I am looking to passively/lump-sum invest stocks. If I averaged the yearly returns of a stock, how many years would I need before I could say with 95% confidence that the averaged value would accurately predict the average yearly return for the next 10 years? If the answer is that "it depends", what factors would it depend on?
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$\begingroup$ For example, it depends on the way you predict the price. It probably also depends on the volatility of stock returns.. $\endgroup$– LazyCatCommented Jan 27, 2021 at 3:56
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It seems implicit in the question that you are happy to assume that the distribution of historical returns is an unbiased and consistent estimator of the distribution of future returns. Else "it depends" (on every variable that affects stock prices, ie all of them) ;-)
You can always come up with such an estimate; but your confidence interval will just be absurdly wide in small sample sizes. Let's say you have 9 years of data, suggesting a mean of say 5% and a vol of 15%. The standard error is your vol divided by root-n = 15% / 3 = 5%. So you can be 95% confident (ie 2 sigma) that the average future return should be between -5% and +15%. Not exactly helpful; and 5% of the time the long-term future should indeed be outside this range!
Broad rule of thumb is you need to quadruple the sample size to halve the confidence bands. So 36 years of data would be +/-5% (ie 2 * 15%/6) around mean 5%, ie somewhere between 0% to 10%. Still not very helpful.
And you would need 144 years to pin this down to say +2.5% to +7.5% - assuming the 15% vol is correct, but that's the long-run average for the US stockmarket. For single companies (more like 25% vols), you would need maybe 100 years of data to be 95% confident the mean was 0-10% (ie 2 * 25%/10 = 5% standard error = same 5% mean).
But once you've been patient enough to find and gather all this data to give you these kinds of levels of confidence, you start to run into an inevitable problem. Is the stockmarket, let alone any company, I'm buying today really the same beast as it was 10, 25, 50 or 100 years ago? That's the real problem here.
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$\begingroup$ Nice one. So statistical representativeness is the issue here; i.e. you simply cannot be certain that the you truly sample from the exact same distribution all the time. Companies come and go, the political framework shifts, tariffs, export bans, new technologies... And then you have a lot of other investors who try to identify valuable investement opportunities, thereby actually increasing the problem of finding/fixing a good (in whatever sense) expected-return-estimate. $\endgroup$ Commented Jan 27, 2021 at 10:24
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$\begingroup$ One typo corrected above. Note that a 95% confidence level AROUND a mean is a 97.5% confidence it will be higher than the lower bound. And the choice of 95% imposes its own constraints. M=5% real, S=15%, N=100, you could argue there was a ONE-TAILED 75% confidence that the average was >4% real; versus almost zero of this from cash or bonds. A lack of academic precision around the underlying return process doesn't alter the uncertain relative likelihoods it generates. $\endgroup$– demullyCommented Jan 27, 2021 at 10:48