Chapter 58
Real Estate Arithmetic and Mathematics
Real estate practitioners perform many types of calculations. These calculations are
more arithmetic than mathematics. Arithmetic deals with real numbers and the
manipulation of these numbers. These real numbers include decimals and fractions.
The manipulation of these real numbers involves the four functions of addition,
subtraction, multiplication and division. Mathematics goes beyond artithmetic with
concepts like geometry, algebra, etc. The calculations in real estate entail a little bit
of geometry when dealing with square feet of a site or structure.
The basic presumption in this chapter is that the reader has a firm grasp of decimals
and fractions and their manipulations. Given existing educational curricula these
arithmetic concepts are taught in grades one through six in most school systems. For
this reason this chapter does not include a review of decimal and fraction arithmetic.
If the reader needs a review of these concepts there are several books on the market
typically entitled “real estate mathematics” that provide such a review.
Specifically, real estate arithmetic in this chapter consists of the following
calculations:
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Commission Calculation
Sales price to list price ratio
Interest payments or receipts
Discount points and origination fees
Appreciation and depreciation
Real Property taxes
Proration
Area measurements
These topics appear in the following sections of this chapter. The order of their
presentation does not signify any opinion about their relative importance.
In addition to these concepts real estate arithmetic also includes the following
calculations that are handled in other chapters.
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Mortgage math
Qualifying the Borrower
Appraisal math
Investment analysis
Closing or settlement statements
Compound interest (six functions of the dollar)
Commission Calculation
Commission rates are expressed as a percentage of the sales price. A specific
commission rate, such as 5%, is not set by law, nor should it be set by custom or
tradition. According to the federal government, commission rates are negotiable. In
the past, brokers have been sued by the federal government for price-fixing as a
violation of the Sherman Anti-Trust Act, with regard to collaboratively setting real
estate commissions in their local market areas.
The arithmetic involved in commissions is the application of a negotiated commission
rate (a percentage) to the sales price of property. Sales price times the commission
rate equals the commission. For example, if the negotiated commission is 5%, and
the property sells for $200,000, the commission is $10,000.
Sales Price x Commission Rate = Commission
Illustrative problem #1:
A property sells for $187,500 and the commission rate is 6.25%.
What is the commission?
$187, 500 x 6.25% = $11,718.75
Illustrative problem #2:
A property sells for $212,450 and the commission rate is 5.85%.
What is the commission?
$212,450 x 5.85% = $12,428.33
Commission Splits
When a real estate transaction involves 2 brokers working on a co-brokered or cooped deal, commissions are typically split between the listing broker and the selling
broker; for example, this could be a 50%/50% split, or as negotiated between the
brokers. So, a $10,000 commission would be split $5,000 to each broker. Often,
the $5,000 commission that went to each brokerage is further split between the
broker and the sales associate. Here the split can vary based on many variables, and
will be documented in a written agreement between the brokerage and the associate.
For example, if the split is 60%/40% to the broker, the broker gets $3,000 ($5,000
times 60%) and the sales associate gets $2,000 ($5,000 times 40%).
Sales price to list price ratio
Brokers and sales associates who list a property will price the property, in
consultation with the seller, so that the property sells in a reasonable amount of time
with adequate market exposure. This is traditionally done using a comparative
market analysis, or CMA. (Note: this process is not an appraisal.) Then to provide
room for negotiation, the list price is set above that anticipated sales price. If the
anticipated sales price is $150,000, the list price might be set at $160,000. If the
property actually sells for $154,000, the sales price to list price ratio is 96.25%
($154,000 / $160,000).
Simple interest payment
A simple interest payment is calculated as the amount of the loan times the interest
rate on that loan. For example if the loan amount is $150,000 and the contract
interest rate on that loan is 6% per year, the simple interest for the first year of the
loan is $ 9,000
Loan amount x interest rate = interest payment
Illustrative problem #1:
A buyer borrows $140,000 at the contract interest rate of 8.25%.
