The mean — what "average return" actually is

Two different "averages"

The mean comes in two flavors that quants need to distinguish:

Arithmetic mean

$\bar{r}_{\text{arith}} = \frac{1}{N} \sum_{t=1}^{N} r_t$

The simple average of returns. This is what most people mean by "average return."

Geometric mean

$\bar{r}_{\text{geom}} = \left(\prod_{t=1}^{N} (1 + r_t)\right)^{1/N} - 1$

The compounded average. Tells you what constant return would produce the same final wealth.

For the 50% / −50% example: arithmetic = 0%, geometric = $\sqrt{1.5 \cdot 0.5} - 1 \approx -13.4\%$ per year, which compounds over 2 years to −25%. Both are correct; they answer different questions.

When to use which

• Arithmetic mean of log-returns = an unbiased estimator of expected log-return per period. Good for forecasting and hypothesis tests.
• Geometric mean of simple returns = the actual realized compounded growth rate. Good for measuring historical performance.
• Arithmetic mean of simple returns = answers "what was the average single-period return" but overstates long-run growth (by approximately half the variance).

A useful identity (for small returns):

$\bar{r}_{\text{geom}} \approx \bar{r}_{\text{arith}} - \frac{\sigma^2}{2}$

This is the famous variance drag. The more volatile your returns, the more the arithmetic mean overstates true growth.

Sample mean as an estimator

When we compute the mean of N observed returns, we're estimating the true (population) mean $\mu$. The sample mean is:

$\hat{\mu} = \frac{1}{N} \sum_{t=1}^{N} r_t$

Two facts that will dominate everything else in this track:

1. The sample mean is unbiased. Its expected value equals the true mean: $E[\hat{\mu}] = \mu$.
2. It's noisy. Its standard error is $\sigma / \sqrt{N}$ — the spread of your estimate shrinks like $\sqrt{N}$.

This matters in trading because estimating the true mean of stock returns is shockingly hard.

Numerical example. Suppose SPY's true monthly mean return is $0.7\%$ with $\sigma = 4.5\%$ (realistic numbers). With 5 years of data (60 monthly observations):

$\text{SE}(\hat{\mu}) = \frac{4.5\%}{\sqrt{60}} \approx 0.58\%$

So a 95% confidence interval around our estimate is about $\hat{\mu} \pm 1.16\%$ — wider than the true mean itself! We can barely tell from 5 years of data whether the true mean is positive.

This is why backtests on a few years of data tell you almost nothing about expected return. The mean is buried in noise. We can measure volatility (variance) reasonably well from short samples. We cannot measure mean returns well from anything less than decades. This asymmetry is foundational.

Code snippet (for laptop session later)

import numpy as np

monthly_returns = np.array([0.012, -0.034, 0.021, ...])  # 60 months
arith_mean = monthly_returns.mean()
std_err = monthly_returns.std(ddof=1) / np.sqrt(len(monthly_returns))

# Annualized arithmetic
arith_annual = (1 + arith_mean)**12 - 1   # one common convention
# Annualized geometric
geom_monthly = np.prod(1 + monthly_returns)**(1/len(monthly_returns)) - 1
geom_annual = (1 + geom_monthly)**12 - 1

Try this on real data in your own environment to build intuition. But first, prove the arithmetic-vs-geometric gap to yourself right here:

Predict before the next lesson

Tomorrow we cover volatility (the sample standard deviation). Predict:

• If the mean is hard to estimate from a small sample, is volatility easier or harder?
• Why?