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Should I include or not the days a strategy has no open positions (thus no returns) in the Sharpe ratio calculation?

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No, don't include them. Otherwise you'll just wind-up with zero-value returns (or worse, forward-filled returns), which will make your Sharpe ratio reflect a performance that didn't actually occur.

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Thank you Chris. Then how should I anualize the Sharpe ratio? Can I still use square_root(250)? –  Victor P Jan 18 '13 at 16:43
    
@VictorP Correct. –  chrisaycock Jan 18 '13 at 16:43
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This is not obvious to me even though I do just that myself (use sqrt(250)). Why couldn't you argue that one should use sqrt(days_in_the_market_per_year)? –  klon Jan 19 '13 at 0:00
    
@klon You are correct; use the number of days in the market that you trade in. Perhaps Victor's market has 250 days. US equities is 252 days, which is what I use. –  chrisaycock Jan 19 '13 at 1:28
    
@chrisaycock Sorry, I wasn't clear. I meant why not sum up the days you actually have open positions over the year and square root that sum. What would be the argument against that? Well obviously it would be a different measure.. –  klon Jan 19 '13 at 15:54
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If you really think about the actual meaning of Sharpe ratios then you should come to the right conclusion yourself:

  • It is a measure of excess risk-adjusted return (whether realized or unrealized)

In that you obviously only want to calculate actual returns. You do not have any actual returns on days with no open positions. Hence, why would you want to include such days?

=> The core thought here is to only measure the nature of the returns not whether returns occurred or not.

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Incorrect, even by your own definition (which happens to be a fine definition of Sharpe ratio): You have forgotten that Sharpe ratio is excess returns. Does the thing they are exceeding also go to zero on days when a book is flat? No. –  feetwet Jul 15 at 0:58
    
@feetwet, I am not sure I understand what you are trying to say. I mentioned explicitly that Sharpe is a measure of excess risk-adjusted return. Because it is risk-adjusted so you should not include days on which you do not take any risk, hence the exclusion of trading days on which no risk is taken. However, you still need to use business days in the market you trade for annualization purposes else you would not end up with an annualized Sharpe measure. So, I am not sure which part you disagree with. –  Matt Wolf Jul 15 at 6:06
    
It depends on what your benchmark rate is. Each day you have to subtract your benchmark return rate from your net realized returns. On days when you aren't in the market you still have to subtract the benchmark rate. So unless your benchmark rate is zero you can't exclude those days. And if it is zero then you're really calculating a signal-to-noise ratio, not a Sharpe ratio. (Though I acknowledge that in the industry we often use "Sharpe ratio" even when our benchmark rate is zero. But that doesn't make it correct, and this question shows why.) –  feetwet Jul 15 at 14:18
    
Call it whatever you want as long as you make the proper disclosure you can define and use whatever risk adjusted return measure pleases you. I merely reflected what from my experience in the market seems to be the accepted practice. You have of course a right to disagree and write up your own answer to reflect how you think the world ticks, which you have done. People care about understanding generated returns in the context of risk taken and if that excludes days on which no returns were generated and everyone knows through disclosures then everyone is happy. –  Matt Wolf Jul 15 at 15:24
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Remember that Sharpe ratio includes a risk-free rate of return ("RFR").

Unless the RFR is zero, then excluding days when you have no position is not correct and will technically overstate your Sharpe ratio.

And if you're using a RFR of zero then what you're actually providing is the signal-to-noise ratio. (Although yes, I acknowledge that in recent years the RFR is practically zero. But that doesn't justify these other incorrect answers.)

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I disagree, because Sharpe is a measure of excess risk-adjusted, returns. It does not matter whether competing investments generate return during days you are not taking any risks. Sharpe is not a comparable performance measure such as the Information Ratio, its focus is on its own performance relative to its own risk taken, which is why a lot of practitioners actually leave out the risk-free rate altogether. I am not arguing here whether it should be included or not, but I try to make a point that days on which risk is not taken should not be included in the Sharpe ratio computation. –  Matt Wolf Jul 15 at 6:11
    
@MattWolf: The risk adjustment is handled by the denominator. The "excess" takes place in the numerator. Sharpe's ratio was originally defined to show performance relative to a benchmark. It is true that the definition has become abused in the manner you describe. And worse, I have noticed that an "information ratio" unique to finance that references a benchmark has come into use. I guess the only solution now is to explicitly clarify whether you mean excess returns (originally Sharpe ratio) or isolated risk-adjusted returns (originally information ratio, a.k.a. signal-to-noise). –  feetwet Jul 15 at 14:27
    
Well you do not generate any returns, excess or not, on days you do not have open posions or close open positions. –  Matt Wolf Jul 15 at 15:10
    
By the way your definition of information ratio is incorrect. Information provides a return in excess of a defined benchmark in the context of risk. It has nothing to do with signal-to-noise, not sure how you make that connection. –  Matt Wolf Jul 15 at 15:29
    
Fair enough: It does appear that "information ratio" is now the accepted term for risk-adjusted returns over a benchmark. Sharpe ratio then is a special case of "information ratio" where the benchmark indicates the "risk-free" rate of return (e.g., treasuries or money markets). Outside of finance I have seen information ratio used as a synonym for mean/stdev, which is a signal-to-noise ratio. Obviously in all of these cases we're looking at a ratio of mean to standard deviation. –  feetwet Jul 15 at 15:56
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