Property Description and Appraisal Math

Chapter 4: Property Description and Appraisal Math

KEY WORDS AND TERMS

• Annuity
• Artificial Markers
• Baseline
• Bench Mark
• Direct Capitalization
• Discounting
• Compound Interest
• Correction Line
• Future Value
• Geodetic Survey System
• Guide Meridian
• Government Lots
• Interest Formula
• Legal Description
• Lot, Block, and Tract System
• Mean
• Median
• Meridian
• Metes and Bounds System
• Mode
• Natural Monument
• Point of Beginning
• Present Value
• Range
• Range Lines
• Reciprocal
• Rectangular Survey System
• Reference Point (Monument)
• Sections
• Standard Deviation
• Standard Parallels
• Townships
• True Point of Beginning

LEARNING OBJECTIVES

After completing this chapter, you should be able to:

1. Name the three major systems of land description used in the United States and explain how land is described under each system.
2. Calculate the area and volume of complex figures.
3. Solve problems involving percentages, interest, capitalization rates, and income multipliers.
4. Use a financial calculator or table to solve problems involving discounting and annuities.

CHAPTER OUTLINE

I. PROPERTY DESCRIPTION

II. METES AND BOUNDS SYSTEM

A. Reference Points

B. Courses and Distances

1. Metes and Bounds Descriptions in Appraisals

III. RECTANGULAR (U.S. GOVERNMENT) SURVEY SYSTEM

A. Base Line and Meridian

B. Townships

C. Sections

D. Partial Sections

E. Adjustments and Government Lots

F. Rectangular Survey System Descriptions

G. Geodetic Survey System

IV. LOT, BLOCK, AND TRACT SYSTEM

V. APPRAISAL MATH

A. Distance, Area, and Volume

B. Area of a Rectangle

C. Units of Area

D. Converting Units

E. Area of a Triangle

F. Right Triangles

G. Areas of Complex Figures

H. Volume

I. Reciprocals

J. Percentages

K. Direct Capitalization

L. Interest

VI. FINANCIAL CALCULATIONS

A. Present and Future Value

B. Interest Compounding

C. “Hoskold” or Sinking Fund Method

D. “Inwood” Method

VII. MEASURES OF CENTRAL TENDENCY

VIII. CHAPTER SUMMARY

IX. CHAPTER QUIZ

I. PROPERTY DESCRIPTION

The orderly process of creating boundaries for land ownership and describing them is referred to as a legal description. The land within the boundaries is often referred to as a “parcel❓❓, lot, plot, or tract.” These terms may refer to all types of improved or unimproved land. A parcel of land generally refers to any piece of land that may be identified by a legal description in one ownership. Thus, every parcel of real estate is unique.

Although an appraiser is not a surveyor, he or she should be able to read and understand a “legal” description of real estate, since most appraisals require a legal description to adequately identify the subject property. Appraisers must also be comfortable applying a variety of different mathematical techniques and formulas used in the valuation process.

In everyday life, real estate is normally identified by its street address (e.g., “111 Main Street” or “1517 Park Avenue”). Some properties may also be known by a common name, such as “Green Acres Ranch.” However, street addresses and common names are not sufficient for legal purposes because they are not precise and may change over time.

The three primary systems of land description used in the United States are:

• Metes and Bounds System: Relies on physical features and measurements to define boundaries.
• Rectangular (U.S. Government) Survey System: Divides land into a grid system based on principal meridians and baselines.
• Lot, Block, and Tract System (Recorded Plat System): Uses recorded plats (maps) to identify properties within a subdivision.

II. METES AND BOUNDS SYSTEM

The Metes and Bounds system is one of the oldest methods of land description. It defines property boundaries by specifying distances (metes) and directions (bounds) from a Point of Beginning (POB). The description proceeds around the property’s perimeter until it returns to the POB, forming a closed shape. This system uses physical features (natural or artificial) as markers or monuments along the boundary line.