What is the borrower’s total interest payment in the first year?
$140,000 x 8.25% = $11,550
Illustrative problem #2:
A lender provides a $215,475 loan to a borrower at a contract interest rate of 6
3/8%.
How much interest does the lender receive in the first year?
3/8 equals 0.375
$215,475 x 6.375% = $13,736.53
Illustrative problem #3:
A lender receives a $9,654.12 interest payment in the first year on a 6% loan that
he made to the borrower. What is the loan amount?
Loan Amount x Interest Rate = Interest Payment
Loan Amount = Interest Payment / Interest Rate
Loan Amount = $9,654.12 / 6%
Loan Amount = $160,902
Discount points and origination fees
Discount points and origination fees are front-end charges that may be imposed by
the lender when making a loan. The lender uses discount points to increase his rate
of return on the front end without raising the interest rate over the life of the loan,
which would raise the monthly payment for the borrower. The lender charges loan
origination fees to cover the costs associated with data gathering and evaluation of
the borrower’s financial ability to repay the loan amount. One discount point is equal
to 1% of the loan amount. An origination fee may also be stated as a percentage of
the loan.
With regard to discount points the explicit wording of the loan agreement dictates
how the points are to be paid. If the statement says "a loan in the amount of
$100,000 at 5% for 30 years plus 2 discount points” the word plus indicates
payment of the points at the time the loan is issued. For example, a loan of
$100,000 at 5% for 30 years plus 2 discount points requires a payment of $2,000 for
discount points ($100,000 x 2% = $2,000) at the time of the loan closing.
Points paid up front increase the amount of funds the borrower requires to
close. This may be a trade-off for keeping his monthly payment lower, which may
even allow the borrower to qualify for the loan that he may not have qualified for at
a higher monthly payment. If the price of the property is $120,000 and the loan
amount is $100,000, the borrower must come to the closing with the $20,000 down
payment, plus the $2,000 in points to close the loan.
In some situations the discount points are netted out of the loan. For example, a
loan of $100,000 at 5% for 30 years with 2 discount points requires a payment of
$2,000 for discount points ($100,000 x 2% = $2,000). But, in this situation, the
borrower signs the promissory note to repay $100,000, but the dollar value of the
discount points is subtracted out of the proceeds, and the borrower receives a net
disbursement of $98,000. The operative word in this transaction is "with", vs. “plus.”
Origination fees are often expressed as a percentage of the loan amount. For
example, a loan amount of $120,000 with a 1% loan origination fee, generates an
origination fee of $1,200. This should be considered an additional cost of financing.
Appreciation and depreciation
Appreciation is an increase in the value of a property. For example, several years
ago the property was worth $120,000; today it is worth $150,000. Over time the
property appreciated by $30,000. On a percentage basis, this is a 25% appreciation
over the time. This is calculated as $150,000 - $120,000 = $30,000. $30,000 /
$120,000 = 25%) If the time period is 10 years, this is a simple annual appreciation
rate of 2.5%.
In practice historical appreciation rates are often used to look forward in time for real
estate investment analysis. An analyst might consider the historic appreciation rate
in a market for a specific type of property and use that as a basis to predict future
values. For example, a property is currently worth $200,000 and the historic
appreciation rate has been 2% per year for the previous five years. What is the
property worth five years in the future? Using the simple annual average of 2% for
five years equals the total appreciation of 10%, so the property would be projected
to be worth $220,000. Analysts will often use a compound appreciation rate which
will be discussed in the later section of this chapter. When a 2% compounded
appreciation rate per year is used, the current $200,000 property would be projected
to increase in value to $220,816.
Depreciation is a decrease in the value of a property. For example, several years ago,
a property was worth $150,000; today it is worth $120,000. Over time, the property
depreciated by $30,000. On a percentage basis this is a 20% depreciation over time.
