When Is One Asset a Substitute for Another?

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Suppose an investor holds an optimized portfolio and wishes to replace one of its constituents with an asset that plays the same role, whether because of a mandate, an access restriction or a preference. Which asset should be offered? The intuitive answer is the one whose inclusion leaves the covariance matrix least disturbed, as measured by a matrix norm. We show that this answer is wrong in three specific and correctable ways. The sharpest is 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, as the first-order change in the value of the optimization problem evaluated at unchanged optimal weights, and we show that it costs no more to compute than the naive one. We then address a limitation that 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 that 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 and show that it grows like √2 log M, so that restricting the candidate set helps only logarithmically, a fact that has been used to justify more than it can support. A simulation study confirms that the first-order criterion is correct to O(ε²), 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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Shehadi Candela, A. (2026). When Is One Asset a Substitute for Another? QUAFI Working Paper No. 2026-06. QUAFI Research.

@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}
}

Historial de versiones

1. v1.113 sep 2026exposition revised throughout; structure and register brought in line with journal practice; no change to results or references
2. v1.026 ago 2026first working-paper draft