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enter image description hereMaybe this is because of my lack of mathematical maturity, but one of the most annoying aspects of mathematics in college or in applied fields (like finance and economics) is that subscripts and superscripts are under the whim of the author and thus are widely different in what they mean.

Is there a universal rulebook on subscript and superscript usage in mathematics that can allay such confusions for the reader?

As far as I am aware the most confusing situations are where indexes are on the superscripts -- is this done because putting many commas in the subscripts is messy, or is this a specific abuse of notation that should be avoided?

PS the picture comes from the Brinson Model(1985). I suppose the superscripts 'b' and 'p' are 'benchmark' and 'portfolio' but they are kind of taken to be granted instead of explained -- i.e. someone could easily just have written j,b and j,p instead, right?

economics
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  • Can you provide an example of an expression you find ambiguous? In the case you wrote, it seems clear that $b,p$ are exponents and $j$, the summation index, is just an index. – lulu Oct 19 '19 at 14:52
  • b and p are not exponents -- they are initials that mean "benchmark" and "portfolio", respectively. – economics Oct 19 '19 at 14:52
  • Note: did you mean $$\sum_j w^b_j,r^p_j$$? If so I will edit your post to embed that expression in MathJax. – lulu Oct 19 '19 at 14:53
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    Ah, in that case I agree the expression is ambiguous. I, personally, would have tried to avoid superscripts there...I think $$\sum_j w_{b,j},r_{p,j}$$ is less ambiguous. Regardless, context is critical and the relevant definitions should be located quite near any potentially ambiguous expression. – lulu Oct 19 '19 at 14:55
  • Note that the Brinson Model(1985) is one of the most influential models in asset management -- can you imagine that such a pivotal model uses less than efficient notation? Why is this situation rampant across finance and economics? – economics Oct 19 '19 at 14:56
  • These notations are ambiguous as they can be confused with exponentiation - and things may get messy when they are actually combined with exponentiation. However, using them as indexes is also not optimal: You want to access the $j$th component of the benchmark-$w$, not the benchmark-and-$j$th component of $w$. If other diacritics such as $w^$, $\tilde w$, $\hat w$, etc. are not feasible either, I would at least use some* typographic hints that these are not exponentiations - for example use upright font to hint that $b,p$ are not variables for numbers: $\sum_jw^{\text b}_jr^{\text p}_j$. – Hagen von Eitzen Oct 19 '19 at 14:58
  • Oh, I have no difficulty at all imagining that people have just stuck with some unfortunate notation because the first practitioners of a model used it. To the broader question, there's always going to be a problem if you need to have several indices in some expression. Nothing will "look good". – lulu Oct 19 '19 at 14:59
  • Personally, if I want to use superscripts that aren't exponents, I might surround them in parentheses., such as when talking about the $i$'th member of the $k$'th member of a sequence of families of sets I might have written $A^{(k)}_i$. That said, different people do different things. – JMoravitz Oct 19 '19 at 15:33
  • As far as I can see here the symbols $b,p$ are not used as summation indices. Indeed, here they've not been defined at all. Exactly what confuses you about the sum? – Allawonder Oct 19 '19 at 16:54

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It's allowed to use superscripts as indices, but usually it is done when it doesn't make sense to regard it as an exponent, such as with a vector. That seems to be the case here, as $w^b$ and $r^p$ are vectors. It still ends up being ambiguous when you take the components of the vector, but hopefully it will be stated beforehand that the superscript is an index and not an exponent.

Matt Samuel
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  • Could you elaborate what you mean here by "vector"? I know that one meaning has to do with the existence of "direction". In the formula the omegas (or ws) represent weights of each sector (financial asset). – economics Oct 19 '19 at 15:33
  • @economics Something with multiple indexed components can be seen as a vector. A vector space has an addition and scalar multiplication operation, and it's general enough that the attempt to characterize vectors by having magnitude and direction is just wrong. But it is commonly quoted in physics and engineering. – Matt Samuel Oct 21 '19 at 17:08