Real Estate Math Instructor Materials PDF

17 Real Estate Math Learning Objectives After completing this lesson, students should be able to... List the four steps involved in solving math problems Convert fractions to percentages and percentages to fractions Distingui...

17 Real Estate Math Learning Objectives After completing this lesson, students should be able to... List the four steps involved in solving math problems Convert fractions to percentages and percentages to fractions Distinguish between area, volume, and percentage problems Recognize the math problems that require percentage calculations, and identify the correct formulas to use for each type of problem Solve capitalization rate problems and tax assessment problems Describe how to calculate the seller’s net Explain how to prorate expenses such as rent, mortgage interest, property taxes, and insur- ance premiums Suggested Lesson Plan 1. Give students Exercise 17.1 to review the previous chapter, “California Real Estate License Law.” 2. Provide a brief overview of Chapter 17, “Real Estate Math,” and review the learning objectives for the chapter. © 2021 Rockwell Publishing  Principles of California Real Estate Instructor Materials 3. Present lesson content: Solving Math Problems – Four-step approach – Decimal numbers Area Problems – Squares and rectangles – Triangles – Odd shapes EXERCISE 17.2 Area problems Volume Problems Percentage Problems – Solving percentage problems – Commission problems EXERCISE 17.3 Percentage and commission problems – Loan problems – Profit or loss problems – Capitalization problems EXERCISE 17.4 Profit/loss and capitalization problems Tax Assessment Problems Seller’s Net Problems EXERCISE 17.5 Tax assessment and seller’s net problems Proration Problems – Property taxes – Insurance premiums – Rent – Mortgage interest EXERCISE 17.6 Proration problems 4. End lesson with Chapter 17 Quiz. 2  Chapter 17: Real Estate Math Chapter 17 Outline: Real Estate Math I. Solving Math Problems A. The four steps to solving every math problem are: 1. Read the question 2. Write down the formula 3. Substitute the relevant numbers 4. Calculate B. If a problem presents you with fractions or percentages, first convert them into deci- mal numbers II. Area Problems A. Problems involving squares and rectangles use the formula Area = Length × Width B. Problems involving triangles use the formula Area = ½ Base × Height C. Odd shapes should be divided into squares, rectangles, and triangles, and then the area of each component should be added up EXERCISE 17.2 Area problems III. Volume Problems A. The formula for volume problems is Volume = Length × Width × Height IV. Percentage Problems A. The basic formula for percentage problems is Part = Whole × Percentage B. This can also be expressed as Whole = Part ÷ Percentage, or Percentage = Part ÷ Whole C. In commission problems, the part is the amount of the commission, the whole is the sales price, and the percentage is the commission rate EXERCISE 17.3 Percentage and commission problems D. In loan problems, the part is the annual interest (you may need to multiply if it is expressed monthly or quarterly), the whole is the loan amount, and the percentage is the annual interest rate E. In profit or loss problems, the formula is stated as Now = Then × Percentage (where the percentage is 100% plus the percentage of profit or minus the percentage of loss) F. In capitalization problems, the formula is stated as Income = Value × Capitalization Rate (you may need to calculate net income by applying an operating expense ratio to gross income) EXERCISE 17.4 Profit/loss and capitalization problems 3  Principles of California Real Estate Instructor Materials V. Tax Assessment Problems A. Tax assessment problems can be solved using the formula Tax = Assessed Value × Tax Rate B. The tax rate may be expressed, instead of as a percentage, as a dollar amount per hun- dred or per thousand dollars of assessed value, or as a number of mills (one-tenth of one cent) per dollar of assessed value VI. Seller’s Net Problems A. A seller’s net problem determines how much a property will have to sell for, if a seller wants to net a specified amount B. First, add the seller’s desired net to the costs of the sale except the commission C. Next, subtract the commission rate from 100% (i.e., 100% – 6% = 94%) D. Divide the total from step 1 by the total from step 2 to find the selling price EXERCISE 17.5 Tax assessment and seller’s net problems VII. Proration Problems A. The three steps in a proration problem are to calculate the per diem (daily) rate of the expense, determine the number of days for which the party is responsible, and multi- ply the per diem rate by the number of days B. Proration may be done using either a 365-day year or a 360-day year (a banker’s year, where every month is considered to have 30 days) C. In property tax proration problems, remember that in California the tax year starts on July 1, and the payments may be divided into installments, some of which may have already been paid D. In insurance proration problems, a seller has usually prepaid, and will be entitled to a refund of the unused portion E. In mortgage interest problems, there are separate calculations for the seller (who will owe interest from the first day of the month when closing occurs, up to the closing date) and the buyer (who will prepay interest from the closing date to the last day of the month when closing occurs) EXERCISE 17.6 Proration problems 4  Chapter 17: Real Estate Math Exercises EXERCISE 17.1 Review exercise To review Chapter 16, “California Real Estate License Law,” read the following True/ False questions aloud to students and have them jot their answers down on a piece of paper; discuss the answers together. 1. It’s legal for an unlicensed person to act as an agent in a real estate transaction if she is the principal’s attorney in fact and is acting without compensation. 2. The state attorney general is the legal advisor for the Real Estate Commissioner. 3. Someone who acts as an agent in the sale of business opportunities, including goodwill, inventory, and other business assets, must have a real estate license or a securities license. 4. If the owner of a small investment firm sold real estate without a real estate broker’s license, he would be prosecuted by the attorney general’s office. 5. A resident manager can manage an apartment house without a real estate li- cense. 6. The Real Estate Commissioner is an elected public official who serves a four-year term. 7. A person with a broker’s license who works for another broker instead of starting her own brokerage is called an affiliated broker. Answers: 1. TRUE. An attorney in fact acting without compensation is exempt from the real estate licensing requirement. 2. TRUE. The Commissioner receives legal advice concerning the Real Estate Law from the state attorney general. 3. TRUE. To sell business opportunities on behalf of others in California, an individual is generally required to have a real estate license, even if there is no real property involved in the transaction. An exception to that requirement is made for licensed securities brokers. 4. FALSE. He would be prosecuted by the district attorney of the county where the violations took place. 5. TRUE. Resident managers are exempt from the broker’s license requirement. 6. FALSE. The Commissioner is appointed by the governor of California. 7. FALSE. A broker who works for another broker is called an associate broker. 5  Principles of California Real Estate Instructor Materials EXERCISE 17.2 Area problems Answer the following multiple choice questions. 1. A building’s dimensions are 35’ × 60’ × 9’. How many square feet does it contain? A. 2,100 B. 3,450 C. 18,900 D. 21,355 2. From the point of beginning, a property’s boundaries run 900 feet in a southerly direc- tion; then due east for 1,250 feet; then in a northerly direction 300 feet; then back to the point of beginning. How many square feet are in the described parcel? A. 675,000 B. 715,000 C. 750,000 D. 805,000 Answers: 1. A. This question asks for square footage, not cubic footage, so you can disregard the fact that the building has nine-foot ceilings. Just use the area formula for rectangles. Area = Length × Width (A = L × W) A = 35 × 60 A = 2,100 square feet 2. C. It’s helpful to draw a picture of this property, since it’s an odd shape. To find the area, break it into two simpler shapes: a triangle above a rectangle. The base of the triangle measures 1,250 feet, and its height is 600 feet. (Determine the height of the triangle by subtracting the length of the property’s eastern boundary from the length of its western boundary: 900 – 300 = 600 feet.) The rectangle measures 1,250 feet by 300 feet. Calculate the area of the rectangle first, by multiplying its length by its width. A = L × W A = 1,250 × 300 A = 375,000 square feet 6  Chapter 17: Real Estate Math Now calculate the area of the triangle using the area formula for triangles. Multi- ply the base times the height times one-half. Area = ½ × Base × Height (A = ½ × B × H) A = ½ × 600 × 1,250 A = 375,000 square feet Finally, add the areas of the triangle and the rectangle together to find the area of the whole property: 375,000 + 375,000 = 750,000 square feet. EXERCISE 17.3 Percentage and commission problems Answer the following multiple choice questions. 