Percentages, Direct Capitalization, and Interest Calculations

This chapter, “Percentages, Direct Capitalisation, and Interest Calculations,” is a vital part of “Fundamentals of Real Estate Mathematics: From Fractions to Valuation.” The chapter aims to provide participants with a deep and integrated understanding of the basic mathematical concepts underlying real estate valuation operations, with a particular focus on percentages, direct capitalisation techniques, and interest calculations of all kinds.

Percentages are an essential tool in financial and real estate analysis, used to express relationships between different values and assess relative changes. Understanding percentages allows real estate analysts to evaluate the share of a specific component of the total value, compare the performance of different investments, or determine growth and inflation rates.

Direct capitalisation is a real estate valuation technique that relies on converting the expected income from the property into present value using the appropriate capitalisation rate. This technique is based on fundamental financial principles and provides a quick and effective estimate of the property’s value, especially in cases where a relatively stable income is available. The use of this technique requires a precise understanding of the relationship between income, value, and the capitalisation rate, and how to choose the appropriate rate based on risks and market conditions.

Interest calculations, whether simple or compound, are necessary to understand the effect of time on the value of money. In real estate, interest calculations are used to evaluate future cash flows, calculate mortgage payments, and determine the return on investment. Understanding these calculations allows real estate analysts to make informed investment decisions and assess the economic feasibility of real estate projects.

Percentages:

Percentages are a mathematical tool to express a part of a whole. They are used in real estate, including property valuation, market analysis, and investment return calculation.

• Definition: A percentage expresses a number as a fraction of 100, represented by the symbol “%” which means “divided by 100”. Example: 25% means 25/100 or 0.25.
• Conversion between Percentages and Decimals:
  • Percentage to decimal: Divide the percentage by 100. Example: 8.5% = 8.5 / 100 = 0.085
  • Decimal to percentage: Multiply the decimal by 100. Example: 0.095 = 0.095 * 100 = 9.5%
• Basic Percentage Formula: Part = Percentage × Whole
• Finding the Unknown Variable:
  1. Finding the Part (A): A = B x C (Part = Percentage × Whole)
  2. Finding the Percentage (B): B = A / C (Percentage = Part / Whole)
  3. Finding the Whole (C): C = A / B (Whole = Part / Percentage)
• Practical Examples:
  1. A house with an area of 1500 square feet is located on a plot of land with an area of 7500 square feet. What percentage of the land area does the house occupy? The house occupies 20% of the land area.
  2. If you have an annual rental income❓❓ of $50,000 and annual operating expenses of $15,000, what is the ratio of expenses to income? The ratio of expenses to income is 30%.

Direct Capitalization:

Direct capitalization is a method of valuing properties that relies on converting the expected income from the property into a present value, based on the relationship between income, value, and the capitalization rate.

• Basic Concepts:
  • Income (I): Net Operating Income (NOI) expected from the property.
  • Value (V): The estimated market value of the property.
  • Capitalization Rate (R): The ratio between the net operating income and the value.
• Basic Direct Capitalization Formulas (IRV):
  1. Finding Income: I = R x V (Income = Capitalization Rate × Value)
  2. Finding Capitalization Rate: R = I / V (Capitalization Rate = Income / Value)
  3. Finding Value: V = I / R (Value = Income / Capitalization Rate)
• Practical Examples:
  1. A property generates an annual income of $40,000, and a capitalization rate of 25% is applied. What is the estimated value of the property? The estimated value of the property is $160,000.
  2. A property with a market value of $500,000 generates an annual income of $60,000. What is the implied capitalization rate? The implied capitalization rate is 12%.
• income multiplier❓: The inverse of the capitalization rate. Income Multiplier = 1 / Capitalization Rate; Capitalization Rate = 1 / Income Multiplier.

Interest Calculations:

Interest calculations are essential in understanding real estate finance, as they affect the cost of loans and the returns on investments.

• Simple Interest: Simple interest is calculated only on the principal amount.

  • Formula: Interest = Principal × Rate × Time (I = P x R x T)
  • Finding the Unknown Variable:
    1. Finding the Interest (I): I = P x R x T
    2. Finding the Principal (P): P = I / (R x T)
    3. Finding the Rate (R): R = I / (P x T)
    4. Finding the Time (T): T = I / (P x R)
  • Practical Examples:
    1. An investment of $1000 earns interest at a rate of 12% per year. What is the value of the interest earned in six months? The value of the interest earned is $60.
    2. What is the capital required to invest at an interest rate of 5% per annum to get an interest of $500 in one year? The capital required is $10,000.