120 Week 8 F /For WIZARD KIM
Chapter 86: Real estate mathematics 567
After reading this chapter, you will be able to:
• understand the fundamental real estate math concepts a real estate agent may be tested on during the state licensing exam and exposed to during their first four years of practice; and
• calculate common real estate math problems, including fee percentages, area calculation, loan and interest payments and general computations for real estate investments.
Learning Objectives
Real estate mathematics
Chapter
86
amortization table
area
capitalization rate (cap rate)
financial calculator
net operating income (NOI)
As a requisite for entry into the real estate profession, a thorough understanding of the formulas commonly used in the real estate industry is necessary.
Once the basic formulas and logic behind them are mastered, most math issues in mortgages, income property, property management and investment are solved with a basic, nonprogrammable calculator. Further, a nonprogrammable calculator is provided during the agent and broker California Bureau of Real Estate (CalBRE) licensing examination. The examinee is well advised to use the calculator supplied to ensure their accuracy on the exam.
Once an individual is licensed, a financial calculator is a wise investment in the real estate industry. Frequently used formulas are pre-programmed into the financial calculator, such as amortization schedules, making it an invaluable asset.
The mathematical tools you need
financial calculator An electronic calculator preprogrammed to perform advanced financial functions needed in real estate transactions.
Key Terms
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A grasp of mathematical basics is helpful when dealing with:
• percentages;
• fractions;
• area;
• loans;
• investments/cost analysis;
• capitalization (cap) rates; and
• all other mathematical concepts common to real estate transactions.
A percentage needs to be converted to a decimal before any mathematical computations can be completed. This is a conversion financial calculators are preprogrammed to make.
When converting percentages to decimals, the basic rule to follow is to move the decimal point two spaces to the left.
Examples:
50% = 0.5
3% = 0.03
115% = 1.15
Math basics
Percentages
Before delving into problems involving land descriptions and areas, here is a review of basic units of land measurement:
Township = 36 square miles broken up into 36 sections;
Section = 640 acres and 1 square mile;
Half section = 320 acres, a quarter section = 160 acres, and a quarter of a quarter = 40 acres;
Acre = 43,560 sq. ft;
Mile = 5,280 ft;
Square mile = 27,878,400 sq. ft (5,280 ft x 5,280 ft) or 640 acres (27,878,400 sq. ft / 43,560 sq ft);
Square acre = approximately 209 ft x 209 ft;
1 yard = 3 ft;
1 square yard = 9 sq. ft (3 ft x 3 ft);
1 mile = 320 rods; 5,280 ft; and
1 rod = 16.5 ft or 5.5 yards (16.5 ft x 3 ft).
Basic units of measurements in real estate
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Alternatively, to convert a decimal to a percentage, move the decimal point two spaces to the right.
Examples:
1.43 = 143%
0.03 = 3%
0.5 = 50%
As the use of a calculator is allowed while taking the CalBRE state licensing exam, it is beneficial to convert all fractions to decimals and then use the calculator to complete the computations.
To understand fractions, it is helpful to understand their basic mechanics. A fraction is composed of a numerator and a denominator. The number on the top of the fraction is the numerator and the number on the bottom is the denominator.
When converting a fraction into a decimal, divide the numerator (the number on top) by the denominator (the number on the bottom).
Examples:
4/5 = 0.8
1/2 = 0.5
3/4 = 0.75
3/100 = 0.03
667/100 = 6.67
The formula for an area of a rectangle is most often used in land measurement. The area of a rectangle equals length multiplied by width, the result being the number of square feet within the parcel.
A = L x W
Problem 1
A rectangular lot is 1,230 ft by 2,340 ft. What is the area of the lot in acres? (Round acres to the nearest whole acre.)
1. Area = 1,230’ x 2,340’ (2,878,200 sq. ft).
2. Conversion of square feet to acres. (One acre equals 43,560 sq. ft.) 2,878,200 / 43,560 = 66 acres.
Solution: 66 acres.
Fractions converted
Basic formulas for area
area The amount of space within the boundaries of a parcel of real estate.
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Problem 2
A rectangular lot is comprised of 10 acres. If the length of the lot is 500 feet, what is its width?
