# Should Sharpe ratio be computed using log returns or relative returns?

I am trying to reconcile some research with some published values of 'Sharpe ratio', and would like to know the 'standard' method for computing the same:

1. Based on daily returns? Monthly? Weekly?
2. Computed based on log returns or relative returns?
3. How should the result be annualized (I can think of a wrong way to do it for relative returns, and hope it is not the standard)?

I think this is a no-brainer. Only log-returns make sense. The average return can only be computed by averaging the sum of individual log returns. Taking the average of standard (relative) returns does not give you an average of the individual returns. Consider a simple case where the value of an investment alternates between 100 and 50 an odd number of times. The standard return series would be: -0.5, 1, -0.5, …1 (-50% and +100%). The average of that sum gives us 0.25 (25%) – nonsense for an investment whose final value is the same as what we started with. The log returns, on the other hand give us alternating log returns of -0.6931, +0.6931, whose average is 0.

The difference between log returns and standard returns goes to zero as we shorten the period over which we evaluate the value of an investment: LN(P(n)/P(n-1)) is approximately equal to P(n)/P(n-1) – 1. Thus there would not be much difference between standard and log returns (and the computed Sharpe Ratio) if daily measurements were made. The scaling of that Sharpe Ratio from daily returns to annual returns is performed by the sqrt of the number of trade days (252), but that, of course assumes the return distribution is iid, which is not really the case.

Andrew W. Lo has a nice paper that considers the scaling of the Sharpe ratio when the return series is correlated ("The Statistics of Sharpe Ratios")

• Don't you think compounding returns (pct changes) can be averaged with geometric average? Even though logreturns have nice mathematical properties, it's hard to interpret them. When I say I have SR of 2, calculated on pct changes, I know historically my average return was 2sd above rf rate. How should I interpret this statistics for log returns, let alone sd of logreturn ??? – Sergey Bushmanov Oct 11 '15 at 22:23
• @Sergey Bushmatov I too like the idea of using the geometric average on say daily returns and then scaling the daily mean to annualised value. The question I have -- is the geometric average equivalent to the expectation of daily return distribution just as arithmetic average usually is for estimating expectation from samples taken from distribution? – A.L. Verminburger Nov 21 '17 at 16:47
• Nonsense. In the reference you gave (Lo's paper), he defines Sharpe ratio using simple returns, not log returns. – James Hirschorn Aug 8 '18 at 18:40

Nowadays most quantitative researchers choose to use Information Ratio, developed and popularized by Grinold and Kahn (1999), as the gold standard for performance evaluation. Generally, though, it is called a Sharpe Ratio if returns are measured relative to the risk-free rate and an Information Ratio if returns are measured relative to some benchmark. Calculations may be done on daily, weekly, or monthly data, but results are always annualized (and typically by a factor of $\sqrt{252}$ for daily equities, $\sqrt{260}$ for daily FX, or $\sqrt{12}$ for any monthly series).

In long-short equities, it's common to use daily returns in $\frac{\mu}{\sigma}$ and then multiply by $\sqrt{252}$ to annualize.

• daily returns as percents or in log returns? – shabbychef Jul 22 '11 at 16:41
• @shabbychef Neither. Just use the dollar returns. – chrisaycock Jul 22 '11 at 18:21
• what exactly do you mean? returns with dollar units? – shabbychef Jul 22 '11 at 18:47
• @shabbychef Correct. Just use the daily P&L in dollars. – chrisaycock Jul 22 '11 at 20:20
• that really makes no sense to me. If the AUM of the fund changes (investments/disbursements), or there is a split in the stock, or a large change in nominal value, you cannot compare dollar returns from one time period to another. Did I misunderstand your comment? – shabbychef Sep 7 '11 at 4:20

I don't feel I can give you an authoritative answer on what the "standard" approach is, maybe someone with more hands-on experience will be able to help. But my quick thoughts.

As to the period, I've seen both daily and monthly returns being used. Weekly probably not that often. But in the end you annualize them either way to make them comparable.

The method I know is to multiply by $\sqrt{12}$ (for monthly data) - as can be seen in Kestner, 2003.

I would go with log returns, but it's rather gut instinct. I haven't really thought about it, so feel free to correct me/validate this statement.

There's one implication to arbitrarily changing your measurement interval - it can (should) alter the deviation. See Spurgin, 2002 for details.

And all this has to be done under the assumption that you can define your performance using only two first moments of the distribution. But the pitfalls of using Sharpe ratio - that's another issue to discuss.

• Yes, daily log (excess) returns are most often used in scholar articles to calculate Sharpe ratio, which then can be annualized. – Dmitrii I. May 2 '11 at 13:49
• Note that annualizing with square root of time implies that the asset returns are i.i.d. – Ram Ahluwalia Jul 20 '11 at 1:00

For fixed income hedge funds, monthly returns are almost always used to calculate the Sharpe ratio, because some securities held are relatively illiquid and the dealers who do the pricing for the hedge funds are only willing to do month-end pricing. Daily returns are not available to be calculated for most such funds.

• This is a good point. For my purposes, I have the daily (or even higher frequency) marks because I am looking at equity portfolios. – shabbychef Jul 22 '11 at 16:43