In several posts I have struggled to articulate the mathematically convenient concept of a historical representation of an index or portfolio as if its past weights had always been equal to its current weights. I have called the returns to such a time series various terms, including “matrix returns,” “imaginary returns,” and “pro forma returns.”
Recently I’ve enjoyed some of the writing of UK finance professorCarol Alexander. For years in her research she’s been using such reconstructed indexes, and that’s what she calls them: “reconstructed.” I like that better than my coinage, so from now on I’ll go with hers.
To review, a reconstructed portfolio return series is the time series of returns that would have occurred had the portfolio always been rebalanced to its present weights. The reason this is mathematically convenient is that regression statistics you estimate versus this time series are identical to what you would estimate if you bothered to grind through all n2 elements of a proper covariance matrix. True historical portfolio returns, since they were generated by a portfolio with different weights from today’s, will produce a distorted set of regression statistics.