Chapter 4: Unlocking Value: Interest, Compounding, and Appraisal Math

B. Interest Compounding

Compound interest is interest that is earned on both the principal amount and any accumulated interest. This is in contrast to simple interest, which is only calculated on the principal. The effect of compounding is that your investment grows at an accelerating rate over time.

• Definition: Compound interest is interest calculated on the initial principal and also on the accumulated interest of previous periods.

• Principle: The core idea behind compound interest rests on the time value of money. Money available in the present is worth more than the same amount in the future due to its potential earning capacity. Compound interest leverages this principle by reinvesting earned interest, allowing it to generate further interest.

• Scientific Theories/Principles: The mathematical foundation of compound interest stems from exponential growth. Each compounding period effectively multiplies the previous balance by a factor greater than 1 (reflecting the interest rate).

1. Compounding Period

A critical factor in calculating compound interest is the compounding period. This is the interval at which interest is actually added to the principal.

• Definition: The compounding period is the frequency with which interest is calculated and added to the principal balance.

• Examples: Common compounding periods include annually, semi-annually, quarterly, monthly, daily, and even continuously. The shorter the compounding period, the more frequently interest is added, and the greater the overall return (all other factors being equal).

2. Compound Interest Formula

The future value (FV) of an investment with compound interest can be calculated using the following formula:

FV = PV (1 + i)^n

Where:

• FV = Future Value
• PV = Present Value (Initial Principal)
• i = interest rate per compounding period❓ (annual interest rate divided by the number of compounding periods per year). i = r/m where r is the annual interest rate and m is the number of compounding periods per year.
• n = Total number of compounding periods (number of years multiplied by the number of compounding periods per year). n = t*m where t is the number of years and m is the number of compounding periods per year.

3. Example

Let’s say you invest $1,000 (PV) in an account that pays an annual interest rate of 5% (r), compounded quarterly (m=4), for 10 years (t=10).

1. Calculate the interest rate per compounding period: i = 0.05 / 4 = 0.0125
2. Calculate the total number of compounding periods: n = 10 * 4 = 40
3. Apply the formula: FV = 1000 (1 + 0.0125)^40
4. FV = 1000 * (1.0125)^40
5. FV = 1000 * 1.6436
6. FV = $1,643.62

Therefore, after 10 years, your investment will have grown to $1,643.62.

4. Continuous Compounding

Continuous compounding represents the theoretical limit of compounding frequency, where interest is calculated and added to the balance infinitely often. While not practically achievable in most real-world scenarios, it serves as a useful benchmark.

The formula for continuous compounding is:

FV = PV * e^(rt)

Where:

• FV = Future Value
• PV = Present Value
• e = Euler’s number (approximately 2.71828)
• r = Annual interest rate (as a decimal)
• t = Number of years

Using the previous example with continuous compounding:

FV = 1000 * e^(0.05*10)
FV = 1000 * e^(0.5)
FV = 1000 * 1.6487
FV = $1,648.72

As you can see, continuous compounding yields a slightly higher return than quarterly compounding.

5. Practical Applications and Related Experiments

• Savings Accounts: Understanding compound interest is crucial for maximizing returns on savings accounts and other investment vehicles.
• Loans: Compound interest also applies to loans. Therefore, understanding the compounding frequency and interest rate is important for minimizing the total amount paid over the life of the loan.

• Appraisal: Discounting is an important tool in appraisal, which relies on the reverse of compounding to determine the present value of future income.

• Experiment: Create a spreadsheet to calculate the future value of a hypothetical investment under different compounding frequencies (annually, quarterly, monthly, daily). Observe how the final amount changes as the compounding period shortens.

6. Visualizing Compound Interest

Graphing the growth of an investment under compound interest demonstrates its exponential nature. The initial growth may seem slow, but as the principal increases, the rate of growth accelerates.

C. “Hoskold” or Sinking Fund Method

The Hoskold method is a valuation technique used to determine the present value of future income streams from wasting assets, such as mineral deposits or timber. It assumes that a portion of the income is set aside each year in a sinking fund that earns interest at a safe rate (typically the U.S. government bond rate) to recapture the initial investment.

• Premise: The Hoskold method acknowledges the depletion of the asset and ensures that the initial investment is recovered through the sinking fund, in addition to providing a return on the investment.

• Formula:

  PV = I / (r + (s / ((1 + s)^n - 1)))

  Where:

  • PV = Present Value of the asset
  • I = Annual Net Income from the asset
  • r = Required rate of return on the investment (discount rate)
  • s = Safe rate of return (sinking fund interest rate)
  • n = Number of years of the income stream

• Use and limitations: The Hoskold method is rarely used nowadays.
  The reasons are:
  1. Modern-day investors prefer to earn a higher rate of return on investment
  2. This method only applies to wasting assets.

D. “Inwood” Method

The Inwood method is a valuation technique used in real estate appraisal to determine the present value of a property based on its expected future income stream and a single discount rate. It assumes a level annual income and a constant discount rate over the investment period.

• Premise: The Inwood method treats the property’s income stream as an annuity and discounts it back to its present value. It considers both the return on the investment (discount rate) and the recapture of the investment over the property’s economic life.

• Concept: It is amortized (paid off) just like a loan.

• Formula:

  PV = I * [1 - (1 + r)^-n] / r

  Where:

  • PV = Present Value of the property
  • I = Annual Net Operating Income (NOI)
  • r = Discount rate
  • n = Number of years in the holding period

• Use and limitations: This method does not consider complex scenarios and is, therefore, less utilized than other modern techniques.

VII. Measures of Central Tendency

Appraisers often use measures of central tendency to analyze market data and determine typical values, rents, or other relevant metrics. These measures provide a single, representative value for a set of data.

• Importance: Appraisers must understand the different measures of central tendency and their appropriate applications to avoid misinterpreting data.

1. Mean

• Definition: The mean is the arithmetic average of a set of numbers. It is calculated by summing all the values and dividing by the number of values.
• Formula:
  Mean = (Sum of all values) / (Number of values)
• Application: Useful for summarizing large datasets when there are no extreme outliers.

2. Median

• Definition: The median is the middle value in a dataset when the values are arranged in ascending or descending order.
• Application: More robust than the mean in the presence of outliers, as it is not affected by extreme values. Often used for housing prices.

3. Mode

• Definition: The mode is the value that appears most frequently in a dataset.
• Application: Useful for identifying the most typical or common value in a distribution.

4. Range

• Definition: The range is the difference between the highest and lowest values in a dataset.
• Application: Provides a simple measure of the spread or variability of the data.

5. Standard Deviation

• Definition: Standard Deviation is a mathematical formula that measures how far the prices vary from the mean.
• Formula: σ = sqrt[ Σ(Xi - μ)^2 / (N-1) ], where σ is the standard deviation, Xi are the data points, μ is the mean, and N is the number of data points.
• Application: a small standard deviation means that most of the prices are approximately the same, while a larger one indicates a bigger variance.