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Say I have historical data of a ticker for the past 5 years. I look at the price on the current date for each of the five years (e.g. today is 20 Jul 2023 so I will look at 20 Jul 2022, etc.) and then consider the returns for 3 months (60-80 business days) from that day.

Can I use this as the sample space to determine how the price will move based on historical data?

I'm thinking to use the return as a way to normalise the historical prices to current price and then work out the probability of the returns to answer this question.

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It might make more sense to include any dividends in your returns. If a stock is \$5 now, you don't care whether in 3 months it's \$5.25, or still \$5 after paying \$0.25 dividend.

Ensure that the prices are adjusted for splits. Otherwise, what looks like a 50% price drop, might just be a 2-for-1 split.

It also might make sense to look at the returns not in isolation, but in comparison to some market index (i.e. CAPM) or factors.

If I understand correctly, you want to use just 5 annual data points per stock. If you were to analyze the historical distribution, calculate historical skewness and kurtosis, then I'd say, 5 is too few. But conversely, if you plan to simply assume that everything is (log)normal with mean 0, and just use the historical data to calculate historical standard deviations, then 5 may be enough. When you backtest how well such historical standard deviations predict the future, then you might observe that using more data points, e.g. quarterly or monthly frquency, predicts better, or you might not.

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