1. A convenience store occupied 30% of a 150’ × 200’ lot (the rest of the space is used for parking). Ten percent of the lot was condemned for a public easement. How many square feet remained for parking after the easement was established? A. 18,000 B. 18,460 C. 19,759 D. 21,346 2. The asking price is $345,000. The property sells for $330,000. The commission is 6%. 60% of the commission goes to the broker and 40% goes to the salesperson. How much did the salesperson receive? A. $7,920 B. $8,280 C. $11,880 D. $12,420 3. Cortezar purchases a property for $255,000 and puts 20% down. The monthly interest payments are $1,742.50. What is the annual rate of interest? A. 9.50% B. 9.90% C. 10.10% D. 10.25% 7  Principles of California Real Estate Instructor Materials Answers: 1. A. The first step is to find the total square footage of the lot. Use the area formula for rectangles. Area = Length × Width (A = L × W) A = 150 × 200 A = 30,000 square feet Next, you need to find what percentage of the lot can be used for parking. The store occupies 30% of the lot, and the easement occupies another 10%, for a to- tal of 40%. Therefore, 60% of the lot remains for parking. Now you can use the percentage formula to find out how many square feet that represents. Part = Percentage × Whole (P = % × W) P =.6 × 30,000 P = 18,000 square feet 2. A. This is a percentage question that must be solved in two steps. First, calculate what the total commission was, by finding 6% of the $330,000 sales price. (The asking price is irrelevant; brokerage commissions are based on the sales price.) P = % × W P =.06 × $330,000 P = $19,800 Next, calculate what the salesperson’s share of the commission was, by finding 40% of the total commission. P = % × W P =.4 × $19,800 P = $7,920 3. D. The first step is to calculate the loan amount. If Cortezar puts 20% down, then the loan amount is 80% of the property’s value. P = % × W P =.8 × $255,000 P = $204,000 Note that the interest is expressed in terms of monthly payments. You’ll need to multiply by 12 to find the annual interest. $1,742.50 × 12 = $20,910 Now you have enough information to calculate the annual rate of interest. You know the “part” (the amount of interest) and the “whole” (the principal), so you need to isolate the percentage rate and switch from multiplication to division. P = % × W becomes P ÷ W = % $20,910 ÷ $204,000 =.1025 Expressed as a percentage, that’s 10.25%. 8  Chapter 17: Real Estate Math EXERCISE 17.4 Profit/loss and capitalization problems Answer the following multiple choice questions. 1. A home is presently appraised at $550,000. Calvin bought it new four years ago. Since then it has depreciated 16%. The home was originally worth approximately: A. $594,420 B. $654,760 C. $683,510 D. $716,975 2. An appraiser determines the property’s net operating income is $125,350. If she ap- plies a capitalization rate of 8%, what is the market value of the property? A. $1,566,875 B. $1,595,675 C. $1,729,580 D. $1,958,740 Answers: 1. B. This problem involves depreciation, which can be tricky. Remember that the house isn’t worth 16% of what it was worth four years ago. Instead, you have to subtract that 16% from 100%. In other words, the house is now worth 84% of what it used to be worth. Now you can apply the Then and Now (Value After) formula, which is just a variation on the basic percentage formula. You know the “part” (its current value, which is only part of the original value) and the percent- age of depreciation, and you need to isolate the “whole” value of the home (in other words, the “then” value, or the value before). P = % × W P ÷ % = W $550,000 ÷.84 = W $654,761.90 = W 2. A. Use the capitalization formula to solve this problem. (Note, however, that the capitalization formula is just a variation on the basic percentage formula.) Income = Rate × Market Value. You know the income and the rate, so you need to isolate the value. I ÷ R = V $125,350 ÷.08 = V $1,566,875 = V 9  Principles of California Real Estate Instructor Materials EXERCISE 17.5 Tax assessment and seller’s net problems Answer the following multiple choice questions. 1. A property recently sold for $390,000. How much will the annual property taxes be if the tax rate is $0.95 per $100 of assessed valuation? A. $837.45 B. $994.67 C. $2,900 D. $3,705 2. Smith bought a home six years ago for $210,000. He wants to sell the property for a 25% profit after paying a 7% commission and $1,750 in other settlement costs. What would he have to sell the home for? A. $234,500 B. $241,216 C. $284,140 D. $350,090 Answers: 1. D. Since the problem doesn’t state otherwise, you can assume that the assessed val- ue is 100% of the market value (as it would be if the property were in California). The first step is to divide the sales price by 100 to determine the number of $100 increments. $390,000 ÷ 100 = 3,900 Next, multiply that figure by the tax rate to determine the amount of the annual taxes. 3,900 × $0.95 = $3,705 2. C. The first step in a seller’s net problem like this one is to calculate what the seller’s desired net is. Here, the seller’s desired net is 25% more than the home’s origi- nal price, or 125% of the original price. Use the percentage formula. Part = Percentage × Whole (P = % × W) P = 1.25 × $210,000 P = $262,500 (desired net) However, you still need to factor in the other costs. The second step is to add all costs other than the commission. $262,500 + $1,750 settlement costs = $264,250 10  Chapter 17: Real Estate Math The next step is to subtract the commission rate from 100%. The reasoning is that if Smith wants to net $262,500, that’s only a portion of what he will actually have to sell the home for if he also has to pay the agent’s commission. 100% – 7% = 93% Finally, use the percentage formula to calculate the entire sales price for the home. You already know the “part” (in other words, the portion of the total sales price that Smith keeps) and the rate, so you need to isolate the “whole” sales price. P = % × W P ÷ % = W $264,250 ÷.93 = W $284,139.78 = W EXERCISE 17.6 Proration problems Answer the following multiple choice questions. 1. The purchase agreement requires the buyer to place four months’ worth of property taxes and hazard insurance in escrow. The applicable tax rate is $1.50 per $100 of assessed value; the assessed value of the property is $400,000. The premium for a three-year hazard insurance policy is $1,800. How much must the buyer place in escrow? A. $1,650 B. $1,750 C. $2,150 D. $2,200 2. A fire insurance policy began on April 1. It cost $1,900.80 for three years of coverage. The insurer canceled the policy on December 15 of the same year it was issued. If you use a 360-day year, which of the following figures is closest to the premium for the unused portion of the policy? A. $448.80 B. $1,399.20 C. $1,452.00 D. $1,504.80 11  Principles of California Real Estate Instructor Materials Answers: 1. D. First, calculate the annual property taxes: $400,000 (assessed value) ÷ $100 = 4,000 4,000 × $1.50 (tax rate) = $6,000 (annual taxes) Next, find the monthly share of the annual taxes and multiply by the number of months required: $6,000 ÷ 12 months = $500 per month × 4 months = $2,000 Now calculate the monthly share of the three-year insurance policy and multiply by the number of months required: $1,800 ÷ 36 months = $50 per month × 4 months = $200 Add the tax and insurance amounts together for the total amount to be placed in escrow: $2,000 + $200 = $2,200 2. C. If the policy began on April 1 and ended on December 15, it was valid for 8.5 months. First calculate the annual premium for the insurance: $1,900.80 ÷ 3 years = $633.60 (annual premium) Then calculate the per diem rate: $633.60 ÷ 360 days = $1.76 per diem Determine the number of days the policy was in effect. Using 30-day months, eight full months (April through November) is 240 days. Add the 15 days in De- cember to get the total number of days: 240 days + 15 days = 255 days. Next, multiply the number of days by the per diem rate: $1.76 per diem × 255 days = $448.80 Bear in mind, though, that we’re looking for the unused portion of the policy, not the used portion. So subtract that figure from the total premium to find the an- swer to the problem: $1,900.80 – $448.80 = $1,452 12  Chapter 17: Real Estate Math Chapter 17 Quiz 1. A two-story commercial building is 50 feet 5. An owner sold a lot for $400,000. He had to wide and 80 feet long. The first story is 16 pay a 6% commission to his broker, as well as feet high; the second story is 14 feet high. The a 1% prepayment penalty on the existing loan replacement cost for the first story is $2.50 per balance