How Leverage Changes a Portfolio’s Market Beta

Summary

The document explains the relationship between borrowing, risk-free investment, and a portfolio’s market beta. Scaling exposure to a risky portfolio scales its market sensitivity: borrowing to enlarge the position raises beta, while holding some wealth at the risk-free rate reduces it.

To target beta one, the examples adjust the risky allocation in proportion to its original beta. A portfolio with beta two can be paired with a risk-free holding by halving the risky exposure; one with beta one-half can be doubled using borrowing at the risk-free rate. These illustrations assume the risk-free asset has zero beta and treat beta as scaling linearly with the position. The discussion is conceptual and does not cover borrowing costs, leverage constraints, or other risks of financing.

Key ideas

• Borrowing to increase exposure to a risky portfolio raises its market beta.
• Combining a risky portfolio with a risk-free asset can reduce the overall beta.
• Under linear scaling, a beta-two portfolio can be reduced to beta one by halving the risky exposure.
• A beta-one-half portfolio can be doubled to target beta one by borrowing at the risk-free rate.

Tags

# What does leverage have to do with beta?


# What does leverage have to do with beta?












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?

## Answer by JeanGuillaume (score 2)

https://quant.stackexchange.com/a/41129

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.