**Linking monthly standard deviation has its root within the AIMR-PPS**

One advantage to being a bit older than most folks in performance measurement is that I have been exposed to a lot of ideas, and quite a bit of history. This includes the predecessor standard to GIPS®) the AIMR Performance Presentation Standards or AIMR-PPS®.

Some very quick background. AIMR was the Association for Investment Management & Research. It was created in 1990, when the Financial Analysts Federation merged with the Institute of Chartered Financial Analysts. Several years ago AIMR became the CFA Institute.

The AIMR-PPS went into effect in January 1993. The prior year, as things were being finalized, the AIMR-PPS Implementation Committee issued a paper that explained how to *link *monthly composite standard deviations to arrive at an annual value. They then issued an *errata* that corrected the earlier formula.

**But, the AIMR-PPS standard deviation linking formula disappeared!**

The AIMR-PPS Implementation Committee did not include this formula with the Standards. I called Sue Martin, CFA , (or it may have been Cindy Kent; please excuse my “senior moment”) to find out why. Sue oversaw the AIMR-PPS at the time, and Cindy worked with Sue. Sue (or Cindy) told me they learned that the idea of linking standard deviations is flawed. She also sent me a paper that explained why. I probably have the original formulas as well as the paper somewhere in my files; suffice it to say: we were told to forget about the idea of linking monthlies to obtain annuals. And so, I put the idea behind me.

**Fast forward: a recent GIPS verification**

I recently conducted a GIPS verification for a new client. It was their first, since they’ve only recently become compliant.

In the course of my review, I asked to see how they arrived at the composite’s dispersion. They provided me with a spreadsheet, and pretty quickly I determined that they were trying to bring monthly standard deviation values together to arrive at an annual. They apparently wanted to somehow average the monthlies. They had used a consultant to do this work for them, and this consultant had familiarity with their portfolio accounting system. The consultant thought that this idea seemed logical.

Well, my initial reaction was “you can’t do this.” But then I thought that perhaps he had discovered a method that could work, so why not be open to it? But after a quick test I found that their result wasn’t close to the actual. I then pondered the idea a bit.

**But why doesn’t it work? Why can’t we link monthly standard deviations?**

I recall that the paper that I was given from AIMR provided a fairly technical reason as to why it’s not possible to link monthly standard deviations. But, it doesn’t have to be difficult to understand why.

**Consider the following composite:**

We see that each account experiences the same result as every other account. They range from 0.01% to 0.12%, or 1 basis point to 12 basis points. We also see that no two accounts have the same return for any given month, and no account has the same return in more than one month.

Pretty simple, right?

And so, for each month we get the same standard deviation (0.03%). Since each account experiences the same 12 monthly returns (just not in the same order), they all get the same annual return (0.78%), and dispersion is 0.00 percent.

**Now, let’s look at a second composite:**

In this scenario, each account has the same return for each month, ranging from 0.01% to 0.12 percent. Again, no two accounts have the same return for any month. As a result, they all have different annual returns, ranging from 0.12% to 1.45 percent.

Because the individual accounts have the same experience each month, the standard deviations remain the same for the full year (0.03%). But because the annual returns differ, the annual standard deviation is much higher than we saw in the first example.

**Now, let’s consider both composites**

Okay, so what’s actually going on here?

For each composite, every month has the exact same standard deviation. This means that whatever method you use to link, combine, or average them, you should end up with the same “annual” value. However, we can also see that in reality the annual standard deviations differ. This means that annual standard deviation has no relationship to monthly, further meaning that to try to link is a waste of time.

**Who else is doing this (“linking, averaging, combining” monthly standard deviation)?**

This is a topic that I have not dealt with in almost 25 years. So imagine my surprise when I saw that someone had implemented it. If one firm has, perhaps others have, too?

For the most part, I think this falls into the “academic” category. That is, it’s kind of interesting, but nothing we really need to be concerned with. However, if there are firms that are doing this, regardless how, guess what: it doesn’t work!

Or, am I missing something? If you think I am, please chime in!

Thank you David for the insight. This a common error that many does while handling composite.

Absolutely, volatility cant be linked like returns within single portfolio. In the case of multiple portfolio composite , contribution of volatility from each single portfolio needs to be added to get the total volatility. The weight times volatility of each single portfolio will be different . Adding them gets the final SD for composite. One definitely cant chain link like cumulative return of single portfolio.The magnitude/strength of volatility for each single portfolio making a composite is different. Some have greater impact than the other depending upon weights. Chain linking return works for single portfolio because the returns are compounded and are cumulative of previous returns. Definitely one cant do that with volatility. In the above chart if contribution to volatility are added up from monthly SD contribution or separately calculating SD on annual return of each portfolio, one gets the same results.

Thanks, Neeraj! I appreciate your comments.

So there is no way to annualize standard deviation ? I have seen formulas where they multiply the monthly standard deviation by the square root of the n periods. Does this work ?

Rene, you’re asking a different question. What this post dealt with is the idea of linking monthly values to arrive at an annual, which you cannot do. Yes, you CAN annualize standard deviation. The common approach is to multiply by the square root of time: e.g., in the case of months, by the square root of 12. While not everyone agrees with this, it’s still the way most do it. Sorry for any confusion.

David,

Well I guess GIPS isn’t helpful for dispersion since it doesn’t specify which method to use – although GIPS does specify annual periods – so to me at least that implies calculating the dispersion of annual returns (why would anyone make this more difficult than they need to). There is also a Q and A I think clarifying only accounts with full year returns are included.

On a related issue even annualising standard deviation over time is not valid. It requires the assumption that each return is independent – which we know not to be true – however everybody does it at least consistently.

Best regards

Carl

Carl, thanks for your comment. GIPS statement that accounts must be present for the full period doesn’t necessarily conflict with the idea of linking monthlies, something that perhaps you don’t recall seeing (don’t know how close you were to the AIMR-PPS in ’91). I also agree that the idea of annualizing is one that some disagree with, but is still common practice (and, in fact, REQUIRED by GIPS!). Looking forward to seeing you later this month, both in London and Boston!

Hi David,

Thanks for this post, which brings to mind something I have seen quite often, that is, many professionals are used to applying concepts such as geometric linking based on a “how-to” approach only and don’t understand the “why” portion of it. Although the issue of linking Standard Deviation is not common and nothing to be concerned about, there are other concepts that practitioners are not applying correctly due to a lack of understanding as to the “why”. Things become a lot simpler and errors are voided when the underlying fundamentals of a concept are well understood and properly applied.

Ioannis

Ioannis, great point. In our Fundamentals and ROR classes, we spell out clearly the “why” behind geometric linking, with an example which, we believe, “drives home” the rational behind it: makes sense out of “compounding.” Linking other things (e.g., attribution effects, risks) is a bit more challenging, but we touch on this, too. Understanding the concepts is critically important. Sadly, too many don’t fully grasp why we use time-weighting, so there is room for a lot of education, we believe.