# portfolio weights based on past returns

In the academic paper Industries and Stock Return Reversals by Hameed and Mian (JFQA,2015) (see picture below), the authors describe a trading strategy based on reversal, which essentially buys past losers and sell past winners.

I do not fully understand how they come up with those portfolio weights. I understand the idea of the 50% margins, but what is not clear to me is how the weights can actually be considered weights as they do not necessarily add up to 1. Take as an example two assets, A and B, with returns $$R_a=0.3$$ and $$R_b = -0.2$$. Assume $$R_m=0$$ for simplicity. Following their rule, the weight for A will be -(0.3)/0.25 = -1.2 and for B +(0.2/1.2)=0.8 because H=0.5*(abs(0.3)+abs(0.2))=0.25. The two weights sum up to -0.4. So how can I actually invest in this? Even if we forget about the 1/2 scaling in H, the weights would still add up to -0.2. How would you actually calculate the weights in practice in this situation?

• I believe $R_m$ in your example would be 0.1. See formula at beginning of second paragraph. The point being that the weights add up to zero, and in this simple example the weights of the two securities will be opposite. It is a so-called arbitrage portfolio, $\sum w_j = 0$ Commented Sep 8, 2023 at 9:14
• Yes I see now, I was trivially not subtracting the average in my example and with that it works fine Commented Sep 8, 2023 at 13:32
• Yes. The text where it says "$R_m$ is the equal weighted market portfolio" is misleading. The formula makes clear that it is the average of the returns of the $N$ stocks you are considering for your portfolio. (It is not necessarily the CRSP EW average or any standard market index like Invesco S&P 500 Eql Wght). Commented Sep 8, 2023 at 15:56