I didn’t anticipate a third posting on this topic, but I received an interesting note from a reader that I wanted to share and comment on:*I disagree with your thoughts on using “arithmetic relative” when considering material differences in portfolio returns. By reporting returns as a percentage, this is already stating a relative figure (to the value of the portfolio). Therefore, I think the “arithmetic absolute” is a better method of determining any material difference.*

*I like to think of it this way: why should this decision be taken on luck? If an error occurred in a month where performance was close to stale, why should this be more material than one where performance was fortunate enough to be high (or unfortunate enough to have a large negative position)?*

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*For example:*

*A fund of $100,000,000 has failed to account for a $500,000 cash flow. It has reported returns of 1% in August 2013 and 15% in September 2013. For simplicity let us assume cash flows occur only on the 1 ^{st}of the month.*

*If the error occurred in August, the actual return would be 0.49751% (modified Dietz).*

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*If the error occurred in September, the actual return would be 14.4335% (modified Dietz).*

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*Using “arithmetic relative,” August would show 50.249% error and September a 3.777% error. So you would probably want to state a material difference if the error occurred in August but not September. However, in monetary terms the error is the same; it’s just luck which month it happened to occur in.*

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*If using “arithmetic absolute” and having a limit of 50 bps a material difference would be stated whichever month the error occurred in. I think this is much more consistent as the error is of the same value.*

*As an investor, I would be more concerned with the monetary impact. Percentages are a nice way to compare between funds, but at the end of day profit or loss is where my concern would lie. Hence, I still believe “arithmetic absolute” is more appropriate in determining material differences.*

This reader raises a very interesting point: yes, both the mistake and the resulting magnitude in the error are identical, at least from an absolute perspective, so why wouldn’t we treat them the same?

My suggestion that arithmetic relative or, better yet (it appears) geometric, is better than arithmetic absolute to determine materiality for errors has to do with **utility theory**. That is, does one experience the same reaction when they see an identical difference in absolute terms (e.g., an error 1.00%) when the returns are low (e.g., 0.25% to 1.25%) versus when they’re high (e.g., 27.35% to 28.35%)? I suspect not. Going from 0.25 to 1.25 is a big jump (in relative terms), while from 27.35 to 28.35 the increase does seem to be as great. **The point,** I believe, rests on what your definition of “materiality” is. Not only do the GIPS(R) standards (rightly) fail to prescribe thresholds for materiality (leaving that properly in the hands of the compliant firm), but it also lacks a definition for the term. My belief is that in the context of errors, it’s a change that would cause the reader to have a different perspective in the information shown. Of course, we all react differently, so it’s impossible to know for sure what this would be in every case, so we base it on our own best judgment; perhaps the “prudent man rule” applies here.

As an analogy, if your child came home and they said they got an A on an exam, but later said they were mistaken, it was actually an A- or A+, would your response be significantly different? But, if they said it was actually a C? Should the policy be consistently applied based on the * magnitude* of the error (50 or 100 bps, for example, in absolute terms) or the

*to the error, using our best judgment?*

**likely response**When I teach our firm’s attribution class I occasionally address the issue of proportionality, and use weight lifting as an example. I used to regularly lift weights, so I have some familiarity with this topic. If, for example, you’re engaged in a particular exercise where you typically begin with 20 lbs, then go to 30, then to 40, you are increasing each time (in absolute terms) by 10 lbs. But, if you do a different exercise where you start with 120 lbs, and go to 130, and then 140, you are again bumping up by 10 lbs each time. But, do you think that you feel the same increase between the different weights? I strongly doubt it. Going from 20 to 40, for example, is a doubling of the weight, while going from 120 to 140 is only a small percentage increase. This analogy isn’t perfect, but hopefully it helps.

In reality, if you prefer arithmetic absolute, that’s fine with me; most firms seem to use this approach. Plus, it is probably easier to implement.

This exercise has allowed me to devote additional time to this rather interesting topic, to provide some examples, and to craft (what I believe is) the first attempt at a geometric approach (as noted a couple days ago, I’m sure Carl Bacon is proud, and perhaps a bit envious that he didn’t think of it first!). I also want to thank our reader for submitting his comments, as they’ve allowed me to ponder this a bit further and offer some additional perspectives.

Care to chime in? Please do!

*p.s.,*Sadly, I had to stop lifting weights some time ago because I was often accused of using steroids.

*p.p.s.,*A more detailed review of this topic will be presented in this month’s newsletters.