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Owing to its overly\n\nbroad formulation, the Ross theorem lacksmeaningful pricing information. By invoking the\n\nlaws of asset prices, we reformulate the theorem to show that the upper bound on the overall\n\npricing error is determined by the pricing kernel and the factor structure. Furthermore, we\n\ndemonstrate that the APT does not preclude arbitrage and allows arbitrarily large pricing\n\nerrors. This reinterpretation shifts the APT’s foundation from systematic risk to relative\n\npricing, which yields deeper insights into asset pricing theory.\n\nKeywords: Arbitrage Pricing Theory (APT), Asymptotic Arbitrage Opportunity, Laws of Asset\n\nPrices, Pricing Proxy, Factor Structure\n\nJEL Classification: G12\n\n\n\nThe Arbitrage Pricing Theory (APT) developed by Ross (1976) is a multibeta representation\n\nthat explains the expected returns1 of securities based on their sensitivities to a small number of\n\nfactors. The key assumptions of the APT are: (1) Asset returns are generated by a linear factor\n\nmodel where returns depend on a small number of factors, plus a disturbance for each asset. (2)\n\nNo asymptotic arbitrage opportunities (equivalently, the pricing function is continuous). Other\n\nassumptions, such as perfect markets, are implicit, conventional assumptions. The fundamental\n\nconclusion of the APT is that the overall pricing error (the sum of the squared pricing errors)\n\nis a finite number. Thus, all but a finite number of securities have squared pricing errors that\n\nare arbitrarily small. Undoubtedly, the APT is designed for markets with an infinite number of\n\nassets, as the overall pricing error of a finite number of assets is always finite.\n\nThe Ross theorem is the main conclusion of the APT. It states that under the assumptions\n\nof factor structure and no asymptotic arbitrage, there always exists an approximate pricing with\n\na beta representation such that the overall pricing error is finite. Let us consider its negation:\n\nFor any beta representation, the overall pricing error is always infinitely large. This negation\n\nis very difficult to establish; therefore, the Ross theorem is almost always true. To clarify the\n\nAPT theory, we first discuss the setting of infinite markets. In infinite markets, asset portfolios\n\nare infinite linear combinations, whereas textbook linear combinations are finite. Moreover, in\n\ninfinite-dimensional spaces, the linearity and continuity of linear functions are separable. These\n\nsituations pose certain challenges for analyzing infinite markets. For this reason, we need clear\n\ndefinitions of the concepts of arbitrage opportunities and asymptotic arbitrage opportunities in\n\ninfinite markets, as well as an axiomatic definition of the pricing function. In a well-defined\n\ninfinite market, we discuss some properties of frontier portfolios.\n\nSection 2 establishes the linear factor model and clarifies the definitions of pricing error and\n\nfactor risk premium. We discuss the mathematical definitions and economic meanings of exact\n\nfactor structure and approximate factor structure. Section 3 dissects the Ross theorem and its\n\nproof to understand why its conclusion is so strong. A concise outline of the process of several\n\nhistorically famous proofs is provided, revealing why these proofs are only statistical analyses\n\n1In the asset pricing literature, pricing is the process of determining the current price of an asset given its future\n\npayoff. When the price is positive, the expected (rate of) return equals the expected payoff divided by the price.