1. 10 acres = 10 x 43,560 sq. ft (435,600 sq. ft).
2. 435,600 sq. ft / 500 ft = 871.2 ft.
Solution: 871.2 ft.
Problem 3
A rectangular lot is comprised of a quarter of a quarter (1/16) of a section of a township. The width of the lot is 800 feet. What is the length of the lot in yards?
3. A quarter of a quarter of a section is equal to 40 acres. (One section of a township equals 640 acres and 1/16 of a section (1/4 x 1/4) is 40 acres.)
4. 40 acres = 40 x 43,560 sq. ft (1,742,400 sq. ft).
5. 1,742,400 sq. ft / 800 ft = 2,178 ft.
6. 2,178 ft / 3 = 726 yards.
Solution: 726 yards.
Problem 4
A rectangular lot measures 1,652 ft by 2,430 ft. The cost per acre for the lot is $120. How much would the lot cost to purchase? (Round to the nearest whole acre.)
1. The lot area = 1,652 ft x 2,430 ft. (4,014,360 sq. ft).
2. 4,014,360 sq. ft = 92 acres (4,014,360 / 43,560).
3. 92 acres x $120 = $11,040.
Solution: $11,040.
Another critical formula is the area of a triangle. This formula is used when determining the area of a triangular shaped lot.
The area of a triangle equals its base multiplied by its height divided by two.
A = (B x H) / 2
Problem 1
A triangular lot features a 200 ft base and a height of 150 ft. How many square feet are contained in the triangular lot?
1. The lot area = (200 ft x 150 ft) / 2.
2. 30,000 ft / 2 = 15,000 sq. ft.
Solution: 15,000 sq. ft.
Area of a triangle
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The percentage formula is a basic calculation used in real estate mathematics. It is typically used to determine the amount of a broker’s fee on a transaction.
As compensation for services, a broker is entitled to a broker’s fee typically stated as a percentage of the:
• sales price;
• loan amount; or
• total rents.
The percentage formula is easily converted to suit a variety of situations. Usually, two of the three variables will be known, leaving the third to be determined.
Understanding the basic formula makes solving problems a question of mechanics.
The percentage formula is as follows:
• Fee = Percent (%) x Principal
Percent (%) = the rate charged.
Principal (P) = the dollar amount of the price, loan or rents.
Given the basic formula and two of the three variables, solutions are determined as follows:
• To determine the fee, multiply the principal by the rate. P x %
• To determine the rate, divide the fee by the principal. Fee / P
• To determine the principal, divide the fee by the rate. Fee / %
Problem 1
A broker earns a 3% fee on the sale of a $100,000 home. How much will the broker earn?
Consider what we know:
% = 3%
P = $100,000
Using the percentage formula: Fee = % x P
1. Fee = 3% x $100,000
2. Fee = 0.03 x $100,000
3. Fee = $3,000
Thus, the broker earns a fee of $3,000.
Solution: $3,000
Percentage formula
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Problem 2
An agent shares in the broker’s fee based on 40% of the 3% fee the broker is paid on a transaction. The home sells for $245,000.
An irregular lot is neither an even square, rectangle or triangle. Instead, it is mathematically broken up into smaller known shapes. Thus, its area is determined by a combination of the above formulas and adding of the results
First, divide the irregular shaped lot into even squares, rectangles and triangles. Compute the area of each shape individually then add them to equal the total area of the lot.
The square portion is 50 ft x 50 ft.
The lot area of the square = (50 ft x 50 ft); thus, 2,500 sq ft.
The rectangular portion is 30 ft x 25 ft.
Editor’s note – Length = 50 ft (the length of the square) less 20 ft. Thus, 30 ft.
Width = 75 ft (the width of the square and rectangle combined) less 50 ft (the length of the square). Thus, 25 ft.
The lot area of the rectangle = (30 ft x 25 ft); thus, 750 sq ft.
The triangular portion is (40 ft x 30 ft) / 2.
Editor’s note – Width = 65 ft (the width of the rectangle plus the bottom outer edge of the triangle) less 25 (the width of the rectangle). Thus, 40 ft.
