# Calculation of Long-Short-Portfolio returns for different holding periods

I have monthly stock returns I want to invest in according to my trading signals. Now I want to figure out the optimal holding period of the long-short-positions. (The same time for both positions).

I invest one unit long and one short. For a holding period of one month I just substract the returns from the short-portfolio of the ones from the long-portfolio and multiply the mean of all of them by 12 for the annual return.

My questions concern the longer holding periods, let's say it is two months. First I have to multiply the return of t=0 with the return of t=1. Do I then invest the dollar every month or do I invest it only every other month so I never invest more than my initial one dollar at a time? (For a holding period of x then every x-th month.)

Or would I only invest 1/2 dollars every month so I don't exceed one dollar at a time? (1/x dollars for x months of holding.)

And do I have to consider anything else for the return calculation afterwards?

• So it sounds that you haven't determined how you will rebalance and reconstitute. You should get a clear methodology for both reconstitution and rebalancing. At any given time, how many securities are there to be invested in? Also, why would you annualize intermittent monthly returns? Would observing the aggregate portfolio over a year be more representative for an annualized return? Jul 14 at 18:36
• I would simply close the position after x months and again invest one dollar short and long, if you mean that. I have not given reconstitution and rebalancing any more thought so far honestly. I have the monthly returns of 19 securtities over 24 years to choose from. The are occasions where some data is missing. The worst performing 20% of the previous month I will sell short, the best 20% I will buy. Jul 14 at 20:36
• So if your strategy is to: long security with largest return in the previous month and short the security with the lowest return in the previous month, you would reconstitute each month with your security universe. It sounds that you are not rebalancing throughout the month. It sounds like your strategy is similar to this, but I don't follow your description too well. Also, you'll never invest more units than you currently have--unless you use some type of leverage (but since you are using leverage, you can borrow those units) Jul 14 at 21:00
• Yes, I don't rebalance. My main problem is how I would calculate the returns if I held the securities for two months. 1. Invest every month and divide the returns by 2. 2. Invest only second month. I found this paper page 68 of Jegadeesh and Titman investing every month and dividing the returns by 2 then if I understand it correctly. Jul 14 at 21:36
• Read that page you referenced. Using their notation of K as the holding period, it seems as though they have K separate portfolios. These K portfolios can include any security in the universe--if a security was in the previous portfolio formed last month, it could be sorted into the next one. Jul 14 at 21:50