10.2 Assessment and Appraisal Ratio Statistics

10.2 Assessment and Appraisal Ratio Statistics

A ratio study compares assessed values (or prior appraised values) to verified, arm's-length market sales prices. It provides empirical answers to two fundamental questions:

1. Assessment Level (Accuracy): Are properties being assessed at the legally mandated statutory standard (e.g., 100% of market value, or a statutory fraction such as 70% or 80%)?
2. Assessment Uniformity (Equity): Are properties assessed equitably relative to one another, both across similar properties (horizontal equity) and across different price tiers (vertical equity)?

---

1. The Sales Ratio and Measures of Assessment Level

The fundamental building block of a ratio study is the sales ratio ($R_i$), calculated for each individual transacted property:

$R_i = \frac{\text{Assessed Value } (AV_i)}{\text{Sale Price } (SP_i)} \quad \text{or} \quad R_i = \frac{\text{Appraised Value } (A_i)}{\text{Sale Price } (SP_i)}$

When a sales ratio equals $1.00$ (or $100\%$), the assessed value perfectly matches the arm's-length market transaction price. A ratio below $1.00$ indicates under-assessment relative to market value, while a ratio above $1.00$ indicates over-assessment.

Measures of Central Assessment Level

To determine the overall level of assessment across an entire jurisdiction or commercial property class, three summary statistics are evaluated:

1. The Median Sales Ratio: The midpoint ratio when all individual ratios are sorted in ascending order. Under the International Association of Assessing Officers (IAAO) Standard on Ratio Studies, the median ratio is the preferred measure of central assessment level. Because ratio distributions are non-normal and often positively skewed, the median is immune to outlier transactions and provides the most reliable indicator of typical assessment practice.
2. The Arithmetic Mean Sales Ratio: The simple unweighted average of all individual ratios: $\bar{R} = \frac{\sum_{i=1}^n R_i}{n}$ Each sale carries equal weight, regardless of whether it is a $500,000 retail condo or a $50,000,000 industrial distribution center. Like all means, it is vulnerable to extreme ratio outliers.
3. The Weighted Mean Sales Ratio (Aggregate Ratio): Calculated by dividing the sum of all assessed values by the sum of all sales prices: $\bar{R}_w = \frac{\sum_{i=1}^n AV_i}{\sum_{i=1}^n SP_i}$ The weighted mean weights each ratio in direct proportion to the property's dollar sale price. A $10,000,000 transaction exerts ten times more influence on the weighted mean than a $1,000,000 transaction. It represents the true aggregate assessment level of the taxable dollar base.

---

2. Measures of Appraisal Uniformity: Horizontal Equity

Horizontal equity requires that properties of equal market value receive equal tax assessments. When assessments lack uniformity, identical commercial properties bear unequal, unfair shares of the municipal tax levy. The primary statistical tool used worldwide to measure assessment uniformity is the Coefficient of Dispersion (COD).

The Coefficient of Dispersion (COD)

The COD measures the average absolute percentage deviation of individual ratios from the median sales ratio:

$\text{COD} = \frac{\frac{\sum_{i=1}^n |R_i - \text{Median } R|}{n}}{\text{Median } R} \times 100$

Where:

• $R_i$ = Individual sales ratio for sale $i$
• $\text{Median } R$ = Median sales ratio of the sample
• $n$ = Total number of sales in the sample
• $\frac{\sum |R_i - \text{Median } R|}{n}$ = Average Absolute Deviation (AAD)

Interpreting COD and IAAO Uniformity Benchmarks

A COD of $0.0\%$ would indicate absolute perfection—every single property sold at the exact same percentage of assessed value. Because real estate markets involve negotiation ranges and heterogeneous physical characteristics, a COD of zero is impossible. The IAAO establishes recognized performance thresholds based on property type:

+---------------------------------------------------------------------------------------------------+
|                         IAAO BENCHMARKS FOR ASSESSMENT UNIFORMITY (COD)                           |
+---------------------------------------------------+-----------------------------------------------+
| PROPERTY CLASSIFICATION                           | ACCEPTABLE COD PERFORMANCE STANDARD           |
+---------------------------------------------------+-----------------------------------------------+
| Single-Family Residential (Newer / Homogeneous)   | 5.0% to 10.0%                                 |
| Single-Family Residential (Older / Heterogeneous) | 10.0% to 15.0%                                |
| Commercial & Industrial (Large Urban / Suburban)  | ≤ 15.0%                                       |
| Commercial & Industrial (Small Rural / Specialized)| ≤ 20.0%                                      |
| Vacant Land / Development Sites                   | ≤ 20.0% (or ≤ 25.0% in rural areas)           |
+---------------------------------------------------+-----------------------------------------------+

The Coefficient of Variation (COV)

The Coefficient of Variation expresses the sample standard deviation of the sales ratios as a percentage of the arithmetic mean ratio:

$\text{COV} = \frac{s}{\bar{R}} \times 100$

While COD is based on absolute deviations from the median, COV is based on squared deviations from the mean. COV is preferred when the ratio distribution is verified to be normally distributed, but COD remains the dominant legal and administrative benchmark because ratio data is frequently skewed.

---

3. Measures of Vertical Equity: Regressivity vs. Progressivity

Vertical equity evaluates whether properties of different values are assessed fairly relative to one another. For an assessment system to be vertically equitable, the assessment-to-sales ratio must remain consistent across all value tiers—from a modest $500,000 flex building to a $25,000,000 corporate campus.