This is calculated as $150,000 - $120,000 = $30,000. $30,000 / $150,000 = 20%.
If the time period is 10 years, this is a simple annual depreciation rate of 2%.
Like the appreciation rate, the depreciation rate can be used to project the expected
future value of a property.
Real estate taxes and millage rates
Real estate taxes are based upon the market value of the property as determined by
the city or county jurisdiction in which the property exists. In the year of sale, the
market value of the property may be presumed to be the sales price of the property.
In subsequent years after sale, the tax assessor of the jurisdiction estimaates the
fair market value.
Georgia uses an “assessment rate” of 40% of the market value to calculate the
assessed value of the property. For example, a property’s sale price is $200,000; the
40% assessment rate of 40% generates an assessed value of $80,000. The
$80,000 times the millage rate which is defined below establishes the property tax
bill.
Real property taxes are a major source of income for local and state governments.
These property tax revenues provide government services to a specific jurisdiction.
Property tax revenues typically pay for local government personnel services (safety,
teachers, judicial and administrative), equipment purchases and maintenance,
infrastructure maintenance and construction, and general supplies and materials.
The property tax calculation involves a series of related actions. The first is the
determination of the local budget, the sum of all funds required to cover the
expenses. Assume that a budget planning process determines that the local
government needs $1 million to operate in the coming year. Secondly and
simultaneously, the tax assessor of the jurisdiction determines the total value of all
real property (residential, retail, office, industrial, agricultural and undeveloped raw
land) is worth $20 million. This is called the tax base.
The third step in the process is the calculation of an assessed value for the property
from its fair market value. This is accomplished by multiplying market value by an
assessment rate. In Georgia, the assessment rate is 40% of the market
value. Market value x 40% = Assessed value. For example, a property worth
$200,000 in the market has an assessed value of $80,000. $200,000 x 40% =
$80,000
The fourth step in Georgia is the calculation of the millage rate. The tax base of $20
million is multiplied by the assessment rate of 40% to obtain the total assessed
value of $8 million ($20,000,000 times 40%). These values yield a tax rate of 12.5%.
(1M / 8M). The tax rate may be expressed as a percentage, but it is often expressed
as a Millage rate. The millage rate is based on a 1000 instead of a 100.
1M/8M = 0.125 = 12.5% or 125 per 1000
The last step is to calculate the individual property tax for a specific property. Having
calculated the assessed value, the tax rate is applied to each property in the
jurisdiction. For example, assume an equalization rate of 40% and a millage rate of
125, what is the property tax bill on a property whose current fair market value is
$200,000?
$200,000 x 40% = $80,000 assessed value
$80,000 assessed value times 125 per 1000 = $80 times 125 = $10,015.63
Outside of GA, some jurisdictions may calculate a different millage rate for
each of the property types.
Illustrative problem #1:
In a city that has a millage rate of 60 what is the property tax on a house with the
current market value of $155,000?
$155,000 x 60/1000 = $930
Illustrative problem #2:
In a city that has a millage rate of 30 mills and an assessment rate of 100% what is
the property tax on a house with the current market value of $220,000?
$220,000 x 30/1000 = $6,600
If you have the real estate tax bill for a property, use the values in hand for
calculating prorations. If you want to project the future tax bill for a buyer, calculate
the projected real estate taxes based on the purchase price.
Proration
As part of the closing process, there are expenses and income to ownership such as
prepaid or unpaid taxes, rent receipts on income property, and insurance that must
be divided between the seller and buyer so that each party only pays for the
expenses incurred or receives the income earned during the time that he or she
actually owned the property. Proration deals with allocating these items between the
buyer and the seller. An important element of this allocation is knowledge about
whether the payment is "in advance" or "in arrears”. Typically, property tax is paid in
arrears, with a tax bill for a current year being due in the last quarter on that
year. For state testing purposes, taxes are assumed to be paid July 1, so that taxes
are paid half in arrears and half in advance, spanning the period from January 1
through December 31. Property insurance and rent receipts are payments in advance.
