Autocorrelation, Maximum Drawdown, and Calmar Ratio Interpretation

# How do you correct Max Draw-Down for auto-correlation?


# How do you correct Max Draw-Down for auto-correlation?












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.

## Answer by Paul H. Lasky (score 6)

https://quant.stackexchange.com/a/604

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.

## Answer by Ralph Winters (score 3)

https://quant.stackexchange.com/a/623

Have you considered a Monte Carlo simulation on your returns? Then you could look at the distribution of Maximum Drawdowns.