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Hedging a Stock Portfolio to Target Zero Beta with Index Futures

Article Quant Q&A · Author: labrynth
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Summary

The document addresses how to keep a stock portfolio beta-neutral when holdings change over time and may consist only of long or only of short positions on a given day. Its proposed approach is to estimate the stock portfolio’s beta against an index, then use futures on that index to offset the portfolio’s market exposure. The hedge size depends on the target beta, the portfolio beta and dollar value, and the beta and price of the futures contract.

For a zero-beta target, the hedge takes the opposite direction to the stock portfolio’s beta exposure. The answer notes that index futures can offer relatively low transaction costs and counterparty risk, while leaving basis risk. It also points to broader treatment of beta hedging with options and swaps. The document gives a formula and a practical outline, but no performance tests or guidance on beta estimation, contract rounding, rebalancing frequency, or how changing exposures affect hedge maintenance.

Key ideas

  • Estimate the stock portfolio’s beta relative to an index before calculating a hedge.
  • Use index futures to offset the portfolio’s beta exposure and reach the chosen target beta.
  • A zero-beta target requires a futures position opposite to the stock portfolio’s beta exposure.
  • The hedge can have basis risk even when futures are relatively inexpensive to trade.

Tags

EquitiesFuturesRisk managementPortfolio construction
Full text
# Creating a Beta-Neutral Portfolio


# Creating a Beta-Neutral Portfolio












Given a portfolio of assets (say 10) and trading signal (1=Hold):

```
      ___________________   Day Count  ______________________

  Asset |0|1|2|3|4|5|6|7|8|9|10|11| ... |30|31|32|33|34|35| ...
--------+------------------------------------------------------
1. IBM  |1|1|1|1|1|1|1|0|0|0| 0| 0| ... | 0|-1|-1|-1| 0| 0| ... 
2. APPL |0|0|0|1|1|1|1|1|0|0|-1|-1| ... |-1| 0| 0| 0| 0| 0| ...
    :                        :                 :   
    :                        :                 :
10.TSLA |0|0|0|0|0|1|1|1|0|0| 0| 0| ... | 0|-1|-1|-1| 0| 0| ...
```

The trading signal can be read as follows:







My question is that, given that the rebalancing time is not fixed and that on some days there are Long only or Short only positions, how can one make this portfolio Beta-Neutral?

## Answer by Bob Jansen (score 8)

https://quant.stackexchange.com/a/29742

There are more ways to approach this but the method I propose should work reasonably well in practice, especially if you increase the number of assets you hold.

- Calculate the beta of the stocks you're holding with respect to an index

- Buy $N_f$ (sell when $N_f$ is negative) future contracts on that index

$N_f$ can be calculated as

$$N_f = \frac{\beta_T - \beta_S}{\beta_f}\frac{S}{f}$$

where $\beta_T$ is your target beta, $\beta_S$ is the $\beta$ of your stock position, $\beta_f$ the $\beta$ of your future, $S$ the value of your portfolio in dollars and $f$ the futures price. In your case, $\beta_T = 0$ and the formula reduces to

$$N_f = -\frac{\beta_S}{\beta_f}\frac{S}{f}.$$

The advantage of this strategy is that you have low counterparty risk and relatively low transaction costs. However, there will be some basis risk.

The CFA Level III curriculum (book 5) has a much broader discussion on this including strategies using options and swaps. The equations given above are taken from there.

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.