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I'm having trouble understanding the intuition of a simple beta hedge using a linear regression.

Assuming an asset has a beta of 0.5 against the market. That implies for a percent move in the market, the asset moves a half of a percent. As an equation:

Asset = 0.5* Market

If one invested 1000 dollars in the market, I would expect one would need to invest 2000 dollars in the asset to remain beta natural.

However, if we plug in 1000 invested into the market in the equation above, it suggests 500 must be invested in the asset to offset the beta.

And that formulation embarrassingly does not make any sense to me. I'm rusty and appreciate any help.

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    $\begingroup$ @nbbo2, your comment seems to answer the question. Why not post it as an answer then? $\endgroup$ Commented Jan 14 at 12:32

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Let's be careful with the variable names we use and everything will be clearer:

As you say, we have Asset_Return = 0.5 * Market_Return

Then

Asset_dollar_p&L = Asset_return * Asset_position_size

Market_dollar_P&L = Market_return * Market_position_size

So to get equal dollar P&L you need:

0.5 * Asset_position size = Market_position_size.

In plain English you need a bigger position in the security that has a smaller percentage return.

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