Correcting Property Taxes on
HighValue Properties©
***
The Variable Tax Rate Phenomenon
In
TaxAdjusted Cap Rate Formulas
By Mark Pomykacz
Prepared: Wednesday, July 30, 2003
Introduction
It is now widely acknowledged that capitalization rates can and should be adjusted for the property tax rate, while the net income should be calculated without the property tax expense, in order to accurately estimate the market value of income producing properties, for property tax assessment purposes. However the standard property tax adjusting methods assume that the property tax rate is invariable. In fact, for highvalue assets, those that represent a substantial portion of the taxing jurisdiction’s taxable base, the property tax rate is dependent on the market and assessed values of the highvalue asset. Applying the standard property tax adjusting methods to highvalue assets will result in overestimated market and assessed values, but underestimated tax rates, and municipal budget short falls [1]. This article explains this phenomenon and presents a formulaic solution that greatly improves appraisal accuracy, and budgeting. The formula is easy to use and cost effective. Additionally this article presents practical steps for employing the analysis.
It is expected that the formulas and processes presented herein will be widely used in the planning, negotiation, litigation and settlement of property tax disputes for highvalue properties to the benefit of both the tax payer and the taxing jurisdiction. The taxing jurisdiction will benefit from the avoidance of previously unexpected changes to the tax rate, allowing more accurate budget planning. Furthermore, it will permit the full capture of the taxable base. For the taxpayer it will provide sound appraisal rationale for relief from underestimated tax rates, which result in overestimated assessments and over taxation.
The Variable Tax Rate Phenomenon
To understand why tax rates are variable and dependent in the assessment analysis process, analysts must recognize that 1) tax rates are based on assessed values but 2) for incomegenerating properties, assessed values are based, in part, on tax rates.
Table I
Basic Tax Rate and Value Formulas [2]
Tax Rate = Budget ¸ Sum of all Assessed Values in the Taxing Jurisdiction
( T = B ¸ Sall )
Assessed Value of Subject HighValue Asset = Income before Property Taxes ¸
(Capitalization Rate + Tax Rate)
( As = I ¸ (C + T) )
When the assessed value of one highvalue asset is wrong, then the sum of all assessed values and the tax rate are computed wrong. When the tax rate is wrong, then the assessed value of the highvalue, incomeproducing assets are computed wrong. Only when both the assessed values of all highvalue assets and the tax rate are correct, will the assessment process work properly. Each depends on the other. This is a circular analysis, or codependent analysis.
The assessment process often proceeds as presented in Table II, for incomegenerating properties.
Table II
Example of Assessment Process
Column/Iteration 

1 
2 



1 
HighValue Asset’s Income Before Taxes 
I 
$7,500,000  $3,750,000 


2 
Capitalization Rate 
C 
10.000000% 
10.000000% 
3 
Assumed Tax Rate 
T 
2.500000% 
2.500000% 
4 
Tax Adjusted Cap Rate 

12.500000% 
12.500000% 


5 
Assumed Assessed Value of HighValue Asset 
A_{s} 
$60,000,000  $30,000,000 


6 
Assessed Value of HighValue Asset 
A_{s} 
$60,000,000  $30,000,000 
7 
Assessed Value of All Other Assets 
S 
$140,000,000  $140,000,000 
8 
Sum of All Assessed Values 
Sall 
$200,000,000  $170,000,000 


Cross Checks 




8 
Budget 
B 
$5,000,000  $5,000,000 
9 
Indicated Tax Rate for Budget 
B 
2.500000% 
2.941176% 
10 
Assumed Tax Rate 
T 
2.500000% 
2.500000% 
11 
Consistency 

Yes 
No 


12 
Sum of All Assessed Values 
Sall 
$200,000,000  $170,000,000 
13 
Assumed Tax Rate 
T 
2.500000% 
2.500000% 
14 
Jurisdiction Revenues 
B 
$5,000,000  $4,250,000 
15 
Budget 
B 
$ 5,000,000  $5,000,000 
16 
Balanced Budget 

Yes 
No 



17 
Indicated Tax Rate for Budget 
T 
2.500000% 
2.941176% 
18 
Taxes on HighValue Asset 

1,500,000  882,353 
19 
Income After Taxes 
I 
$6,000,000  $2,867,647 
20 
Capitalization Rate 
C 
10.000000% 
10.000000% 
21

Indicated Value of HighValue Asset 
A_{s} 
$60,000,000  $28,676,471 
22

Assumed Assessment of HighValue Asset 
A_{s} 
$60,000,000  $30,000,000 
23 
Consistency 