A. Reference Points

Reference points (monuments) serve as starting and ending points for the land description. They can be natural monuments, artificial monuments, or a combination of both.

• Natural Monuments: These are permanent natural features such as trees, rocks, rivers, and lakes.
• Artificial Monuments: These are man-made objects such as iron pins, concrete markers, roads, fences, and buildings.

A Bench Mark (BM) is a permanently affixed marker indicating elevation above sea level. These are used for elevation reference but may also serve as a reference point in the metes and bounds system.
When using natural monuments, it’s important to recognize that their permanency can be challenged by natural processes. Floods, erosion, or decay can displace or eliminate natural monuments, leading to boundary disputes. Thus, artificial monuments are favored for their relative stability.

The description begins at the Point of Beginning (POB), which can be either a natural or artificial monument. Ideally, the POB should be easily identifiable and permanent. In some instances, the POB is not directly on the property’s boundary, leading to the concept of a True Point of Beginning (TPOB). The description may first describe the location of the POB and then provide directions to the TPOB before delineating the property’s actual boundaries.

B. Courses and Distances

Once the POB is established, the description proceeds by specifying the course and distance of each boundary line.

• Course: The direction of a line, usually expressed in degrees, minutes, and seconds, measured from North or South. For example, “North 45 degrees 30 minutes East.”
• Distance: The length of the line, typically measured in feet, chains, or rods.

1. Metes and Bounds Descriptions in Appraisals

In an appraisal report, the metes and bounds description is crucial for accurately identifying the subject property. Appraisers must carefully examine the description to ensure it is complete, accurate, and unambiguous. Any discrepancies or ambiguities should be noted and resolved before proceeding with the appraisal.

Example:

“Beginning at an iron pin at the intersection of the east right-of-way line of Elm Street and the south right-of-way line of Oak Avenue; thence North 45 degrees 00 minutes East, a distance of 200 feet to an iron pin; thence South 45 degrees 00 minutes East, a distance of 150 feet to a concrete monument; thence South 45 degrees 00 minutes West, a distance of 200 feet to an iron pin on the south right-of-way line of Oak Avenue; thence North 45 degrees 00 minutes West, a distance of 150 feet along said right-of-way line to the Point of Beginning.”

III. RECTANGULAR (U.S. GOVERNMENT) SURVEY SYSTEM

The Rectangular Survey System, also known as the U.S. Government Survey System, was established to provide a systematic and uniform method for describing and dividing land, particularly in the western states. This system is based on a grid of lines running North-South and East-West.

A. Base Line and Meridian

The system relies on two primary reference lines:

• Principal Meridian: A North-South line designated by a name (e.g., Principal Meridian). Each principal meridian has a unique name and serves as the reference line for a specific area.
• Base Line: An East-West line that intersects the Principal Meridian.

There are 37 Principal Meridians in the United States, each with its own designated Base Line. These meridians are not equally spaced.
Guide Meridians: These are lines running North-South, located every 24 miles east and west of the Principal Meridian. They are used to compensate for the curvature of the Earth.
Standard Parallels (Correction Lines): These are lines running East-West, located every 24 miles north and south of the Base Line. They are used to compensate for the curvature of the Earth and ensure the grid system remains relatively accurate.

B. Townships

The intersection of range lines and township lines creates squares called Townships. A township is approximately six miles square, containing 36 square miles.

• Township Lines: East-West lines parallel to the Base Line, located six miles apart. Townships are numbered north and south of the Base Line (e.g., Township 1 North, Township 2 South).
• Range Lines: North-South lines parallel to the Principal Meridian, located six miles apart. Ranges are numbered east and west of the Principal Meridian (e.g., Range 1 East, Range 2 West).

Townships are identified by their township number and range number (e.g., Township 2 North, Range 3 East, abbreviated as T2N, R3E).

