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A mathematical study of the excess growth rate

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Pith's one-line read Three axiomatic characterizations force the excess growth rate — the Jensen gap of the logarithm — to be the unique measure of diversification return up to a multiplicative constant.

desk verdict Three genuinely new characterization theorems, the first two self-contained and sound; the third leans on the authors' prior results and needs a patch, but the paper deserves refereeing. read the letter →

arxiv 2510.25740 v2 pith:7HUUZGBB submitted 2025-10-29 cs.IT math.ITmath.PRq-fin.MFq-fin.PM

Steven Campbell , Ting-Kam Leonard Wong This is my paper · ORCID
classification cs.ITmath.ITmath.PRq-fin.MFq-fin.PM MSC 94A1791G1060F10
keywords excessgrowthrateaxiomaticcharacterizationrelativeentropyJensengaplogarithmicdivergenceportfoliotheorylargedeviationsRényi
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper aims to show that the excess growth rate — the Jensen gap log(ΣπᵢRᵢ) − Σπᵢ log Rᵢ used in portfolio theory as a measure of diversification return — is not one ad hoc functional among many. It gives three axiomatic characterizations: any measurable, permutation-invariant, support-dependent, constant-vanishing functional satisfying a general chain rule must be a constant multiple of Γ; any measurable generator satisfying the gap axioms plus numeraire invariance must be c·log; and any regular logarithmic divergence invariant under perturbations must come from negative cross-entropy and hence equal Γ. The paper also proves that the deterministic maximizer of Γ puts positive weight only on the largest and smallest log-returns, with an explicit closed form, and derives a first-order condition for the expected version. These results pull together portfolio theory, information theory, geometry, and large deviations around a single logarithmically defined object.

What carries the argument

The central object is the family Γn(π,R)=log Σ_{i∈supp π} πᵢRᵢ − Σ πᵢ log Rᵢ, a Jensen gap for the logarithm and the difference between exponential and arithmetic means. Three mechanisms carry the argument: (i) the algebraic identity Γ(π,r)=H(π∥π⊕π r) that rewrites Γ as a relative entropy under the simplex's multiplicative perturbation structure; (ii) the general chain rule Γ(π∘p,a∘R)=Γ(π,a⟨⟨p,R⟩⟩)+ΣπᵢΓ(pᵢ,Rᵢ), which lets the proof reduce to a known characterization of relative entropy; (iii) for the divergence characterization, the portfolio map π(p)=pᵢ(1+∂_{eᵢ−p}φ(p)) and the information-geometric identity Γᵏᵢⱼ(θ)=δᵢⱼₖ−δᵢₖπⱼ−δⱼₖπᵢ, which turns perturbation invariance into constancy of the

What would settle it

Produce a regular exponentially concave φ (C⁴ with Φ=e^φ strictly concave in every tangent direction) whose logarithmic divergence satisfies Lφ(q⊕h∥p⊕h)=Lφ(q∥p) for all p,q,h in the open simplex but whose portfolio map π(p) is nonconstant; the paper's Theorem 3.20 predicts no such φ exists. A direct calculation of the portfolio map and Christoffel symbols for any candidate φ would settle the claim.

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Extended reading notes

Core claim

On its own terms, the paper establishes that the family Γn is the unique (up to a common multiplicative constant) solution to three separate sets of natural axioms. First, from Lebesgue measurability, permutation invariance, dependence only on the support, vanishing on constant returns, and a general chain rule for composite portfolios, the only functionals are cΓ. Second, among gap functions φ(⟨π,R⟩)−Σπᵢφ(Rᵢ) that vanish on constant returns and are affine on constant-mean slices, numeraire invariance forces φ = c log plus an affine term, hence the gap is cΓ. Third, within the class of regular exponentially concave functions on the open simplex, the logarithmic divergence is perturbation inv

Load-bearing premise

The load-bearing premise is that the C⁴, strictly-concave-Φ regularity assumption (3.21) is enough to make two cited prior results hold — the information-geometry identity for the Christoffel symbols and the fact that a constant portfolio map forces φ = −H×(π∥·)+c; if those cited results need extra boundary hypotheses beyond (3.21), the 'if and only if' in Theorem 3.20 collapses to the easy direction.

