The Improved Net Rate Analysis

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: s.shi@massey.ac.nz; 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 1

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 2

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 3

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. 4

2. Methodology 2.1 The traditional sales comparison approach For the traditional sales comparison method, the ith property s sale price at time period t 1 can be written as: (1) Where represents the ith property s sale price at time period t 1, is itsland value and is its structure value at time t 1. Based on equation (1), the sale price for the jth property at time t 2 can 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 t 1 and t 2. 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. 5

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 6

is the estimated building value of the ith comparable property at time t 1 ; isthe sale price of the ith comparable property at time t 1 ; 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 t 2 ; isthe subject property s market land value at time t 2 ( ) 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 t 1 to t 2. 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 land values. represents the estimated market value of subject property using assessed represents for the subject property s assessed land value at time t 2. is the derived building value for the ith comparable property using the assessed land value at time t 1. A denotes for the assessed value. 7

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 it 1. 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). 8

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 9

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 10

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 11

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 dwellings 2. 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. 12

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> 13

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> 14

<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> 15

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. 16

Appendix: The proof of equation (8) ( ( ) ) ( ( ) ) Since, t 1 t 2 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: 17

Since Therefore 18

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. 19

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. 20

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 (M 2 ) Land area (M 2 ) 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 21

γ: the ratio of property's sale price to its assessed land value Table 2a: Estimated appraisal errors, when and β: the ratio of property's assessed land value to its market land value 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 Notes: -- line denotes for 10% margin of errors 22

γ: the ratio of property's sale price to its assessed land value Table 2b: Estimated appraisal errors, when and β: the ratio of property's assessed land value to its market land value 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 Notes: -- line denotes for 10% margin of errors 23

Table 3: The required minimal values of γ, at 10% margin of errors β: the ratio of property's assessed land value to its market land value 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 α=1.0 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 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 24

Table 4a: Theoretical simulation results for β=(0.20-2.60) β: ratio of assessed land value to market land value 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 Table 4b: Theoretical simulation results for β=(2.70 5.00) 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 25

Table 5a: Empirical simulation results for β=(0.30-2.60) β: ratio of assessed land value to market land value 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 Table 5b: Empirical simulation results for β=(2.70 5.00) 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 26

Figure 1: Theoretical simulations results 20% 16% 12% 8% 4% 0% 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 sales 10 comparable sales 27

Figure 2: Empirical simulation results 14% 12% 10% 8% 6% 4% 2% 0% 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 sales 10 comparable sales 28