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arxiv: 2411.19444 · v4 · submitted 2024-11-29 · 💱 q-fin.MF · math.PR· q-fin.ST

Capital Asset Pricing Model with Size Factor and Normalizing by Volatility Index

Abraham Atsiwo , Andrey Sarantsev This is my paper

Pith reviewed 2026-05-23 16:58 UTC · model grok-4.3

classification 💱 q-fin.MF math.PRq-fin.ST
keywords CAPMsize factorvolatility indexdiscrete-time modelstochastic portfolio theorystock returns normalizationasset pricing
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The pith

A discrete-time model extends CAPM by adding a size factor after dividing returns by the volatility index.

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

The paper constructs a new discrete-time model that merges the Capital Asset Pricing Model with a size factor, where small stocks show higher average risk and return than large ones. Central to the construction is dividing stock index returns by the Volatility Index, which the authors state renders the returns independent and normally distributed. With this step in place the model incorporates volatility, relative size, and CAPM, is fitted to real data, shown to be stable over long horizons, and linked to Stochastic Portfolio Theory. A sympathetic reader would care because the approach supplies a concrete way to embed observed size differences into standard asset-pricing relations while handling volatility explicitly.

Core claim

Dividing stock index returns by the Volatility Index makes them independent and normal. This property enables a discrete-time model that simultaneously includes volatility, relative size, and the Capital Asset Pricing Model. The resulting model is fitted to real-world data, its long-term stability is established, and the construction is connected to Stochastic Portfolio Theory, thereby filling gaps left by earlier work on CAPM with the size factor.

What carries the argument

Normalization of returns by division with the Volatility Index, which produces independent normal variables that then support the joint inclusion of volatility, relative size, and CAPM inside one discrete-time process.

If this is right

  • The combined model fits observed stock-return series.
  • The fitted process remains stable over long time horizons.
  • The model establishes an explicit link to Stochastic Portfolio Theory.
  • The construction closes specific gaps identified in prior CAPM-with-size work.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the normalization step holds across markets, the same framework could be applied to sector or country indices without re-deriving the size adjustment.
  • Stability proofs in discrete time suggest the model may generate reliable long-horizon forecasts once parameters are estimated from a given window.
  • The link to Stochastic Portfolio Theory opens the possibility of using the size-volatility terms inside dynamic rebalancing rules.

Load-bearing premise

Dividing stock index returns by the Volatility Index makes them independent and normal.

What would settle it

New data in which returns divided by the Volatility Index remain serially dependent or fail standard normality tests would undermine the modeling step.

Figures

Figures reproduced from arXiv: 2411.19444 by Abraham Atsiwo, Andrey Sarantsev.

Figure 1
Figure 1. Figure 1: The quantile-quantile (QQ) plot for equity premia P0, and the empirical autocorrelation function (ACF) for P0 and for |P0| (a) QQ (b) ACF (c) ACF [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The quantile-quantile (QQ) plot for equity premia P0/V , and the empirical autocorrelation function (ACF) for P0/V and for |P0/V | market exposure from the CAPM. This model (12) has good fit, as shown in [22, Section 3]. Innovations W are IID but not Gaussian, with some finite exponential moments. The point estimate βˆ = 0.88, and we reject the unit root hypothesis β = 1. Methodology of [22] stands in cont… view at source ↗
Figure 3
Figure 3. Figure 3: Split the US stock market into 20 equal parts based on market size: top 5%, next 5%, etc up to bottom 5%. Take the 19 breakpoints between these parts and use these as a substitution for market size Sk. We can compute market weights µk from these quantities Sk, for k = 1, . . . , 19. We plot only 5 curves here for clarity but other months and years give very similar curves. If the model is stable, the capit… view at source ↗
Figure 4
Figure 4. Figure 4: The set of (µ, σ) such that ξ ∼ N (µ, σ2 ) satisfies E [ln |ξ|] < 0. This domain cannot be described in closed form. Assume Z(t) ∼ N (0, σ2 ) IID, and V (t) = 1. This violates Assumption 1 (positive density). But this does not affect the stationarity proof. Let us check Assumption (9). We have: R0(t) = g + Z(t). Then we can rewrite the left-hand side of Assumption 2: E [ln |ξ|] < 0, ξ := 1 + a + bR0(t) ∼ N… view at source ↗
Figure 5
Figure 5. Figure 5: Two simulations of ranked relative size terms (ln k, C(k)), k = 1, . . . , N. We pick (c, ρ) to be (0.1, 0), (0.2, 0), (0.1, −0.5), (0.2, −0.5) In our previous article [13], we captured the observation that well-diversified portfolios of small stocks have higher risk but higher return than that of large stocks. We reproduced this shape of capital distribution curve in [13] using simulation. From (37) and (… view at source ↗
Figure 6
Figure 6. Figure 6: Three simulations of the standard normal market curve (ln k, Z(k)) for k = 1, . . . , N if Z1, . . . , ZN ∼ N (0, 1) IID sample, for N = 100. S 2 := σ 2 ε [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Left panel: (− ln k, − ln τk) for k = 1, . . . , 100 and y = −x for x ∈ [0, ln 100]. Right panel: (ln(501 − k), − ln τk) for k = 1, . . . , 100 and y = ln(500−e x ) for x ∈ [ln 401, ln 500]. For each panel, we do three simulations (solid) and plot the deterministic function (dotted). Similarly, recall (43) and plot this Poisson point process with x-axis log scale. This represents (up to shift by −bN and re… view at source ↗
read the original abstract