The lot area of the triangle = 1,200 ft / 2; thus, 600 sq ft.
Lastly, combine the lot area of each shape.
2,500 ft (the square) + 750 ft (the rectangle) + 600 ft (the triangle) = 3,850 sq ft.
Solution: 3,850 sq ft.
75’
50’
50’
20’ 65’
Figure 1
Area of an irregular lot
Chapter 86: Real estate mathematics 573
Editor’s note — Alternatively, the purchase price is analogous to the total loan amount in a mortgage loan brokerage situation, or the total amount of all rents due during the initial term of a lease agreement in a leasing agent situation.
How much will the agent earn?
Start with what we know to determine the broker’s fee:
% = 3%
P = $245,000
Using the percentage formula:
1. Fee = 3% x $245,000
2. Fee = 0.03 x $245,000
3. Fee = $7,350
Thus, the broker will earn $7,350. The sales person will then receive 40% of $7,350.
Using the percentage formula:
1. Fee = 40% x $7,350
2. Fee = 0.4 x $7,350
3. Fee = $2,940
The agent receives $2,940 on the close of the transaction.
Solution: $2,940
Problem 3
A real estate transaction involves both a seller’s broker and a buyer’s broker. The brokers agree to split a 6% fee paid by the seller 50-50.
You are the buyer’s broker’s agent and will receive 70% of the fee your broker receives.
If the home sells for $149,500, how much will you receive?
First, compute the total fee to be received by both brokers.
1. Total fee = 6% x $149,500
2. Total fee = 0.06 x $149,500
3. Total fee = $8,970
Thus, the total brokerage fee is $8,970.
Second, calculate the seller’s broker’s fee:
1. Seller’s broker’s fee = 50% x $8,970
2. Seller’s broker’s fee = 0.5 x $8,970
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3. Seller’s broker’s fee = $4,485
$4,485 is the amount the seller’s broker will receive on the close of the transaction.
Finally, calculate your share of the fee on in the deal:
1. Your fee = 70% x $4,485
2. Your fee = 0.7 x $4,485
3. Your fee = $3,139.50
You will receive $3,139.50 from your broker on the completion of the sale.
Solution: $3,139.50
Problem 4
A broker lists an office building for sale. The listing agreement calls for a graduated fee payment computation. The broker agrees to accept a fee of 5% on the first $150,000 and a smaller percent on the remaining sales price.
The broker sells the office building for $240,000 and earns a total fee of $11,100.
What is the percent of the sales price the broker has agreed they are to be paid on amounts over $150,000?
The broker earns 5% on the first $150,000 and an unknown percentage on the remaining $90,000 ($240,000 sales price - $150,000). The total fee received is $11,100.
Using the percentage formula (Fee = % x P), determine the fee on the first $150,000.
1. $11,100 = (5% x $150,000) + (?% x $90,000)
2. $11,100 = (0.05 x $150,000) + (?% x $90,000)
3. $11,100 = $7,500 + (?% x $90,000)
Next, determine the percentage the fee amount is of the remainder of the price.
1. $11,100 - $7,500 = ?% x $90,0000
2. $3,600 = ?% x $90,000
3. $3,600 / $90,000 = 0.04
Convert the amount to a percent by moving the decimal two spaces to the right. Therefore, the fee is 4% on the amount over $150,000.
Solution: 4%
The percentage formula can also be used in loan calculation problems.
Cost of using the lender’s money = Cost Loan
problems
Chapter 86: Real estate mathematics 575
Interest rate = Percent
Principal amount of the loan = Principal
When calculating problems involving simple interest, the interest rate is the rate of interest over one year.
Interest can be further broken down into months by dividing the annual interest rate by 12, and into days by dividing the interest rate by 360 (months are uniformly considered to be 30 days to avoid awkward numbers).
Financial calculators and amortization tables make interest calculations easier. An amortization table is a schedule of monthly loan payments which show the amount of principal and the amount of interest which comprises each constant payment until it is paid in full by the end of the term.
Early in the amortization schedule, a majority of the monthly payment is applied to interest. However, towards the end of the schedule, most of the monthly payment is applied towards the diminishing principal balance.