\n\nConversely, when the expected return is nonzero, the price equals the expected payoff divided by the expected\n\nreturn. Therefore, determining the expected return is often referred to as pricing.\n\n1\n\n\n\nand lack economic meaning. Section 4 argues that APT is actually a relative pricing theory.\n\nWe first give the upper bound of APT’s pricing error, then explain how factor loadings and\n\nfactor risk premiums jointly accomplish the pricing proxy. APT is relative pricing, with factor\n\nrisk premiums as exogenous inputs. From an equilibrium pricing perspective, the entire pricing\n\nprocess is divided into two stages: determining the factor risk premiums is the first stage of\n\npricing, while the APT linear pricing completes the second stage pricing task. Section 5 provides\n\nnumerical examples demonstrating that although pricing errors are finite, they can be arbitrarily\n\nlarge. Furthermore, APT only requires continuity of the pricing function, not positivity, and\n\ntherefore allows for the existence of arbitrage opportunities. Section 6 summarizes the text. We\n\nhave also compiled a mathematical appendix providing more detailed proofs and mathematical\n\nsupplements.\n\n1 An Infinite Market\n\nIn a finite market, however large the number of assets, we never face ∞ · 1\n∞ , or ∞ − ∞.\n\nBut in an infinite market, it is an unavoidable reality. For example, the equal-weighted portfolio\n\ntends to be zero (nothing in the portfolio in ℓ∞), or may not even exist in an infinite market (no\n\nlimit in ℓ1).\n\nIn an infinite market, we have infinitely many assets with the payoff sequence\n\nx = (X1, X2, · · · , Xn, · · · )\n\nConsequently, the payoff of the market portfolio is XM = ∑∞\nn=1 Xn. Infinite sums are unnec-\n\nessary in finite markets but are indispensable fundamental operations in infinite markets. An\n\ninfinite sum is defined by the limit of its partial sums, where the limit should be properly defined\n\nby a specific convergence form. For this reason, if the convergence form is not specified, the\n\ninfinite sum ∑∞\nn=1 Xn = X defaults to almost sure convergence such that ∑N\n\nn=1 Xn\na.s.−→ X as\n\nN → ∞.\n\n1.1 Market Payoff Space\n\nNote that a portfolio is an infinite linear combination; the market portfolio is\n\nh = 1 = (1, 1, 1, · · · ) ∈ ℓ∞\n\n2\n\n\n\nwhere we standardize the number of shares outstanding to 1 for each asset. For any portfolio\n\nh ∈ ℓ∞, its payoff is X = h′x = ∑∞\nn=1 hnXn. There exists a risk-free asset; it has a constant\n\npayoff X0 > 0 and a constant return R0 > 1, so the price of the risk-free asset is P0 = X0/R0.\n\nWhen the portfolio includes the risk-free asset, the payoff is X = h0X0 + h′x.\n\nNotations: The following variables are auto-sized\n\n1 I ik 0\n\nwhere 1 stands for conforming vector of ones, I for identity matrix, ik for the k-th column of\n\nI, and 0 for the scalar 0 or a matrix with all elements 0. Note that the sizes of these variables\n\nare conformable to the operation in the context. For matrices A and B, we shall say A ⩾ B if\n\nA − B is a positive semi-definite matrix.\n\nSpace of the p-th power integrable functions on some probability space (Ω, F , P) is denoted\n\nby Lp(Ω, F , P), p ⩾ 1. Without first specifying some additional structure, vector spaces only\n\nguarantee that the sum of finitely many vectors is a vector. Thus, we assume that ∑∞\nn=1 |Xn|\n\nconverges in Lp for some p ⩾ 1, say\n\nMp ≡\n∥∥∥∥∥\n\n∞∑\nn=1\n\n|Xn|\n∥∥∥∥∥\n\np\n\n= E\n(( ∞∑\n\nn=1\n|Xn|\n\n)p)1/p\n\n< ∞ p ⩾ 1 (1.1)\n\nCondition (1.1) assures that∑∞\nn=1 hnXn ∈ Lp for any h ∈ ℓ∞. Thus, we have the market payoff\n\nspace\n\nX = {h0X0 + h′x : h0 ∈ R, h ∈ ℓ∞} ⊆ Lp\n\nUnder condition (1.1), we have\nN∑\n\nn=1\nhnXn\n\na.s.