Yes 
No 
If the budget is $5 million and the sum of all assessed values is correct, at $200 million, then the tax rate is correct at 2.5 percent (See Line 11, Column 1, Table II). If the assessed value of the highvalue assets are estimated based on the 2.5 percent tax rate, then all elements of the assessment process are consistent (See Lines 11, 16 and 23, Column 1, Table II). . If the value of one highvalue asset should need a substantial correction, then the sum of the assessed values for the jurisdiction would be substantially impacted (See Line 8, Column 2, Table II). Consequentially the tax rate would need to be corrected in order to maintain the budget (See Lines 11, 16 and 23, Column 2, Table II). The failure to correct the tax rate will result in a failure for the revenues to meet the budget, and an improper assessment on the highvalue asset.
The Iteration Solution
The appraiser can solve this problem using an iteration process. The appraiser estimates a corrected tax rate, calculates the assessed value of the highvalue asset, the sum of all assessed values and cross checks the conclusions with the assumptions. When the conclusions and assumptions do not match, then the appraiser attempts another estimate. The example from Table II is solved using iteration, in Table III below.
Table III
Example of Iteration Process
Column/Iteration 

1 
2 
3 
4 
5 



1  HighValue Asset’s Income Before Taxes 
I 
$ 7,500,000  $ 3,750,000  $ 3,750,000  $ 3,750,000  $ 3,750,000 


2  Capitalization Rate 
C 
10.000000%  10.000000%  10.000000%  10.000000%  10.000000% 
3  Assumed Tax Rate 
T 
2.500000%  2.500000%  2.958978%  2.959702%  2.959703% 
4  Tax Adjusted Cap Rate 

12.500000%  12.500000%  12.958978%  12.959702%  12.959703% 


5  Assumed Assessed Value of HighValue Asset 
A_{s} 
$ 60,000,000  $ 30,000,000  $ 28,937,468  $ 28,935,850  $ 28,935,847 


6  Assessed Value of HighValue Asset 
A_{s} 
$ 60,000,000  $ 30,000,000  $ 28,937,468  $ 28,935,850  $ 28,935,847 
7  Assessed Values of All Other Assets 
S 
$ 140,000,000  $ 140,000,000  $ 140,000,000  $ 140,000,000  $ 140,000,000 
8  Sum of All Assessed Values 
Sall 
$ 200,000,000  $ 170,000,000  $ 168,937,468  $ 168,935,850  $ 168,935,847 




Cross Checks 




8  Budget 
B 
$ 5,000,000  $ 5,000,000  $ 5,000,000  $ 5,000,000  $ 5,000,000 
9  Indicated Tax Rate for Budget 
B 
2.500000%  2.941176%  2.959675%  2.959703%  2.959703% 
10  Assumed Tax Rate 
T 
2.500000%  2.500000%  2.958978%  2.959702%  2.959703% 
11  Consistency 

Yes 
No 
No 
No 
Yes 



12  Sum of All Assessed Value 
Sall 
$ 200,000,000  $ 170,000,000  $ 168,937,468  $ 168,935,850  $ 168,935,847 
13  Assumed Tax Rate 
T 
2.500000%  2.500000%  2.958978%  2.959702%  2.959703% 
14  Jurisdiction Revenues 
B 
$ 5,000,000  $ 4,250,000  $ 4,998,822  $ 4,999,998  $ 5,000,000 
15  Budget 
B 
$ 5,000,000  $ 5,000,000  $ 5,000,000  $ 5,000,000  $ 5,000,000 
16  Balanced Budget 

Yes 
No 
No 
No 
Yes 


17  Indicated Tax Rate for Budget 
T 
2.500000%  2.941176%  2.959675%  2.959703%  2.959703% 
18  Taxes on HighValue Asset 

1,500,000  882,353  856,455  856,415  856,415 
19  Income After Taxes 
I 
$ 6,000,000  $ 2,867,647  $ 2,893,545  $ 2,893,585  $ 2,893,585 
20  Capitalization Rate 
C 
10.000000%  10.000000%  10.000000%  10.000000%  10.000000% 
21  Indicated Value of HighValue Asset 
A_{s} 
$ 60,000,000  $ 28,676,471  $ 28,935,450  $ 28,935,847  $ 28,935,847 
22 
Assumed Assessed Value of HighValue Asset

A_{s} 
$ 60,000,000  $ 30,000,000  $ 28,937,468  $ 28,935,850  $ 28,935,847 
23  Consistency 

