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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?

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  • $\begingroup$ Hi: What you say is correct so if you want to only rebalance monthly, then you have to accept that the ranking, technically, will be somewhat incorrect during the month. Note though that the horizon is not arbitrary and really depends on where you think the arbitrage exists. For example, maybe you think it's weekly in which case you should be rebalancing at the beginning of every week. My point is that the frequency of rebalancing is intimately connected to the PNL of the strategy and getting more frequent does not necessarily bring you closer to optimality. $\endgroup$
    – mark leeds
    Apr 2 at 17:19
  • $\begingroup$ Yes, to reduce transactions costs, you allow weights to drift during the month as market prices change, then (usually on last day of the month) you reset them to the weights calculated afresh for your investment strategy. In real life you will own a certain number of shares of each stock, and this number will not change during the month. (It may be helpful intramonth to think about share amounts rather than weights). $\endgroup$
    – noob2
    Apr 2 at 17:34
  • $\begingroup$ So you agree that I should not just keep the weights constant based on the strategy? I should let them increase and decrease with their returns during the month, and begin from scratch again after a month? 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: 0.5*1.02/(0.5*1.02+0.5*1.03) = 0.4975 Stock 2: 0.5*1.03/(0.5*1.02+0.5*1.03) = 0.5025 Next day: stock 1 has 1% and stock 2 has 2% return Stock 1: 0.4975*1.01/(0.4975*1.01+0.5025*1.02) ... Is this correct? $\endgroup$
    – theone
    Apr 3 at 11:46
  • $\begingroup$ I updated the question with the example, so it is easier to see! $\endgroup$
    – theone
    Apr 3 at 11:54
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    $\begingroup$ Hi: As noob2 said, if you really think your arbitrage is monthly, then letting the weights change slightly during the month is fine. There's nothing you can do about that and there's nothing that you should want to do because, if you re-balance to handle that, then your changing the hopeful monthly arbitrage to some other frequency. $\endgroup$
    – mark leeds
    Apr 3 at 14:19

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