# Questions on the calculation of time series momentum

I read Moskowitz, Ooi, Pedersen's Time series momentum (2012).

The ex-ante volatility estimate (equation (1) in the paper) is 1. I am not sure about the period of the return reflected in the volatilty calculation above. Since the Time series momentum consists of a portfolio based on 12 months of return, does it mean to calculate geometrical mean and weighted volatility based on daily returns over the past 12 months? Then why is the infinity symbol above the sigma attached? It's written as if to add an infinite number of daily returns. 2.Let me ask you a question on delta. The paper says that the delta value is set so that the center of mass of the weight of the delta is 60 days. Does it means delta=60/61?

• The article says δ/(1−δ)=60 days. So indeed δ=60/61=0.9836. The formula is a moving average formula, which in theory depends on all past daily returns, but in practice the most recent 2*60 =120 days really matter (60 is the so called half life), the rest have little effect on the result. Standard practice would be to use all the available data from the start of the sample to the time you need the volatility estimate (no need to start at $-\infty$). 261 is the number of trading days in a year so you can see the formula is indeed based on daily returns and annual variances. HTH. May 18, 2022 at 11:57