Hi,Hedge fund industry practice is to compute standard deviation (and thus Sharpe ratio + myriad of other statistics) using simple returns, not log returns. A colleague believes this practice is "seriously flawed" and "statistically biased". We don't disagree that log returns are in many contexts superior due to limited liability, additive property arguments, etc. however I take exception to his characterization, and ask the stats gurus out there for advice and perhaps a more formal verdict on the matter.Taking it a step further, suppose empirically we find that hedge fund returns are quite a different animal than stock returns, i.e. the assumption of geometric brownian motion itself is suspect. Given that returns may be neither normal nor iid, and can often be best represented not by (log)normal distributions but, say, something fat-tailed (Stable Paretian or Extreme Value or what have you), then again, what is the correct way to compute variance and standard deviation?

Well, one issue is consistency. In the Sharpe ratio, thereis a numerator and a denominator. Presumably you wantto use the same series for both.Now suppose this is the series: an extended run of however many years you want (an even number) with the risk-free rate = 0 and the hedge fund returns: (+100%, -50%, +100%, -50%, etc.)How would you compute the Sharpe ratio and interpret it?regards,

sharpe ratio makes maximum sense, when the time-series is iid and continuous (enforcing normality).so its better not to use sharpe in hedge fund analytics, because as you said, hedge fund returns aretotally different from gbm. do you want a replacement for sharpe ratio, or hedge fund analytics in general.there is a good doc on hedge fund analytics lying on web. ---------amit7ul

QuoteOriginally posted by: AlanNow suppose this is the series: an extended run of however many years you want (an even number) with the risk-free rate = 0 and the hedge fund returns: (+100%, -50%, +100%, -50%, etc.)How would you compute the Sharpe ratio and interpret it?QuoteAnd how would that change if you invest 10% of your assets in that fund and rebalance annually?

Hi, thanks for the pathological cases here, but what about the original question. Is computing standard deviation using simple returns = committing a statistical crime? Would a statistician approve using simple returns if applied consistently and conscientiously, or disapprove with a more rigorous proof demonstrating bias?If the underlying population is known to be fat-tailed, is the application of log returns less justified?Basically want authoritative proof that computing simple returns is/is not "seriously flawed" and "statistically biased." Thanks!

Given a series of numbers {x1, x2, ..., xN}, thestandard deviation is a well-defined concept. The onlyroom for ambiguity is using N or N-1 in the denominator.A statistician would not object whether the x(i) are simple returnsor log-returns. He would simply expect a proper description of whatwas used.The objections would come in on how the numbers were used.One can construct a whole list of possible misuses:1. The presumption that the series predicted anything about futurerisks. Since hedge funds thrive on non-transparency, this wouldbe the top error.2. If indeed the underlying distribution is far from normal or lognormal, then standard confidence intervals interpretations are wrong.3. If indeed the underlying distribution is heavily skewed, theninterpretations of sigma as a risk measure may be suspect.Most of the historical issues related to 2. and 3. can easilybe addressed by disclosure: show the full series, plot the distribution,measure the upside and downside deviations, measure the confidence quartiles,do chi-squared tests for normality, etc. In other words, justsimply fully describe the actual data. Do this with both simplereturns and log-returns. Are there any significant differences?If you do all this, then the statistician will be happy. The problems associated with 1. are the heart of the matter.For example, if the hedge fund is knowingly pursuing variousrisks that have simply not materialised (yet), then no amountof analysis of the historical returns will show this. Perhaps worseis the over-confidence that comes from -not- knowing the variousrisks that have not materialized (some classic blow-ups come tomind). But this is a completely separate issue from simple returns or log-returns. The latter is really a non-issue easily cured by disclosure. ---------------------------------------------------------------------------------------------p.s. Another disclosure item that I think any investor should insiston is the dollar-weighted return (internal rate of return). How the historical returns have been affected by the assets under management is a vital statistic. This is true even if the manager has little control over the timing of these cash flows. Sorry for the off-topic parts

I have a closely related query on the Sharpe ratio:It seems to be market pratice to take a monthly standard deviation and multiply it by sqrt12, calling it "annualized" standard deviation. Sometimes this number is recorded as a stand alone statistic and sometimes just used in the Sharpe ratio. My query is, conceptually, what does this statistic represent? It doesn't make any sense to me to multiplty a stdev by a constant. ThanksMaursh

Alan has it all down but let me try to put in a different way.There are two issues here, (a) simple returns and log returns and (b) non-normality and returns not being i.i.d .Difference between log returns and simple returns is the additional taylor series terms which become significant when returns are large.We all know that under normality and i.i.d returns assumption the sample variance is unbiased which is a first moment property.Also, variance of the variance (or consequently std dev) is very tractable under normality and i.i.d. Now, if we only drop the assumption of normality the unbiased result still holds but the variance of the variance involves higher moments. As Alan has pointed out standard confidence intervals are not valid. but still can be estimated. The largest blow i think is when the assumption of independent returns is removed.Tthen everything is out of the window and all the plots about asymptotic normality, your chi-squared tests everything is seriously flawed.this is only with std dev if we take one step further and find the statistical properties of sharpe ratio. again, its just more and more flawed if you use wrong assumtions. but with sharpe ratio or other statistics the error in bias and efficiency ofthe variance is more complicated to estimate given its intractable form. And then the statistician is right There are some other things to try, its not completely hopeless though I think K.