Mathematical representation of the area of a township:

Area_Township = side * side = 6 miles * 6 miles = 36 square miles

C. Sections

Each township is divided into 36 sections, each approximately one mile square. Sections are numbered 1 through 36, starting in the northeast corner and proceeding west, then south in an alternating pattern.

Sections are the basic unit of the Rectangular Survey System.

Mathematical representation of the area of a section:

Area_Section = side * side = 1 mile❓ * 1 mile = 1 square mile = 640 acres

D. Partial Sections

Due to the curvature of the Earth and surveying inaccuracies, some sections are not perfectly square. These are called partial sections, and they are typically found along the north and west sides of a township.

The most common way to describe a section is using fractional parts: quarter-section, half-section, etc. For instance:

• Quarter-Section: A section divided into four equal parts (160 acres each).
• Half-Section: A section divided into two equal parts (320 acres each).

E. Adjustments and Government Lots

Correction Lines and Guide Meridians correct for the Earth’s curvature. These create irregularities in some sections, leading to the formation of Government Lots. These lots are irregular parcels of land, typically smaller than a quarter-section, located along the north and west sides of a township. They are assigned lot numbers and their acreage is officially recorded.

F. Rectangular Survey System Descriptions

A Rectangular Survey System description typically reads from the smallest division to the largest, specifying the section, township, range, and principal meridian.

Example:

“The northwest quarter❓❓ of the Southeast Quarter of Section 10, Township 2 North, Range 3 East, Principal Meridian.”

This description is abbreviated as:

“NW 1/4, SE 1/4, Sec. 10, T2N, R3E, PM”

G. Geodetic Survey System

The Geodetic Survey System is a highly precise surveying method used to establish a network of control points (benchmarks) across a large area. It accounts for the curvature of the Earth and uses sophisticated instruments and mathematical techniques to determine the exact latitude, longitude, and elevation of these points. This system is often used as a basis for more localized surveys.

IV. LOT, BLOCK, AND TRACT SYSTEM

The Lot, Block, and Tract System, also known as the Recorded Plat System, is commonly used in urban and suburban areas. This system relies on recorded plats (maps) to identify individual properties within a subdivision.

A plat is a map of a subdivision showing the boundaries of individual lots, blocks, streets, easements, and other features. The plat is recorded in the county recorder’s office, becoming a public record.

• Lot: An individual parcel of land within the subdivision.
• Block: A group of lots bounded by streets or other features.
• Tract: A larger area encompassing multiple blocks and lots.

A Lot, Block, and Tract description typically includes the lot number, block number (if applicable), subdivision name, and the recording information (e.g., book and page number) of the plat.

Example:

“Lot 12, Block 3, Sunny Acres Subdivision, as recorded in Plat Book 45, Page 123, of the County Recorder’s Office.”

V. APPRAISAL MATH

Appraisal math involves a variety of calculations used to estimate value and analyze property characteristics. These include calculations of area, volume, percentages, capitalization rates, interest, and financial analysis tools.

A. Distance, Area, and Volume

Appraisers frequently need to calculate distances, areas, and volumes to accurately describe and compare properties.

• Distance: The linear measurement between two points. Common units include feet, inches, yards, and miles.

• Area: The two-dimensional space enclosed by a boundary. Common units include square feet, square yards, acres, and square miles.

• Volume: The three-dimensional space occupied by an object. Common units include cubic feet, cubic yards, and gallons.

B. Area of a Rectangle

The area of a rectangle is calculated by multiplying its length (L) by its width (W).

Formula:

Area_Rectangle = L * W

Example:

A rectangular lot measures 100 feet by 150 feet. What is the area of the lot in square feet?

Area = 100 feet * 150 feet = 15,000 square feet

C. Units of Area

Common units of area and their relationships are:

• 1 square foot (sq ft) = 144 square inches (sq in)
• 1 square yard (sq yd) = 9 square feet (sq ft)
• 1 acre = 43,560 square feet (sq ft)
• 1 square mile = 640 acres

D. Converting Units

To convert between units of area, use conversion factors.