Proration for payments “in arrears”:
Property Tax: Consider a property tax payment situation. Property tax payments are
due on December 31 for the current year. Property taxes are $5,000 for the year.
The property is sold on June 30. The owner and seller of the property used the
property for six months, sold it, and the buyer owns it for the remaining six months
in the year. The proration of property tax payment in this situation requires the seller
to compensate the buyer $2,500, as the buyer will pay the entire tax bill on
December 31. In this simple example, the seller owned the property for half a year
and the buyer owned the property for half a year, so the $5,000 is prorated between
the two of them.
(Annual Tax Bill / 365) x days in the proration period = Proration $
Consider the following illustrations of more complicated tax proration.
Illustration #1:
Property tax payments are due on December 31 for the fiscal year. Property taxes
are $4,800 per year. The property is sold on April 30. What is the proration?
The seller owns the property for 120 days (Jan = 31; Feb= 28, March = 31; April =
30) while the buyer owns the property for the rest of the year, which is 245 days..
The seller needs to compensate the buyer, as he will pay the entire year’s tax bill on
December 31.
$4,800/365 = $13.15 tax per day
$13.15 x 120 days = $1,578 seller to buyer proration
stration #3:
Illustration #2:
Property tax payments are due on December 31 for the year. Property taxes are
$4,800 per year. , The closing is August 15. What is the proration?
The seller owns the property for 227days, (count days in Jan, Feb, March, April, May,
June, July, and 15 days into August) while the buyer owns the property for 138 days
for a total of 365 days in the year. The seller needs to compensate the buyer.
$4,800/365 = $13.15 tax per day
$13.15 x 227 days = $2,985.05 tax proration, seller to buyer
The property tax payment is due on November 1 for the fiscal year. Property taxes
are $6,400 per year. The closing is November 10. What is the proration? The seller
just paid the tax bill for the entire year, including the 10 months + 10 days he
owned the property, and the rest of the year, that the buyer will own the property
through to December 31 when the tax year ends. November has 30 days. The
buyer owns the property 20 of those days, plus the 31 in December, for a total of 51
days in the tax year. The buyer needs to compensate the seller.
$6,400/365 = $17.53 tax per day
$17.53 X 51 days = $894.03 buyer to seller tax proration
Accrued Interest: Accrued interest payments occur at the closing when the seller
has a mortgage payment that was paid on the first day of the month and the closing
occurs during the month with the buyer assuming the loan. For example, the interest
payment for the month is $900 paid in advance and the closing is on the 20th of the
month. The buyer owes the seller 10 days of interest. The amount of the proration is
$900 divided by 30 days = $30 per day times 10 days yielding $300 as the proration.
Proration of payments “in advance”
Insurance: Property insurance is paid in advance. However, in most cases the buyer
obtains his or her own property insurance and proration of insurance payment is not
an issue at closing. If the buyer accepts or assumes the seller’s property insurance,
the proration process involves reimbursing the seller for the prepaid insurance
premiums. The proration is for any 365 day period. So, one must determine the
starting and ending dates of the policy, and calculate how many days of the policy
period the seller has used while owning the property, and how many the buyer will
use until the end of the policy period. Then the buyer must reimburse the seller for
the pre-paid insurance amount that the buyer will use. Here is an example:
You have a closing July 26. The buyer will be assuming the seller’s homeowner’s
insurance policy, on which coverage began October 20. The annual premium is
$2,690. What is the insurance proration?
The seller will receive a refund for $626.44. The insurance bill is paid in advance for
365 days starting Oct 20, and ending Oct 19 the following year. Buyer will use the
remainder of the policy period after closing day July 26. That is 5 days in July, plus
August, Sept, and 19 days in Oct, or a total of 85 days.
