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# Does the order matter?

by | May 7, 2013

In mathematics we are occasionally confronted with the questionable applicability of the commutative law. You’ll recall that it basically states that you can move (as in commuting or moving about) values around in a mathematical expression without causing the result to change; e.g., A x B = B x A (just as A + B = B + A). This law doesn’t always work, right? For example, with limited exceptions A – B ≠ B – A nor will A/B typically equal B/A.

Since the commutative law applies to multiplication, there’s often an assumption that it always applies; but it doesn’t. For example, some folks discover problems when using percentages; or, more correctly, when NOT using them when they should! We often see firms remove the percentage sign (%) from their reports (e.g., show 3.03 rather than 3.03%), which is fine, but knowing how the number is stored is important when attempting calculations. If you multiply percentages together AS percentages you’ll get different results than if you multiply the values as if they weren’t percentages (e.g., 3.03% x 2.14% ≠ 3.02 x 2.14). Whether multiplying or dividing, we can get problems (usually addition and subtraction are okay). For example, the Sharpe ratio, which involves division, will be a problem if you don’t treat the values as percentages.

We occasionally discover other issues. I recently conducted a GIPS(R) (Global Investment Performance Standards) verification for a client who often uses blended benchmarks (that is, their benchmarks are made up of two or more indices). They chose to link the monthly returns of the individual indexes and then take the ratios, as defined for the benchmark allocation. However, this is incorrect. That is, the commutative law does not apply.

We can look at the math from two perspectives:

1) Link, THEN take the ratios
2) Take the ratios, THEN link.

We will get differences, and they can be material. You can try this yourself or wait until this month’s newsletter, when I’ll have more to say on this matter. The issue is partly attributable to the challenges we often face with compounding, which deserves a fair amount of treatment itself, which I hope to provide in the coming months.

# The Journal of Performance Measurement

The Performance Measurement Resource.