−→ X ⇐⇒\nN∑\n\nn=1\nhnXn\n\nLp\n\n−→ X\n\nwhich states that Lp convergence and almost sure convergence are equivalent in this market\n\nsetting.\n\n1.2 Asymptotic Arbitrage\n\nAn arbitrage opportunity is said to exist if there exists a negative-cost portfolio with zero\n\npayoff, a zero-cost portfolio with payoffs which are never negative and sometimes positive, or a\n\ncombination of the above two situations. More exactly, these three cases are (disjoint subsets):\n\n1. The immediate arbitrage opportunity turns something into nothing: P < 0, X = 0. The\n\nprofit is realized immediately, and investors can walk away without any obligations.\n\n3\n\n\n\n2. The zero-investment arbitrage opportunity turns nothing into something: P = 0, X ⪈ 0.\n\nIf X = a > 0 for some constant a, it is called a naïve arbitrage opportunity.\n\n3. The strong arbitrage opportunity, P < 0, X ⪈ 0, is a positive combination of immediate\n\nand zero-investment arbitrage opportunities.\n\nThe immediate arbitrage opportunities and naïve arbitrage opportunities and their positive com-\n\nbinations are called deterministic arbitrage opportunities, or risk-free arbitrage opportunities.\n\nExcept for deterministic arbitrage opportunities, other arbitrage opportunities are called non-\n\ndeterministic arbitrage opportunities, or risky arbitrage opportunities.\n\nAn asymptotic arbitrage opportunity is said to exist if there exists a sequence of portfolios\n\napproaching a risk-free arbitrage opportunity. Ross (1976) does not provide an explicit definition\n\nof asymptotic arbitrage, but he covers the no asymptotic arbitrage condition by assumptions on\n\nagents’ bounded relative risk aversion: His quadratic programming (19) minimizes the variance\n\nof a zero-investment portfolio (the weights are orthogonal to the factor loadings, so the variance\n\nof the portfolio is equal to the variance of the residuals), subject to the expected value of the zero-\n\ninvestment portfolio being a positive number. The utility function requires that the variance of\n\nthe portfolio cannot approach zero (as the number of assets grows) and must be no less than\n\nsome positive number. He implicitly employs the condition 2 of our Proposition 4. Huberman\n\n(1982, p.186) explicitly defines the asymptotic arbitrage opportunities in the form of condition\n\n4 of our Proposition 4.2\n\nProposition 1 of Chamberlain and Rothschild (1983, p.1288) shows that the absence of\n\nasymptotic arbitrage opportunity implies that the pricing function is continuous. It is straightfor-\n\nward that a continuous pricing function rules out any asymptotic arbitrage opportunities. Thus,\n\nin an infinite market, the market is free of asymptotic arbitrage iff the pricing function is contin-\n\nuous (see Proposition 5 in the appendix).\n\n1.3 Laws of Asset Prices\n\nThe pricing function maps future payoffs to current prices. A reasonable pricing function\n\nsatisfies three laws: Let the primitive assets be {Xn}∞\nn=1 and M = {0, 1, X1, X2, · · · }.\n\n2Note that he calls the zero-investment portfolio an arbitrage portfolio, and he calls the asymptotic arbitrage\n\nsimply arbitrage. The term asymptotic was added by Dybvig (1983).\n\n4\n\n\n\n1. Law of Unique Price: This stipulates that assets can be priced and that there exists a raw\n\npricing function Υ : M → R such that Υ(0) = 0, Υ(1) = 1/R0 > 0, and\n\nPn = Υ(Xn) > 0 n = 1, 2, · · ·\n\n2. Law of Financial Portfolio: (1) The price of a portfolio equals the linear combination of\n\nprices of its constituent assets. The payoff of a portfolio equals the linear combination of\n\npayoffs of its constituent assets. (2) If a payoff sequence converges to zero, then its price\n\nsequence tends to zero.