of $320,000, escrow fees of $172, cubic foot; the replacement cost for the second title insurance of $628, and four points on the story is $2 per cubic foot. What would be the buyer’s new loan for $380,000. What percent- replacement cost for the entire building? age of the seller’s equity will be eaten up by his selling costs? A. $18,000 B. $268,000 A. 40% C. $270,000 B. 54% D. $272,000 C. 63% D. 78% 2. An appraiser values an investment property at $800,000 using a capitalization rate of 9%, 6. A borrower obtained a loan secured by real based on its annual income of $72,000. What property for $200,000. The borrower paid four would the value of the property be if a 12% discount points when the loan was originated. capitalization rate is applied? The borrower made monthly payments of $1,630 per month, including 8% interest. When A. $500,000 the borrower sold the property after five years, B. $600,000 the bank imposed a prepayment penalty of 2% C. $846,000 of the face amount. The average loan balance D. $900,000 during those five years was $185,000. What amount did the lender make on the loan over 3. Marla borrowed $25,000 on a straight note. She that five-year period? paid $2,625 in interest over 14 months. What A. $29,800 is the annual interest rate on her loan? B. $62,000 A. 8% C. $82,000 B. 9% D. $86,000 C. 10% D. 11% 7. Fergus purchased a property for $500,000, us- ing a 12% downpayment. To finance the rest of 4. A high-rise office building has an effective the purchase, he obtained a 30-year amortized gross income of $530,000 each quarter. The loan with monthly payments of $4,192 at an annual expenses are 32% of the building’s 8.75% interest rate. He sold the property for effective gross income. What is the building’s $559,000 before making his first mortgage annual net income? payment. What was his equity at the time of A. $678,400 the sale? B. $1,441,600 A. $59,000 C. $1,632,400 B. $60,000 D. $2,120,000 C. $119,000 D. None of the above 13  Principles of California Real Estate Instructor Materials 8. A tract of rural land is 395,340 square feet. The 12. If a parcel is 110 yards long and 264 yards wide, county owns a small corner of the land, measur- how many acres does it contain? ing 30 feet by 110 feet. The private portion of A. 2/3 acre the land sold for $10,500 per acre. What was B. 2 acres the amount realized from the sale? C. 5 acres A. $10,500 D. 6 acres B. $94,500 C. $96,090 13. A property is reassessed at $600,000. The ap- D. $95,295 plicable tax rate is 1.15%. What will the new monthly property tax bill be? 9. A rectangular parcel of land is 1,980 feet wide A. $57.50 and 1,980 feet long. How many acres does it B. $575 contain? C. $5,750 A. 40 acres D. $6,900 B. 72.7 acres C. 82 acres 14. A property was sold on February 1. The seller D. 90 acres had paid the first installment of property taxes for the year. The annual property tax bill was 10. A landlord and tenant enter into a two-year $2,400. How much more does the seller have commercial lease. The lease states that the to pay? monthly payment for the second year will be A. $200 whatever was the average monthly payment B. $1,000 during the first year. During the first year, the C. $2,200 monthly payment was $820 per month, plus D. Nothing; the seller will be reimbursed 2.5% of all gross annual receipts over $42,000. $1,000 The tenant’s average gross receipts were $51,000 per month. What will the monthly pay- ments be during the second year of the lease? 15. The Culebras purchased a home for $536,800. Their broker tells them that it will cost 12% of A. $1,045 the selling price to sell the property, assuming B. $1,255.83 all other variables remain constant. How much C. $2,007.50 will the property need to have appreciated in D. $2,125 order for the Culebras to resell it without a loss? 11. Vera bought a house for $250,000. She obtained A. 6.8% a loan for 85% of the purchase price, subject B. 12% to a 7% interest rate. Before making the first C. $64,416 payment on the loan, though, she sold the D. $73,200 house to a friend for $275,000. At the time of the sale, what was the amount of her equity in the property? A. $12,500 B. $25,000 C. $37,500 D. $62,500 14  Chapter 17: Real Estate Math Answer Key 1. D. Multiply the width by the length of 4. B. First, multiply the quarterly gross in- the building to calculate each floor’s come to determine the annual gross square footage (80 feet × 50 feet = income ($530,000 × 4 = $2,120,000). 4,000 square feet). For the first story, Next, determine the annual expenses calculate the cubic footage (4,000 ($2,120,000 ×.32 = $678,400) and square feet × 16 feet = 64,000 cubic subtract the annual expenses from feet) and multiply that by $2.50 to the gross income to arrive at the net calculate the first story’s replacement income ($2,120,000 – $678,400 = cost (64,000 × $2.50 = $160,000). $1,441,600). For the second story, calculate the cubic footage (4,000 square feet × 14 5. B. First, calculate the value of the seller’s feet = 56,000 cubic feet) and multi- equity ($400,000 sale price – $320,000 ply that by $2 to calculate the second loan balance = $80,000 equity). Next, story’s replacement cost (56,000 × calculate the value of all the costs of $2 = $112,000). Add the two replace- sale (commission: $400,000 ×.06 ment costs to calculate the total cost = $24,000) (prepayment penalty: ($160,000 + $112,000 = $272,000). $320,000 ×.01 = $3,200) (discount points: $380,000 ×.04 = $15,200) 2. B. All that needs to be done to solve this and add them all together ($24,000 + question is divide the annual income $3,200 + $15,200 + $628 + $172 = by the new capitalization rate ($72,000 $43,200). Divide the total loan costs ÷ 12% = $600,000). Because the by the seller’s equity to determine desired rate and annual income are what percentage the loan costs repre- known, there’s no need to know the sent ($43,200 ÷ $80,000 =.54). value at a 9% rate. 6. D. First, calculate the amount the bor- 3. B. First, calculate how much interest rower paid in interest; use the average is earned each month ($2,625 ÷ 14 loan amount during the loan term, months = $187.50 per month). That rather than the original face amount will enable you to calculate the inter- ($185,000 ×.08 = $14,800) ($14,800 est payment in one year ($187.50 × 12 × 5 years = $74,000). Calculate the months = $2,250). Then, calculate the amount received from the discount interest rate; you know the part and the points ($200,000 ×.04 = $8,000). Cal- whole, so use the formula P ÷ W = % culate the amount received from the ($2,250 ÷ $25,000 =.09). prepayment penalty ($200,000 ×.02 = $4,000). Add the three components together to determine the total received by the lender ($74,000 + $8,000 + $4,000 = $86,000). 15  Principles of California Real Estate Instructor Materials 7. C. First, determine the amount of the 11. D. First, calculate the amount of the loan ($500,000 selling price ×.12 = downpayment (100% – 85% = 15%) $60,000 downpayment) ($500,000 ($250,000 ×.15 = $37,500). Next, – $60,000 = $440,000 loan). Then sub- calculate the amount of apprecia- tract the amount of the loan from the tion in value ($275,000 – $250,000 selling price to find the equity at the = $25,000). Both of these form time of the sale ($559,000 – $440,000 Vera’s equity, so add the two together = $119,000). You can disregard all the ($37,500 + $25,000 = $62,500). information about the interest rate and monthly payments. 12. D. First, multiply the length and width of the parcel to calculate the square yard- 8. B. First, determine the area of the county- age (110 yards × 264 yards = 29,040 owned portion of the land (30 feet × square yards). Then, convert square 110 feet = 3,300 square feet). Subtract yardage to square footage (29,040 that portion from the total area of square yards × 9 square feet per square the land (395,340 – 3,300 = 392,040 yard = 261,360 square feet). Finally, square feet). Convert that area into convert square footage into acreage acres (392,040 square feet ÷ 43,560 (261,360 square feet ÷ 43,560 square square feet per acre = 9 acres). Fi- feet per acre = 6 acres). nally, multiply the number of acres by $10,500 (9 acres × $10,500 = 13. B. Multiply the property value by the tax $94,500). rate to find the annual tax ($600,000 ×.0115 = $6,900). Divide the annual 9. D. First, multiply the length and the tax by 12 to find the monthly payment width to determine the square footage ($6,900 ÷ 12 = $575). (1,980 feet × 1,980 feet = 3,920,400 square feet). Convert the square foot- 14. A. Don’t forget that in California the age to acreage (3,920,400 square feet property tax year runs from July 1 to ÷ 43,560 square feet per acre = 90 June 30, with the tax payable in two acres). installments, one from July to Decem- ber, one from January to June. If the 10. C. First, calculate the total gross receipts first installment has been paid but the over one year ($51,000 × 12 months second has not, the seller will still be = $612,000). Subtract the exempted responsible for the period of January 1 amount ($612,000 – $42,000 = to January 31: one month. Here, since $570,000). Multiply the result by 2.5% the period is only one month, the only ($570,000 ×.025 = $14,250). Divide step is to determine the monthly rate that by 12 to find the monthly amount ($2,400 ÷ 12 months = $200). ($14,250 ÷ 12 = $1,187.50). Finally, add the fixed monthly amount to de- termine the total monthly payment ($1,187.50 + $820 = $2,007.50). 