In loan problems, time is an important factor and is part of the percentage formula. Thus, the percentage formula is modified slightly for loan problems as follows:
Cost of borrowing money (C) = Interest Rate (%) x Time (T) x Principal (P)
C = % x T x P
When dealing with interest rates for different periods, remember the following:
• for annual interest, the interest rate is the percent given;
• for monthly interest, the interest rate is equal to the interest rate given divided by 12 (for the 12 months in the year); and
• for daily interest, the interest rate is equal to the interest rate given divided by 360.
Problem 1
A borrower owes $15,000 on a straight note payable at the end of the quarter. The borrower pays $225 in total interest costs with the timely payoff of the principal. What is the interest rate on the loan?
Editor’s note – A quarter is equal to 3 months.
Using the modified percentage formula and the information given above:
1. $225 = (?%/12 months) x 3 months x $15,000
2. $225 = ?%/4 (one quarter) x $15,000
3. $225 x 4/$15,000 = ?%
4. .06 = ?%
Thus, the interest rate on the loan is 6%.
amortization table A tabular schedule detailing the apportionment of principal and interest on each periodic payment due on an amortizing loan.
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Solution: 6%
Problem 2
If an investor earns $360 on a straight note with an interest rate of 8% payable in 60 days, what is the principal amount of the loan?
C = % x T x P
1. $360 = 8% x 60 days x P
2. $360 = .08 x (60 days/360) x P
3. $360/0.08 x 360/60 = P
4. 4,500 x 6 = P
5. $27,000 = P
The principal amount of the loan is $27,000.
Solution: $27,000
Problem 3
Consider a $200,000 interest-only straight note with monthly interest payments with a 6% annual interest rate due in 15 years. What is the amount of the final/balloon payment due on the end of the loan term?
Editor’s note — Some loan problems don’t require the computation of time, and thus this variable is omitted from the equation.
C = % x P
C = 6% x 200,000
The final/balloon payment equals the total loan balance, plus the last monthly interest payment. Start by determining the total annual interest charged for the loan.
1. C = 6% x $200,000
2. C = .06 x $200,000
3. C = $12,000
Next, determine the monthly interest and add this amount to the loan balance.
4. $12,000/12 months = $1,000
5. $1,000 + $200,000 = $201,000.
The final/balloon payment is $201,000.
Solution: $201,000
Calculations in income property transactions determine the profit from a sale or the amount an investor will pay for the net operating income (NOI) produced by the property.
Investment and cost problems
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Problem 1
A seller wants an 8% return (profit) on a $125,000 investment they have in an unencumbered property they own.
What is the minimum net selling price the seller needs to receive to realize the 8% return?
First, consider that at $125,000, the seller has a 100% return of their investment. Thus, the seller needs to realize a 108% total return to receive an 8% profit on their investment.
Using the percentage formula and the information given:
1. Net price = 108% x $125,000
2. Net price = 1.08 x $125,000
3. Net price = $135,000
For an 8% profit on the original investment, the seller needs to net $135,000 on the sale.
Solution: $135,000
Problem 2
A lender purchases a straight note at an 18% discount. The note has a remaining principal balance of $60,000 and a 10% interest rate.
What is the lender’s annual yield on its investment in the straight note?
First, compute the amount the lender paid to purchase the principal remaining due on the note:
1. $60,000 x 18% = the amount of the discount.
2. $60,000 x 0.18 = $10,800
The principal amount of the note minus the amount of the discount equals the investment the lender has in the note:
3. $60,000 - $10,800 = $49,200
Next, determine the dollar amount of the annual interest accruing on the $60,000 principal remaining due on the note.
4. Annual interest = 10% x $60,000
5. Annual interest = 0.10 x $60,000
6. Annual interest = $6,000
Lastly, divide the dollar amount of the annual interest ($6,000) accruing on the note’s principal by the amount invested in the note, i.e., the purchase price ($49,200):
7. $6,000 / $49,200 = 0.1219
8. 0.1219 = ?%
578 Real Estate Principles, Second Edition
Thus, the lender realized a 12.19% rate of return on the investment in the straight note.