The Capital Asset Pricing Model (CAPM) relates a well-diversified stock portfolio to a benchmark portfolio. We insert size effect in CAPM, capturing the observation that small stocks have higher risk and return than large stocks, on average. Dividing stock index returns by the Volatility Index makes them independent and normal. In this article, we combine these ideas to create a new discrete-time model, which includes volatility, relative size, and CAPM. We fit this model using real-world data, prove the long-term stability, and connect this research to Stochastic Portfolio Theory. We fill important gaps in our previous article on CAPM with the size factor.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a discrete-time CAPM extension that incorporates a size factor and normalizes returns by the Volatility Index (VIX), asserting that this normalization renders returns independent and normal. The authors fit the resulting model to real-world data, prove long-term stability, and relate the framework to Stochastic Portfolio Theory while addressing gaps from their prior CAPM-with-size work.

Significance. If the VIX-normalization property is empirically validated and the stability result holds without circular dependence on fitted parameters, the model could supply a volatility-aware, size-adjusted discrete-time asset pricing framework with direct links to Stochastic Portfolio Theory, offering potential utility for long-horizon risk assessment and portfolio construction.

major comments (2)
  1. [Abstract] Abstract: the foundational claim that 'Dividing stock index returns by the Volatility Index makes them independent and normal' is asserted without derivation, statistical tests (e.g., autocorrelation, normality diagnostics), or citations to prior verification. This assumption is load-bearing for the subsequent model construction, data fitting, and stability proof.
  2. [stability proof] The long-term stability proof (referenced in the abstract) must be checked against the free parameters for size and volatility scaling; if the result reduces to quantities defined by the fitted parameters rather than holding parameter-free, the proof would be circular with the data-fitting step.
minor comments (2)
  1. [Abstract] Abstract should specify the exact dataset, time period, and functional form of the size factor used in fitting.
  2. [conclusion] The connection to Stochastic Portfolio Theory is stated at a high level; a concrete mapping (e.g., to a specific SPT result or growth-rate functional) would improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. We address the major comments point by point below, providing clarifications and committing to revisions where appropriate to strengthen the paper.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the foundational claim that 'Dividing stock index returns by the Volatility Index makes them independent and normal' is asserted without derivation, statistical tests (e.g., autocorrelation, normality diagnostics), or citations to prior verification. This assumption is load-bearing for the subsequent model construction, data fitting, and stability proof.

    Authors: We acknowledge that the claim regarding VIX normalization requires additional empirical support and justification in the manuscript. In the revised version, we will add a dedicated subsection or appendix presenting statistical evidence, including autocorrelation analysis, Ljung-Box tests for independence, and normality tests such as Jarque-Bera and QQ-plots on the VIX-normalized returns using the dataset. We will also include relevant citations to prior work on volatility normalization in asset returns if applicable, or provide a brief theoretical motivation based on the properties of the VIX as a volatility measure. This addresses the load-bearing nature of the assumption. revision: yes

  2. Referee: [stability proof] The long-term stability proof (referenced in the abstract) must be checked against the free parameters for size and volatility scaling; if the result reduces to quantities defined by the fitted parameters rather than holding parameter-free, the proof would be circular with the data-fitting step.

    Authors: The stability proof is constructed to be independent of the specific fitted parameter values. It relies on the general form of the model incorporating the size factor and VIX normalization, demonstrating convergence or stability properties over long horizons through mathematical analysis that holds for the model structure rather than particular estimates. The data fitting serves to illustrate practical applicability but is not input to the proof. In the revision, we will expand the stability section to explicitly note the parameter-free nature of the result and provide the key steps of the proof to make this clear, avoiding any potential perception of circularity. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper introduces a discrete-time model combining CAPM, size factor, and VIX normalization as an enabling assumption, then fits parameters to data and derives long-term stability from the resulting stochastic process. No equation or result is shown to reduce by construction to the fitted values themselves (e.g., no 'prediction' that is definitionally identical to an input parameter). The reference to prior work by the same authors is used only to note gap-filling and does not serve as the sole justification for any load-bearing uniqueness claim or ansatz. The chain therefore retains independent mathematical content from the stability proof and data fit.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central construction rests on the VIX normalization assumption and on parameters fitted to market data; no invented entities are introduced.

free parameters (1)
  • model parameters for size and volatility scaling
    The model is explicitly fitted to real-world data, implying parameters are chosen or estimated to match observations.
axioms (1)
  • domain assumption Dividing stock index returns by the Volatility Index makes them independent and normal
    Directly stated in the abstract as the enabling step for the model.

pith-pipeline@v0.9.0 · 5637 in / 1239 out tokens · 27665 ms · 2026-05-23T16:58:27.013226+00:00 · methodology

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Reference graph

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