16  Chapter 17: Real Estate Math 15. D. First, calculate the selling price if the Culebras sell the property at 12% more than its purchase price (100% – 12% = 88%) ($536,800 ÷.88 = $610,000). Subtract the actual purchase price to determine the required amount of ap- preciation ($610,000 – $536,800 = $73,200). Double-check your answer by finding out what percentage ap- preciation is necessary ($73,200 ÷ $536,800 =.136 or 13.6%, which is not one of the choices listed as per- centages). 17  Principles of California Real Estate Instructor Materials PowerPoint Thumbnails Use the following thumbnails of our PowerPoint presentation to make your lecture notes. Principles of California Real Estate Lesson 17: Real Estate Math © 2021 Rockwell Publishing 1 Solving Math Problems Four steps ⚫ Read the question. ⚫ Write down the formula. ⚫ Substitute the numbers in the problem into the formula. ⚫ Calculate the answer. © 2021 Rockwell Publishing 2 Solving Math Problems Using formulas Each of these choices expresses same formula, but in way that lets you solve it for A, B, or C: A=B×C B=A÷C C=A÷B © 2021 Rockwell Publishing 3 18  Chapter 17: Real Estate Math Solving Math Problems Using formulas Isolate unknown figure. ⚫ The unknown is the element that you’re trying to determine. ⚫ The unknown should always sit alone on one side of the equals sign. ⚫ All the information that you already know should be on the other side. © 2021 Rockwell Publishing 4 Decimal Numbers Converting fraction to decimal Calculators use only decimals, not fractions. ⚫ If problem contains fraction, convert it to decimal. ⚫ Divide top number (numerator) by bottom number (denominator). 1/4 = 1 ÷ 4 = 0.25 1/3 = 1 ÷ 3 = 0.333 5/8 = 5 ÷ 8 = 0.625 © 2021 Rockwell Publishing 5 Decimal Numbers Converting decimal to percentage To convert decimal to percentage, move decimal point two numbers to the right and add percent sign. 0.02 = 2% 0.80 = 80% 1.23 = 123% © 2021 Rockwell Publishing 6 19  Principles of California Real Estate Instructor Materials Decimal Numbers Converting percentage to decimal To convert percentage to decimal, reverse process: ⚫ Move decimal point two numbers to left and remove percent sign. 2% = 0.02 80% = 0.8 123% = 1.23 © 2021 Rockwell Publishing 7 Summary Solving Math Problems Read problem Fractions Write formula and Decimal numbers isolate the unknown Percentages Substitute Conversion Calculate © 2021 Rockwell Publishing 8 Area Problems Formula: A = L × W To determine area of rectangular Area or square space, use this formula: Area Length Length A=L×W Width Width © 2021 Rockwell Publishing 9 20  Chapter 17: Real Estate Math Area Problems You might also be asked to factor other elements into area problem, such as: ⚫ cost per square foot ⚫ rental rate ⚫ amount of broker’s commission © 2021 Rockwell Publishing 10 Area Problems Example An office is 27 feet wide by 40 feet long. It rents for $2 per square foot per month. How much is the monthly rent? © 2021 Rockwell Publishing 11 Area Problems Example An office is 27 feet wide by 40 feet long. It rents for $2 per square foot per month. How much is the monthly rent? ⚫ Part 1: Calculate area A = 27 feet × 40 feet A = 1,080 square feet ⚫ Part 2: Calculate rent Rent = 1,080 × $2 Rent = $2,160 © 2021 Rockwell Publishing 12 21  Principles of California Real Estate Instructor Materials Area Problems Square yards Some problems express area in square yards rather than square feet. Remember: 1 square yard = 9 square feet ⚫ 1 yard is 3 feet ⚫ 1 square yard measures 3 feet on each side ⚫ 3 feet × 3 feet = 9 square feet © 2021 Rockwell Publishing 13 Area Problems Triangle formula: A = ½ B × H To determine area of a right triangle, use this formula: A=½B×H Right triangle: a triangle with a 90º angle © 2021 Rockwell Publishing 14 Area of a Triangle Visualize a rectangle, then cut it in half diagonally. What’s left is a right triangle. ⚫ If you’re finding area of a right triangle, it doesn’t matter at what point in formula you cut the rectangle in half. ⚫ In other words, any of these variations will reach the same result: A=½B×H A=B×½H A = (B × H) ÷ 2 © 2021 Rockwell Publishing 15 22  Chapter 17: Real Estate Math Triangles Example A triangular lot is 140 feet long and 50 feet wide at its base. What is the area? ⚫ Do the calculation in any of the following ways to get the correct answer. © 2021 Rockwell Publishing 16 Triangles Example, continued A triangular lot is 140 feet long and 50 feet wide at its base. What is the area? Variation 1: A = (½ × 50) × 140 A = 25 × 140 A = 3,500 sq. feet © 2021 Rockwell Publishing 17 Triangles Example, continued A triangular lot is 140 feet long and 50 feet wide at its base. What is the area? Variation 2: A = 50 × (½ × 140) A = 50 × 70 A = 3,500 sq. feet © 2021 Rockwell Publishing 18 23  Principles of California Real Estate Instructor Materials Triangles Example, continued A triangular lot is 140 feet long and 50 feet wide at its base. What is the area? Variation 3: A = (50 × 140) ÷ 2 A = 7,000 ÷ 2 A = 3,500 sq. feet © 2021 Rockwell Publishing 19 Area Problems Odd shapes To find the area of an irregular shape: ⚫ Divide the figure up into squares, rectangles, and right triangles. ⚫ Find the area of each of the shapes that make up the figure. ⚫ Add the areas together. © 2021 Rockwell Publishing 20 Odd Shapes Example A lot’s western side is 60 feet long. Its northern side is 100 feet long, but its southern side is 120 feet long. To find the area of this lot, break it into a rectangle and a triangle. © 2021 Rockwell Publishing 21 24  Chapter 17: Real Estate Math Odd Shapes Example, continued Area of rectangle: A = 60 × 100 A = 6,000 sq. feet © 2021 Rockwell Publishing 22 Odd Shapes Example, continued To find the length of the triangle’s base, subtract length of northern boundary from length of southern boundary. 120 – 100 = 20 feet Area of triangle: A = (½ × 20) × 60 A = 600 sq. feet © 2021 Rockwell Publishing 23 Odd Shapes Example, continued Total area: 6,000 + 600 = 6,600 sq. feet © 2021 Rockwell Publishing 24 25  Principles of California Real Estate Instructor Materials Odd Shapes Avoid counting same section twice A common mistake when working with odd shapes is to calculate the area of part of the figure twice. This can happen with a figure like this one. © 2021 Rockwell Publishing 25 Odd Shapes Avoid counting same section twice Here’s the wrong way to calculate the area of this lot. 25 × 50 = 1,250 40 × 20 = 800 1,250 + 800 = 2,050 By doing it this way, you measure the middle of the shape twice. © 2021 Rockwell Publishing 26 Odd Shapes Avoid counting same section twice Avoid the problem by breaking the shape down like this instead. Find height of smaller rectangle by subtracting height of top rectangle (25 feet) from height of the whole shape (40 feet). 40 – 25 = 15 feet © 2021 Rockwell Publishing 27 26  Chapter 17: Real Estate Math Odd Shapes Avoid counting same section twice Now calculate the area of each rectangle and add them together: 25 × 50 = 1,250 sq. ft. 20 × 15 = 300 sq. ft. 1,250 + 300 = 1,550 sq. ft. © 2021 Rockwell Publishing 28 Odd Shapes Avoid counting same section twice Here’s another way to break the odd shape down into rectangles. To find width of the rectangle on the right, subtract width of left rectangle from width of whole shape: 50 – 20 = 30 feet © 2021 Rockwell Publishing 29 Odd Shapes Avoid counting same section twice Now calculate the area of each rectangle and add them together: 40 × 20 = 800 sq. ft. 30 × 25 = 750 sq. ft. 800 + 750 = 1,550 sq. ft. © 2021 Rockwell Publishing 30 27  Principles of California Real Estate Instructor Materials Odd Shapes Narrative problems Some area problems are expressed only in narrative form, without a visual. In that case, draw the shape yourself and then break the shape down into rectangles and triangles. © 2021 Rockwell Publishing 31 Odd Shapes Example A lot’s boundary begins at a certain point and runs due south for 319 feet, then east for 426 feet, then north for 47 feet, and then back to the point of beginning. To solve this problem, first draw the shape. © 2021 Rockwell Publishing 32 Odd Shapes Example, continued Break it down into a rectangle and a triangle as shown. Subtract 47 from 319 to find the height of the triangular portion. 319 – 47 = 272 feet © 2021 Rockwell Publishing 33 28  Chapter 17: Real Estate Math Odd Shapes Example, continued Calculate the area of the rectangle. 426 × 47 = 20,022 sq. ft. © 2021 Rockwell Publishing 34 Odd Shapes Example, continued Calculate the area of the triangle. (½ × 426) × 272 = 57,936 sq. feet © 2021 Rockwell Publishing 35 Odd Shapes Example, continued Add together the area of the rectangle and the triangle to find the lot’s total square footage. 