Editorial extensions

If this is right

  • Any quantity that satisfies the five properties in Assumption 3.1 must be proportional to Γ; so the rebalancing premium of a constant-rebalanced portfolio is, up to scale, the only possible such measure.
  • The deterministic maximizer of Γ is supported on the two assets with the largest and smallest log returns, and the max value has the closed form in Theorem 4.3; this gives an explicit 'volatility harvesting' portfolio.
  • For the expected version, a portfolio maximizes E[γ(π,r)] iff it satisfies (4.14); in the special case where all expected log returns are equal, the EGR-maximizing portfolio coincides with the growth optimal portfolio.
  • Γ emerges as the rate function for a large-deviation principle for scaled Dirichlet distributions and equals a Rényi divergence between members of that family, so information-theoretic tools such as Sanov-type bounds and Rényi divergences can be applied to portfolio volatility questions.
  • The gap axioms plus numeraire invariance imply that among all gap generators, only the logarithm yields a gap that is invariant under rescaling returns; hence the functional form is forced by economics rather than by convention.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: Because all three uniqueness theorems only pin Γ up to the same multiplicative constant, the constant carries no information; in practice it can be absorbed into the length of the rebalancing period. This suggests that any empirical calibration of an 'excess growth' parameter is testing the time scale, not the functional form.
  • Editorial inference: The two-point support result points toward a testable portfolio rule: when one asset has the best expected log return but also the highest volatility, the EGR-maximizing portfolio deliberately holds the worst asset as a risk offset. One could backtest whether such max-min portfolios empirically harvest rebalancing premia better than equal-weighting.
  • Editorial inference: The perturbation-invariance characterization could be turned into a model diagnostic: given a candidate divergence on returns, check numerically whether it depends on initial prices or only on returns; any non-logarithmic divergence that passes the test would refute the paper's uniqueness, while a log divergence that fails would suggest the regularity conditions are needed.
  • Editorial inference: The large-deviation connection suggests using the excess growth rate as a divergence for compositional data beyond finance; a straightforward extension is to test the scaled Dirichlet rate function empirically on simplex-valued data such as market shares or species abundances.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper provides a mathematical study of the excess growth rate Γ(π,R) = log⟨π,R⟩ − Σπ_i log R_i, a quantity used in stochastic portfolio theory. It first collects properties of Γ, including permutation invariance, support dependence, numéraire invariance, a general chain rule, and connections to the Helmholtz free energy, Campbell's coding measure, a large-deviation principle for scaled Dirichlet distributions, and Rényi divergence. The main results are three axiomatic characterizations: Theorem 3.2 shows that Γ is, up to a multiplicative constant, the unique family satisfying measurability, permutation invariance, support dependence, vanishing on constants, and the general chain rule; Theorem 3.13 characterizes Γ among Jensen-gap functions as the unique one satisfying homogeneity (numéraire invariance) plus a constant-mean affinity condition; Theorem 3.20 characterizes Γ as the unique perturbation-invariant logarithmic divergence. The paper also solves the deterministic maximization of γ(π,r) explicitly (Theorem 4.3), gives a variational/perspective interpretation, and derives a first-order condition for maximizing the expected excess growth rate (Theorem 4.11).

Significance. If the results are correct, this is a substantial contribution: it places a finance-motivated functional on the same axiomatic footing as entropy and relative entropy, proves three complementary characterizations, and provides a pleasing link between portfolio theory and information geometry. Theorems 3.2 and 3.13 are proven in the paper with only standard functional-equation ingredients; the explicit two-support solution of the deterministic maximization problem is useful and clearly stated. Theorem 3.20 is the most ambitious, as it identifies the excess growth rate within the family of logarithmic divergences; however, as discussed below, its converse depends on external results whose hypotheses are not checked against the paper's own regularity condition. The connections to Campbell's measure, Dirichlet large deviations, and Rényi divergence are interesting and broaden the significance beyond finance. Overall the paper is well within the scope of cs.IT and would be a solid contribution once the proof gap in Theorem 3.20 is addressed.

