Chapter 18 - real estate math

The percentage goes in the middle, but it's not 30%, it's 130%. Sue received 100% of what she originally paid, plus a 30% profit.

$227,500 divided by 130% is $175,000. That's how much Sue originally paid for the house.

On the state exam, this kind of problem is often stated in terms of appreciation or depreciation, instead of profit or loss. But our Then and Now formula will solve the problem either way.

*A typical question might read like this: Jerry's home was recently appraised for $720,000. It has depreciated in value by 22% since he purchased it five years ago. How much was it originally worth?

Some problems that require you to calculate a property's operating expenses present irrelevant information along with the pertinent information.

*Costs that aren't classified as operating expenses may be mingled with the property's actual operating expenses, and it's up to you to know the difference. *Only the actual operating expenses should be added together before subtracting them from gross income. *Costs that are often confused with a property's operating expenses include depreciation, income taxes paid on the property's net income, and the principal and interest payments on the mortgage. (The mortgage payments are frequently referred to as the debt service.) *These should be ignored in NET INCOME CALCULATION = Depreciation, income taxes, and P & I PAYMENT (DEBT SERVICE)

When property is assessed at 100% of its market value, the assessed value is the same as the market value. In this case, it's $396,000.

*Divide by 1,000 to determine how many thousand dollar increments there are in $396,000. $396,000 ÷ 1,000 = 396. *Next you multiply the number of thousand dollar increments by the tax rate of $9.75, to arrive at a tax of $3,861.

In these problems, you're asked to compare the cost or value of something at an earlier point in time with its cost or value at a later point.

*For instance, Sue sold her house for $240,000, which represents a 30% loss. How much did Sue originally pay for the house? *In the middle of the equation, you need to put the appropriate percentage. Notice that we've put down 70%, not 30%. ? x 70% = 240,000 *That's because Sue didn't sell the home for 30% of what she paid for it. She sold it for 70% of what she paid, which amounted to a 30% loss. 240,000 divided by 0.70% is 342,857 is what sue originally paid for her house.

Sometimes a profit or loss problem involves appreciation or depreciation that has accrued at an annual rate over a specified number of years.

*For instance, you might be told that a property's current value is $240,000, but it has lost value at a rate of 4½% per year for the past five years. You are then asked what the property was worth five years ago. *The difference between this type of profit and loss problem and those discussed previously is that you must solve it one year at a time. So the first step in solving this problem is determining what the property was worth last year. SET UP THE THEN AND NOW FORMULA* ? X 95.5% = 240,000 SUBTRACT THE 4.5 PERCENT FROM 100 PERCENT TO ARRIVE AT 95.5%

In the real world, the assessment ratio depends on the tax rules in the place where the property is located.

*It isn't unusual for a tax assessment ratio to be quite low. For example, property might be assessed at only 25% of its market value. *But a 25% assessment ratio doesn't necessarily mean the tax will be lower than it would be if the ratio were 75%, or even 100%. As a rule, when the assessment ratio is set low, the tax rate is set high, so it balances out.

You may be wondering how you're supposed to decide where to place the known components in the equation. Once again, it's very simple.

*The percentage (the interest rate, the commission rate, or rate of return) always goes in the middle. It's always in the 3 position. meaning 2 X 3 = 6 *The larger number (the whole) goes on the left side, in the 2 position. And the smaller number (the part) belongs on the right, in the 6 position.

A typical volume question that involves cubic yards concerns pouring concrete, which is almost always measured in cubic yards. A representative question might read something like this: "A contractor is building a sidewalk that will be 100 feet long, 6 feet wide, and 6 inches deep. How many cubic yards of concrete will be needed to complete the sidewalk?" When you've completed your calculations, advance to the next frame for the answer.

*To find the answer to this problem, first make sure that the units of measurement match up. Six inches is half a foot, or 0.5 feet. *100 feet times 6 feet times 0.5 feet equals 300 cubic feet of concrete. *Divide 300 cubic feet by 27, the number of cubic feet in a cubic yard. You'll determine that the contractor will require 11.1 cubic yards of concrete to complete the sidewalk.

When you want to change a decimal number into a percentage, you just reverse the process. Move the decimal point two places to the right and add a percent sign.

0.06 = 6% 0.17 = 17% 0.212 = 21.2%

Now that you know the dimensions of the triangle, here's the simplest way to determine Lot 13's area. Using the formula for a rectangle (A = W × L), multiply the lot's 60-foot base by its 120-foot height, as if its eastern boundary didn't angle off at all.

60 x 120 = 7,200 Sq. Ft *Next, use the triangle formula (A = ½ B × H) to calculate the area of the triangle that cuts into Lot 13. 10x20 = 200 sq. ft. * Finally, subtract the area of the triangle from the area of the rectangle to arrive at Lot 13's total area. 7,200 - 200 = 7,000

Calculating Net Income ( OER ) OPERATING EXPENSE RATIO

A convenience store grosses $758,000 annually. It has an OER of 87%. With a capitalization rate of 9¼%, what is the capitalized value of the property? Try to solve the problem, and then advance to the next frame for the answer.

As we discussed earlier, a yard is 3 feet long, and a square yard measures 3 feet by 3 feet, or 9 square feet

A cubic yard measures 3 feet by 3 feet by 3 feet, or 27 cubic feet.

Now let's consider how to find the area of a triangle. The formula is: Area = ½ Base × Height.

A=1/2B X H

A property with no outstanding liens sold. The seller paid $2,450 in discount points, $1,236 in other closing costs, and a 6% commission. She received a closing check for $112,000. What was the selling price?

Add the settlement costs to the seller's net: $112,000 + $3,686 (settlement costs of $2,450 + $1,236) = $115,686. This amount is 94% of the sales price (100% - 6% commission = 94%). $115,686 ÷ 94% = $123,070.21 (sales price).

Sellars Net Problem

Another type of problem commonly tested on the state exam concerns a property seller's net proceeds from the sale of a property.

Do you understand why we used 70% instead of 30% as the percentage in the equation? If Sue sold her house for the same amount that she paid for it, she'd receive 100% of what she paid. Since she sold the home for 30% less than she paid, she received 30% less than 100%, or 70%.

But now suppose Sue sold the house for 30% more than she paid for it. Instead of subtracting 30% from 100%, you'll add 30% to 100%. Unlike other types of percentage problems, like commission problems or loan problems, it's possible you may use a percentage over 100% when talking about a profit. So let's say Sue sold her house last week for $227,500, and that represented a 30% profit. What did she originally pay for the property? Try to solve the problem and then advance to the next frame for the answer and explanation.

ASSESSED VALUE X TAX RATE = TAX

But you'll need to become familiar with a few extra steps and some special terminology to do these problems. The test questions can sound more difficult than they are.

The whole is a larger figure, such as the property's sales price. The part is a smaller figure, such as the broker's commission. Multiply the whole by the percentage to find the part.

Calculate the size of the pet deposit by determining what one-quarter of a month's rent is ($1,000 × 0.25 = $250). Add that to the ordinary security deposit to find the total amount ($1,000 + $250 = $1,250).

Capitalization Problems

Capitalization is the mathematical process of converting income generated by a property into the value of the property. *The formula for a capitalization problem is: Value x Cap Rate = Income.

Area Problems

Concern the measurement of a surface or space

A brokerage sells a house and charges a 6% commission. The amount of the commission is 17,000 What was the sales price?

Divide to get the answer. 17,000 divided by 0.06 Change the percent into a decimal

Finding the area of a square

Earlier we used the formula for finding the area of a rectangle: Area = Width x Length. This same formula is used to find the area of a square.

PQR Store enters into a lease agreement in which it agrees to pay a fixed $500 monthly rent, plus 5% commission on all of its sales over $200,000. If its gross sales for the year total $230,000, what is PQR's total annual rent?

First determine the fixed monthly rent total: $500 × 12 = $6,000. Next, determine by how much the sales exceeded $200,000. $230,000 - $200,000 = $30,000. Multiply that amount by 5%. $30,000 × .05 = $1,500 (additional commission). Add $1,500 to $6,000 for a total of $7,500.

The owner of a lot that is 99' by 110' would like to sell it. Similar properties sell for $180,000 per acre. What is the likely selling price for this property?

First, find the square footage of the lot (99 feet × 110 feet = 10,890 square feet). Convert that to acreage (10,890 square feet ÷ 43,560 square feet per acre = 0.25, or one-quarter of an acre). Multiply that by the price per acre to find the selling price ($180,000 × 0.25 = $45,000).

A square parcel of farmland measures 1,320 feet from north to south. In the center of the parcel is eight acres of swampland that is not tillable. If the property sells for $2,000 per acre of tillable land, what is the selling price?

First, multiply the property's boundaries to find its square footage (1,320 feet × 1,320 feet = 1,742,400 square feet). Convert it to acreage (1,742,400 square feet ÷ 43,560 square feet per acre = 40 acres), subtract the unusable portion (40 acres - 8 acres = 32 acres), and multiply by the price per acre (32 acres × $2,000 = $64,000).

On the state exam, you're more likely to see the tax rate expressed as a dollar amount per hundred dollars of assessed value, or per thousand dollars of assessed value.

For instance, a property assessed at $125,000 might be taxed at a rate of $2.10 per hundred dollars of assessed value. *To find the tax amount, you first divide the assessed value, $125,000, by 100 to determine how many hundred dollar increments make up the value. $125,000 ÷ 100 = 1,250.

Converting Units of Measurement

For the area formula to work, the units of measurement must match up. If W and L are stated in feet, then the area (A) will be stated in square feet: 10 feet × 15 feet = 150 square feet If W and L are stated in yards, then the area will be stated in square yards: 10 yards × 15 yards = 150 square yards

Here's an example. Jerry is selling his home, and he wants to net $360,000. *He will have to pay $1,950 in attorney's fees, $4,800 in closing costs, $1,650 for repairs, and a 6% commission to the real estate brokerage.

How much will he have to sell the property for, to net $360,000? Try to solve the problem, and then advance to the next frame for the answer.

VALUE X CAP RATE =

INCOME

The process is the same if you're calculating compound appreciation. Instead of subtracting an annual rate from 100%, as you do with depreciation, you add the rate to 100%.

If a home is currently worth $300,000 and it has appreciated in value at a rate of 4% annually for four years, how much was it originally worth? *Once again, you start with the most recent year, then go backwards.

In this formula, "Income" refers to the net income generated by investment property. Net income is sometimes called net operating income, or NOI.

In certain problems, you may be given the gross income, from which you will have to deduct expenses to find net income. You may also be given monthly income, which you'll need to turn into annual income. Simply multiply the monthly income by 12.

Interest Calculation

Instead of calling for semiannual payments, a question might give you interest payments every 90 days. Those are quarterly payments in a banker's year. (For the sake of simplicity, a banker's year is 360 days, and every month has 30 days, including February.) The 90-day interest payment will be made four times annually.

The capitalization rate

Often called the cap rate, refers to the percentage of return that an investor wants to realize on the investment. *And "Value" in this formula is the value of the property, or the price that an investor would be willing to pay for it. The higher the desired rate of return, the higher the capitalization rate, and the less an investor would be willing to pay for a property generating the specified income.

Through careful management, a property manager increases an apartment building's monthly income of $3,000 by an additional $500/month. Assuming a cap rate of 8%, what is the increase in the building's value?

One way to solve the problem is to calculate the value before the increase ($3,000 x 12 = $36,000; $36,000 ÷ .08 = $450,000), calculate the value after the increase ($3,500 x 12 = $42,000; $42,000 ÷ .08 = $525,000), and then calculate the difference ($525,000 - $450,000 = $75,000). Another way would be simply to multiply $500 x 12 ($6,000) and then factor in the capitalization rate ($6,000 ÷ .08 = $75,000).

A ten-year-old home cost $120,000 to build and the land it's on cost $20,000. If an appraiser finds that depreciation has averaged 1.5% per year, what is the total amount of depreciation on the property?

Only the building itself is depreciated, rather than the land, which cannot lose value due to depreciation. Multiply 1.5% by 10 years (since the question says "an average of" 1.5% per year, rather than saying it depreciated 1.5% each year), which is 15%. 15% of $120,000 is $18,000.

CHANGING THE CAP RATE

Selecting a capitalization rate is an investor's personal decision. The investor who was willing to pay $2,125,000 for the Concord Plaza was satisfied with a 9.5% return on her investment. $201,875 is 9.5% of $2,125,000.

Note that there are two other simple ways to calculate the area of a triangle.

Since a triangle is half of a rectangle, you can draw a line, like the dotted line on your screen, and turn the triangle into a rectangle. Then multiply the base times the height and divide by two. *Another alternative for determining the area of a triangle is to multiply the base times half the height.

To find out how many square yards there are in 252 square feet, divide 252 by 9. The answer is 28.

So in order to carpet his living room, Stan needs to buy 28 square yards of carpet. If he pays $22 per square yard, he'll spend $616 to carpet the room.

Calculating Net Income

Some capitalization questions give you the property's gross income (gross earnings) and require you to calculate its net income. You then go on to capitalize the net income to arrive at the property's value. *There are a couple of ways these problems can be presented. In one version, you're given the property's gross income and told that the property has a certain OER, which stands for operating expense ratio. The OER tells you what percentage of the gross income goes to pay the operating expenses. What remains is the net income.

Amortization

Some questions about loans and interest rates ask you to determine a loan's current principal balance. For example, you might be asked what a loan's balance is prior to the fifth payment, if the interest rate is 7% and the interest portion of that payment is $2,566.67. You'll solve this type of problem the same way as the others.

A seller wants to net $60,000 from a transaction, but will have to pay off a mortgage and other fees, at a total cost of $181,800. The seller will also need to pay a 7% commission. What will the property need to sell for?

Start by adding the desired net and the other costs, including the mortgage ($60,000 + $181,800 = $241,800). Subtract the commission percentage from 100% (100% - 7% = 93%), and then divide the total by that percentage ($241,800 ÷ 0.93 = $260,000). Since the agent bases her commission on the total selling price, you can't simply add 7% of the costs and profit to the selling price or you won't quite have her full commission. You need a price that the agent can take 7% of and still satisfy the seller's goals. A fuller explanation is found in the section on seller's net problems in ch.18 of Fundamentals.

Section 9: TAX ASSESSMENT PROBLEMS

Tax assessment problems are essentially another type of percentage problem. The basic formula is Assessed Value x Tax Rate = Tax.

Figuring out a lot or building with an irregular shape

The easiest way to do this is to divide the figure up into squares, rectangles, and triangles. *Once you've found the area of each of the shapes that make up the irregular one, add them up to get the total area of the original figure.

Let's try it out. Sheila is listing Lot 13 in a subdivision, and she needs to know its area in square feet.

The first step is to analyze the shape and the dimensions of the lot. Lot 13's eastern boundary angles off in a northwesterly direction near the back of the lot. By drawing a dotted line as indicated, you create a small triangle. The dimensions of this triangle (which is part of Lot 14) can easily be determined using the boundary measurements given on the plat map.

A parcel measures 198 feet by 330 feet. It sold for $30,000. What was the price per acre?

The first step would be to find the square footage of the lot (198 ft. x 330 ft. = 65,340 square feet). Convert that to acreage (65,340 square feet / 43,560 square feet per acre = 1.5 acres). Divide the cost by the acreage to find the cost per acre ($30,000 / 1.5 acres = $20,000 per acre).

Art sold his house, which was not encumbered with a mortgage. Closing expenses were $5,264, and he paid a commission of 7% of the selling price. He received a check at closing for $372,316. What did the house sell for?

The house sold for $372,316 plus whatever was subtracted from the sales proceeds at closing (that is, the seller's selling expenses). Thus, to get the sales price (so we can determine the commission), you first add the selling costs to the sale proceeds. $5,264 + $372,316 = $377,580. Then use the seller's net formula. That is, divide the sales price by the commission rate subtracted from 100 (100% - 7% = 93%). $377,580 ÷ 0.93 = $406,000.

An owner sells a property, and the buyer is going to make a 20% downpayment. The lender requires the buyer to pay two discount points, which turn out to total $1,000. What was the purchase price?

The purchase price was $62,500. This question is simply two separate percentage problems. First, find the loan amount. Two discount points is 2% of the loan amount, so divide the value of the discount points by 2% to determine that this was a $50,000 loan ($1,000 ÷ .02 = $50,000). The 20% downpayment means this is an 80% loan. Now you can use the loan amount and the loan-to-value ratio to find the purchase price. To calculate the purchase price, divide the loan amount by 80% ($50,000 ÷ .80 = $62,500).

To solve a seller's net problem you start with the seller's desired net, then add the costs of sale, excluding the commission.

The second step is to subtract the commission rate from 100%. The third and last step is to divide the result of step one by the result of step two. *Here's the reasoning. For Jerry to receive his desired net, he must be left with $360,000, plus enough to pay the other selling expenses, after paying the commission. *The commission is 6% of the sales price. This means that $368,400 needs to be 94% of the sales price to meet Jerry's goal. *The sales price is the unknown component. You need to find out what times 94% equals $368,400 *The 2 x 3 model tells you this is a division problem. Jerry must sell the property for at least $391,915 to achieve his desired net.

Most of us learned our multiplication tables in the third grade. We learned that 2 × 3 = 6. That's a straightforward multiplication problem. But what if you were asked what times 3 equals 6? You would still know the answer (2), but in this case the solution involves division instead of multiplication:

The trouble is, you didn't really divide 6 by 3, because you long ago memorized the answer. You knew the answer was 2 without considering whether you needed to multiply or divide.

The basic formula for solving all the different types of percentage problems can be stated as shown here.

The whole is a larger figure, such as the property's sales price. The part is a smaller figure, such as the broker's commission. Multiply the whole by the percentage to find the part.

Since a cubic yard is 27 cubic feet, divide 2,520 by 27 to make the conversion from cubic feet to cubic yards.

There are 93.33 cubic yards of storage space in the truck's trailer

A borrower is purchasing a $300,000 home. She will be using a loan with an LTV of 80%. She will also be paying 3 discount points. What is the total amount of cash that she will need to bring to closing?

There are two components that you'll need to calculate. First, calculate the size of the loan: $300,000 x .8 = $240,000. She'll need to bring $60,000 to cover the downpayment ($300,000 - $240,000 = $60,000). Second, calculate the amount of the discount points, using the loan amount and not the purchase price ($240,000 x .03 = $7,200). Add the two together for the total amount of cash needed ($60,000 + $7,200 = $67,200).

Here's a pitfall to watch out for with profit or loss problems. Don't make the mistake of thinking that if someone sells something for 125% of what they paid for it, they made a 125% profit.

They received 100% of what they paid, plus 25%. The profit was 25%

There's one final variation. You may encounter questions that give you a sales price and ask how much a seller netted from a sale, after accounting for closing costs and the listing agent's commission.

This is a lot like a standard seller's net problem, only with the steps performed backwards. *In a way, this is much easier, because it's more intuitive; it's essentially what an escrow agent would do at closing. *First you calculate the amount of the commission by multiplying the sales price by the commission rate. *Second, subtract the commission from the sales price. And third, subtract the other costs from the sales price.

A property's assessed value is the value assigned by the government for taxation purposes. The assessed value is ordinarily based on the property's market value.

To calculate the assessed value of a particular piece of real estate, you must know its market value and the assessment ratio. *The assessment ratio states the relationship between the assessed value and the market value as a percentage. *For instance, if a property's market value is $100,000 and the assessment ratio is 80%, the assessed value of the property is $80,000. *Then if the tax rate is 2.5%, you multiply the assessed value by the tax rate to determine the amount of the tax. $80,000 x 2.5% = $2,000.

First multiply $2,566.67 by 12 for an annual interest figure. Then divide by 7% (the interest rate) for the loan balance. It doesn't matter whether it's the fifth, the tenth, or the fiftieth payment; the interest portion of the payment ($2,566.67) multiplied by 12 is 7% of the loan balance.

With an amortized loan the principal balance changes each month, but in problems like this the interest portion of the monthly payment always represents one-twelfth of the annual interest. *The annual interest figure is a percentage (equal to the interest rate) of the loan balance at that point, regardless of which payment is about to be made.

When you solve percentage problems, you'll need to convert the percentages to decimal numbers in order to do the math.

You may also need to change the numbers back to percentage form, if that's how the multiple choice answers are expressed on the license exam.

Subtracting Operating Expenses

You're given the gross income and a list of specific operating expenses, which you subtract from the gross income to determine the net income.