20,022 + 57,936 = 77,958 sq. feet © 2021 Rockwell Publishing 36 29  Principles of California Real Estate Instructor Materials Volume Problems Area: A measurement of a two-dimensional space. Volume: A measurement of a three- dimensional space. ⚫ Width, length, and height. ⚫ Cubic feet instead of square feet. © 2021 Rockwell Publishing 37 Volume Problems Formula: V = L × W × H To calculate volume, use this formula: V=L×W×H Volume = Length × Width × Height © 2021 Rockwell Publishing 38 Volume Problems Cubic yards If you see a problem that asks for cubic yards, remember that there are 27 cubic feet in a cubic yard: 3 feet × 3 feet × 3 feet = 27 cubic feet © 2021 Rockwell Publishing 39 30  Chapter 17: Real Estate Math Volume Problems Example A trailer is 40 feet long, 9 feet wide, and 7 feet high. How many cubic yards does it contain? 40 × 9 × 7 = 2,520 cubic feet 2,520 ÷ 27 = 93.33 cubic yards © 2021 Rockwell Publishing 40 Summary Area and Volume Area of a square or rectangle: A = L × W Area of a right triangle: A = ½ B × H Divide odd shapes into squares, rectangles, and triangles Volume: V = L × W × H Square feet, square yards, cubic feet, cubic yards © 2021 Rockwell Publishing 41 Percentage Problems Many math problems ask you to find a certain percentage of another number. This means that you will need to multiply the percentage by that other number. © 2021 Rockwell Publishing 42 31  Principles of California Real Estate Instructor Materials Percentage Problems Working with percentages Percentage problems usually require you to change percentages into decimals and/or decimals into percentages. Example: What is 85% of $150,000? © 2021 Rockwell Publishing 43 Percentage Problems Working with percentages Percentage problems usually require you to change percentages into decimals and/or decimals into percentages. Example: What is 85% of $150,000?.85 × $150,000 = $127,500 © 2021 Rockwell Publishing 44 Percentage Problems Example One common example of a percentage problem is calculating a commission. Example: A home sells for $300,000. The listing broker is paid a 6% commission on the sales price. The salesperson is entitled to 60% of that commission. How much is the salesperson’s share? $300,000 ×.06 = $18,000 $18,000 ×.60 = $10,800 © 2021 Rockwell Publishing 45 32  Chapter 17: Real Estate Math Percentage Problems Formula: W × % = P Basic formula for solving percentage problems: Whole × Percentage = Part W×%=P © 2021 Rockwell Publishing 46 Percentage Problems Formula: W × % = P The “whole” is the larger figure, such as the property’s sale price. The “part” is the smaller figure, such as the commission owed. Depending on the problem, the “percentage” may be referred to as the “rate.” ⚫ Examples: a 7% commission rate, a 5% interest rate, a 10% rate of return. © 2021 Rockwell Publishing 47 Percentage Problems Interest and profit problems Note that you’ll also use the percentage formula when you’re asked to calculate interest or profit. Example: A lender makes an interest-only loan of $140,000. The interest rate is 6.5%. How much is the annual interest? W×%=P $140,000 ×.065 = $9,100 © 2021 Rockwell Publishing 48 33  Principles of California Real Estate Instructor Materials Percentage Problems Interest and profit problems Example: An investor makes an $85,000 investment. She receives a 12% annual return on her investment. What is the amount of her profit? W×%=P $85,000 ×.12 = $10,200 © 2021 Rockwell Publishing 49 Percentage Problems Isolating the unknown If you need to determine the percentage (the rate) or the amount of the whole, rearrange the formula to isolate the unknown on one side of the equals sign. A=B×C P=W×% A÷B=C P÷W=% A÷C=B P÷%=W © 2021 Rockwell Publishing 50 Percentage Problems Finding the percentage or rate Example: An investor makes an $85,000 investment and receives a $10,200 return. What is the rate of return? P÷W=% $10,200 ÷ $85,000 =.12 (or 12%) © 2021 Rockwell Publishing 51 34  Chapter 17: Real Estate Math Percentage Problems Finding the whole Example: An investor receives a $10,200 return on her investment. This is a 12% return on her investment. How much did she invest? P÷%=W $10,200 ÷.12 = $85,000 © 2021 Rockwell Publishing 52 Percentage Problems Multiply or divide? Knowing when to divide or to multiply can be the hardest part of solving percentage problems. Rule of thumb: ⚫ If missing element is the part (the smaller number), it’s a multiplication problem. ⚫ If missing element is either the whole (the larger number) or the percentage, it’s a division problem. © 2021 Rockwell Publishing 53 Multiply or Divide? Finding the percentage or rate Example: A lender makes an interest-only loan of $140,000. The annual interest is $9,100. What is the interest rate? You know the part (the interest) and the whole (the loan amount). The percentage (the interest rate) is the missing element, so this is a division problem. P÷W=% © 2021 Rockwell Publishing 54 35  Principles of California Real Estate Instructor Materials Multiply or Divide? Finding the part Example: A home sells for $300,000. The listing broker is paid a 6% commission on the sales price. The salesperson is entitled to 60% of that commission. How much is the salesperson’s share? You know the whole (the sale price) and the rate. The part (the commission) is the missing element, so this is a multiplication problem. W×%=P © 2021 Rockwell Publishing 55 Summary Percentage Problems Percentage formula: Whole × Percentage (Rate) = Part W × % = P P ÷ W = % P ÷ % = W Types of percentage problems: commission problems, interest problems, and profit problems. © 2021 Rockwell Publishing 56 Loan Problems Interest You’ve already learned how to solve interest problems where the interest is given as an annual figure. Let’s look at how to solve problems where interest is given in semiannual, quarterly, or monthly installments. ⚫ In each case, the first step is to convert the interest into an annual figure. © 2021 Rockwell Publishing 57 36  Chapter 17: Real Estate Math Loan Problems Semiannual interest Example: A real estate loan calls for semiannual interest-only payments of $3,250. The interest rate is 9%. What is the loan amount? Semiannual: two payments per year. $3,250 × 2 = $6,500 annual interest You know the part (the interest) and the rate. You need to find the whole (the loan amount). P÷%=W $6,500 ÷.09 = $72,222.22 © 2021 Rockwell Publishing 58 Loan Problems Quarterly interest Example: A real estate loan calls for quarterly interest-only payments of $2,371.88. The loan balance is $115,000. What is the interest rate? Quarterly: 4 payments per year. $2,371.88 × 4 = $9,487.52 (annual interest) You know the part (the interest) and the whole (the loan amount). You need to find the rate. P÷W=% $9,487.52 ÷ $115,000 =.0825, or 8.25% © 2021 Rockwell Publishing 59 Loan Problems Monthly interest Example: The interest portion of a loan’s monthly payment is $517.50. The loan balance is $92,000. What is the interest rate? Monthly: 12 payments per year $517.50 × 12 = $6,210 (annual interest) You know the part (the interest) and the whole (the loan amount). You need to find the rate. P÷W=% $6,210 ÷ $92,000 =.0675, or 6.75% © 2021 Rockwell Publishing 60 37  Principles of California Real Estate Instructor Materials Loan Problems Amortization Some problems will tell you the interest portion of a monthly payment and ask you to determine the loan’s current principal balance. ⚫ Solve these in the same way as the problems just discussed. © 2021 Rockwell Publishing 61 Loan Problems Amortization Example: The interest portion of a loan’s monthly payment is $256.67. The interest rate is 7%. What is the loan balance prior to the fifth payment? $256.67 × 12 = $3,080.04 (annual interest) You know the part (the interest) and the rate, and you need to find the whole (the loan amount). P÷%=W $3,080 ÷.07 = $44,000 © 2021 Rockwell Publishing 62 Loan Problems Amortization Some problems may tell you the monthly principal and interest payment (instead of just the interest portion of the monthly payment). ⚫ These require several additional steps. © 2021 Rockwell Publishing 63 38  Chapter 17: Real Estate Math Loan Problems Amortization Example: The balance of a loan is $96,000. The interest rate is 8%. The monthly principal and interest payment for a loan is $704.41. How much will this payment reduce the loan balance? © 2021 Rockwell Publishing 64 Loan Problems Amortization Loan balance: $96,000 Monthly P&I: $704.41 Interest rate: 8% Step 1: Calculate the annual interest. W×%=P $96,000 ×.08 = $7,680 (annual interest) Step 2: Calculate the monthly interest. $7,680 ÷ 12 = $640 © 2021 Rockwell Publishing 65 Loan Problems Amortization Loan balance: $96,000 Monthly P&I: $704.41 Interest rate: 8% Monthly interest: $640 Step 3: Subtract monthly interest from total monthly payment to determine monthly principal. $704.41 – $640 = $64.41 Step 4: Subtract monthly principal