Proration calculation is $2,690/ 365 days X 85 days = $626.44
Rent: Rents are also paid in advance and typically on a monthly basis. The owner of
an income property receives rent payments on the first of the month every month
during the fiscal year. If the owner received $5,000 in monthly rents for the
apartments or retail space and then sells the property on the 12th day of the month,
proration must take place for the income, so that it is divided between the buyer and
the seller according to their ownership dates. The seller must compensate the buyer.
For example, the owner/seller owns the property for 12 days in April while the buyer
owns the property for 18 days or a total of 30 days in the month of April. The seller
must provide 18 days of rental income to the buyer.
$5,000 times (18/30) = $3,000
Pre-paid Interest: Pre-paid interest occurs at the closing when the buyer/borrower
must pay for the use of the new loan amount from the closing date to the end of the
month. If the closing is on March 15 the first mortgage payment is generally
scheduled for May 1. This timing makes sense when you realize that mortgage
payments are customarily paid in arrears in Georgia. The May 1 payment covers
April. The borrower owes the interest for March 15 to March 30; this is the pre-paid
interest. If the loan amount is $100,000 at 5% per annum, the interest payment is
$5000 for the first year. The pre-paid interest is $5000 divided by 360 equaling
$13.89 per day times 15 days yielding $208.33 for the pre-paid interest.
Transfer Tax
At the closing the property ownership transfers from the seller to the buyer, from the
grantor to the grantee. The county taxes this transfer. The formula for the transfer
tax is:
(Sales Price – Assumed Debt) times 0.1%
This formula simplifies to:
(Sales Price – Assumed Debt) / 1000 =
Transfer Tax
For example, consider a property sale at $200,000 with no loan assumption. What is
the transfer tax payment?
$200,000 / 1000 = $200
In other words, the transfer tax is currently $1 per each $1000 of sales price.
What is the transfer tax payment if the transfer tax rate increases? Consider a
property sale at $200,000 with no loan assumption. What is the transfer tax
payment if the transfer tax rate is $2/$1000? Intuitively, it is twice what it is a
$1/$1000 which would be $400. If the tax increases to $1.50 / $1000, it is 1.5 times
the amount which is $300. Arithmetically the formula should be written as:
(Sales Price – Assumed Debt) / (1000/T) = Transfer Tax
Where T is the transfer tax rate.
Intangible Recording Tax
As expressed by the Georgia Department of Revenue, an intangible recording tax is
due and payable on each instrument (security deed) securing one or more long-term
notes at the rate of $1.50 per each $500.00 or fraction thereof of the face amount of
all notes secured thereby in accordance with O.C.G.A. Section 48-6-61
(www.gabar.org/public/pdf/SECTIONS/.../TitleStandards.pdf). This tax is assessed
on the security instrument securing one or more long-term notes secured by real
property, to be paid upon the recording thereof, and must be paid within 90 days
from the date of the instrument executed to secure the note or notes. The maximum
tax on a single security instrument is $25,000. Authority O.C.G.A. Secs. 48-2-12,
48-6.61.
For example, if the security deed and the accompanying note are for a $200,000
loan, the formula for the intangible tax is:
($ Loan/500) times $1.50 = tax
($200,000/500) x $150 = $400 x $150 = $600
Several exemptions from payment of the intangible recording tax are provided by
Georgia and federal law. The Georgia Department Revenue has interpreted
applicable statutes concerning exemptions in the department’s Rules and Regulations
(Chapter 560-11-8-.14),
The lender can pass this tax on to the borrower.
Area measurement
Area measurement occurs for sites and structures. First consider sites.
Area measurement of sites requires a little knowledge of basic geometry. In the vast
majority of situations, knowledge of the calculation procedure for a rectangle and for
a triangle will be sufficient. For a rectangle you need to know length times width. A
square is a special case of a rectangle, with all sides being equal. For a triangle, you
need to know base times height.