major comments (2)
  1. [Section 3.3, Theorem 3.20, Eq. (3.24)-(3.25)] The converse of Theorem 3.20 is the load-bearing step of the third characterization, but its proof invokes [49, Proposition 6] and [50, Theorems 4.5 and 4.7] to assert that the portfolio map π(p) in (3.24) lies in the open simplex, that the metric (g_ij) is strictly positive definite, and that the Christoffel identity (3.25) holds. The regularity condition stated in the paper is (3.21): C^4 and strict concavity of Φ=e^φ in every tangent direction. It is not demonstrated that these cited theorems apply under exactly this hypothesis; if 'regular exponentially concave' in [50] includes additional boundary or nondegeneracy conditions, then the proof does not establish the converse for every function satisfying (3.21), and the if-and-only-if may reduce to the easy direction. Since this is the central claim of Section 3.3, the authors should either prove the needed identities from (3.21) alone
  2. [Section 3.2, Assumption 3.12 and proof of Theorem 3.13] There is a domain inconsistency in the statement and proof of the Jensen-gap characterization. The theorem states g:A_n→R with A_n = Δ_n × Δ_n, but Assumption (D3) quantifies over (π,R)∈D_n, and the proofs of Lemma 3.11 and Theorem 3.13 use R=(u,v,1,...,1) with arbitrary u,v>0, which need not lie in the simplex. For example, in Step 1 of the proof of Theorem 3.13(ii), π=(1−t,t,0,...,0) and R=(u,v,1,...,1) are used, but (π,R)∉A_n unless u=v=...=1. The argument implicitly extends g to D_n via scaling. This extension should be stated explicitly (or g should be defined on D_n throughout), otherwise the derivation of φ=clog is not formally justified as written. This is fixable but needs to be corrected for the proof to be complete.
minor comments (5)
  1. [Section 3.3, Eq. (3.24)] In the definition of the portfolio map, 'π_i(p) := x_i (1 + ...)' should read 'p_i' instead of 'x_i'; as written, the symbol x_i is undefined. This appears in the load-bearing part of Theorem 3.20 and should be corrected.
  2. [Introduction, Section 3.1/3.3] The bullet list in the introduction says 'Our first characterization (Theorem 3.20), proved in Section 3.1' but Theorem 3.20 is in Section 3.3 and is the third characterization (via logarithmic divergence). The first characterization is Theorem 3.2 in Section 3.1. The numbering/labeling should be fixed.
  3. [Section 3.3, proof of Theorem 3.20] Just before the exponential-coordinate change, the text writes 'θ=(θ_1,...,θ_n)∈R^{n−1}' and defines θ_i for i=1,...,n−1. The vector should be (θ_1,...,θ_{n−1}), not (θ_1,...,θ_n). This is a typo but it is confusing in a key proof.
  4. [Theorem 2.15(ii)] The proof of the large deviation principle is omitted with 'we omit the details.' While this result is not used later, the uniform convergence in (i) alone does not automatically yield the LDP for all open and closed sets unless goodness of the rate function and exponential tightness are also verified. A brief justification or a precise reference for the implication would make the theorem self-contained.
  5. [Eq. (1.8), Definition 1.1 area] In several places, the notation uses supp(p) where p is not defined (e.g., Eq. (1.8) and the surrounding discussion); these should be supp(π). Also, in the proof of Lemma 2.13, Γπ(y∥x) and Γπ(y|x) are used interchangeably; unify the notation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity found: the characterization theorems are derived from explicit axioms, and the self-citations used in Theorem 3.20 are to prior published theorems with stated assumptions that do not include the target result.

full rationale

The paper's central claims are three axiomatic characterizations of the excess growth rate. Characterization I (Theorem 3.2) is proved from its axioms by reducing to a characterization of relative entropy (Proposition 3.6), whose proof in Appendix A uses external functional-equation results; no fitted parameters or assumed conclusions appear. Characterization II (Theorem 3.13) is self-contained and reduces to a Cauchy functional equation after deriving that the generator is measurable and affine up to a logarithmic term; this is a genuine derivation, not a restatement. Characterization III (Theorem 3.20) is the only theorem whose converse (perturbation invariance implies cross-entropy generator) uses prior results of the authors: [49, Proposition 6] and [50, Theorems 4.5 and 4.7]. These are published, parameter-free theorems with explicit regularity hypotheses (C4 and strict concavity as in (3.21)); they establish general metric and Christoffel identities and the fact that a constant portfolio map determines the exponentially concave function up to a constant. They do not assume the target result (perturbation invariance), so by the review rules they count as independent support rather than circularity. The footnote that the result was previously claimed in [50, Example 3.10] without proof flags provenance and a possible proof-completeness gap (the imported theorems may carry hidden boundary assumptions), but this is a correctness concern, not a circular reduction. The omitted LDP details in Theorem 2.15(ii) are incidental and do not affect the characterization results. No step in the paper exhibits a constructed identity of the form 'prediction = input by definition', and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

The central claims rest on (i) standard functional-equation theory, (ii) a cited characterization of relative entropy, and (iii) the authors' own prior information-geometry theorems in [49, 50]. The third pillar is the main structural debt: the converse of Characterization III imports its conclusion from [49, Proposition 6] and the Christoffel identity from [50, Theorem 4.7]. There are no fitted free parameters — the multiplicative constant c in the characterizations is an output of the functional equations, not an adjusted input — and no invented entities; the axiom systems of the three theorems are stated hypotheses, transparently.

