How would one interpret the below turnover formula ignoring the average from each time period i.e., what is the meaning of the term inside the brackets?

Reference: Empirical Asset Pricing via Machine Learning, Shihao Gu, Bryan Kelly,Dacheng Xiu, RevFinStud, 2020 (link)

On Page 2268:

We define the strategy’s average monthly turnover as

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where $w_{i,t}$ is the weight of stock $i$ in the portfolio at time $t$. (And $r_{i,t+1}$ is the return realized by stock $i$ in the month from $t$ to $t+1$)

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    $\begingroup$ Pherhaps you can give a more exact reference to the formula, so we do not have to look through the entire paper to find it. (e.g page or numbering) $\endgroup$ Mar 3 at 8:35

2 Answers 2


This is a very standard approach for measuring turnover from portolio weights.

First, assume there are no buys or sales during the month. Then we can predict the weights at the end of the month from the weights at the beginning of the month and the stock returns. Intuitively stocks that go up more than the average see their weights increase, while stocks that go up less (or go down) see their weights in the portfolio shrink.

The predicted weight of stock $i$ at the end of month $[t,t+1]$ is

$\hat{w}_{i,t+1} =\frac{w_{i,t}(1+r_{i,t+1})}{1+\sum_j w_{j,t}r_{j,t+1}}$

The numerator is the effect of performance of stock $i$ "all by itself" and the denominator is the average performance of all stocks in the portfolio.

At the end of the month we compare the actual weight of the stock $w_{i,t+1}$ with the predicted weight $\hat{w}_{i,t+1}$. A smaller actual weight means that some of the stock was sold during the month, and if the weight is larger it must have been bought during the month (recall that in making the prediction we assumed no buys or sells).

So the difference $w_{i,t+1}-\hat{w}_{i,t+1}$ is a measure of the buys or sell for stock $i$ during the month. And by adding over all $i$ we have a measure of turnover during the month.

Of course some buying/selling may escape detection (buying something and immediately selling it before the month end). But it is generally considered an acceptable approximation of annual turnover when you don't have detailed information about all buys and sells that occurred and their dates, only monthly weight data. (It is also exact if you do all your buying selling at the end of the month, as in this paper).

  • $\begingroup$ What is the possible range of the following formula? My understanding is that when it is above 1, our portfolio take the opposite direction of the previous one on average. I also get the values above 2 occasionally which I still don't understand how to interpret it. Perhaps this is due to the number of stocks that is in the long/short position at each time period are unequal. $\endgroup$
    – PrinceZard
    Mar 4 at 9:03
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    $\begingroup$ A monthly turnover of 2 (or 200% in common usage) is when the portfolio completely changes during the month. Everything you had has been sold and every stock you have now is new. For ex. at the beginning you have w=[1,0] i.e. only Stock1, at the end of month you have [0,1] i.e. all your money is in stock 2. $\endgroup$
    – nbbo2
    Mar 4 at 9:11
  • $\begingroup$ So the Turnover as defined here ranges from 0 to 2. $\endgroup$
    – nbbo2
    Mar 4 at 11:25

To add onto Julien's comments, you should add the labels for the variables and what they mean. The turnover (or change in weight per asset $i$) is due to this term:

$Retained\:Weights=\frac{w_i (1+r_{i,t+1})}{1+\sum_{j}w_{j,t}r_{j,t+1}}$

The difference between $w_i$ and retained weights is the turnover (summed over all assets and across all time). For an asset that has greater (lower) return $r_{i,t+1}$, it has less (more) turnover as the retained weight term is higher (lower), leading to a lower (higher) difference between $w_i$ and retained weights.


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