Example:

Convert 15,000 square feet to acres.

Acres = 15,000 sq ft / 43,560 sq ft/acre = 0.344 acres

E. Area of a Triangle

The area of a triangle is calculated by multiplying one-half times the base (b) by the height (h). The height is the perpendicular distance from the base to the opposite vertex.

Formula:

Area_Triangle = 0.5 * b * h

Example:

A triangular lot has a base of 8❓0 feet and a height of 60 feet❓❓. What is the area of the lot in square feet?

Area = 0.5 * 80 feet * 60 feet = 2,400 square feet

F. Right Triangles

A right triangle is a triangle with one angle of 90 degrees. The side opposite the right angle is called the hypotenuse (c). The other two sides are called legs (a and b).

The Pythagorean Theorem states that the square of the hypotenuse is equal to the sum of the squares of the legs.

Formula:

c² = a² + b²

Example:

A right triangle has legs of 3 feet and 4 feet. What is the length of the hypotenuse?

c² = 3² + 4² = 9 + 16 = 25
c = √25 = 5 feet

G. Areas of Complex Figures

Complex figures can be divided into simpler shapes (rectangles, triangles) to calculate their areas. Calculate the area of each simpler shape and then add them together to find the total area.

H. Volume

The volume of a rectangular prism (box) is calculated by multiplying its length (L), width (W), and height (H).

Formula:

Volume = L * W * H

Example:

A room measures 12 feet long, 10 feet wide, and 8 feet high. What is the volume of the room in cubic feet?

Volume = 12 feet * 10 feet * 8 feet = 960 cubic feet

I. Reciprocals

The reciprocal of a number is 1 divided by that number. Reciprocals are useful for converting between capitalization rates and income multipliers.

Formula:

Reciprocal of x = 1 / x

Example:

What is the reciprocal of 0.10?

Reciprocal = 1 / 0.10 = 10

J. Percentages

Percentages are used to express a proportion of a whole.

• To find a percentage of a number, multiply the number by the percentage (expressed as a decimal).
• To find what percentage one number is of another, divide the first number by the second number and multiply by 100.

Formulas:

• Percentage of a number: Part = Percentage * Whole
• Percentage one number is of another: Percentage = (Part / Whole) * 100

Example:

A property sold for $200,000, and the broker’s commission was 6%. What was the amount of the commission?

Commission = 0.06 * $200,000 = $12,000

Example:

A property has an operating expense of $10,000 and revenue of $50,000. What percentage of the revenue is the operating expense?

Percentage = ($10,000 / $50,000) * 100 = 20%

K. Direct Capitalization

Direct capitalization is a valuation technique that estimates the value of a property by dividing its net operating income (NOI) by a capitalization rate (cap rate).

Formula:

Value = NOI / Cap Rate

Where:

• Value = Estimated value of the property
• NOI = Net Operating Income (annual income after operating expenses)
• Cap Rate = Capitalization Rate (the rate of return an investor requires)

Example:

A property has an NOI of $20,000 and a capitalization rate of 8%❓. What is the estimated value of the property?

Value = $20,000 / 0.08 = $250,000

L. Interest

Interest is the cost of borrowing money, expressed as a percentage of the principal amount.

• Simple Interest: Interest calculated only on the principal amount.

Formula:

Simple Interest = Principal * Rate * Time

Where:

• Principal = The initial amount borrowed
• Rate = The annual interest rate (expressed as a decimal)
• Time = The length of the loan (in years)

Example:

A loan of $100,000 has a simple interest rate of 5% for 3 years. What is the total interest paid?

Interest = $100,000 * 0.05 * 3 = $15,000

VI. FINANCIAL CALCULATIONS

Financial calculations are essential for analyzing the time value of money, which is the concept that money available today is worth more than the same amount of money in the future due to its potential earning capacity.

