Some Things I've Learned Recently about the Standard Deviation of Returns [updates]

…or rather "Some things I've learned about calculating and projecting standard deviation of returns from a short frequency measurement (monthly) to a longer frequency horizon (year)." 

[ Note: since I clearly don't know this material nearly as well as I think I do, this post will continue to be a work in progress which I will update as I learn and assimilate more over time.]

If you asked me how many people out of, say, 400 million in the US were interested in standard deviation projections I'd have to guess the sub-population is relatively small, maybe 10s of thousands and that may be generous though it may be more.  If, of that sub-population, one were to ask how many are really, truly interested in whether using the square root of 12 (SQ12) is "good enough" to project a standard deviation using monthly return data to an annual horizon or if there is something better or more correct, well then, we are down, if I am not being overly dramatic, to maybe hundreds (maybe more). I know there is at least one because he is writing this sentence and probably more than one because I am reacting to an article or two on this subject. 

But the stakes seem small for me.  I am not competing against other hedge fund managers for clients or a track record[1], it's just me.  On the other hand I am competing against myself with respect to capital allocation choices, like: do I keep doing what I am doing with my systematic alt risk strategy (it's like a hedge fund, perhaps)? or do I stop? or do I transform the allocation to something similar but better and cheaper and run by someone else and to whom with what record?  To ask those questions means that I am, in fact, doing a little bit of performance evaluation on the margin where the kind of questions about variance implied in this post might really matter. In addition, since I have been trading for only around 12 years, coherently for five, and doing it with a consistent strategy for only something like 36-44 months, I don't have the luxury of using a long-running annual series to make my evaluations and my choices, I can only really credibly use monthly data. And since my choices do affect me, my family and my "future me" in big ways in terms of what will I bequeath to him (and them) after decades of managing returns, allocations, spending, etc. it probably does matter to some extent. It is at least worth paying attention to the question.

Based on some recent reading I've been doing it appears that, in order to calculate things like Sharpe ratios or to look at things like mean variance optimization, calculating annualized standard deviation the right or wrong way[2] might have a real impact on comparative decision making and thus long term results.  In fact, this is exactly the point that Will Kinlaw et al. made in two of the articles I was reading: "The Divergence of High- and Low-Frequency Estimation: Implications for Performance Measurement" and "…Causes and Consequences" Journal of Portfolio Management. 

But let's back up a little here first before I get to Kinlaw.  I've known for a while, only because someone pointed it out to me, that not everyone has liked the SQ12 rule in the past.  The article to which I am often directed when this point is made is the Morningstar paper "What’s Wrong with Multiplying by the Square Root of Twelve" by Paul D. Kaplan, Ph.D., CFA.  He describes there what he calls the "mathematically correct" way to do it while also pointing back to other people as the origin of the math (…Tobin 1965, Levy and Gunthorpe 1993, and Morningstar 2012).  Among other things, he explains to us that: a) if a random variable Y is the sum of 12 independently and identically and symmetrically distributed random variables, SQ12 is correct, but b) annual returns are not typically symmetric and are not a sum but a product so SQ12 is incorrect.  He then, by some derivations you can read on your own, provides the "correct" answer for arithmetic returns which looks like this: 

He then also helpfully points out that annualized standard deviation using this formula will now also depend on the expected value of monthly returns and that lower returns will over-estimate standard deviation and higher will under-estimate.  So far so good.  After noting the flaws of measuring return volatility over fixed time horizons for non normal distributions like regular arithmetic returns, he goes on to discuss alternative measures of return volatility such as using log returns (std dev of (ln(1+Rn)).  Since these supposedly independent and identically and normally distributed returns are now summable, the SQ12 rule works! (and he does, in fact, use an exclamation point at this juncture) or more formally and familiarly:

 So it's one or the other. Or, he says, maybe one can just assume the returns follow a lognormal distribution and go the SQ12 route anyway, if I read him right.  I might have to re read that section again. 

