13.3 Uses of the Geometric Mean

The Geometric Mean (G.M.) is the most appropriate measure of average when data involves multiplicative relationships or follows an exponential law of growth or decline. It is particularly useful in fields like finance, economics, and business, where it helps to determine average growth rates, interest rates, and ratios over multiple periods.

Common Use Cases

The Geometric Mean finds its application in various scenarios, including:

• Population Growth Analysis: Averaging population increases over several years.
• Compound Interest Calculation: Determining the average interest rate when interest is compounded.
• Depreciation Calculations: Applying the diminishing balance method for asset depreciation.
• Average Growth/Decline Percentages: Calculating the average rate of change over time.
• Averaging Ratios or Percentage Returns: Finding the mean of investment returns or financial ratios.

Relation with Compound Interest

The fundamental formula for compound interest is:

A = P(1 + r / 100)ⁿ

Where:

• A = Final Amount
• P = Principal (initial value)
• n = Number of periods
• r = Rate of interest

To find the average interest rate (r) for a single, constant rate over n periods, we can rearrange the formula:

r = [(A / P)^(1/n) – 1] × 100

Variable Rate Scenario – Using G.M.

When the interest rate varies over time (e.g., r₁, r₂, r₃, ... rₙ), the final amount A after n periods is calculated as:

A = P × (1 + r₁/100) × (1 + r₂/100) × ... × (1 + rₙ/100)

To find the average interest rate (r) in such a variable rate scenario, we use the Geometric Mean. The formula becomes:

r = {[(1 + r₁/100) × (1 + r₂/100) × ... × (1 + rₙ/100)]^(1/n) – 1} × 100

This can be expressed more concisely using the Geometric Mean:

r = G.M.(100 + r₁, 100 + r₂, ..., 100 + rₙ) – 100

Explanation: We add 100 to each percentage rate to convert them into growth factors (e.g., 5% becomes 105% or 1.05). We then calculate the geometric mean of these growth factors and subtract 100 to convert the resulting average growth factor back into an average percentage rate.

Example: Average Percentage Increase in Net Worth

Let's calculate the average annual percentage increase in a company's net worth over a 6-year period.

Steps to Calculate the Average Percentage Increase:

1. Calculate the Geometric Mean of the Growth Factors: The formula for the Geometric Mean using logarithms is: log(G.M.) = (1/n) × ∑log X

   Using the data from the table (where X represents 100 + % Increase): log(G.M.) = 12.10653 / 6 = 2.017755

2. Find the Antilog to get the Geometric Mean: G.M. = Antilog(2.017755) G.M. = 104.175

3. Subtract 100 to find the Average Percentage Increase: The G.M. of 104.175 represents the average growth factor. To get the average percentage increase, subtract 100: Average % Increase = 104.175 – 100 = 4.175% per annum

Conclusion

The Geometric Mean is an invaluable tool for calculating the average percentage change over multiple time periods. It is particularly suited for scenarios involving compound growth, variable interest rates, and the analysis of multiplicative relationships, providing a more accurate representation of the average rate than the arithmetic mean in these contexts.

Formula Recap

• Compound Rate (constant): r = [(A / P)^(1/n) – 1] × 100

• Average Rate (variable rates): r = G.M.(100 + r₁, 100 + r₂, ..., 100 + rₙ) – 100

• Geometric Mean via Logarithms: G.M. = Antilog[(1/n) × ∑log X]

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Interview Questions

• What is the geometric mean and when is it most appropriately used?
• How do you calculate the geometric mean, particularly using logarithms?
• Why is the geometric mean often preferred over the arithmetic mean for compound growth calculations?
• Explain the relationship between the geometric mean and compound interest.
• Describe how to determine the average interest rate when rates vary over time, using the geometric mean.
• Can you walk through an example of calculating an average percentage increase using the geometric mean?
• What are some typical real-world applications where the geometric mean is applied?
• How should one interpret the result of a geometric mean calculation in a financial context?
• What are the key differences between the arithmetic mean and the geometric mean?
• How is the geometric mean formula applied to handle concepts like depreciation or diminishing balances?