Yes 
No 
No 
No 
Yes 
The Algebraic Solution [3]
The iteration technique, while intuitive, is cumbersome. As an alternative, the appraiser may wish to employ “Value/Tax Rate Algebraic Solution”. The algebraic proof follows.
Table IV
Algebraic Proof
Definitions
Property Tax Rate = T
Taxing Jurisdiction Budget = B
Subject Property Net Operating Income before Property Taxes = I
Subject Property Capitalization Rate = C
Subject Property Assessed Value = As
Sum of all Other Assessed Values = S
Known Relationships
Tax Rate, Budget, Assessment Base Formula
T = B ¸ ( As + S )
Income Capitalization, using a Property TaxWeighted Capitalization Rate
As = I ¸ ( C + T )
Given Data
Supplied by Taxing Jurisdiction: B and S
Estimated by Appraiser: I and C
Derivation [4]
T = B ¸ ( As + S )
Substitute I ¸ ( C + T ) for As
T = B ¸ [{ I ¸ ( C + T )} + S ]
Transform using Algebra
T = B ¸ [{ I + S ( C + T )} ¸ ( C + T )]
T = { B ( C + T )} ¸ ( I + S C + S T )
T ( I + S C + S T ) = B ( C + T )
T I + T S C + S T2 = B C + B T
T I + T S C + S T2 – B C – B T = 0
S T2 + T I + T S C – B T – B C = 0
S T2 + T ( I + S C – B ) – B C = 0
Use the quadratic formula to solve for T
T = [ B – I – S C + {( I + S C – B )2 + 4 S B C}½ ] ¸ 2 S
The Value/Tax Rate Algebraic Solution
T = [ B – I – S C + {( I + S C – B )2 + 4 S B C}½ ] ¸ 2 S
Cross Check
Use data from Column 2, Table II;
T = [ B – I – S C + {( I + S C – B )2 + 4 S B C}½ ] ¸ 2 S
T = [ 5,000,000 – 3,750,000 – 140,000,000 x .10 + {( 3,750,000 + 140,000,000 x .10 – 5,000,000 )2 + 4 x 140,000,000 x 5,000,000 x .10}½ ] ¸ 2 x 140,000,000
T = 2.959703%
Thus,
As = I ¸ ( C + T )
As = 3,750,000 ¸ ( .10 + .02959703 )
As = 28,935,848
And
T = B ¸ ( As + S )
T = 5,000,000 ¸ ( 28,935,779 + 140,000,000 )
T = 2.959703%
Therefore,
The assumed tax rate is consistent with the indicated tax rate and the Value/Tax Rate Algebraic Solution works!
Practical Considerations
Now that the theory and mathematics have been established, there remain only a few practical considerations before the appraisal community can widely deploy this new technique, the “Value/Tax Rate Algebraic Solution”. The appraiser must estimate the given inputs to be entered into the formula. The appraiser should continue to estimate the net operating income before property taxes and the capitalization rate as the appraiser has always done.
New to the appraiser’s research and analysis work is the need to estimate the budget for the taxing jurisdiction and sum of the assessed values for all other properties in the jurisdiction. To complete this work, the appraiser will need to research and analyze the historical trends and future expectations within the taxing jurisdiction. Historical data will provide the customary basis for a projection of both the budget for the taxing jurisdiction and the sum of the assessed values for all other properties in the jurisdiction. Of course, adjustments to the historical trends may be needed to stabilize trends, to account for abnormal trends, or to account for anticipated changes to historical trends for inflation, structure change and social and political activity. Interviews the authorities at taxing jurisdiction and reviews of marketwide general economic and real estate industry market forecasts will be useful.
This new research and analysis may not be mindlessly simple, but often the most basic forecasting techniques will suffice to greatly improve the accuracy of the assessment process. Furthermore, the difficulties with precisely forecasting budget and general assessed values are mitigated by recognizing the fact the between the tax rate and the assessed values, there must always be enough municipal revenue generated to support the budget, which often grows year after year, regardless of the health of the real estate market. Between the three, budgets, assessed values and tax rates, all else remaining constant, budgets increase perennially, assessed values oscillate up and down, but are generally up year by year, and tax rates make up the difference.
Conclusion
In conclusion, now that the theory and mathematics have been established, the appraiser can estimate assessed values and forecast budgets with a new and greater accuracy. Given the simplicity and the ease of use of the “Value/Tax Rate Algebraic Solution”, the appraisal community should have little difficulty adding this tool to its repertoire. The formulas and processes will need to be integrated into automated valuation models and mass appraisal systems. It is expected that the formulas and processes presented herein will be widely used in the planning, negotiation, litigation and settlement of property tax disputes for highvalue properties to the benefit of both the tax payer and the taxing jurisdiction. The taxing jurisdiction will benefit from the avoidance of previously unexpected changes to the tax rate and budget, allowing more accurate budget planning. Furthermore, it will permit the full capture of the taxable base. For the taxpayer it will provide sound appraisal rationale for relief from underestimated tax rates, which result in overestimated assessments and over taxation.
[1] Strictly speaking, the error with the standard formulas occurs on properties of all sizes, but the magnitude of the error is not material on smaller properties, unless the error occurs throughout a group of smaller properties.
[2] This article assumes that the assessed value to market value ratio (AKA equalization rate) is 100 percent. Thus assessed values equal market values, and the nominal tax rate equals the effective tax rate. The formula and processes proposed in this article work equally well with assessed value to market value ratios of other than 100 percent, but only after proper adjustments are made for the assessed value to market value ratio. For details on the proper adjustments for the assessed value to market value ratio, see “The Next Generation in Weighted Cap Rates”, (add link when ready), by Mark Pomykacz, MAI.
[3] © 2003. The contents of this article, and specifically the Value/Tax Rate Algebraic Solution are copyrighted by Mark Pomykacz. All rights reserved. This article, and any part thereof, may not be used, copied, modified, or distributed by any means without the express written permission of Mark Pomykacz. Also the use of the procedures and specifically the “Value/Tax Rate Algebraic Solution” are prohibited without the express written permission of Mark Pomykacz.
[4] Special acknowledgement and thanks to Professor Xiaochun Rong, Rutgers University, New Brunswick, NJ, Math Department, for his assistance with the completion of the algebra.