Simple mean reversion strategy portfolio construction

Question

I had a quick idea I wanted to test, but am not sure of the correct way to size bets. Basically, I think that for a given index (say S&P), I want to be long under performers and short over performers (hence the mean reversion). Here is my approach

1. To simplify things, lets take an approach where I simply sort from highest to lowest the past period return for all the tickers.

2. I then want to "pair off" each of the tickers in that index. So pairing most over-perform with most under-perform, second most, third most, etc.

3. For each pair, create a beta neutral position where I am long the under performer and short the over performer

4. Weight each pair in a linear ramp fashion. So what I mean is if I have 100 dollars and 50 pairs, I would have $\sum_{i=1}^{50}D_i = 100$, where $D_1$ would represent the dollar weight on the most under/over perform pair and the $D_i$ satisfy $D_i = (51-i) \times x$ so a linear ramp and if we solved for $50x + 49x + ... x = 100 \implies x = \frac{100}{51\times25}$ so $D_1 \approx \$3.92$ or 3.92% in portfolio weight terms.

5. However, my question then lies on the right approach to breaking apart the long and short side for each pair so that combined exposure is 3.92% while ensuring the pairs beta is also zero.

My initial thoughts on (5) is that for $s$ short and $l$ long, the positions need to satisfy $p_s \beta_s + p_l \beta_l = 0$ and $p_s + p_l = D_i$, but in this case with two equations and two unknowns, there will only be one solution (assuming linear independence). However, that solution could mean that $p_l <0$ which is taking a short position in the ticker I want to be long. How do I account for this?

Is there anything else with my assumptions and thoughts for 1-4 that might be wrong? Any help would be much appreciated!

Lets take a concrete example where I have 50 dollars to allocate to this pair $p_s\beta_s + p_l \beta_l = 0$ and $p_s + p_l = 50$, where $p_l >0$ and $p_s <0$. If the following happen:

1. $\beta_s = 2, \beta_l = 1$, then we have a clear solution where $p_s = -50$ and $p_l = 100$
2. If we flip betas instead to be $\beta_s = 1, \beta_l = 2$, then solving gives us that $p_l = -50 < 0$ which is a short position.
3. Other signs combinations of $\beta_s,\beta_l$ make it so that if $\beta_s$ and $\beta_l$ are opposite signs, then I cannot generate a beta neutral position (either it will be always beta positive or beta negative)

How should I think about this portfolio construction?

Answer 1

How about thinking about each trade in terms of gross exposure instead of long? I would pick a gross leverage ratio at the portfolio level and size this way. Concretely, change the formula to be |Ps| + |Pl| = Di

The issue here is that for each pair to have net long exposure at all, the beta of the long has to be lower than that of the short, which is not guaranteed.

Another issue you may run into is a singularity. If the betas are very close together of two securities within a pair, you’ll have to buy and sell a ton of shares of each to achieve your target long exposure, ending up with really high gross.