How Arithmetic Average Returns Mislead Investors
Suppose a mutual fund records a +50% return in year one and a -50% return in year two. The arithmetic average return is (50% + (-50%)) / 2 = 0%. However, tracking the actual asset trajectory of a 1 million yen investment shows 1.5 million yen after year one and 750,000 yen after year two - a 25% loss. The arithmetic average return is 0%, yet an actual loss occurred. This discrepancy is the most important problem created by the difference between arithmetic and geometric average returns.
The geometric average return (CAGR: Compound Annual Growth Rate) is the metric that accurately reflects the actual compound growth rate. In the example above, the geometric average return is sqrt(1.5 x 0.5) - 1 = sqrt(0.75) - 1, which is approximately -13.4%, correctly representing the actual asset decline. The arithmetic average is simply the average of each year's returns and ignores the compounding effect. When fund reports or advertisements use "average return," checking whether it is the arithmetic or geometric average is the first step in investment decision-making.
Volatility Drag - How Fluctuations Erode Compound Growth
The divergence between arithmetic and geometric averages is caused by a phenomenon called "volatility drag" (variance drain). Mathematically, the approximation geometric average is roughly equal to arithmetic average minus sigma-squared divided by 2 holds (where sigma is the standard deviation of returns). In other words, the higher the volatility, the more the geometric average diverges downward from the arithmetic average. Even with the same arithmetic average return of 10%, the geometric average drops to approximately 9.5% with a standard deviation of 10%, and to approximately 5.5% with a standard deviation of 30%.Books analyzing the mathematical relationship between volatility and returns also provide detailed analysis of this relationship.
The practical implications of volatility drag are significant. One of the main reasons leveraged ETFs are considered unsuitable for long-term holding is this phenomenon. A 2x leveraged ETF doubles the daily return, but volatility also doubles, causing volatility drag to increase approximately fourfold. In a sideways market, the value of the leveraged ETF gradually decays even though the underlying asset remains unchanged. This "decay" is volatility drag itself, and it is an essential concept for understanding leveraged products.
Choosing the Right Return Metric and Applying It to Investment Decisions
Which return metric to use in investment decisions depends on the purpose. When evaluating past investment performance, you should use the geometric average (CAGR), which reflects the actual compound growth rate. On the other hand, when estimating future expected returns, the arithmetic average is considered the theoretically correct estimator. This is because, assuming future returns follow an independent and identically distributed pattern, the expected return for any single year is estimated by the arithmetic average.
As a practical note, when fund reports state "annualized return," always check whether it is the arithmetic or geometric average.Practical books on investment performance measurement explain that fund advertisements tend to use the more flattering arithmetic average, but the return investors actually receive is closer to the geometric average. The divergence between the two is particularly large for high-volatility asset classes (emerging market equities, cryptocurrencies, etc.), making evaluation using the geometric average essential.
Next Actions for Using Return Metrics Correctly
To apply the difference between geometric and arithmetic averages to your investing, start by checking whether the past returns displayed for your funds are arithmetic or geometric averages. The "performance" figures in fund reports are typically the geometric average (CAGR) for each period, but advertisements and ranking sites sometimes use arithmetic averages. The divergence is larger for higher-volatility funds, so particular caution is needed for emerging market equity funds and sector-specific funds.
As a next step, use our compound interest calculator to experience firsthand how final asset values differ when the arithmetic average return is the same but volatility differs. For example, comparing a case with an arithmetic average of 8% and volatility of 10% against one with 25% volatility reveals a significant difference in assets after 20 years. Through this experience, realize how important volatility suppression is - not just returns - and use it as motivation to continue low-cost diversified investing.