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It is common to price things at their discounted expected value; for the example of a stock, over what time horizon is this thought to occur in academic finance?

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One would normally use the discounted expected cash flows over the lifetime of the asset. Companies don't die so it's customary to take cash flows into infinity. Due to discounting, cash flows very far in the future don't contribute much to the sum.

This method might be split into two parts where explicit cash flows are forecast and discounted for the first $n$ years + a residual value. However, this residual value would again be based on expected future cash flows properly discounted.

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