from loan balance. $96,000 – $64.41 = $95,935.59 © 2021 Rockwell Publishing 66 39  Principles of California Real Estate Instructor Materials Loan Problems Amortization You might see a question like this where you’re asked how much the second or third payment will reduce the loan balance. In that case, you would calculate the first payment’s effect and then repeat the four steps again, using the new balance. © 2021 Rockwell Publishing 67 Loan Problems Amortization Step 1: $95,935.59 ×.08 = $7,674.85 Step 2: $7,674.85 ÷ 12 = $639.57 Step 3: $704.41 – $639.57 = $64.84 Step 4: $95,935.59 – $64.84 = $95,870.75 The second payment would reduce the loan balance to $95,870.75. To see how much the third payment would reduce the loan balance, repeat the four steps yet again. © 2021 Rockwell Publishing 68 Summary Loan Problems Use the percentage formula for loan problems. Whole × Percentage (Rate) = Part Convert semiannual, quarterly, or monthly interest into annual interest before substituting numbers into formula. Amortization problems ask you to find a loan’s principal balance. © 2021 Rockwell Publishing 69 40  Chapter 17: Real Estate Math Profit or Loss Problems Another common type of percentage problem involves a property owner’s profit or loss over a period of time. ⚫ Here the “whole” is the property’s value at an earlier point (which we’ll call Then). ⚫ The “part” is the property’s value at a later point (which we’ll call Now). © 2021 Rockwell Publishing 70 Profit or Loss Problems “Then” and “Now” formula The easiest way to approach these problems is by using this modification of the percentage formula: Then × Percentage = Now Of course, this can be changed to: Now ÷ Percentage = Then Now ÷ Then = Percentage © 2021 Rockwell Publishing 71 Profit or Loss Problems Calculating a loss Example: A seller sells her house for $220,000, which represents a 30% loss. How much did she originally pay for the house? ⚫ You know the Now value and the percentage of the loss. ⚫ You need to find the Then value (the original value of the house). ⚫ Rearrange the basic formula to isolate Then: Now ÷ Percentage = Then © 2021 Rockwell Publishing 72 41  Principles of California Real Estate Instructor Materials Profit or Loss Problems Calculating a loss Now ÷ Percentage = Then $220,000 ÷.70 = $314,286 The key to solving this problem is choosing the correct percentage to put into the formula. ⚫ Here the correct percentage is 70%, not 30%. ⚫ The house didn’t sell for 30% of its original value. It sold for 30% less than its original value. 100% – 30% = 70% © 2021 Rockwell Publishing 73 Profit or Loss Problems Calculating a loss When dealing with a loss, you can determine the rate using this formula: 100% – Percentage Lost = Percentage Received It’s the percentage received that must be used in the formula. © 2021 Rockwell Publishing 74 Profit or Loss Problems Calculating a gain To calculate a gain in value, add the percentage gained to 100% to find the percentage received: 100% + Percentage Gained = Percentage Received Returning to the example, if the sale had resulted in a 30% profit instead of a 30% loss, that would mean the house sold for 130% of what the seller originally paid for it: 100% + 30% = 130% © 2021 Rockwell Publishing 75 42  Chapter 17: Real Estate Math Profit or Loss Problems Calculating a gain Example: A seller sells her house for $220,000, which represents a 30% gain. How much did she originally pay for the house? $220,000 ÷ 1.30 = $169,231 Now ÷ Percentage Received = Then © 2021 Rockwell Publishing 76 Profit or Loss Problems Calculating a gain Note that if a seller sells a house for 130% of what she paid for it, she didn’t make a 130% profit. She received 100% of what she paid, plus 30%. She received a 30% profit. © 2021 Rockwell Publishing 77 Profit or Loss Problems Appreciation and depreciation A profit or loss problem may also be expressed in terms of appreciation or depreciation. ⚫ If so, the problem is solved the same way as an ordinary profit and loss problem. © 2021 Rockwell Publishing 78 43  Principles of California Real Estate Instructor Materials Profit or Loss Problems Compound depreciation You may see problems where you’re told how much a property appreciated or depreciated per year over several years. ⚫ This requires you to repeat the same calculation for each year. © 2021 Rockwell Publishing 79 Profit or Loss Problems Compound depreciation Example: A property is currently worth $220,000. It has depreciated four and a half percent per year for the past five years. What was the property worth five years ago? © 2021 Rockwell Publishing 80 Profit or Loss Problems Compound depreciation The house is losing value, so first subtract the rate of loss from 100%. 100% – 4.5% = 95.5%, or.955 You know the Now value and the rate. The missing element is the Then value: Now ÷ Percentage = Then $220,000 ÷.955 = $230,366.49 The house was worth $230,366 one year ago. © 2021 Rockwell Publishing 81 44  Chapter 17: Real Estate Math Profit or Loss Problems Compound depreciation Now repeat the calculation four more times, to determine how much the house was worth five years ago: $230,366 ÷.955 = $241,221 (value 2 years ago) $241,221 ÷.955 = $252,587 (value 3 years ago) $252,587 ÷.955 = $264,489 (value 4 years ago) $264,489 ÷.955 = $276,952 (value 5 years ago) © 2021 Rockwell Publishing 82 Profit or Loss Problems Compound appreciation If you’re told that a property gained value at a particular rate over several years, you’ll use the same process. ⚫ The difference is that you’ll need to add the rate of change to 100%, instead of subtracting it from 100%. © 2021 Rockwell Publishing 83 Summary Profit or Loss Problems Then × Percentage = Now To find the percentage received: – If there’s been a loss in value, subtract the rate of change from 100%. – If there’s been a gain (a profit), add the rate of change to 100%. Compound appreciation and depreciation: repeat the profit or loss calculation as needed. © 2021 Rockwell Publishing 84 45  Principles of California Real Estate Instructor Materials Capitalization Problems Capitalization: The process used to convert a property’s income into the property’s value. ⚫ In the appraisal of income property, the property’s value depends on its income. ⚫ Thevalue is the price an investor would be willing to pay for the property. ⚫ The property’s annual net income is the return on the investment. © 2021 Rockwell Publishing 85 Capitalization Problems Formula: V × % = I Capitalization problems are another type of percentage problem. Whole × Percentage = Part Here the “part” is the property’s income, and the “whole” is the property’s value: Value × Capitalization Rate = Income or Income ÷ Rate = Value or Income ÷ Value = Rate © 2021 Rockwell Publishing 86 Capitalization Problems Capitalization rate The capitalization rate represents the rate of return an investor would be likely to want on this investment. ⚫ An investor who wants a higher rate of return would not be willing to pay as much for the property as an investor who’s willing to accept a lower rate of return. © 2021 Rockwell Publishing 87 46  Chapter 17: Real Estate Math Capitalization Problems Calculating value Example: A property generates an annual net income of $48,000. An investor wants a 12% rate of return on his investment. How much could he pay for the property and realize his desired rate of return? Income ÷ Rate = Value $48,000 ÷.12 = $400,000 The investor could pay $400,000 for this property and realize a 12% return. © 2021 Rockwell Publishing 88 Capitalization Problems Calculating value Example: An investment property has a net income of $40,375. An investor wants a 10.5% rate of return. What would the value of the property be for her? Income ÷ Rate = Value $40,375 ÷.105 = $384,524 She could pay $384,524 for this property and realize a 10.5% return. © 2021 Rockwell Publishing 89 Capitalization Problems Finding the cap rate Example: An investment property is valued at $425,000 and its net income is $40,375. What is the capitalization rate? Income ÷ Value = Rate $40,375 ÷ $425,000 =.095, or 9.5% © 2021 Rockwell Publishing 90 47  Principles of California Real Estate Instructor Materials Capitalization Problems Changing the cap rate The capitalization rate is up to the investor: depends on how much risk she is willing to assume. ⚫ One investor might be satisfied with a 9.5% cap rate. ⚫ A more aggressive investor might want a 10.5% return on the same property. Some problems ask how a property’s value will change if a different cap rate is applied. © 2021 Rockwell Publishing 91 Capitalization Problems Changing the cap rate Example: Using a capitalization rate of 10%, a property is valued at $450,000. What would its value be using an 11% capitalization rate? © 2021 Rockwell Publishing 92 Capitalization Problems Changing the cap rate Step 1: Calculate the property’s net income. You know the value and the rate, so use the formula Value × Rate = Income. $450,000 ×.10 = $45,000 Step 2: Calculate value at the higher cap rate. Income ÷ Rate = Value $45,000 ÷.11 = $409,091 The property would be worth $40,909 less at the higher cap rate. © 2021 Rockwell Publishing 93 48  Chapter 17: Real Estate Math Capitalization Problems Changing the cap rate Example: Property with a net income of $16,625 is valued at $190,000. If its cap rate is increased by 1%, what would its new value be? © 2021 Rockwell Publishing 94 Capitalization Problems Changing the cap rate Step 1: Find the current capitalization rate. Income ÷ Value = Rate $16,625 ÷ $190,000 =.0875 Step 2: Increase the cap rate by 1%. 