\n\nThis law asserts that pricing functions are linear and continuous. If the first two laws of\n\nasset prices are satisfied, the pricing function is termed packable. Let X̄ ⊆ Lp be the\n\nclosure of the market payoff space generated by {Xn}. Given a packable pricing function\n\nΠ : X̄ → R, there exists a pricing kernel Ψ ∈ (Lp)∗ such that ℘(X) = E(ΨX) = Π(X)\n\nfor any X ∈ X̄, and\n\n℘(X) = Υ(X) ∀X ∈ M\n\nThus, ℘ : Lp → R is a continuous linear function on Lp. The packable pricing function\n\n℘(X) = E(ΨX) extends Υ(·), from M to X̄, and then to Lp.\n\n3. Law of Positive Price: The price of any attainable weak-positive3 payoff is positive.\n\nThe raw pricing function Υ(·) is usually derived from equilibrium pricing, such as mean-\n\nvariance equilibrium pricing. Therefore, the raw pricing function Υ(·) is usually nonlinear, e.g.\n\nΥ(2X1) ̸= 2Υ(X1). Based on the prices of the primitive assets given by Υ(·), the packable pric-\n\ning function ℘(·) computes relative price through the Law of Financial Portfolio. A packable\n\npricing function excludes asymptotic arbitrage opportunities and risk-free arbitrage opportuni-\n\nties.\n\n1.4 Efficient Frontier\n\nLet the Ψ-mimicking payoff be the projection of Ψ onto X̄, Xπ = Pj(Ψ | X̄), then Xπ ̸= 0\n\n(otherwise, Pn = E(XπXn) = 0). Define 1̄ = Pj(1 | X̄), then E(1̄) = 1, and if there exists a\n\nrisk-free asset, 1̄ = 1. Moreover, for any X ∈ X̄, there is E(1̄X) = E(X).\n\n3A random variable X is termed weak-positive (nonnegative and nonzero), written X ⪈ 0, if X ⩾ 0 and\n\nP(X > 0) > 0.\n\n5\n\n\n\nTheorem 1: Given any X ∈ X̄, let Xp = Pj(X | 1̄, Xπ) = x1̄ + yXπ be the orthogonal\n\nprojection of X onto the span of 1̄ and Xπ for some x, y ∈ R. Then Xp solves the following\n\nproblem:\n\nmin\n{\nvar(Y ) : Y ∈ X̄, E(Y ) = E(X), ℘(Y ) = ℘(X)\n\n}\n\nIf X = x1̄+yXπ and ℘(X) = 1, then X is a return and it is on theMarkowitz mean-variance\n\nfrontier. Define Xs = 1̄ − E(Xπ)\nE(X2\n\nπ)Xπ and\n\nS = E(Xs)\nstd(Xs)\n\nwhere std(·) =\n√\n\nvar(·) is the standard deviation. Then\n\nS = sup\n{\n\nE(X)\nstd(X)\n\n: X ∈ X̄\\0, ℘(X) = 0\n}\n\nattains the highest Sharpe ratio.\n\nLet Rπ = 1\nE(X2\n\nπ)Xπ. Define\n\nβn = cov(Rn, Rπ)\nvar(Rπ)\n\nthen we have\n\nβn = E(Rn) − R0\n\nE(Rπ) − R0\nn = 1, 2, · · ·\n\nProjecting Rn onto R0 and Rπ, we get\n\nRfn = Pj(Rn | R0, Rπ) = (1 − βn)R0 + βnRπ\n\nwhich is on the mean-variance frontier, and E(Rfn) = E(Rn). Thus, the projection error Run =\n\nRn − Rfn represents the excess return between asset n and the frontier portfolio with the same\n\nmean. Furthermore, we have the following representation\n\nRn = Rfn + Run = E(Rn) + βn(Rπ − E(Rπ)) + Run n = 1, 2, · · · (1.2)\n\n2 Linear Factor Model\n\nWe are working in the L2(Ω, F , P), which is a Hilbert space. Given a sequence of payoffs\n\nx = (X1, X2, · · · , Xn, · · · ), with Xn ∈ L2(Ω, F , P) for each n, and K × 1 vector of factors\n\nf = [F1, F2, · · · , FK ] with var(f) = S > 0. We assume that Pn > 0 for each asset n, and\n\np = (P1, P2, · · · , Pn, · · · ) ∈ ℓ1, thus PM = ∑∞\nn=1 Pn < +∞.\n\n6\n\n\n\nTo make sure that the payoff of any portfolio h ∈ ℓ∞ is in L2, we assume that condition (1.1)\n\nis satisfied such that∑∞\nn=1 |Xn| converges in L2. Accordingly, the market payoff space is\n\nX = {h0X0 + h′x : h0 ∈ R, h ∈ ℓ∞} ⊆ L2\n\nWe also have\n\ncsp({Xn}) = X̄ ⊆ L2\n\nwhere csp({Xn}) is the closure of the linear span (the linear subspace consisting of all linear\n\ncombinations of {Xn}, the word span always refers to finite linear combinations) of {Xn}, and\n\nX̄ = cl(X) is the closure of market payoff space X.