 4.50 >5.00 >5.00 >5.00 >5.00 >5.00 >5.00 >5.00 >5.00 >5.00 >5.00 >5.00 >5.00
δ=-0.15 2.75 1.00 1.75 3.75 5.00 >5.00 >5.00 >5.00 >5.00 >5.00 >5.00 >5.00 >5.00 >5.00 >5.00 >5.00 >5.00 >5.00
δ=-0.30 3.75 1.00 2.25 5.00 >5.00 >5.00 >5.00 >5.00 >5.00 >5.00 >5.00 >5.00 >5.00 >5.00 >5.00 >5.00 >5.00 >5.01
β: the ratio of property's assessed land value to its market land value
25
Table 4a: Theoretical simulation results for β=(0.20 - 2.60)
Table 4b: Theoretical simulation results for β=(2.70 – 5.00)
0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60
10 comparable sales 0.152 0.128 0.100 0.069 0.048 0.031 0.018 0.008 0.000 0.006 0.012 0.018 0.022 0.027 0.030 0.033 0.037 0.041 0.042 0.046 0.046 0.049 0.050 0.055 0.056
5 comparable sales 0.198 0.161 0.129 0.096 0.064 0.043 0.025 0.011 0.000 0.009 0.017 0.024 0.031 0.035 0.041 0.044 0.050 0.052 0.057 0.059 0.062 0.066 0.067 0.070 0.074
β: ratio of assessed land value to market land value
2.70 2.80 2.90 3.00 3.10 3.20 3.30 3.40 3.50 3.60 3.70 3.80 3.90 4.00 4.10 4.20 4.30 4.40 4.50 4.60 4.70 4.80 4.90 5.00
10 comparable sales 0.058 0.058 0.058 0.067 0.067 0.065 0.066 0.068 0.068 0.067 0.073 0.073 0.073 0.077 0.076 0.077 0.079 0.078 0.079 0.081 0.079 0.079 0.082 0.083
5 comparable sales 0.076 0.074 0.082 0.081 0.084 0.085 0.087 0.087 0.090 0.090 0.092 0.095 0.097 0.093 0.093 0.097 0.099 0.095 0.100 0.098 0.102 0.108 0.103 0.107
β: ratio of assessed land value to market land value
26
Table 5a: Empirical simulation results for β=(0.30 - 2.60)
Table 5b: Empirical simulation results for β=(2.70 – 5.00)
0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60
10 comparable sales 0.112 0.121 0.093 0.079 0.061 0.037 0.017 0.000 0.014 0.025 0.035 0.043 0.051 0.057 0.063 0.068 0.072 0.077 0.081 0.084 0.087 0.091 0.092 0.096
5 comparable sales 0.119 0.134 0.103 0.086 0.063 0.040 0.018 0.000 0.015 0.027 0.036 0.045 0.052 0.059 0.064 0.070 0.075 0.079 0.082 0.086 0.089 0.092 0.095 0.098
β: ratio of assessed land value to market land value
2.70 2.80 2.90 3.00 3.10 3.20 3.30 3.40 3.50 3.60 3.70 3.80 3.90 4.00 4.10 4.20 4.30 4.40 4.50 4.60 4.70 4.80 4.90 5.00
10 comparable sales 0.099 0.101 0.103 0.104 0.107 0.109 0.110 0.113 0.113 0.116 0.116 0.117 0.117 0.119 0.122 0.123 0.123 0.125 0.125 0.126 0.126 0.128 0.130 0.129
5 comparable sales 0.099 0.102 0.104 0.108 0.110 0.109 0.113 0.113 0.115 0.116 0.118 0.117 0.120 0.121 0.122 0.122 0.123 0.125 0.126 0.126 0.128 0.129 0.129 0.132
β: ratio of assessed land value to market land value
27
Figure 1: Theoretical simulations results
0%
4%
8%
12%
16%
20%
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
Ratios of assessed land value to market land value
5 comparable sales10 comparable sales