Movement of both would mirror the movement of the market haina ta?
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Beta is just a measure of past correlation. it is not indicative of what the stock might do in the future.
theoretically, that is what should happen.
I think beta is calculated within a sector. So if beta stays at 1 then its equivalent to investing in the sector index fund if such a thing exist (not in nepal ofcourse). But mostly index fund i see are a best performing stocks in a exchange that are mix of different sectors.