\n\nWe say the asset market has a factor structure if\n\nXn = E(Xn) + b′\nn(f − E(f)) + ϵn n = 1, 2, · · ·\n\nwhere E(ϵn) = 0, and the coefficient bn = [bn1, bn2, · · · , bnK ] ∈ RK . Note that ϵn need not\n\npossess variances, need not be uncorrelated with the factors f and need not even be uncorrelated\n\nwith each other. In matrix form, we have\n\nx = E(x) + B(f − E(f)) + ϵ (2.1)\n\nwhere\n\nx =\n\n\n\nX1\n\nX2\n...\n\nXn\n\n...\n\n\nB =\n\n\n\nb′\n1\n\nb′\n2\n...\n\nb′\nn\n...\n\n\n=\n\n\n\nb11 b12 · · · b1K\n\nb21 b22 · · · b2K\n\n... ... . . . ...\nbn1 bn2 · · · bnK\n\n... ... ... ...\n\n\nϵ =\n\n\n\nϵ1\n\nϵ2\n...\n\nϵn\n\n...\n\n\nwith E(ϵ) = 0. Note that B has full column rank (otherwise we can throw away some factors).\n\nHence, let BN be the first N rows of B, BN has full column rank for N sufficiently large.\n\nThe factor model (2.1) is well-defined in L2. We assume that there is no asymptotic arbitrage\n\nin the market. Thus, the pricing function is continuous. There exists a pricing kernel Ψ ∈ L2\n\nsuch that Pn = E(ΨXn) for all n, and E(Ψ) = 1/R0, where R0 is the risk-free return.\n\n2.1 Pricing Error\n\nLet the disturbance be en = ϵn\n\nPn\nand factor loading (factor beta) be\n\nβn = bn/Pn\n\nthen\n\nRn = Xn\n\nPn\n\n= E(Rn) + β′\nn(f − E(f)) + en n = 1, 2, · · · (2.2)\n\n7\n\n\n\nFrom\n\n1 = E(ΨRn) = 1\nR0\n\nE(Rn) + β′\nn E(Ψ(f − E(f))) + E(Ψen)\n\nwe obtain\n\nµn = E(Rn) = R0 + β′\nnλ + dn n = 1, 2, · · · (2.3)\n\nwhere\n\ndn = −R0 E(Ψen)\n\nis the pricing error, and\n\nλ = −R0 E(Ψ(f − E(f))) = −R0cov(f , Ψ) = E(f) − R0 E(Ψf)\n\nis the factor risk premium. If some Fk is an excess return, E(ΨFk) = 0, then\n\nλk = E(Fk) − R0 E(ΨFk) = E(Fk)\n\nWe see that the factor risk premium corresponding to an excess return is simply the mean of the\n\nexcess return. If some Fk is a portfolio return, E(ΨFk) = 1, then\n\nλk = E(Fk) − R0 E(ΨFk) = E(Fk) − R0\n\nthe factor risk premium of a return is its expected excess return. Since λ =\n(\n\nE(f)\nR0\n\n− E(Ψf)\n)\n\nR0,\n\nthe factor risk premium is the deviation from the risk-neutral price compounded at risk-free rate.\n\nWhen there is no risk-free asset, we can rewrite (2.3) with 1\nE(Ψ) in place of R0, i.e.,\n\nE(Rn) = λ0 + β′\nnλ + dn\n\nwhere λ0 = 1\nE(Ψ) , λ = E(f) − E(Ψf)\n\nE(Ψ) , and dn = −E(Ψen)\nE(Ψ) .\n\n2.2 Exact Factor Structure\n\nIf dn = 0 for all n, i.e., E(Ψen) = 0 for all n, we call (2.2) an exact APT model. As\n\nE(Rn) = R0 + β′\nnλ n = 1, 2, · · ·\n\nthe asset prices are determined exactly by the given factors.4 Note that e = 0 is a special case of\n\nexact factor structure, called a noiseless factor structure, where the payoffs are fully determined\n\nby the factors.\n\n4E(Ψe) = 0 means that the pricing kernel Ψ is orthogonal to the disturbances. For the exact factor model, when\n\nE(ef ′) = 0, the pricing kernel lies in the factor span (LeRoy and Werner, 2014, p.231).\n\n8\n\n\n\n2.3 Approximate Factor Structure\n\nWhen there exists dn ̸= 0 for some n, if\n\nA2 ≡ sup\ny∈ℓ2,∥y∥2=1\n\nE\n\n( ∞∑\nn=1\n\nenyn\n\n)2\n < ∞ (2.4)\n\nwe say the factor model (2.2) has an approximate factor structure. Condition (2.4) is equivalent to\n\nthe condition that the variance of the residual of any feasible portfolio is bounded. Furthermore,\n\nfor any y ∈ ℓ2 with ∥y∥2 = 1, we have\n∞∑\n\nn=1\nyn E(Ψen) = E\n\n(\nΨ\n\n∞∑\nn=1\n\nenyn\n\n)\n(2.5)\n\nwe can interchange the expectation operator and the infinite summation under condition (2.4).\n\nThe approximate factor structure has the following special cases\n\n1. Ross’s factor structure: Uncorrelated disturbances with E(emen) = 0 if m ̸= n, and there\n\nexists A > 0 such that supn E(e2\nn) = A2 < ∞.