geometric_mean [CapiPedia]

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======Geometric Mean======
The Geometric Mean is a type of average that reveals the typical growth rate of an investment over time. While most of us are familiar with the simple average (the `[[arithmetic mean]]`), the geometric mean is far more useful for investors because it accounts for the magic and tyranny of compounding. Think of it this way: the arithmetic mean tells you the average of a set of numbers, but the geometric mean tells you the central tendency of a //process// of growth. For anyone looking to accurately measure their `[[portfolio]]` performance, understanding this concept is non-negotiable. It is the mathematical engine behind one of the most important metrics in finance: the `[[Compound Annual Growth Rate (CAGR)]]`. It provides a true, smoothed-out `[[rate of return]]` that, if achieved consistently each year, would result in your portfolio's final value.
===== Why the Geometric Mean Matters in Investing =====
The biggest reason to use the geometric mean is that it tells the truth about performance, especially when `[[volatility]]` is involved. The simple arithmetic mean can be dangerously misleading, making volatile investments look much better than they actually are.
==== The Illusion of the Arithmetic Mean ====
Let’s imagine a "hot" stock that has a fantastic year, followed by a terrible one.
  * Year 1: Your investment of $1,000 soars by 100%, becoming $2,000.
  * Year 2: The stock crashes, losing 50% of its value. Your $2,000 is now back to $1,000.
What was your average annual return?
An arithmetic mean calculation would be: (100% gain - 50% loss) / 2 years = 25% average annual return. This sounds amazing! But wait a minute... you started with $1,000 and ended with $1,000. Your actual return was 0%. The arithmetic mean lied to you by completely ignoring the effects of compounding on a fluctuating asset value.
==== The Truth-Teller: Geometric Mean in Action ====
Now let’s run the same numbers using the geometric mean. The formula multiplies the returns for each period together and then takes the nth root, where n is the number of periods.
The formula is: //nth root of [(1 + Return1) x (1 + Return2) x ... x (1 + ReturnN)] - 1//
For our example:
  - First, convert percentages to decimals: 100% = 1.0; -50% = -0.50.
  - Next, add 1 to each return: (1 + 1.0) = 2.0; (1 - 0.50) = 0.5.
  - Multiply them: 2.0 x 0.5 = 1.0.
  - Take the square root (since it's 2 periods): The square root of 1.0 is 1.0.
  - Finally, subtract 1: 1.0 - 1 = 0.
The geometric mean is 0%. This number accurately reflects the reality that after two years of gut-wrenching volatility, you ended up exactly where you started. It reveals the true, compounded experience of your investment journey.
===== Practical Takeaways for the Value Investor =====
For a `[[value investing]]` practitioner, the geometric mean isn't just a better calculator; it's a tool that reinforces a sound investment philosophy.
  * **It Punishes Volatility:** As the example shows, wild swings up and down are penalized by the geometric mean. A portfolio with steady returns of 10% and 10% will have the same geometric and arithmetic mean (10%). A portfolio with returns of 40% and -15% will have a much lower geometric mean than its arithmetic mean. This mathematically validates the value investor's preference for stable, predictable businesses over speculative gambles.
  * **It Promotes Honesty:** When you evaluate a mutual fund manager or a stock's historical performance, always look for the CAGR. If a fund only advertises its "average return," be skeptical. They may be using the flattering arithmetic mean. The geometric mean (via CAGR) is the honest scorecard of long-term wealth creation.
  * **It Fosters Realistic Expectations:** Calculating the geometric mean of your own returns gives you a sober, realistic view of how you are actually doing. It cuts through the emotional highs of good years and the lows of bad years to provide a single, actionable number that represents your true progress.
===== The Bottom Line =====
In short, the arithmetic mean is what you would have earned in a //single, representative year//, assuming your capital wasn't compounding. The geometric mean is what you //actually did earn// on your capital each year on a compounded basis. For any investor serious about measuring long-term performance, the geometric mean is not just an alternative—it is the only average that tells the whole story.