The article was certainly articulate and well argued and it was enough to educate and convince me about the proper use of the SQ12 rule or it's arithmetic Morningstar alternative.  But I guess this approach does not go far enough (queue Kinlaw now).  In fact, a quip in a Michaud Article about something other than standard deviation (geometric mean analysis) might have tipped me off (since Morningstar made the emphatic point about the compatibility of returns, lognormality and SQ12) if I had known more about the statistics of returns, which I still really don't. He says, in exploring an aspect of geometric returns unrelated to the particular standard deviation question at hand: "An important consequence of this result is to highlight the often-critical limitations of the lognormal assumption for applications of geometric mean analysis. While it is easy to show that empirical asset return distributions are not normal, if only because most return distributions in finance have limited liability, it is just as easy to show that empirical asset returns are not lognormal, if only because most assets have a non-zero probability of default. Unless empirical returns are exactly lognormal, important properties of the geometric mean are ignored with a lognormal assumption. In general, lognormal distribution approximations of the geometric mean are not recommendable." [emphasis added] 

And this is where I think the Morningstar paper is left behind and the Kinlaw and company paper comes in when it comes to the question of standard deviation if I have understood all of this correctly so far [3].   To quote Kinlaw: "Most financial analysts assume implicitly that standard deviations scale with the square root of time and that correlations are constant across the time intervals [all lags] over which they are measured. However these properties hold only if asset returns are independently distributed across time, which means that auto-correlations and lagged-cross correlations are zero. As we have just seen, this is hardly the case." (he uses log returns in the paper, btw).  And continuing "However, to the extent the lagged auto-correlations differ from zero, extrapolating the single period standard deviation [think month] to the longer horizon standard deviation [think year] may provide a poor estimate of the actual longer-horizon standard deviation."  (note that at this point we have this --- from Morningstar: use log returns, assume normality, and one can use SQ12; from Michaud: returns are not necessarily always lognormal and so [I have to infer] SQ12 might be suspect on log returns in some cases; and from Kinlaw: there are auto- and cross-correlation effects that can lead to underestimated volatility using SQ12 even when and if returns are lognormal). So any way you cut it probably behooves us to pay attention to how long-horizon standard deviation is calculated when using a high-frequency series to make horizon projections. 

With respect to only the auto correlation effects, for example, this is how Kinlaw describes the "correct" formula for standard deviation:  

where sigma(x…) is the standard deviation of x measured over single period intervals, q is 12 in our case, and everything to the right of q is a factor for a series of increasingly lagged auto-correlations.  One can see that according to the math, if the autocorrelation factor is zero then the SQ12 rule still stands.

He/they (Kinlaw) attribute the difference between short and far horizons estimation failures not to non-normality, though that could be part of it perhaps if we were to explore it a bit more ala Michaud -- normal is easily solved, btw, as we saw with Morningstar -- but to nonzero auto-correlations and lagged cross-correlations. Those correlation effects they attribute in turn to the broader economic causes related to changes in discount rates (for short horizon intervals) and changes in cash flows (for far horizons).  Of course the math is mostly impenetrable to me but the basic idea makes sense I guess.  In the end all it means, I gather, is: 1) we might if not should use Morningstar math for highly asymmetric arithmetic returns, 2) we might use SQ12 for log returns or if we don't care or, 3) if there are big auto- or cross-correlative effects, we would use the Kinlaw math…or maybe use longer intervals in the first place.   Fwiw, I tried to conjure here, in my own terms, a good visual example at this point in the post but I couldn't quite pull it off.  Mostly the math was fine but I had trouble when it came to getting a properly comparable longer horizon standard deviation that made sense for the example, something that I thought would be simple but maybe I was trying too hard.  Maybe another post another time.

Any conclusions from any of this?

There is still a lot I don't know about this stuff and how I would implement it in anything I currently do.  That means I'd describe this post as less of an answer and more of both an unfolding process/question and a "reporting" of what I have been reading lately.  If I was forced at gunpoint to summarize anything at all it might be the following.

