## No Parity like a Risk Parity.

## Portfolio Selection and Exchangeability

Consider the problem of *portfolio selection*, where you observe
some historical data on \(p\) assets, say \(n\) days worth in an \(n\times p\)
matrix, \(X\), and then are required to construct a (dollarwise)
portfolio \(w\).
You can view this task as a function \(w\left(X\right)\).
There are a few different kinds of \(w\) function: Markowitz,
equal dollar, Minimum Variance, Equal Risk Contribution ('Risk Parity'),
and so on.

How are we to choose among these competing approaches? Their supporters can point to theoretical underpinnings, but these often seem a bit shaky even from a distance. Usually evidence is provided in the form of backtests on the historical returns of some universe of assets. It can be hard to generalize from a single history, and these backtests rarely offer theoretical justification for the differential performance in methods.

One way to consider these different methods of portfolio
construction is via the lens of *exchangeability*.
Roughly speaking, how does the function \(w\left(X\right)\) react
under certain systematic changes in \(X\) that "shouldn't" matter.
For example, suppose that the ticker changed on
one stock in your universe. Suppose you order the columns of
\(X\) alphabetically, so now you must reorder your \(X\).
Assuming no new data has been observed, shouldn't
\(w\left(X\right)\) simply reorder its output in the same way?

Put another way, suppose a method \(w\) systematically
overweights the first element of the universe
(This seems more like a bug than a feature),
and you observe backtests over the 2000's on
U.S. equities where `AAPL`

happened to be the
first stock in the universe. Your \(w\) might
seem to outperform other methods for no good reason.

Equivariance to order is a kind of exchangeability condition. The 'right' kind of \(w\) is 'order …

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