# 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| ...


1. IBM : Buy on Day0 and Sell on Day7; then Short on Day31 and Buy back on Day34, and so on.
2. APPL: Buy on Day3 and Sell on Day8; then Short on Day10 and Buy back on Day31, and so on
3. TSLA: Buy on Day5 and Sell on Day8; then Short on Day31 and Buy back on Day34, and so on.

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?

• Looks like in the example you are only Long stocks. Don't you also have to Short to be beta neutral? – noob2 Aug 19 '16 at 20:04
• A stock could have negative beta but in practice short selling would be necessary. – Bob Jansen Aug 19 '16 at 20:52
• @noob2 Thanks for pointing that out. I have modified the portfolio and rephrased the question. – labrynth Aug 19 '16 at 21:24
• @Bob Jansen Thanks for pointing that out. I have modified the portfolio and rephrased the question – labrynth Aug 19 '16 at 21:24

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.

1. Calculate the beta of the stocks you're holding with respect to an index
2. 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.

• Thank you Bob, this looks like the solution I was looking for. And thank you for the references, I'll follow them through. – labrynth Aug 20 '16 at 18:18