Why arithmetic and geometric returns differ

Arithmetic and geometric averages differ because returns compound multiplicatively. A sequence like a five percent gain then a negative five percent loss has a zero arithmetic average but still turns 100 into 99.75, so realized growth is below the simple average. For multi period portfolio growth, the geometric mean is the relevant measure because volatility creates a gap between average period returns and compounded wealth growth.

Arithmetic and geometric averages answer different questions about returns. The arithmetic mean is the simple average of periodic returns. The geometric mean is the compound rate that links starting wealth to ending wealth across multiple periods. When the goal is to describe growth through time, compounding requires the geometric mean, not the arithmetic mean.

The difference comes from how multi period returns are formed. Wealth evolves by multiplying period by period gross returns, where each period contributes a factor of one plus the return. Because these factors multiply, gains and losses do not offset symmetrically. A loss reduces the capital base on which future gains are earned, and a gain increases the base on which future losses are applied.

A simple example shows the mechanism. Start with 100. A five percent gain raises wealth to 105. A five percent loss then applies to 105, which gives 99.75. The arithmetic average of the two returns is zero percent because five percent and negative five percent sum to zero. But the compounded result is negative because 1.05 multiplied by 0.95 equals 0.9975. The geometric average is therefore below the arithmetic average.

The same asymmetry appears in larger moves. Start with 100. A fifty percent gain takes wealth to 150. A 40 percent loss then applies to 150, leaving 90. The arithmetic average of fifty percent and negative 40 percent is positive five percent. The compounded outcome is a decline from 100 to 90. This is why averaging periodic returns can overstate growth when returns are volatile.

Standard return measurement reflects this distinction. The compound annual growth rate is based on the geometric mean because it is the constant rate that reproduces the same final wealth over the full horizon. By contrast, the arithmetic mean is appropriate as a one period expectation under repeated sampling, but it is not a direct measure of realized multi period growth.

Volatility is the reason the gap opens between the two averages. In the Sonar Sciences fundamentals material, compounding and growth are framed around the multiplicative nature of returns rather than simple addition. That same logic implies a volatility drag on compounded growth: dispersion in periodic returns lowers the geometric mean relative to the arithmetic mean when the average return is held fixed.

Backtest summaries and evaluation metrics should be interpreted carefully, especially when they aggregate evidence across many trials or periods. Overfitting audits and the deflated Sharpe ratio are tools for correcting overly optimistic inference in backtests. Summary statistics can overstate what is likely to persist unless they account for the structure of the data and the number of opportunities for bias.

So the core teaching point is straightforward. Arithmetic averaging gives a simple period by period snapshot. Geometric averaging gives the realized compound growth rate. When returns vary over time, the arithmetic mean will generally sit above the geometric mean because losses and gains act on changing capital bases under compounding. Understanding both averages helps traders evaluate performance accurately consistently.

Drafted with AI assistance from cited sources. Reviewed and approved by Sonar Sciences Quant & Research Team.