The Trap of CAGR vs. Simple Average
Year 1: +50%. Year 2: -50%. Average them and you get 0%, so you're back to even, right? Do the math and you actually end up at -25%—your principal has shrunk. This illusion is the 'trap of the average.'
The illusion of the simple average
Simply adding up several years of returns and dividing gives you the simple (arithmetic) average. The simple average of +50% and -50% is 0%. But actual money moves differently.
About $740 becomes about $1,110 in year 1 with +50%, and then in year 2 with -50% it becomes about $555. It started at $740 and ended at $555. In other words, that's -25% over the two years. It's completely different from the simple average of 0%. The larger the swings, the wider this gap grows.
CAGR — the actual compound average
CAGR (compound annual growth rate) is the value you get by tracing back, with compounding, 'what percentage it would have been if it had grown at the same rate every year.' In the example above, since about $740 became about $555 two years later, CAGR = (555/740)^(1/2) - 1 ≈ -13.4%.
In other words, the annual average actually experienced was not 0% but roughly -13.4% every year. Because CAGR is calculated backward from the actual result like this, there is no illusion. This is the geometric mean, and it is the true average of compounding.
The simple average is always greater than or equal to CAGR. The gap between them is created by volatility, and this is called 'volatility drag.'
A habit for reading numbers
When you see the phrase 'past average return of X%,' check whether it's a simple average or CAGR. Especially for high-swing assets (individual stocks, crypto assets, etc.), the simple average is inflated to look far better than reality.
What honestly expresses long-term performance is CAGR. And even CAGR can give the impression that 'it rose smoothly,' so you have to look at the maximum drawdown along the way to truly understand the investing experience.
よくある質問
Q. Is the simple average completely useless?
Not at all. When dealing with the expected returns of multiple assets at a single point in time, or when calculating a statistical expected value, the arithmetic mean is needed. However, when summarizing 'actual performance over multiple years,' CAGR (the geometric mean) is the correct one. They serve different purposes.
Q. Why does the simple average always look larger?
Mathematically, as long as there is any variation, the arithmetic mean is larger than the geometric mean (CAGR). This is because recovering from a loss requires not the same %, but a larger %. To undo -50%, you need not +50% but +100%. This asymmetry inflates the simple average.
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