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When returns are auto-correlated, calculating a Sharpe ratio := $\frac {mean(x)}{\sqrt{var(x)}}$, (where $x$ are the returns) is complicated, but basically solved (see, e.g. Lo (2005)). Without the correction, the Sharpe ratio is too large, b/c auto-correlation reduces the variance of the returns.

However calculating the Calmar ratio := $\frac{mean(x)}{Maxdrawdown(x)}$ with auto-correlated returns gives too small an answer b/c $Mdd(x)$ is too large. How do you correct Mdd(x) for auto-correlation? The answer must be quite novel, because unlike the Sharpe ratio case (where $var(x)$ is a linear statistic), $Mdd(x)$ is not a linear statistic, so the delta method cannot be employed.

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    $\begingroup$ Could you provide the full reference for Lo (2005)? $\endgroup$ – Joshua Ulrich Feb 10 '11 at 18:07
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    $\begingroup$ Wouldn't that imply that an asset or strategy with negatively auto-correlated returns has a sharpe ratio that is too low? Is there a logical reason why returns should be "adjusted" for auto-correlation when using sharpe ratio? And if so, why not adjust for negative auto-correlation? $\endgroup$ – Joshua Chance Feb 11 '11 at 9:41
  • $\begingroup$ @Joshua Ulrich - I think he's referring to Autocorrelation, Bias, and Fat Tails - Are Hedge Funds Really Attractive Investments? $\endgroup$ – Joshua Chance Feb 12 '11 at 4:21
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    $\begingroup$ The reference paper is actually "The Statistics of the Sharpe Ratio" by A.Lo. Financial Analysts Journal, Vol. 58, No. 4, July/August 2002. And, as it is the statement in the question is incorrect: when returns are positively autocorrelated, the SR is too large. When they are negatively autocorrelated however, it is too small. Also, it is Calmar ratio, not Calamar ratio. And finally, it is not heavily used in practice because, being based on an extremal statistics, it has very high variance. $\endgroup$ – gappy Feb 13 '11 at 23:49
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Thanks gappy for your precise response. However the answer to this auto-correlation is much more important than an academic discussion of which portfolio performance ratio is best. Auto-correlation distorts max draw-down calculations raising the question of whether the (positive) auto-correlation will continue in the future producing large draw-downs, or whether it will subside to normally low levels. [ Incidentally I have never seen a negative auto-correlation in real-world monthly publicly traded asset returns.]

For example take two MLP's: the well known and large cap KMP ( a pipe-line operator) and NRGY ( a mid-cap retail propane distributor.) On data (post Lehman) from 2/2009 to 2/2011 KMP's monthly returns are not auto-correlated, while NRGY's are highly correlated. The two Calmar ratios are: KMP= 0.1304 (StndErr=0.009); NRGY = 0.1472 (StndErr=0.25), i.e. risk-adjusted returns for the 2 assets are statistically equivalent. But if NRGY's auto-correlation is expected to subside then it's past mdd is overstated and it will be a better risk-adjusted investment than KMP in the future.

I've done some research and have been able to calculate the theoretical maxdd's for 2 models: No auto-correlation ( the much more difficult calculation) and complete auto-correlation($\rho=1$) for a no drift, normal dist. vol model, Irrespective of the size of the returns and volatility, $\rho=1$ $mdd / \rho=0$ mdd is 4.35 - - - a large difference!

In other words if period (e.g. monthly ) returns are auto-correlated you can expect a future maxdd of 4.35 times that for a normal- no auto-correlation return within the same horizon.

Auto-correlation of returns can appear in low-volume traded assets, Hedge funds, Preferred stocks, etc. In common stocks it occurs in high-momentum assets. In all cases of auto-correlation, BEWARE, the maxdd's will be large. There is an easy test to determine if the returns are auto-correlated: the Ljung-Box test (Please Gappy correct my misspelling of the names if incorrect.) I have a simple R script to calculate the LB if anyone is interested.

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  • $\begingroup$ I agree that the question is not academic, but I think you make a bold assumption when you say the maxdrawdown would be too high in autocorrelated returns, because autocorrelation would subside in the future. I think it would be important to analyse the evolution of autocorrelation in every case to see if there is a reversion to a low mean in autocorrelation or not. $\endgroup$ – Owe Jessen Feb 25 '11 at 15:28
  • $\begingroup$ I think I was misunderstood: all I am saying that with assets that exhibit autocorrelated returns, my limited experience has been that the autocorrelation condition can and does cease at some time, basically because I believe it is a temporary condition generally caused by heavy buying or demand. In these cases historical high maxdd's will not continue in the future. $\endgroup$ – Paul H. Lasky Feb 26 '11 at 20:12
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Have you considered a Monte Carlo simulation on your returns? Then you could look at the distribution of Maximum Drawdowns.

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