• Trying to come up with a good example exposed a deep conceptual or methodological flaw in the way I think about this. For example, you try, given only 36 months of data (about what I have for my own strategy) to annualize, say, the standard deviation of SPY by any method of any kind. You may get something like 10 or 11 or 12 % but you know from experience that the long term annual std dev is really closer to 18 or 19 or 20%.  That just means that short-ish time series of any interval are probably too limited and they will create flawed expectations for the future no matter how you do the math. But I knew this already. I was just trying to maybe rationalize the fact that my strategy only has about 40 months of data and the fact that I was trying, without justification or rational thought, to make forward judgments about the strategy's volatility that will never stand the test of experience.
  
  
• There are "outcome" errors to be made in all directions. The more conservative Morningstar method, for example, might look conservative and therefore "good" or "better" but there are optimization opportunity costs in every choice and there will be one here too.  The real outcomes can only really be known ex-post when it might be too late.
  
  
• The previous couple of points notwithstanding, if one has billions of dollars to manage and/or one knows that there are big tracking errors coming from auto- or cross-correlation and/or one has made promises to people based on predictions of the future on which one must then deliver, then all of this might matter very much.
  
  
• If, on the other hand, one is like me, where the stakes are relatively low and the willingness to assume lognormality is, right or wrong, relatively high and the summed auto-correlation lags in one's own strategy are close to zero (which they are) then the SQ12 approach is more than likely good enough.  That makes my personal math in the future probably look something more like this (I made this up)

      Where "A" stands for "awareness that there are other ways to do it if it is important or necessary."

[updates 4/2/17]

some additional comments:

• For retirees this could theoretically be one more thing (minor, probably) to think about since higher volatility -- in this case maybe it's underestimated in a particular instrument or strategy or even asset class -- can have a long term impact via sequence risk or through return mis-estimation using vol-dependent geometric return estimates over long time frames. Although in practice I'm guessing that very few retirees are projecting vol with the square root of 12 or geometric returns for that matter. But their advisor or strategy manager might be…
  
  
• Like I mentioned above, I'm not going to lose any sleep over this kind of thing. It's just one more of the many weird finance nuances that come up when you dig into this stuff that are mostly meaningless unless you are managing billions.
  
  
• I tried again to come up with a self-rolled example.  This time, rather than naively try to imagine that a 36 month track record is predictive of anything, I looked at a longer time series like I probably should have in the first place.  I'm not sure if this is the right way to do it but it is probably representative of how a layman might, right or wrong, do it.  I took 22 years of monthly SPY data for the analysis with a little extra older data to do the lagged auto-correlation. For the auto-correlation I could have used R which will do it correctly or I could have done the math right from scratch but I used an Excel hack using correl() that is a close proxy.  This is what I came up with but if I were you I would not place too much confidence in this:
• 14.8%  - Standard deviation of monthly arithmetic x SQ12
• 16.3% - Standard deviation of monthly arithmetic with Morningstar math
• 14.9% - Standard deviation of monthly log ln(1+r) x SQ12
• 15.1% - Stdev of monthly log ln(1+r) x SQ12 with the autocorr adj
• 20.9% - Stdev of annual march-to-march arithmetic returns*
• 20.5% - Stdev of annual march-to-march log returns* 

The net? "If" I did the math right and have understood and framed it correctly, three big ifs, then projecting a monthly series looks like it underestimates risk and no adjustment looks like it does anything reasonable to fix the problem. I wish I had a better command of statistics to make sure I knew this stuff better.

* I have to go back an check again. I think I incorrectly included an extra year in the annual series. Plus I want to look at a rolling return version that will take a little extra effort to see if the timings of the annual measurement will affect the result which it will.

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[1] In the Kinlaw article, he starts with examples where re-evaluating hedge fund sharpe ratios after factoring in autocorrelation, the order changes and former winners are less so and former so-so strategies move up the chain with serious consequences for manager, investors, etc

[2] Frankly, since I am an amateur, I did not originally know the math of the x sqrt(12) or why it was supposed to be right for an annual horizon. It just seemed like another derivative statistical thingy -- which I guess it is -- that might or might not be the right answer for whatever task was in question. 

[2] actually I take it that the author's interest in this came from seeing that assets otherwise similar in return and variance with supposedly similar simple single period correlations had such divergent outcomes over long timeframes.  A worthy topic, that.