When Is One Asset a Substitute for Another?

Agustín Shehadi Candela · QUAFI Research · 2026-08-26 · v1.0

Abstract

Suppose an investor holds an optimized portfolio and wishes to replace one of its constituents — because of a mandate, an access restriction, or a preference — with an asset that plays the same role. Which asset should be offered? The intuitive answer is the one whose inclusion leaves the covariance matrix least disturbed, measured by a matrix norm. We show this answer is wrong in three specific and correctable ways, the sharpest being that a matrix norm is sign-blind: it penalises a candidate that strictly improves the portfolio exactly as much as one that strictly damages it. We derive the correct criterion from the envelope theorem — the first-order change in the value of the optimization problem, evaluated at unchanged optimal weights — and show it costs no more to compute than the naive one. We then address a limitation the covariance formulation cannot see at all: a covariance of one-period returns is a single-horizon object, and two assets can co-move at high frequency while their cumulative returns diverge permanently. We propose the h-period tracking variance and its associated spread variance ratio as the horizon-aware criterion, prove that its boundedness in h is equivalent to cointegration of the log prices with a unit cointegrating vector, and argue this continuous statistic is preferable to hypothesis testing when thousands of candidates are screened. We formalise the selection bias induced by minimising a noisy criterion over a large universe, showing it grows like the square root of 2 log M — so restricting the candidate set helps only logarithmically, a fact that has been used to justify more than it can support. A simulation study confirms the first-order criterion is correct to second order in the perturbation, that it fails badly at discrete substitutions near a binding constraint, and that it nonetheless ranks well enough for a two-stage procedure to recover the exact optimum.

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Cite

@techreport{shehadi2026substitution,
  author      = {Shehadi Candela, Agust\'in},
  title       = {When Is One Asset a Substitute for Another?},
  institution = {QUAFI Research},
  type        = {QUAFI Working Paper},
  number      = {2026-06},
  year        = {2026}
}

Preliminary working paper; circulated for discussion. The views are the author's own. Not investment advice.