Solution: 12.19%
Problem 3
A subdivider purchases three lots for $150,000. The three lots are subdivided into nine parcels. The parcels are all sold for $25,000 each.
What is the subdivider’s return on their investment?
First, calculate the total amount received for the nine parcels:
1. (9 x $25,000) = $225,000
Next, subtract the owner’s cost:
2. $225,000 - $150,000 = $75,000
Thus, $75,000 is the amount the subdivider profited on their investment.
Next, use the percentage formula to determine the percentage return on the subdivider’s investment:
3. $75,000 = ?% x $150,000
4. $75,000/$150,000 = ?%
5. 0.50 = ?%
Thus, the percentage return on the subdivider’s investment is 50%.
Solution: 50%
Problem 4
An owner sells their property for $230,000. The owner takes a profit of 15% over the amount they originally invested in the property. How much has the owner invested in the property?
The way to conceptualize the problem is as follows: the owner will receive a 100% return of their investment in the property, plus an additional return on the investment of 15%. Thus, the owner’s total return will equal 115%.
Use the percentage formula to determine how much the owner originally paid for the property:
1. $230,000 = 115% x P
2. $230,000 = 1.15 x P
3. $230,000/1.15 = P
4. $200,000 = P
Thus, the investor originally paid $200,000 for the property.
Solution: $200,000
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Problem 5
An investor purchased property listed at $100,000. The investor paid 15% less than the property’s listed price. The investor then resold the property at the prior owner’s listed price.
What is the investor’s percentage return?
First, determine the price the investor paid for the property. If the investor received a 15% discount on the listed price, the investor paid 85% (100% - 15%) of $100,000, the listed price.
1. Price paid = 85% x $100,000
2. Price paid = 0.85 x $100,000
3. Price paid = $85,000
Since the investor sold the property for $100,000, the investor made a $15,000 profit on the sale.
Using the percentage formula:
4. $15,000 = ?% x $85,000
5. $15,000/$85,000 = ?%
6. 0.1765 = ?%
The investor received a 17.65% return on the purchase and sale of the investment property.
Solution: 17.65%
Problem 6
A seller wishes to net a 14% profit over the original $240,000 purchase price of real estate.
What sales price does the property need to receive to yield a 14% return over the original price after the broker is paid a 6% fee out of the net proceeds? (Round answer to nearest whole number.)
Using the percentage formula, determine the amount of net proceeds the seller needs to receive to realize a gain of 14% over the original purchase price. The seller’s net proceeds need to be 114% of their original investment.
1. Net proceeds = 114% x $240,000
2. Net proceeds = 1.14 x $240,000
3. Net proceeds = $273,600
The net proceeds from the sale of the property to yield a 14% return would be $273,600.
To determine the sales price after paying the 6% broker’s fee, consider that $273,600 equals 94% (100% - 6%) of the total sales proceeds needed to net 14% after the 6% fee.
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Using the percentage formula:
4. $273,600 = 94% x P
5. $273,600/0.94 = P
6. $291,063.83 = P
Rounding to the nearest whole number, the price needed on a sale to yield 14% over the original purchase price after paying a 6% broker’s fee is $291,064.
Solution: $291,064
Problem 7
A borrower took out a $300,000 loan which contained a 3% prepayment penalty. The borrower paid two points to the lender to buy down the interest rate to 4.5%. The borrower sold the property and paid off the mortgage after seven years.
If the loan had an average balance of $250,000 during the seven years the borrower owned the property, what was the lender’s total gross profit on the loan?
Editor’s note — Solve this problem by individually calculating the lender’s profit on each component of the loan and totaling these amounts.
1. Total paid for points = $300,000 x 2% = $300,000 x 0.02 = $6,000
2. Total prepayment penalty = $300,000 x 3% = $300,000 x. 0.03 = $9,000
3. Total interest paid per year = $250,000 x 4.5% = $11,250
4. Total interest paid when loan was paid off = $11,250 x 7 years = $78,750
5. $6,000 + $9,000 + $78,750 = $93,750.
The lender’s total gross profit on the loan $93,750.
Solution: $93,750.
