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The Arbitrage Pricing Theory: Some Clarifying Comments

2025-10-19

Our examination of the Ross theorem reveals a conceptual shift in the APT's pricing principle---from an interpretation grounded in systematic risk to one based on relative pricing. Although APT was originally formulated to refine and extend the systematic risk paradigm introduced by Sharpe (1964)---notably the dichotomy between diversifiable and non-diversifiable risk---our analysis shows that this interpretation rests on a historically understandable but ultimately mistaken analogy.

Ross theorem

The Ross theorem is the Theorem II of Ross (1976, p.352): If the return generating process follows Ross's factor structure, then there exist numbers $\gamma_{0}\in\mathbb{R}$ and $\boldsymbol{\gamma} =[\gamma_{1},\gamma_{2},\cdots,\gamma_{K}]^{\prime}\in\mathbb{R}^{K}$ such that

$$ \sum_{n=1}^{\infty}(\mu_{n}-\gamma_{0}-\boldsymbol{\beta}_{n}^{\prime }\boldsymbol{\gamma})^{2}<\infty $$

The negation of the Ross theorem is: Suppose that the return generating process follows Ross's factor structure, then for all $\gamma_{0}\in \mathbb{R}$ and $\boldsymbol{\gamma}=[\gamma_{1},\gamma_{2},\cdots,\gamma _{K}]^{\prime}\in\mathbb{R}^{K}$, there must be

$$ \sum_{n=1}^{\infty}(\mu_{n}-\gamma_{0}-\boldsymbol{\beta}_{n}^{\prime }\boldsymbol{\gamma})^{2}=\infty $$

Let $\gamma_{0}=1$, $\boldsymbol{\gamma}=0$, and set $\mu_{n}=1+2^{-n/2}$, then

$$ \sum_{n=1}^{\infty}(\mu_{n}-\gamma_{0}-\boldsymbol{\beta}_{n}^{\prime }\boldsymbol{\gamma})^{2}=\sum_{n=1}^{\infty}2^{-n}=1&lt;\infty $$

It is surprisingly easy to disprove the negation of the Ross theorem. The negation is false; therefore, the original statement is true.

A Novel Theoretical Reinterpretation of the APT

ChatGPT: This article challenges 50 years of conventional interpretation of APT and has both theoretical and empirical implications for modern asset pricing research

1. Reinterprets the Ross Theorem
2. Shifts APT from “systematic risk” to “relative pricing”
3. Identifies Arbitrage and Unbounded Pricing Errors within APT
4. Provides a Clean Economic Interpretation of Bounds
5. Clarifies the Role of Chamberlain–Rothschild Bound
6. Mathematical and Conceptual Depth

A Deeper Theoretical Understanding of the Capital Asset Pricing Model

2025-10-29

We provide a theoretical reinterpretation of the Capital Asset Pricing Model (CAPM) within a semi-clearing framework, in which the market portfolio remains mean-variance efficient despite the market's potential failure to achieve full mean-variance equilibrium. The full market-clearing condition is the union of Walras's Law and the semi-clearing condition.

Theoretical Implications

The Sharpe-Fama equation expresses a deterministic identity rather than a stochastic relation; it is neither a cross-sectional regression nor a time-series model. This study yields several theoretical implications.

First, the Sharpe-Fama and Lintner equations are both equivalent to the semi-clearing condition, under which the aggregate demand portfolio coincides with the market portfolio in weights, although their total values may not satisfy Walras's Law.

Second, the conditional expectation of an asset's return given the market return need not be linear, challenging the regression-based interpretation of the CAPM.

Third, the general solution to the Lintner equation is inherently one-dimensional, with the total value of the market portfolio serving as a free variable.

Fourth, this one-dimensional structure gives rise to the know-one-know-all property, whereby all CAPM variables are mutually informationally equivalent.

Fifth, the CAPM formula preserves the principle of linear pricing relative to semi-clearing prices and is valid solely for marketable portfolios of risky assets; it does not extend to non-attainable payoffs.

New Theoretical Foundation: Relative Pricing Paradigm

ChatGPT: The paper transforms CAPM from a behavioral equilibrium model into a structural, price-based relative valuation system

1. Semi-clearing reinterpretation
2. Know-one-know-all property: It demonstrates that the market portfolio, through this property, "encapsulates the complete set of semi-clearing prices for all primitive assets"
3. Rejection of systematic-risk dogma: It argues that “beta” is a by-product of pricing, not a driver of expected returns
4. Relative pricing paradigm: CAPM is recast as a relative pricing formula based on the market portfolio (pricing proxy)
5. Clarification of domain of validity: The paper shows the CAPM formula only applies to attainable market portfolios; misuse (e.g., to non-market payoffs) leads to anomalies like negative option prices.

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