\n\n2. Bounded sum of variances: ∑∞\nn=1 E(e2\n\nn) < ∞.\n\n3. Chamberlain-Rothschild factor structure: Let yN = [y1, y2, · · · , yN ]′ ∈ RN , eN = [e1, e2,\n\n· · · , eN ]′ and EN = var(eN) = E(eNe′\nN). Then\n\nA2\nN ≡ max\n\n∥yN∥=1\ny′\n\nNENyN\n\nis the largest eigenvalue of EN . Chamberlain and Rothschild (1983) assume that the fac-\n\ntors are uncorrelated with the disturbances, E(fe′) = 0, and define the condition as (their\n\nDefinition 2) the maximum eigenvalue of EN is bounded:\n\nA2 = sup\nN\n\nA2\nN < ∞ (2.6)\n\nWithout the uncorrelation assumption, condition (2.6) alone is equivalent to condition\n\n(2.4).\n\nWhen the factor model (2.2) has an approximate factor structure (2.4), we have the following\n\napproximate factor pricing\n\nE(Rn) ≈ R0 + β′\nnλ n = 1, 2, · · ·\n\nin the sense that the overall pricing error (sum of the squared pricing errors) is finite, or more\n\nprecisely, as stated in equation (4.1).\n\n9\n\n\n\n3 Ross Theorem\n\nTheRoss theorem is the Theorem II of Ross (1976, p.352). The result obtained is surprisingly\n\nstrong, due to its overly broad formulation.\n\nTheorem 2 (Ross): If the return generating process follows Ross’s factor structure, then there\n\nexist numbers γ0 ∈ R and γ = [γ1, γ2, · · · , γK ]′ ∈ RK such that\n∞∑\n\nn=1\n(µn − γ0 − β′\n\nnγ)2 < ∞\n\nIf there is a risk-free asset, then γ0 = R0 is its rate of return.\n\nThe negation of the Ross theorem is: Suppose that the return generating process follows\n\nRoss’s factor structure, then for all γ0 ∈ R and γ = [γ1, γ2, · · · , γK ]′ ∈ RK , there must be\n∞∑\n\nn=1\n(µn − γ0 − β′\n\nnγ)2 = ∞\n\nLet γ0 = 1, γ = 0, and set µn = 1 + 2−n/2, then\n∞∑\n\nn=1\n(µn − γ0 − β′\n\nnγ)2 =\n∞∑\n\nn=1\n2−n = 1 < ∞\n\nIt is surprisingly easy to disprove the negation of the Ross theorem. The negation is false; there-\n\nfore, the original statement is true.\n\n3.1 Proofs\n\nLet\n\nrN =\n\n\nR1\n\nR2\n...\n\nRN\n\n µN =\n\n\nµ1\n\nµ2\n...\n\nµN\n\n BN =\n\n\nβ′\n\n1\n\nβ′\n2\n...\n\nβ′\nN\n\n eN =\n\n\ne1\n\ne2\n...\n\neN\n\n\nthen for the first N assets\n\nrN = µN + BN(f − E(f)) + eN\n\nNote that we do not require E(eN f ′\nN) = 0, generally we do not have BN = cov(rN , fN)S−1.\n\nLet CN = [1, BN ], XN = [µN , CN ], and qN = [1; −γN0; −γN ] where γN0 ∈ R and\n\nγN ∈ RK , then we can write\n\ndN ≡ XNqN = µN − γN01 − BNγN\n\nIf we choose a factor-neutral portfolio zN (weight vector) such that z′\nNCN = 0, then for the\n\n10\n\n\n\nportfolio return RzN = z′\nNrN\n\nvar(RzN) = var(z′\nN(µN + BN(f − E(f)) + eN)) = var(z′\n\nNeN) = z′\nNDNzN\n\nwhere DN = var(eN) is a diagonal matrix.\n\nSketch of the proof by Ross (1976), see §A.5 for a full proof.\n\nProof (Ross, 1976): Given Ross’s factor structure E(e2\nn) ⩽ A2, the minimum variance of\n\nRzN at an expected return µ > 0 is voN . No asymptotic arbitrage opportunity requires that voN\n\ncannot approach zero as N → ∞, thus there exists vo > 0 such that voN ⩾ vo > 0 for any N .\n\nRoss takes\n\nqN = (X′\nND−1\n\nN XN)−1i1\n\ni′\n1(X′\n\nND−1\nN XN)−1i1\n\nand shows that\nN∑\n\nn=1\n(µn − γN0 − β′\n\nnγN)2 = ∥dN∥2 ⩽ µ2\n\nvo\n\nA2\n\nfor each N . Then he picks a subsequence qNm → q = [1; −γ0; −γ], and obtains\nN∑\n\nn=1\n(µn − γ0 − β′\n\nnγ)2 ⩽ µ2\n\nvo\n\nA2\n\nTaking the limit as N → ∞ completes the proof.\n\nSketch of the proof by Huberman (1982), see §A.5 for a full proof.