A convention used in the real estate industry is to specify the width of the property
along the street or road first. This measurement is known as frontage. Consider a
rectangle that is 120' x 180'. The area of this site is 21,600 square feet. This can be
converted to acres by remembering that an acre contains 43,560 square feet; the
acreage of this site is 0.496 acres.
120’ x 180’ = 21,600 sq ft
21,600 = .496 acres
43,560
Consider a site that is 225' x 410'. Calculate the following measurements.
ft
acres.
What is the frontage?
What is the square footage of the site?
Answer 225 feet.
Answer 225 x 410 =92,250 sq
What is the acreage of the site?
Answer 92,250/43,560 = 2.12
Consider the area in the chart above.
What is the frontage? Answer 100 feet.
What is the square footage?
Answer: to obtain the square footage the site needs to be divided into a rectangle
and a triangle. The rectangle is 100 feet times 80 feet equals 8000 sq ft The triangle
has a base of 40 feet and a height of 80 feet; the area is “one half times the base
times the height of” which is (1/2) times 40x80 = 1600 sq ft. The square footage is
8000 + 1600 = 9600 sq ft.
What is the acreage? Answer 9,600/43,560 = 0.22 acres.
Consider the site in the chart below.
Part A is 100 feet times 250 feet = 25,000 sq ft
Part B is 200 feet times 180 feet = 36,000 sq ft
Total square footage = 61,000 sq ft
Total acreage = 61,000 / 43,560 = 1.4 acres
More complex site shapes can exist. For example in a residential subdivision cul-desac sites have slightly curved frontage and it irregular sidelines and back lines. Raw
land or agricultural land can have frontage along a winding road and could have
sidelines and/or back lines determined by a river or a stream. In such situations the
estimation of square footage and acreage will require the work of an expert, most
likely a surveyor.
Area measurement also occurs for structures. Structures are typically composed of
rectangles, and in many instances these rectangles are stacked to form two-story
houses, two-level regional malls and office buildings. So, you can calculate one level
and multiply by the number of levels to obtain the total square feet.
The different property types have different area measurement definitions.
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Single-family detached houses use "gross living area”
Retail properties use “gross building area” and “gross leasable area”.
Office buildings use “gross building area”, “rentable area” and “usable area”.
Gross Building Area
Several definitions of gross building area exist. The first one comes from the FNMA
Guidelines, Chapter 4, Section 405.7 and focuses on residences.
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Gross building area, which is the total finished area (including any interior
common areas, such as stairways and hallways) of the improvements based
on exterior measurements, is the most common comparison for two- to fourfamily properties. The gross building area must be consistently developed for
the subject property and all comparables that the appraiser uses. It should
include all finished above- and below-grade living areas, counting all interior
common areas (such as stairways, hallways, storage rooms, etc.), but not
counting exterior common areas (such as open stairways).
The second definition of gross building area comes from InvestorWords.com
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The sum of areas at all floor levels, including the basement, mezzanine,
and penthouses included in the principal outside faces of the exterior walls
without allowing for architectural setbacks or projections.
The third definition comes from the Appraisal Institute:
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Total floor area of a building, excluding unenclosed areas, measured from the
exterior of the walls of the above-grade area. This includes mezzanines and
basements if and when typically included in the region. Source: Appraisal
Institute, The Dictionary of Real Estate Appraisal, 5th ed. (Chicago: Appraisal
Institute, 2010).
Also see the many industry definitions prepared and sold by the Building Owner and
Managers Association (BOMA) (http://www.boma.org/MeasurementStandards)
Closing exercises
Consider the following single family residential structures (see chart below):
Part A 40’ x 30 = 1200’
Part B 20’ x 50 = 1000’
What is the gross living area of the house?
Answer: 2,200 square feet
Consider the following chart below.
1st Story (45’ x 30’) + (20’ x 10’) = 1,350 + 200 = 1,550
2nd Story 65’ x 30’ = 1,950
What is the gross living area of the house?
Answer: 3,500 square feet