Understanding the Relationship Between Geometric Mean and Arithmetic Mean

Key Points

1. Geometric Mean vs. Arithmetic Mean:
• The geometric mean is always less than or equal to the arithmetic mean.
• The only time they are equal is when there is no variability in the observations, meaning all observations are the same.
2. Jensen’s Inequality:
• This inequality states that for a concave function, the average value of the function is less than or equal to the function evaluated at the mean.
• For the natural logarithm function (ln), which is concave, this implies the relationship between the geometric and arithmetic means.

Example to Illustrate the Concept

Scenario: Investment Returns

Suppose you have investment returns for three years, each being 10%.

1. Calculate the Arithmetic Mean:

2. Calculate the Geometric Mean:

Explanation

• Equal Returns: Since all returns are the same (10%), both the arithmetic and geometric means are equal at 10%.
• Variability: If the returns were different (e.g., 10%, 20%, 5%), the arithmetic mean would be higher than the geometric mean. This is because the geometric mean takes into account the compounding effect over multiple periods, which is more accurate for varying returns.

Detailed Example with Variability

Scenario: Investment Returns with Variability

Suppose you have investment returns of 10%, 20%, and 5% over three years.

1. Calculate the Arithmetic Mean:

2. Calculate the Geometric Mean:

Conclusion

• No Variability: When there is no variability in the returns, the arithmetic mean equals the geometric mean.
• With Variability: When returns vary, the geometric mean is less than the arithmetic mean, accurately reflecting the compounded growth rate.

This principle helps in understanding how the means behave in different scenarios, providing a clearer picture of investment performance and other similar data sets.