How Short-Stock Hedges Produce the Return Formula rA − h rB

Summary

The document explains the return of a hedged position that holds one unit of stock A and shorts h units of stock B. The portfolio’s return is written as rA − h rB: stock A contributes its return, while the short position contributes the negative of stock B’s return.

It then frames hedge selection as a minimum-variance problem. The hedge ratio h is the number of shares of B to short, chosen to reduce the variance of the combined portfolio. The explanation is conceptual and does not derive the optimal hedge ratio or discuss practical details such as transaction costs, borrow availability, or changes in the hedge over time.

Key ideas

• A portfolio holding one unit of A and shorting h units of B has return rA − h rB.
• The short position contributes a negative return exposure to stock B.
• A minimum-variance hedge chooses h to reduce the portfolio return variance.
• The document does not derive an optimal hedge ratio or cover implementation costs.

Tags

# Hedge by shorting stock


# Hedge by shorting stock












Is it possible to explain to me why the formula is $r_A-hr_B$?

My interpretation is that, you short stock B (by selling it), and then you use the money to buy stock A. Thus we have a $r_A$ term there.

Then, we have to buy back stock B. so $r_A-hr_B$.

Is this interpretation correct?

## Answer by JejeBelfort (score 1)

https://quant.stackexchange.com/a/34351

This is entirely correct.

Basically, the initial portfolio you hold is one unit of stock A that you want to hedge.

The hedge portfolio is therefore this unit of stock A that you bought, minus $h$ units of stock B that you have short for the hedge. Of course, shorting stock B means that you will have to buy it back at some point, hence the "-" sign.

Therefore, the return of your hedging portfolio is:

$$r_{port} = r_A - h r_B$$.

In the following, you want to minimize the risk of this hedging portfolio. In this framework, risk is determined by the variance of your portfolio, and the goal is to find the optimal number of shares $h$ of stock B you should own to reach this minimum variance (or optimal) portfolio.

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.