A. Present and Future Value

• Present Value (PV): The current value of a future sum of money or stream of cash flows, discounted at a specific rate of return.

• Future Value (FV): The value of an asset or investment at a specified date in the future, based on an assumed rate of growth.

Formulas:

• FV = PV * (1 + i)ⁿ
• PV = FV / (1 + i)ⁿ

Where:

• i = Interest rate per period
• n = Number of periods

Example:

What is the future value of $1,000 invested today at an annual interest rate of 6% for 5 years?

FV = $1,000 * (1 + 0.06)⁵ = $1,338.23

Example:

What is the present value of $1,000 to be received 5 years from now, discounted at an annual rate of 6%?

PV = $1,000 / (1 + 0.06)⁵ = $747.26

B. Interest Compounding

Compound Interest is interest calculated on the principal amount and also on the accumulated interest from previous periods.

The frequency of compounding (e.g., annually, semi-annually, quarterly, monthly) affects the total amount of interest earned.

Formula:

FV = PV * (1 + i/m)^nm

Where:

• m = Number of compounding periods per year

Example:

What is the future value of $1,000 invested today at an annual interest rate of 6%, compounded monthly, for 5 years?

FV = $1,000 * (1 + 0.06/12)^(5*12) = $1,349.01

C. “Hoskold” or Sinking Fund Method

The “Hoskold” method is used to determine the present value of a depleting asset, such as a mine or oil well. It considers the capital recovery rate and the safe rate of return.
This method is less frequently used in modern real estate appraisal.

D. “Inwood” Method

The “Inwood” method is used to determine the present value of a series of future income streams (an annuity). It considers the discount rate and the term of the annuity. This is also less frequently used as most prefer the annuity tables or calculators.

Formula:

PV = CF * [1 - (1 + r)^-n] / r

Where:

• CF = Cash flow per period
• r = Discount rate per period
• n = Number of periods

Example:

What is the present value of an annuity that pays $10,000 per year for 10 years, discounted at an annual rate of 8%?

PV = $10,000 * [1 - (1 + 0.08)^-10] / 0.08 = $67,100.81

VII. MEASURES OF CENTRAL TENDENCY

Measures of central tendency are statistical measures that describe the typical or central value of a dataset. These measures are used to analyze and interpret appraisal data.

• Mean: The average of a set of numbers. Calculated by summing all the numbers and dividing by the total number of values.

Formula:

Mean = (Sum of all values) / (Number of values)

• Median: The middle value in a sorted dataset. If there is an even number of values, the median is the average of the two middle values.

• Mode: The value that occurs most frequently in a dataset.

Example:

Consider the following sales prices of comparable properties: $250,000, $275,000, $260,000, $280,000, $260,000

• Mean = ($250,000 + $275,000 + $260,000 + $280,000 + $260,000) / 5 = $265,000
• Median = $260,000 (after sorting: $250,000, $260,000, $260,000, $275,000, $280,000)
• Mode = $260,000

Range: The difference between the highest and lowest values in a dataset.

• Range = Highest Value - Lowest Value = $280,000 - $250,000 = $30,000

Standard Deviation: A measure of the dispersion or spread of data around the mean. It indicates how much individual values deviate from the average. A lower standard deviation indicates that the data points are clustered closely around the mean, while a higher standard deviation indicates a wider spread.

VIII. CHAPTER SUMMARY

This chapter has covered the fundamental concepts of property description and appraisal math. Understanding legal descriptions is essential for accurately identifying the subject property and ensuring a sound appraisal. The metes and bounds, rectangular survey, and lot, block, and tract systems are the primary methods used in the United States.

Appraisal math involves a variety of calculations, including area, volume, percentages, capitalization rates, interest, and financial analysis tools. These calculations are used to estimate value, analyze property characteristics, and make informed appraisal decisions.

IX. CHAPTER QUIZ

(Chapter quiz questions will be provided separately)