\n\nProof (Huberman, 1982): Let dN = µN − γN01 − BNγN be the residuals from the projec-\n\ntion of µN on to CN . Assume that ∥dN∥ < ∞ is false; Huberman constructs an asymptotic\n\narbitrage opportunity by a sequence of zero-investment portfolios whose return goes to infinity\n\nand whose variance tends to zero. Thus, there exists a real number B such that\nN∑\n\nn=1\n(µn − γN0 − β′\n\nnγN)2 ⩽ B < +∞\n\nfor each N . By Cantor’s Intersection Theorem, he shows that there exists [γ0; γ] such that\nN∑\n\nn=1\n(µn − γ0 − β′\n\nnγ)2 ⩽ B\n\nTaking the limit as N → ∞ completes the proof.\n\n11\n\n\n\n3.2 Chamberlain-Rothschild Bound\n\nThe Chamberlain-Rothschild factor structure requires that the residuals satisfy condition\n\n(2.6), and that E(eN f ′) = 0. In addition, they also require that var(f) = I.5 Chamberlain\n\nand Rothschild (1983) approach an infinite market through a sequence of finite markets. Define\n\nXN = {h0X0 + h′\nNxN : h0 ∈ R, h ∈ RN}\n\nand the Ψ-mimicking payoff in XN\n\nXπN = Pj(Ψ | XN)\n\nLet XsN = 1 − E(XπN )\nE(X2\n\nπN )XπN , then we have E(XsN) > 0. Define\n\nSN = E(XsN)\nstd(XsN)\n\nthen\n\nSN = sup\n{\n\nE(X)\nstd(X)\n\n: X ∈ XN\\0, ℘(X) = 0\n}\n\nN = 2, 3, · · ·\n\nis the highest Sharpe ratio of the N risky assets and the risk-free asset. We have SN ⩽ SN+1 ⩽ S\n\nfor any N . Furthermore\n\nS2\nN = (µN − R01)′V−1\n\nN (µN − R01) (3.1)\n\nwhere µN = E(rN) and VN = var(rN). By Lemma 6, we have S = limN→∞ SN .\n\nConsider the minimum sum of squared deviations\n\nDN = min\ny∈RN\n\n(µN − R01 − BNy)′(µN − R01 − BNy)\n\nthe optimal solution is γN = (B′\nNBN)−1B′\n\nN(µN −R01). Thus, let dN = µN −R01−BNγN ,\n\nthere is\n\nDN = d′\nNdN = (µN − R01)′MN(µN − R01)\n\nwhere MN = I − BN(B′\nNBN)−1B′\n\nN .\n\nChamberlain and Rothschild (1983) show that (see §A.6 for more detail)\n\nDN ⩽ A2\nNS2\n\nN ⩽ A2S2\n\nwhere A2\nN is the largest eigenvalue of EN . Thus, their Theorem 3 shows that\n\n∥d∥2\n2 =\n\n∞∑\nn=1\n\n(µn − R0 − β′\nnγ)2 ⩽ A2S2\n\nwhere γ = [γ1, γ2, · · · , γK ]′ = limm→∞ γNm\nis the limit of a subsequence of γN .\n\n5The restriction var(f) = I is not a real constraint. If var(f) = S, by Cholesky factorization S = LL′, we have\n\nvar(L−1f) = I. An S−1/2-transform can also be used, such that var(S−1/2f) = I.\n\n12\n\n\n\nChamberlain and Rothschild (1983) also consider the market without a risk-free asset, they\n\nobtain the same upper bound, except that S is the supremum of the Sharpe ratio on the efficient\n\nfrontier (the slope of the asymptote of the frontier).\n\n3.3 Deviations and Bounds\n\nIn the above proof, the finite market series computes the minimum sum of squared deviations\n\nin a certain sense. The Chamberlain-Rothschild bound was probably inspired by Ross (1976).\n\n3.3.1 Ross (1976)\n\nRoss (1976) obtains qN = [1; −γN0; −γN ] as the optimal solution to\n\nmin\nq∈RK+2\n\n(XNq)′D−1\nN (XNq)\n\ns.t. i′\n1q = 1\n\nAs we can write q = [1; −y0; −yN ], the programming can be reformed as\n\nDN = min\n[y0;y]∈RK+1\n\n(µN − y01 − BNy)′D−1\nN (µN − y01 − BNy)\n\nWe see that the deviations dN = XNqN = µN − γN01 − BNγN are the residuals of the\n\nGeneralized Least Squares (GLS) of µN on CN = [1, BN ]. From §A.7, we have the optimal\n\nsolution\n\nγN0 = (1′HN1)−1 1′HNµN\n\nγN =\n(\nB′\n\nND−1\nN BN\n\n)−1\nB′\n\nND−1\nN (µN − γN01)\n\nwhere HN = D−1\nN − D−1\n\nN BN\n\n(\nB′\n\nND−1\nN BN\n\n)−1\nB′\n\nND−1\nN .\n\nWhen there exist risk-free asset, Ross (1976) employs\n\nDN = min\ny∈RK\n\n(µN − R01 − BNy)′D−1\nN (µN − R01 − BNy)\n\nand as γN0 = R0 is fixed, we have γN =\n(\nB′\n\nND−1\nN BN\n\n)−1\nB′\n\nND−1\nN (µN − R01).\n\n3.3.2 Huberman (1982)\n\nHuberman (1982) uses the orthogonal projection of µN into the linear subspace spanned by\n\n1 and the columns of BN . Since an orthogonal projection (on a finite-dimensional space) is the\n\nsame thing as a linear regression. The deviations dN = µN − γN01 − BNγN are the residuals\n\n13\n\n\n\nof the Ordinary Least Squares (OLS) of µN on CN = [1, BN ]. Thus,\n\nd′\nNdN = min\n\n[y0;y]∈RK+1\n(µN − y01 − BNy)′(µN − y01 − BNy)\n\nFrom §A.7, we have\n\nγN0 = (1′MN1)−1 1′MNµN\n\nγN = (B′\nNBN)−1\n\nB′\nN(µN − γN01)\n\nwhere MN = I − BN (B′\nNBN)−1\n\nB′\nN .