## Thursday, January 19, 2012

### The many faces of standard deviation

Confusion abounds when it comes to standard deviation. Some of the issues include:
• Equal-weighted or asset-weighted?
• Divide by "n" or "n-1"?
• Is it a measure of variability, volatility, or dispersion?
• Is it a measure of risk?
• What's the best way to measure relative to the composite's average return?
I'll be brief, but promise to expound further upon this subject in this month's newsletter.

Equal or asset-weighted?

If you've been reading my stuff for any length of time, chances are you know the answer: EQUAL! Okay, so you're allowed to do asset-weighted, but why would you? What does the number mean or represent? This was an idea that some folks thought made sense almost 20 years ago ("since returns are asset-weighted, shouldn't standard deviation?"), but didn't and doesn't. But if you insist on doing asset-weighted, be my guest.

Divide by "n" or "n-1"?

By "n" we mean the number of accounts. I recall that the AIMR-PPS® flip flopped on this one (the first edition (1993) had one form, the second (1997) a different one [perhaps someone was planning to enter politics, and wanted practice]).

We're supposed to use "n" when we're measuring against the population, and "n-1" when against a sample. Dividing by "n" makes standard deviation a bit smaller. Most firms seem to use "n," so I say "why not join them?" We can debate which is appropriate, but why bother?

Is it a measure of variability, volatility or dispersion?

The short answer: yes!

Bill Sharpe, in his 1966 paper used the term "variability" to describe standard deviation (he referred to what we know as the "Sharpe Ratio" as the "reward to variability" (recall it has standard deviation in the denominator) and Jack Treynor's risk-adjusted measure as the "reward to volatility" (it has beta in the denominator)). However, in an email to me not long ago, he said using either the term "variability" or "volatility" is fine. Both of these are used in the context of standard deviation being a measure of risk; what some call "external dispersion."

As for "dispersion," I usually mean this in the same context as some do for "internal dispersion," meaning how the composite's returns compare / vary.

The GIPS® standards (Global Investment Performance Standards) now require both (a) a measure of dispersion (and standard deviation is just one way to accomplish this) and (b) the 36- month, annualized standard deviation for both the composite and benchmark. The former is for a single time period (standard deviation of annual portfolio returns for 2011, for example) and the other across time; a longitudinal measure, if you will (e.g., the 36-month standard deviation of the composite for the period ending 31 December 2011).

Is it a measure of risk?

It depends who you speak to. Since many consider risk to be either (a) the failure to meet the client's objective or (b) losing money, it wouldn't qualify, because it does neither. However, Spaulding Group research has shown that it's the most common measure of risk. And, the GIPS standards now require it (although they've shied away from calling it a "risk measure"). And so, regardless of its detractors, most folks do consider it a measure of risk.

What's the best way to measure relative to the composite's average return?

I saved the best for last. I am conducting a GIPS verification and was validating the client's measure of dispersion; in this case, equal-weighted standard deviation. Because I couldn't match what they had, I tried comparing it to the composite return; let me explain.

If you use Excel, for example, and run the "STDEVP" function against the returns of all account's present for the full year, you're measuring standard deviation against the average of these returns, which in almost all cases will not be the same as the composite's return, meaning it's telling us how disparate the returns are around this average, not the average reported in the presentation. I believe that ideally it should be run against the composite's return. However, this would require several more steps, and couldn't be invoked by simply running a similar function like STDEVP. Too bad.

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And so, standard deviation isn't really so simple, is it?