8.75% + 1% = 9.75%, or.0975 Step 3: Calculate the new value. Income ÷ Rate = Value. $16,625 ÷.0975 = $170,513 © 2021 Rockwell Publishing 95 Capitalization Problems Calculating net income In some problems, you’ll be given the property’s annual gross income and a list of the operating expenses instead of the annual net income. ⚫ Before you can use the capitalization formula, you’ll have to subtract the expenses from the gross income to get the net income. © 2021 Rockwell Publishing 96 49  Principles of California Real Estate Instructor Materials Capitalization Problems Calculating net income Example: A six-unit apartment building rents three units for $650 a month and three units for $550 a month. The annual operating expenses are $4,800 for utilities, $8,200 for property taxes, $1,710 for insurance, $5,360 for maintenance, and $2,600 for management fees. If the capitalization rate is 8¾%, what is the property’s value? © 2021 Rockwell Publishing 97 Capitalization Problems Calculating net income Step 1: Calculate the gross annual income. $550 × 3 × 12 = $19,800 $650 × 3 × 12 = $23,400 $19,800 + $23,400 = $43,200 (gross income) © 2021 Rockwell Publishing 98 Capitalization Problems Calculating net income Step 2: Subtract expenses from gross income. $43,200 -$4,800 -$8,200 -$1,710 -$5,360 -$2,600 $20,530 (net income) © 2021 Rockwell Publishing 99 50  Chapter 17: Real Estate Math Capitalization Problems Calculating net income Step 3: Calculate the value. You know the net income and the rate, so use the formula Income ÷ Rate = Value. $20,530 ÷.0875 = $234,629 © 2021 Rockwell Publishing 100 Capitalization Problems Calculating net income: OER Some problems give you the property’s operating expense ratio (OER) rather than a list of the operating expenses. ⚫ The OER is the percentage of the gross income that goes to pay operating expenses. ⚫ Multiply the gross income by the OER to determine the annual operating expenses. Then subtract the expenses from the gross income to determine the net income. © 2021 Rockwell Publishing 101 Capitalization Problems Calculating net income: OER Example: A store grosses $758,000 annually. It has an operating expense ratio of 87%. With a capitalization rate of 9¼%, what is its value? © 2021 Rockwell Publishing 102 51  Principles of California Real Estate Instructor Materials Capitalization Problems Calculating net income: OER Step 1: Multiply the gross income by the OER. $758,000 ×.87 = $659,460 (operating expenses) Step 2: Subtract the expenses from gross income. $758,000 – $659,460 = $98,540 (net income) Step 3: Use the capitalization formula to find the property’s value. Income ÷ Rate = Value $98,540 ÷.0925 = $1,065,297 © 2021 Rockwell Publishing 103 Summary Capitalization Problems Value × Capitalization Rate = Net Income Capitalization rate: the rate of return an investor would want from the property. The higher the cap rate, the lower the value. Subtract operating expenses from gross income to determine net income. OER: Operating expense ratio © 2021 Rockwell Publishing 104 Tax Assessment Problems Tax assessment problems are another type of percentage problem. Whole × % = Part Assessed Value × Tax Rate = Tax © 2021 Rockwell Publishing 105 52  Chapter 17: Real Estate Math Tax Assessment Problems Assessment ratio Some problems simply give you the assessed value. Others give you the market value and the assessment ratio, and you have to calculate the assessed value. Example: The property’s market value is $100,000 and the assessment ratio is 80%. $100,000 ×.80 = $80,000 The assessed value is $80,000. © 2021 Rockwell Publishing 106 Tax Assessment Problems Assessment ratio Example: The property’s market value is $200,000. It is subject to a 25% assessment ratio and an annual tax rate of 2.5%. How much is the annual tax the property owner must pay? © 2021 Rockwell Publishing 107 Tax Assessment Problems Assessment ratio Step 1: Calculate the assessed value by multiplying the market value by the ratio. $200,000 ×.25 = $50,000 (assessed value) Step 2: Calculate the tax. Assessed Value × Tax Rate = Tax $50,000 ×.025 = $1,250 (tax) The property owner is required to pay $1,250. © 2021 Rockwell Publishing 108 53  Principles of California Real Estate Instructor Materials Tax Assessment Problems Tax rate per $100 or $1,000 In some questions, the tax rate will not be expressed as a percentage, but as a dollar amount per hundred dollars or per thousand dollars of assessed value. Divide the value by 100 or 1,000 to find the number of $100 or $1,000 increments. Then multiply that number by the tax rate. © 2021 Rockwell Publishing 109 Tax Assessment Problems Tax rate per $100 Example: A property is assessed at $125,000. The tax rate is $2.10 per hundred dollars of assessed value. What is the annual tax? © 2021 Rockwell Publishing 110 Tax Assessment Problems Tax rate per $100 Step 1: Determine how many hundred dollar increments are in the assessed value. $125,000 ÷ 100 = 1,250 ($100 increments) Step 2: Multiply the number of increments by the tax rate. 1,250 × $2.10 = $2,625 (annual tax) © 2021 Rockwell Publishing 111 54  Chapter 17: Real Estate Math Tax Assessment Problems Tax rate per $1,000 Example: A property is assessed at $396,000. The tax rate is $14.25 per thousand dollars of assessed value. What is the annual tax? © 2021 Rockwell Publishing 112 Tax Assessment Problems Tax rate per $1,000 Step 1: Determine how many thousand dollar increments are in the assessed value. $396,000 ÷ 1,000 = 396 ($1,000 increments) Step 2: Multiply the number of increments by the tax rate. 396 × $14.25 = $5,643 (annual tax) © 2021 Rockwell Publishing 113 Tax Assessment Problems Tax rate in mills One other way in which a tax rate may be expressed is in terms of mills per dollar of assessed value. ⚫ A mill is one-tenth of a cent, or one-thousandth of a dollar. ⚫ To convert mills to a percentage rate, divide by 1,000. © 2021 Rockwell Publishing 114 55  Principles of California Real Estate Instructor Materials Tax Assessment Problems Tax rate in mills Example: A property is assessed at $290,000 and the tax rate is 23 mills per dollar of assessed value. What is the annual tax? © 2021 Rockwell Publishing 115 Tax Assessment Problems Tax rate in mills Step 1: Convert mills to a percentage rate. 23 mills/dollar ÷ 1,000 =.023, or 2.3% Step 2: Multiply the assessed value by the tax rate to determine the tax. $290,000 ×.023 = $6,670 © 2021 Rockwell Publishing 116 Summary Tax Assessment Problems Assessed Value × Tax Rate = Tax To find assessed value, you may have to multiply market value by the assessment ratio. Tax rate may be given as a percentage, as a dollar amount per $100 or $1,000 of value, or in mills. Divide mills by 1,000 to get a percentage rate. © 2021 Rockwell Publishing 117 56  Chapter 17: Real Estate Math Seller’s Net Problems This type of problem asks how much a seller will have to sell the property for to get a specified net amount from the sale. © 2021 Rockwell Publishing 118 Seller’s Net Problems Basic version In the basic version of this type of problem, you’re told the seller’s desired net and the costs of sale. Start with the desired net proceeds, then: ⚫ add the costs of the sale, except for the commission ⚫ subtract the commission rate from 100% ⚫ divide the results of Step 1 by the results of Step 2 © 2021 Rockwell Publishing 119 Seller’s Net Problems Basic version Example: A seller wants to net $220,000 from the sale of his property. He will pay $1,650 in attorney’s fees, $700 for the escrow fee, $550 for repairs, and a 6% brokerage commission. How much will he have to sell the property for? © 2021 Rockwell Publishing 120 57  Principles of California Real Estate Instructor Materials Seller’s Net Problems Basic version 1. Add the costs of the sale to the desired net: $220,000 + $1,650 + $700 + $550 = $222,900 2. Subtract the commission rate from 100%: 100% - 6% = 94%, or.94 3. Calculate the necessary sales price: $222,900 ÷.94 = $237,127.66 The sales price will have to be at least $237,128 for the seller to get his desired net. © 2021 Rockwell Publishing 121 Seller’s Net Problems Variations There are some variations on this type of problem. Variation 1: You’re told the original purchase price and the percentage of profit the seller wants from the sale. ⚫ This requires an additional step, calculating the seller’s desired net. © 2021 Rockwell Publishing 122 Seller’s Net Problems Variation 1 Example: A seller bought land two years ago for $72,000 and wants to sell it for a 25% profit. She’ll have to pay a 7% brokerage fee, $250 for a survey, and $2,100 in other closing costs. For what price will she have to sell the property? © 2021 Rockwell Publishing 123 58  Chapter 17: Real Estate Math Seller’s Net Problems Variation 1 1. Use the “Then and Now” formula to calculate the desired net. Then × Rate = Now $72,000 × 1.25 = $90,000 desired net Or calculate the profit and add it to the original value to get the desired net: $72,000 × 25% = $18,000 + $72,000 = $90,000 © 2021 Rockwell Publishing 124 Seller’s Net Problems Variation 1 2. Next, add the costs of sale, except for the commission. $90,000 + $250 + $2,100 = $92,350 3. Subtract the commission rate from 100%. 