Arithmetic and Geometric Averages for Measuring Returns
Summary
The answer explains the distinction between averaging periodic simple returns and calculating a compounded return over multiple periods. Because wealth evolves multiplicatively, the geometric average gives the constant per-period return equivalent to the realized compounded growth. It can be expressed by taking the average of log gross returns and converting back to a simple return.
The answer says arithmetic averaging is often used as an approximation when periodic returns are small, citing daily returns as a common case. Log returns can instead be averaged directly to summarize compounding. The discussion is brief and does not fully address why empirical asset-pricing studies often use arithmetic returns, the role of expected one-period returns, or how portfolio construction affects the choice. The practical interpretation depends on whether the goal is expected single-period performance or realized multi-period growth.
Key ideas
- Simple returns compound multiplicatively across periods.
- The geometric average describes the constant periodic return equivalent to observed compounded growth.
- Arithmetic averages can approximate geometric growth when periodic returns are small.
- Log returns add across time, so their average directly summarizes average log growth.
- The appropriate return measure depends on whether the analysis concerns one-period expectations or multi-period compounding.
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# Measuring expected returns
# Measuring expected returns
Most papers in the literature measure expected returns using the simple average of past returns. Why is this? When is it more correct to use geometric returns instead? Any good references?
I know that using log returns solves many of the issues with arithmetic/geometric returns, but not ideal to construct portfolios.
In any case most empirical asset pricing papers tend to use arithmetic returns. Why is that?
## Answer by Count (score 3)
https://quant.stackexchange.com/a/61494
When using arithmetic returns the right way to calculate an average is via the geometric average. The reason is that there is a multiplicative relationship between the returns. Example: Let $P_t$ denote the stock price at time $t$, then the simple (arithmetic) net return is defined as: \begin{equation} r_t=\frac{P_t-P_{t-1}}{P_{t-1}}=\frac{P_t}{P_{t-1}}-1 \end{equation} Now look at: \begin{align} r_t[k]&=\frac{P_t}{P_{t-k}}-1=\frac{P_t}{P_{t-1}}\cdot \frac{P_{t-1}}{P_{t-2}} \dots \frac{P_{t-k+1}}{P_{t-k}}-1=(1+r_t)\cdot(1+r_{t-1})\dots(1+r_{t-k+1})-1\\&=\prod_{j=0}^{k-1}(1+r_{t-j})-1 \end{align} The annualized average return (geometric average of $k-1$ period returns) is given by \begin{align} \overline{r_t[k]}&=\left(\prod_{j=0}^{k-1}(1+r_{t-j})\right)^{\frac{1}{k}}-1\\&=\exp\left(\ln\left(\prod_{j=0}^{k-1}(1+r_{t-j})\right)^\frac{1}{k} \right)-1\\&=\exp\left(\frac{1}{k} \sum_{j=0}^{k-1}\ln(1+r_{t-j}) \right)-1 \end{align} However, notice that this expression is quite difficult to compute. So it is often approximated by the arithmetic average of the $k-1$-period returns \begin{equation} \overline{r_t[k]} \approx \frac{1}{k}\sum_{j=0}^{k-1}r_{t-j} \end{equation} This linear approximation obtains from a 1-order Taylor expansion around zero. So if the returns $r_{t-j}$ are small you get almost the same result with less effort. Especially when looking at daily returns this approximation is very good in most cases, because $r_{j-i} \approx 0$.
When you are using log-returns it is even simpler, because there is no multiplicative relationship and the "right" average is the simple average.
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