With income producing property, the value of the property is determined by capitalizing the net operating income (NOI) generated by the property (rental income minus operating expenses).
The rate of return an investor expects on their investment after rental operating expenses are subtracted from rental income is called the capitalization rate (cap rate).
Returning to the percentage formula:
Net operating income or loss = NOI
Capitalization rate = %
Purchase price = P
Capitalization rate
capitalization rate (cap rate) The annual rate of return on investment produced by the operations of an income property or sought by an investor on the investment of capital. The cap rate is calculated by dividing the net operating income by the price asked or offered for income property.
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Problem 1
An apartment building produces an NOI of $24,000. A buyer seeks an annual rate of return of 12%.
What price does the buyer pay for the property?
Using the percentage formula:
1. $24,000 = 12% x P
2. $24,000/0.12 = P
3. $200,000 = P
Thus, the buyer is to pay no more than $200,000 to yield a 12% rate of return.
Solution: $200,000
Problem 2
Rent on each unit in a four-unit apartment building is $545 per month. The owner has been receiving a rate of return equal to an 8% cap rate on their investment in the property.
If the rent for each unit drops to $500 per month, what is the owner’s loss in value over the year?
First, determine the rental loss for all units over the year.
1. $545 - $500 = $45 loss per unit per month
2. $45 x 12 = $540 loss per year per unit
3. $540 x 4 = $2,160 loss for the entire complex
Using the percentage formula:
4. $2,160 = 8% x P
5. $2,160/0.08 = P
6. $27,000 = P
Thus, the owner suffers a $27,000 loss in value over the year.
Solution: $27,000
Editor’s note – This can also be calculated by subtracting the total annual rent at $500 per month from the total annual rent at $545 per month. Problem 3 below illustrates this alternative method of calculation using a different set of facts.
Problem 3
An owner owns a 5-unit apartment building. The owner uses a cap rate of 10% on their investment. The owner usually realizes an NOI of $650 per month per unit. However, due to an increase in rents, the owner now nets $700 per month per unit.
What is the corresponding increase in value for the apartment building?
net operating income (NOI) The net revenue generated by an income producing property as the return on capital, calculated as the sum of a property’s gross operating income less the property’s operating expenses. [See RPI Form 352 §4]
582 Real Estate Principles, Second Edition
First, compute the value of the apartment building when the net income was $650 per unit.
The total net income over the year for the apartment building is $39,000 ($650 x 5 units x 12 months).
Using the percentage formula:
1. $39,000 = 10% x P
2. $39,000/0.1 = P
3. $390,000 = P
Next, compute the value of the apartment building at the new net income.
The new total net income over the year for the apartment building is $42,000 ($700 x 5 units x 12 months).
Using the percentage formula:
4. $42,000 = 10% x P
5. $42,000/0.1 = P
6. $420,000 = P
Thus, the increase in value for the apartment building is $30,000 ($420,000 - $390,000).
Solution: $30,000
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As a requisite for entry into the real estate profession, a thorough understanding of the formulas commonly used in the real estate industry is necessary.
The formula for an area of a rectangle is most often used in land measurement. The area of a rectangle equals length multiplied by width.
A = L x W
Another critical formula is the area of a triangle. This formula is used when determining the area of a triangular or irregular shaped lot. The area of a triangle equals its base multiplied by its height divided by two.
A = (B x H) / 2
The percentage formula is a basic formula typically used to determine the amount of a broker’s fee on a transaction. The broker’s fee equals the percentage multiplied by the principal amount.
F = % x P
The percentage formula is also used in loan calculation problems. However, in loan problems, time is an important factor and is added to the percentage formula. Thus, the cost of borrowing money equals the interest rate on the loan, multiplied by the term of the loan (time), multiplied by the principal amount.
C = % x T x P
amortization table ..................................................................... pg. 575 area ................................................................................................ pg. 569 capitalization rate (cap rate) ................................................... pg. 580 financial calculator ................................................................... pg. 567 net operating income (NOI) .................................................... pg. 581
Chapter 86 Summary
Chapter 86 Key Terms
No quiz or exam questions are based on this chapter.
Notes: