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Percentages, Direct Capitalization, and Interest
Percentages
A percentage expresses a number as a fraction of 100, denoted by “%”. Mathematically, it means “divided by 100”. For example, 10% = 10/100 = 0.1, and 50% = 50/100 = 0.5.
The basic formula for percentages is: Part = Percentage ร Whole, or A = B ร C, where A is the Part, B is the Percentage (as a decimal), and C is the Whole.
To calculate the percentage, rearrange the formula: Percentage = Part รท Whole, or B = A รท C.
Example: A house with an area of 1500 square feet is on a plot of land with an area of 7500 square feet. What percentage of the land does the house occupy?
- Part (A): House area = 1500 sq ft
- Whole (C): Land area = 7500 sq ft
Percentage (B) = 1500 รท 7500 = 0.2. Convert to percentage: 0.2 ร 100 = 20%. The house occupies 20% of the land area.
To convert percentage to decimal: divide by 100 (e.g., 8.5% = 0.085). To convert decimal to percentage: multiply by 100 (e.g., 0.095 = 9.5%).
Practical applications in real estate appraisal include: calculating occupancy rate, sales analysis (increase/decrease in property prices), expense assessment (percentage of operating expenses vs. total income), and comparing properties based on key indicators (e.g., price-to-rent ratio).
Direct Capitalization
Direct capitalization is a method for valuing properties by estimating the property’s value based on its expected net income.
Key concepts:
- Net Operating Income (NOI): Income after deducting operating expenses (taxes, insurance, maintenance, management) but before debt service.
- Capitalization Rate (Cap Rate): The ratio between NOI and property value, reflecting the expected return on investment.
The basic formula is: Income = Rate ร Value, or I = R ร V, where I is NOI, R is Cap Rate, and V is Property Value.
To calculate property value: V = I รท R. To calculate cap rate: R = I รท V.
Example: A property generates an NOI of $40,000 annually, and a cap rate of 25% is used. What is the property value?
- Income (I) = $40,000
- Rate (R) = 25% = 0.25
Property Value (V) = 40,000 รท 0.25 = $160,000. The estimated property value is $160,000.
Income Multiplier: The inverse of the cap rate, used to estimate property value by multiplying annual income by the multiplier. Income Multiplier = 1 รท Cap Rate.
In the previous example, Income Multiplier = 1 รท 0.25 = 4. This means the property value is four times its annual income.
Practical applications include: valuing commercial properties (offices, retail, warehouses), investment analysis (evaluating expected return), and making investment decisions (comparing properties based on cap rates and income multipliers).
Interest
Interest is the amount paid by a borrower to a lender for the use of money.
Key concepts:
- principalโโ: The initial amount borrowed or invested.
- Interest Rate: The percentage paid as interest on the principal.
- Time: The period over which interest is calculated.
Simple Interest: Calculated only on the principal.
The basic formula for simple interest is: Interest = Principal ร Rate ร Time, or I = P ร R ร T, where I is Interest, P is Principal, R is Interest Rate (as a decimal), and T is Time (in years).
To calculate the principal: P = I รท (R ร T). To calculate the interest rate: R = I รท (P ร T). To calculate the time: T = I รท (P ร R).
Example: A person invests $1000 at a simple interest rate of 12% per year for six months. What is the amount of interest earned?
- Principal (P) = $1000
- Rate (R) = 12% = 0.12
- Time (T) = 6 months = 6/12 year = 0.5 year
Interest (I) = 1000 ร 0.12 ร 0.5 = $60. The person will earn $60 in interest.
Important notes: the time period must be consistent with the unit of measure of the interest rate, and the interest rate must be converted to a decimal.
Practical applications in real estate finance include: calculating mortgage payments, evaluating real estate investments, and analyzing the cost of financing from different sources.
Chapter Summary
The chapter covers percentages, direct capitalization, and interest and their application in real estate appraisal.
Percentages:
- Express a part of a whole as a number divided by 100. “Percent of” means “percentage multiplied by”.
- Formula: Part = Percentage ร Whole. This can be rearranged to solve for any of the three variables.
- Conversion: Percentages are converted to decimals by dividing by 100 before calculations. Decimals are converted to percentages by multiplying by 100.
- Importance: Used in real estate appraisal techniques such as allocation and direct sales comparison.
Direct Capitalization:
- Estimates property value based on net operating incomeโโ (NOI) and a capitalization rateโ.
- Formulas:
- Income = Rate ร Value (I = R ร V)
- Rate = Income รท Value (R = I รท V)
- Value = Income รท Rate (V = I รท R)
- Relation to Percentages: Direct capitalization equations are similar in form to percentage problems (A = B ร C).
- The capitalization rate and income multiplierโ are reciprocals of each other.
- Importance: A tool for appraisers to estimate the value of income-producing properties.
Interest:
- The cost of borrowing money or the return on investment.
- Formula: Interest = principalโ ร Rate ร Time (I = P ร R ร T)
- Time must be expressed in the same units as the interest rate’s period (usually annually). If the rate is annual, time must be converted to years (e.g., 6 months = 0.5 years).
- Importance: Understanding interest calculations is necessary for evaluating the financial feasibility of real estate investments and mortgages.
Conclusions:
- Percentages, direct capitalization, and interest are interrelated mathematical concepts providing a foundation for real estate appraisal.
- Understanding these concepts enables appraisers to make accurate and reliable calculations for estimating property value.
- The basic formulas can be adapted to solve for any unknown variable if the other variables are known.