100% - 7% = 93%, or.93 4. Finally, calculate the necessary sales price. $92,350 ÷.93 = $99,301 © 2021 Rockwell Publishing 125 Seller’s Net Problems Variation 2 In another variation on this type of problem, you’re asked to factor in the seller’s mortgage balance. ⚫ This is more realistic, since most sellers have a loan to pay off. ⚫ Just add the loan balance as one of the closing costs. © 2021 Rockwell Publishing 126 59  Principles of California Real Estate Instructor Materials Seller’s Net Problems Variation 2 Example: A seller wants to net $24,000 from selling his home. He will have to pay $3,300 in closing costs, $1,600 in discount points, $1,475 for repairs, $200 in attorney’s fees, and a 6% commission. He will also have to pay off the mortgage balance, which is $46,050. How much does he need to sell his home for? © 2021 Rockwell Publishing 127 Seller’s Net Problems Variation 2 1. Add the costs of sale and the mortgage balance to the desired net. $24,000 + $3,300 + $1,600 + $1,475 + $200 + $46,050 = $76,625 2. Subtract the commission rate from 100%. 100% - 6% = 94%, or.94 3. Finally, calculate the necessary sales price. $76,625 ÷.94 = $81,516 © 2021 Rockwell Publishing 128 Summary Seller’s Net Problems 1. Desired Net + Costs of Sale + Loan Payoff 2. Subtract commission rate from 100% 3. Divide Step 1 total by Step 2 rate. Result is how much property must sell for. © 2021 Rockwell Publishing 129 60  Chapter 17: Real Estate Math Proration Problems Prorating an expense means dividing it proportionally, when someone is responsible for only part of it. Items often prorated in real estate transactions include: ⚫ property taxes ⚫ insurance premiums ⚫ mortgage interest © 2021 Rockwell Publishing 130 Proration Problems Closing date is proration date Seller’s responsibility for certain expenses ends on closing date. Buyer’s responsibility for certain expenses begins on closing date. © 2021 Rockwell Publishing 131 Proration Problems In advance or in arrears If seller is in arrears on a particular expense, seller will be charged (or debited) for a share of the expense at closing. ⚫ Buyer may be credited with same amount. If seller has paid an expense in advance, seller will be refunded a share of the overpaid amount at closing. ⚫ Buyer may be debited for same amount. © 2021 Rockwell Publishing 132 61  Principles of California Real Estate Instructor Materials Proration Problems 365 days or 360 days You will be told whether to use a 365-day or 360-day year. ⚫ In a 365-day year, use the exact number of days in each month. ⚫ In a 360-day year, each month has 30 days. © 2021 Rockwell Publishing 133 Proration Problems 3 Steps Prorating an expense is a three-step process: 1. Calculate the per diem (daily) rate of the expense. 2. Determine the number of days the party is responsible for. 3. Multiply per diem rate by number of days. © 2021 Rockwell Publishing 134 Proration Problems Property taxes Remember that in California, the property tax year runs from July 1 through June 30. Taxes are paid in two installments: ⚫ first installment due November 1 (covers July through December) ⚫ second installment due February 1 (covers January through June) © 2021 Rockwell Publishing 135 62  Chapter 17: Real Estate Math Prorating Property Taxes Taxes in arrears Example: The annual property taxes on the house are $2,860, and the seller has paid the first installment, but not the second installment. The sale closes on March 10. The buyer becomes responsible for the taxes on the closing date. How much will the seller have to pay in taxes at closing? (Use a 360-day year.) © 2021 Rockwell Publishing 136 Prorating Property Taxes Taxes in arrears Step 1: Calculate the per diem rate. $2,860 ÷ 360 = $7.94 Step 2: Count the number of days. 30 (Jan.) + 30 (Feb.) + 9 (March) = 69 days Step 3: Multiply rate by number of days. $7.94 × 69 = $547.86 The seller will be debited $547.86 at closing. The buyer will be credited for the same amount. © 2021 Rockwell Publishing 137 Prorating Property Taxes Taxes paid in advance Example: A buyer is purchasing a home. Closing will be on Oct. 20. The buyer is responsible for the closing date. Annual property taxes are $4,924, and they’ve already been paid through the end of the year. How much does the buyer owe at closing for property taxes? (Use a 360-day year.) © 2021 Rockwell Publishing 138 63  Principles of California Real Estate Instructor Materials Prorating Property Taxes Taxes paid in advance Step 1: Calculate the per diem rate. $4,924 ÷ 360 = $13.68 Step 2: Count the number of days. 11 days (Oct.) + 240 days (Nov.–June) = 251 days Step 3: Multiply per diem rate by number of days. $13.68 × 251 = $3,433.68 Buyer will be debited $3,433.68 at closing. Seller will be credited for the same amount. © 2021 Rockwell Publishing 139 Proration Problems Insurance Example: The sellers of a house have a one- year prepaid hazard insurance policy with an annual premium of $1,350. The policy has been paid for through March of next year, but the sale of their house will close on November 12 of this year. The buyer’s responsibility for insuring the property begins on the day of closing. How much will be refunded to the sellers at closing? (Use a 360-day year.) © 2021 Rockwell Publishing 140 Proration Problems Insurance Step 1: Calculate the per diem rate. $1,350 ÷ 360 = $3.75 Step 2: Count the number of days. 19 (Nov.) + 120 (Dec.–March) = 139 days Step 3: Multiply per diem rate by number of days. $3.75 × 139 = $521.25 Sellers will be refunded $521.25 by the insurer. © 2021 Rockwell Publishing 141 64  Chapter 17: Real Estate Math Proration Problems Mortgage interest For interest prorations, don’t forget that mortgage interest is almost always paid: ⚫ on a monthly basis ⚫ in arrears (at end of the month in which it accrues) If you aren’t given the amount of annual interest, first use the loan amount and interest rate to calculate it. ⚫ Then do the other proration steps. © 2021 Rockwell Publishing 142 Proration Problems Mortgage interest Two types of mortgage interest usually have to be prorated at closing: ⚫ seller’s final interest payment ⚫ buyer’s prepaid interest © 2021 Rockwell Publishing 143 Prorating Mortgage Interest Seller’s final interest payment Example: A seller is selling her home for $275,000. She has a mortgage at 7% interest with a balance of $212,500. The sale closes on May 14, and the seller will owe interest for the day of closing. At closing, how much will the seller’s final interest payment be? (Use a 360- day year.) © 2021 Rockwell Publishing 144 65  Principles of California Real Estate Instructor Materials Prorating Mortgage Interest Seller’s final interest payment Step 1: Calculate the annual interest. $212,500 ×.07 = $14,875 Step 2: Calculate the per diem rate. $14,875 ÷ 360 = $41.32 Step 3: Count the number of days. May 1 through May 14 = 14 days Step 4: Multiply per diem by number of days. $41.32 × 14 = $578.48 © 2021 Rockwell Publishing 145 Prorating Mortgage Interest Buyer’s prepaid interest Prepaid interest: At closing, buyer is charged interest for closing date through the end of the month in which closing occurs. Also called interim interest. ⚫ Example: Sale is closing on April 8. ⚫ Buyer’s first loan payment, due June 1, will include May interest, but not April interest. ⚫ At closing, buyer will pay interest for April 8 through April 30. © 2021 Rockwell Publishing 146 Prorating Mortgage Interest Buyer’s prepaid interest Example: A buyer purchased a house with a $350,000 loan at 5.5% annual interest. The transaction closes Jan. 17. The buyer is responsible for the day of closing. How much prepaid interest will the buyer have to pay? (Use a 360-day year.) © 2021 Rockwell Publishing 147 66  Chapter 17: Real Estate Math Prorating Mortgage Interest Buyer’s prepaid interest Step 1: Calculate the annual interest. $350,000 ×.055 = $19,250 Step 2: Calculate the per diem rate. $19,250 ÷ 360 = $53.47 Step 3: Count the number of days. Jan. 17 through Jan. 30 = 14 days Step 4: Multiply per diem rate by days. $53.47 × 14 = $748.58 Buyer will owe $748.58 in prepaid interest at closing. © 2021 Rockwell Publishing 148 Summary Proration Problems 1. Calculate per diem rate. (365-day or 360-day year?) 2. Count number of days. 3. Multiply per diem rate by number of days.