I am to replicate the Betting against beta strategy by Pedersen and Frazzini. We use daily returns of the stocks and construct two portfolios based on their ranked betas. Weights is also based on the rank and you can think of it as given for this problem.
So the problem is that when the portfolios is to rebalance monthly, it means that at the first date of each month we take the already calculated weights and assign them to each stock and multiply it with the returns each day of the month. OR, and this is what I struggle to understand: Will not the weights of the stocks change based on the stocks returns within the month? Because if we invest 1 dollar in the portfolio upon construction and one of the stocks has a weight of 0.05 the first day of the month, we will invest 0.05 dollar in that stock. If the stock increases by 10% this day, we have 0.05*(1+0.1)=0.055 dollar in that stock after that day. So this should represent what we have invested for day two of the month. This continous for the rest of the month for all stocks.
Should this be the procedure of all the stocks within each month after it is rebalanced? So you rebalance at the 1 day of the month, invest 1 dollar in the portfolio, and it develops cumulatively throughout the month as stated above? This continues to the next month, where it starts over with the orginal weights that is calculated beforehand. Please help with the understanding :D
Example: You have two stocks in the portfolio, with equal weights 0.5 at the start of the month. Day 1, stock 1 has 2% and stock 2 has 3% return. The weights day 2 is then:
Stock 1: 0.51.02/(0.51.02+0.5*1.03) = 0.4975
Stock 2: 0.51.03/(0.51.02+0.5*1.03) = 0.5025
Next day: stock 1 has 1% and stock 2 has 2% return
Stock 1: 0.49751.01/(0.49751.01+0.5025*1.02) =
Stock 2: 0.50251.02/(0.49751.01+0.5025*1.02) =
And so it continues until next month, where the original weights the first day is used calculate for day 2 and so on. Is this correct?