The Price-Related Differential (PRD)

The primary statistical index of vertical equity is the Price-Related Differential (PRD), defined as the arithmetic mean ratio divided by the weighted mean ratio:

$\text{PRD} = \frac{\text{Mean Ratio}}{\text{Weighted Mean Ratio}} = \frac{\bar{R}}{\bar{R}_w} = \frac{\frac{\sum R_i}{n}}{\frac{\sum AV_i}{\sum SP_i}}$

+---------------------------------------------------------------------------------------------------+
|                             INTERPRETING THE PRICE-RELATED DIFFERENTIAL                           |
+-------------------+-----------------------+-------------------------------------------------------+
| PRD VALUE         | EQUITY DIAGNOSIS      | REAL-WORLD TAX IMPACT                                 |
+-------------------+-----------------------+-------------------------------------------------------+
| 0.98 to 1.03      | Vertical Equity       | Acceptable equity; fair across all price tiers.       |
| > 1.03            | Regressivity          | High-value properties under-assessed; low-value over. |
| < 0.98            | Progressivity         | High-value properties over-assessed; low-value under. |
+-------------------+-----------------------+-------------------------------------------------------+

1. Assessment Regressivity ($\text{PRD} > 1.03$)

• Diagnostic Condition: When the arithmetic mean ratio is significantly greater than the weighted mean ratio, $\text{PRD} > 1.03$.
• Mechanism: Because the weighted mean gives heavy weight to expensive properties, a lower weighted mean indicates that higher-value properties are assessed at lower ratios than lower-value properties.
• Tax Consequence: Regressivity places an unfair, disproportionate tax burden onto owners of lower-valued commercial real estate. Assessors often under-value large, complex commercial properties due to fear of high-stakes tax litigation or lack of specialized valuation expertise.

2. Assessment Progressivity ($\text{PRD} < 0.98$)

• Diagnostic Condition: When the weighted mean ratio exceeds the arithmetic mean ratio, $\text{PRD} < 0.98$.
• Mechanism: Higher-value properties are assessed at systematically higher ratios than lower-value properties.
• Tax Consequence: High-value property owners carry a disproportionate share of the tax burden relative to true market value.

---

4. Comprehensive Worked Commercial Ratio Study

To master ratio statistics for the Certified General examination, work through the following complete ratio study analyzing 10 commercial property sales in an industrial park submarket.

Transaction Dataset and Ratio Calculations

Step 1: Determine the Median Sales Ratio

1. Arrange the 10 sales ratios in ascending rank order: $0.800, \ 0.860, \ 0.900, \ 0.920, \ \mathbf{0.960}, \ \mathbf{0.980}, \ 1.000, \ 1.020, \ 1.060, \ 1.100$
2. Since $n = 10$ is an even number, average the 5th and 6th observations: $\text{Median } R = \frac{0.960 + 0.980}{2} = \mathbf{0.970} \quad (97.0\%)$ Interpretation: The central assessment level is 97.0%, demonstrating that properties in this class are assessed slightly below full market value.

Step 2: Compute the Arithmetic Mean and Weighted Mean Ratios

1. Arithmetic Mean Ratio ($\bar{R}$): $\bar{R} = \frac{\sum R_i}{n} = \frac{9.600}{10} = \mathbf{0.9600} \quad (96.00\%)$
2. Weighted Mean Ratio ($\bar{R}_w$): $\bar{R}_w = \frac{\sum AV}{\sum SP} = \frac{\$23{,}465{,}000}{\$25{,}550{,}000} \approx \mathbf{0.918395} \approx \mathbf{0.9184} \quad (91.84\%)$

Step 3: Calculate the Coefficient of Dispersion (COD)

1. Average Absolute Deviation (AAD): $\text{AAD} = \frac{\sum |R_i - \text{Median } R|}{n} = \frac{0.720}{10} = 0.0720$
2. Coefficient of Dispersion (COD): $\text{COD} = \frac{\text{AAD}}{\text{Median } R} \times 100 = \frac{0.0720}{0.970} \times 100 \approx \mathbf{7.4227\%} \approx \mathbf{7.42\%}$ Uniformity Evaluation: The COD is $7.42\%$. According to IAAO benchmarks for commercial and industrial real estate (which require a COD of $\le 15.0\%$), this assessment jurisdiction demonstrates excellent horizontal uniformity.

Step 4: Calculate and Diagnose the Price-Related Differential (PRD)

1. Price-Related Differential (PRD): $\text{PRD} = \frac{\text{Mean Ratio}}{\text{Weighted Mean Ratio}} = \frac{0.9600}{0.918395} \approx \mathbf{1.0453} \approx \mathbf{1.045}$
2. Vertical Equity Diagnosis:
   • Acceptable IAAO boundary: $0.98 \le \text{PRD} \le 1.03$.
   • Because $\text{PRD} = 1.045 > 1.03$, the PRD indicates assessment regressivity (formal statistical significance testing is a separate step).
   • Physical Inspection of the Data: Notice that the $1,000,000 building (Sale 1) is assessed at a $1.100$ ratio (110% of market value), while the $5,000,000 building (Sale 10) is assessed at an $0.800$ ratio (80% of market value). Higher-value commercial properties are systematically under-assessed relative to smaller commercial properties, shifting an inequitable tax burden onto smaller local businesses.