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Given a portfolio $P$ with return $R$ and market-beta $\beta$, we have

$$E (R - R_f) = \beta (E R_M - R_f)$$

Now, what does leveraging $P$ have to do with $\beta$? How it is affected if we leverage the portfolio up or down?

For example, say I want a portfolio of beta 1. Then I divide through by $\beta$ to get $$E(R - R_f)/\beta$$

... but what exactly is this? How do I obtain a portfolio using leverage that gives a beta of one?

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    $\begingroup$ You leverage by borrowing or lending money. If $\beta$ is less than one you have to borrow money, if greater than one you lend. For example if $\beta$ of the stock is 2 you put half your money in treasury bills (or a bank account) the other half in the stock; that gives you a beta of 1 overall (because the bank account has a beta of zero and the overall beta is the weighted average of 2 an 0). $\endgroup$
    – Alex C
    Aug 6, 2018 at 15:45

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Your leverage will be the amount of money you borrow to buy the risky portfolio P. Intuitively, the more you borrow money to buy P, the more you are exposed to market behaviour and so β will be high.

As explained in the comment, if for instance your portfolio has a β of 2, you sell half of it and put your money at the risk free rate and get a general β of 1. Conversely, if your portfolio has a β of 0.5, you will borrow at the risk free rate the current value of your portfolio and double your position.

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