How many years does it take to know a market's average return?

The equity premium is the most quoted number in finance and the least precisely known. What follows is the arithmetic of why: the standard error of a mean return, measured on nineteen calendar years of SPY.US, and what it says about every claim of the form "this earns X% a year".

The measurement, 2026-09-04

Calendar-year total returns of SPY.US from 2007 to 2025, nineteen observations on adjusted closes: mean 12.25% a year, standard deviation 17.73%. Three of the nineteen were negative — 2008 at −36.8%, 2018 at −4.6%, 2022 at −18.2%.

The standard error of the mean is the standard deviation divided by the square root of the number of observations: 17.73% over √19, about 4.07 percentage points. A rough 95% band for the true average is the mean plus or minus two standard errors: from about 4% to about 20% a year. Nineteen years of the most-watched market in the world cannot say whether its average return is four percent or twenty.

Why it does not get better quickly

The standard error shrinks with the square root of the sample. To halve it you need four times the years; to bring it to one percentage point at this volatility you need about three hundred years of data, which no market has and which would not be one market if it did. Sampling more often does not help the way it seems to: daily data gives thousands of observations, but they are slices of the same years, not new years. Under independent daily returns the standard error of the mean annualises to the same figure as the calendar-year one; with the clustering and autocorrelation the later lessons measure, a daily estimate that respects that needs a long-run variance correction and comes out wider, not narrower. Precision about a mean return comes from calendar time, and calendar time is the one thing a dataset cannot be given more of.

Volatility is different. The standard deviation is estimated from the spread of the observations, not their level, and daily data does help: a year of daily returns pins down a volatility to within a few percent of itself. This asymmetry — means are unknowable, volatilities are knowable — is the central fact of market statistics, and most bad conclusions come from treating a mean as if it were a volatility.

What it does to claims

"This strategy returns 15% a year." Over how many years, with what standard deviation? Ten years at 20% volatility is a standard error of 6.3 points; the claim is consistent with a true mean of 2%. "Value earned 4% a year over the market." Over ninety years at a spread standard deviation near 12%, the standard error is about 1.3 points — a claim with three standard errors behind it, which is why value's premium was believed and why its three-year droughts do not refute it. "Emerging markets outperform." Twenty-five years at 22% volatility: a standard error of 4.4 points on a mean that has itself moved by more than that as the window shifted.

The habit this lesson installs: every mean return arrives with its standard error, or it does not arrive.

In the data

Nineteen calendar-year returns are twenty year-end closes: /eod/SPY.US?from=2006-12-20&to=2025-12-31&period=m&fmt=json gives month-end rows, and the December rows divided by the previous December's give each year. The standard error is one division after the standard deviation.

Try it now

1. Here are the twenty year-end rows of that monthly series. Reproduce the nineteen calendar-year returns from them and compute their mean, standard deviation and standard error; you should land on the figures above. Then drop 2008 and recompute all three. Write down how much one year moved the mean — that is the sample size talking.

2. The period, so the three negative years have faces:

Measure each calendar year that was negative and the one after it. 3. Someone tells you a fund returned 11% a year over five years with 15% volatility. Compute the standard error and write the 95% band. Then answer: could this fund's true average be zero?