- LineOfBestFit
**Posts:**60**Joined:**

Given a series of returns with less than annual frequency, is there a well-known way to annualize the median of this series in the same way we may annualize the mean?

Just thinking out loud, I suppose you could first annualize the returns in any way that makes sense for your data, and then take the median of those.

- Traden4Alpha
**Posts:**23951**Joined:**

Tricky!Whereas means are additive so that mean(year1+year2) = mean(year1) + mean(year2), medians are not. The medians of the individual years can be arbitrarily different from the median of the multi-year aggregate.Consider this example of three instruments measured annually for two years (Year1,Year2), biannually (BothY), and then annualized (AnnY):Year1: 2%, 4%, 18%; Mean = 8%; Median = 4%Year2: 36%, 8%, 2%; Mean = 16%; Median = 8%BothY: 38%, 12%, 20%; Mean = 24%; Median = 20%AnnY: 19%, 6%, 10%; Mean = 12%, Median = ??? The annualized median is not 10% although Alan's suggestion does work in this case. Whether Alan's suggestion works with higher numbers of instruments and years is less clear, but it seems like a usable approximation.The deeper issue here is that medians, which are robust to extreme outliers within a time series, are NOT robust to anti-correlated patterns of outliers that aggregate over time in a more sparsely-sampled time series. You might want to think about why you are using the median function and how fat-tail outliers in individual years and individual instruments will affect the median. If returns are anti-correlated across the years, the median of infrequently sampled series will be markedly different from the medians of an annually-sampled series.

Last edited by Traden4Alpha on March 29th, 2015, 10:00 pm, edited 1 time in total.

- LineOfBestFit
**Posts:**60**Joined:**

Thanks for the responses so far. Yes, T4A, the additive issue is proving difficult to conceptualize. I have an application that accepts sub-annual series and annualizes them; I wanted an apples-to-apples comparison for the median but it doesn't seem to be a common thing. I suppose this is also why there is a dearth of material covering annualization of skewness and kurtosis, although of course it is very straightforward to annualize standard deviation.

We scale VaR, which is a percentile just like the median, with [$]\sqrt t[$] rule. So, intuitively it seems that the median should scale too.

Last edited by Buran on March 29th, 2015, 10:00 pm, edited 1 time in total.

- LineOfBestFit
**Posts:**60**Joined:**

Thanks, Buran. Intuitively, I'd like to think that is the case too...but the reason you can scale VaR with the square root of time is because it is an extension of standard deviation, and standard deviation scales no problem due to assumptions of random walk pattern of returns and normal distribution. I'm looking to annualize the median as the expectation is that it will be consistently different than the mean; hence, Gaussian assumptions of scaling, I presume, will not work.

When you're scaling (annualizing) the mean, you're making some distributional assumptions. For instance, you're assuming that the asset prices generally grow exponentially, so that if you hold on to a security for a year, you're going to get 12 times more return. If you don't assume a drift, you shouldn't be annualizing. So, the same drift should get into the median, and the same annualizing should be applied, intuitively.All that annualizing does is quotes your monthly return on "per year" basis. It would represent annual drift only if you make a strong assumption of a constant drift. I don't think this has much to do with the distribution being Gaussian.

- LineOfBestFit
**Posts:**60**Joined:**

Outrun, thanks for the link and steps. I've gotten roped into some nonsense at work but I want to make sure I understand the underpinnings of this method before implementing. Looks interesting!

Yes, most people don't annualize beyond mean and the std for good reason. For higher moments since you mentioned skewness and kurtosis, see this one.

- LineOfBestFit
**Posts:**60**Joined:**

It was this exact paper I had in mind when I mentioned "dearth of material"...it seems all roads lead to Attilio Meucci on this subject.

There is also the method proposed by Wingender in The Analytics of the Intervaling Effect on Skewness and Kurtosis of Stock Returns outlined here Lessons from the Credit crisis the importance of Liquidity with spreadsheet implementation here.

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