assumptions (9)
  • standard math Lebesgue-measurable solutions to Cauchy's equation ψ(x+y)=ψ(x)+ψ(y) on R are ψ(x)=cx.
    Used in Theorem 3.13 Step 3 and Appendix Lemma A.4 (via [38, Theorem 1.1.8]) to force log out of the functional equation (3.12).
  • standard math Ebanks–Kannappan–Ng solution of the generalized fundamental equation of information ([23], [34, Cor. 10.7c]) characterizes B(x,y) satisfying (A.1) on the restricted open domain.
    Load-bearing for Proposition 3.6 and hence Theorem 3.2; the paper explicitly substitutes [23] for [35] to handle the open-domain restriction (Remark A.1).
  • standard math Stirling's approximation; standard implication 'uniform convergence of scaled log-densities implies LDP' on the compact simplex.
    Theorem 2.15(i) uses Stirling; (ii) is asserted from (i) with 'we omit the details'.
  • domain assumption Identity Γᵏᵢⱼ(θ) = δᵢⱼₖ − δᵢₖπⱼ(θ) − δⱼₖπᵢ(θ) for the primal connection of a logarithmic divergence [50, Theorem 4.7].
    Converse of Theorem 3.20; prior result by the second author, cited and not reproven. The theorem's statement was previously claimed without proof in [50, Example 3.10].
  • domain assumption Portfolio-map properties: π(p) ∈ ∆◦ₙ and, when the portfolio map is constant, φ(p) = −H×(π∥p) + c [49, Proposition 6].
    Used in Theorem 3.20's converse to convert 'constant portfolio map' into the cross-entropy generator; self-cited prior work.
  • domain assumption Regularity of φ in Theorem 3.20: C⁴ on ∆◦ₙ and strict concavity of Φ=e^φ in every tangent direction (condition (3.21)).
    Stated hypothesis guaranteeing the induced metric and Christoffel symbols are well-defined; the authors say it 'can be partially relaxed' but do not pursue it.
  • standard math Scaled Dirichlet density (2.22) from [44]; identity SD(α,β) = C[β⁻¹] ⊕ Z with Z ∼ D(α) (Lemma 2.11).
    Background for the large-deviation and Rényi-divergence results (Theorems 2.15 and 2.17).
  • standard math Convex-analysis tools: Slater/KKT, Sion's minimax theorem, Danskin's theorem, perspective functions.
    Used throughout Section 4 (Lemmas 4.6, 4.7; Propositions 4.5, 4.9; Theorem 4.11).
  • domain assumption Integrability Assumption 4.10 (E|r_i| < ∞ for all i).
    Ensures J(π) = E[γ(π,r)] is finite and well-defined in Section 4.4.

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Pith. "Pith review of A mathematical study of the excess growth rate." pith.science (2026). https://pith.science/paper/7HUUZGBB

@misc{pith2026251025740,
  author       = {Pith},
  title        = {Pith review of: A mathematical study of the excess growth rate},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7HUUZGBB}},
  note         = {Machine review of arXiv:2510.25740}
}
read the original abstract

The excess growth rate, defined as the gap in Jensen's inequality for the logarithm, is a fundamental functional in portfolio theory. In this paper, we present a mathematical study motivated by information theory. We begin by establishing its properties and showing that it has rich connections with information theoretic concepts such as the Helmholtz free energy, L. Campbell's measure of average code length and large deviations. Our main results consist of three axiomatic characterization theorems of the excess growth rate, in terms of (i) the relative entropy, (ii) the gap in Jensen's inequality, and (iii) the logarithmic divergence that generalizes the Bregman divergence. Furthermore, we study maximization of the excess growth rate and compare it with the growth optimal portfolio. Our results not only provide theoretical justifications of the significance of the excess growth rate, but also establish new connections between information theory and quantitative finance.

Figures

Figures reproduced from arXiv: 2510.25740 by the authors.

Figure 1
Figure 1. Excess growth rates of the largest 500 stocks, over con￾secutive 20-day periods and equally weighted, of the US stock mar￾ket from 1962 to 2024. We show both the per period excess growth rate and its aggregate through time. The log return, which is additive over time, is given by log Y T t=1 ⟨π, R(t)⟩ ! = X T t=1 log⟨π, R(t)⟩ = X N t=1 (log⟨π, R(t)⟩ − ⟨π, r(t)⟩ + ⟨π, r(t)⟩) = * π, X T t=1 r(t) + + X T t=1 Γ(π, R(t))… view at source ↗
Figure 2
Figure 2. A path p(t) (in black) on ∆◦ n and its perturbation q(t) = p(t) ⊕ h (in grey). is an ordinary translation in the Aitchison vector space (∆◦ n , ⊕, ⊗). Now, (3.22) implies that X T t=0 Lφ(p(t + 1) ∥ p(t)) = X T t=0 Lφ(q(t + 1) ∥ q(t)). That is, the two paths have the same cumulative (relative) volatility. Perturba￾tion invariance is closely related to num´eraire invariance. Observe that (3.22) is equivalent to the id… view at source ↗

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