\n\nWhen there exist risk-free asset, Huberman (1982) uses\n\nDN = min\ny∈RK\n\n(µN − R01 − BNy)′(µN − R01 − BNy)\n\nand as γN0 = R0 is fixed, we have γN = (B′\nNBN)−1\n\nB′\nN(µN − R01).\n\n3.3.3 Chamberlain and Rothschild (1983)\n\nLet aN = 1′V−1\nN 1 > 0 and b = 1′V−1\n\nN µN , define µoN = bN\n\naN\n. Chamberlain and Rothschild\n\n(1983) employ the OLS residuals of µN − µoN1 on BN (assume E(eN f ′) = 0 and var(f) = I)\n\ndN = µN − µoN1 − BNγN = MN(µN − µoN1)\n\nwhere γN = (B′\nNBN)−1\n\nB′\nN(µN − µoN1).\n\nWhen there exist risk-free asset, they employ the OLS residuals of µN −R01 on BN similar\n\nto Huberman (1982). They take\n\ndN = µN − R01 − BNγN = MN(µN − R01)\n\nwith γN = (B′\nNBN)−1\n\nB′\nN(µN − R01).\n\n3.3.4 Bounds\n\nRoss (1976) does not provide a bound for the overall deviations (sum of squared deviations).\n\nHe claims that there exists vo > 0 such that the pricing error has an upper bound µ2\n\nvo\nA2. Huber-\n\nman (1982) proves the existence of an upper bound but no clues were given. Chamberlain and\n\nRothschild (1983) provide the specific form of the upper bound for the first time, A2S2, where\n\nS is the highest Sharpe ratio. As µ2\n\nvo\n⩽ S2, the upper bounds they obtained make sense.\n\n14\n\n\n\n3.4 Factor Risk Premium\n\nRecalling that a pricing error dn = −E(Ψen)\nE(Ψ) takes another form dn = µn − λ0 − β′\n\nnλ,\n\nwhere λ0 = 1\nE(Ψ) and the factor risk premium λ = E(f) − E(Ψf)\n\nE(Ψ) . Thus, the deviations used in\n\nRoss (1976), Huberman (1982), and Chamberlain and Rothschild (1983) are not pricing errors,\n\nbecause the parameters γ0 and γ used in dn = µn −γ0 −β′\nnγ are not factor risk premiums: they\n\nare the limit (no explicit solution) of some subsequence of GLS or OLS regression coefficients,\n\nwhich are computed from the factor sensitivity matrix B but not derived from the pricing kernel\n\nΨ. When there exist a risk-free asset, they set γ0 = R0, and\n\n• Ross (1976): γR0 = limm→∞(B′\nNm\n\nD−1\nNm\n\nBNm)−1B′\nNm\n\nD−1\nNm\n\n(µNm\n− R01).\n\n• Huberman (1982): γH0 = limm→∞(B′\nNm\n\nBNm)−1B′\nNm\n\n(µNm\n− R01).\n\n• Chamberlain and Rothschild (1983) (assume E(fe′) = 0, and var(f) = I): γC0 = γH0\n\nWhen there is no risk-free asset, γ0 is optimized\n\n• Ross (1976): γR0 = limm→∞ (1′HNm1)−1 1′HNmµNm\n, and\n\nγR = lim\nm→∞\n\n(B′\nNm\n\nD−1\nNm\n\nBNm)−1B′\nNm\n\nD−1\nNm\n\n(µNm\n− γR01)\n\n• Huberman (1982): γH0 = limm→∞ (1′MNm1)−1 1′MNmµNm\n, and\n\nγH = lim\nm→∞\n\n(B′\nNm\n\nBNm)−1B′\nNm\n\n(µNm\n− γH01)\n\n• Chamberlain and Rothschild (1983) (assume E(fe′) = 0, and var(f) = I): γC0 =\n\nlimm→∞ µoNm , and\n\nγC = lim\nm→∞\n\n(B′\nNm\n\nBNm)−1B′\nNm\n\n(µNm\n− γC01)\n\nNote that when there exist a risk-free asset, let d1n = µn − R0 − β′\nnγC0, and d2n = µn − γC0 −\n\nβ′\nnγC , we have A2S2 ⩾ ∑∞\n\nn=1 d2\n1n ⩾ ∑∞\n\nn=1 d2\n2n. Both d1n and d2n have finite sum of squares.\n\nThere are similar results for the versions of Ross (1976) and Huberman (1982). Therefore, the\n\nγ0 and γ in Ross theorem can have infinitely many solutions, such as lR0 + (1 − l)γC0 and\n\nlγC0 + (1 − l)γC for any l ∈ [0, 1].\n\nThe statement of the Ross theorem cannot simply require the existence of the parameter γ,\n\nbut should set γ to the factor risk premiums and then discuss the overall pricing error. If we use\n\na set of true factor risk premiums, does the Ross theorem still hold? This is the question we will\n\nanswer in the coming section.\n\n15\n\n\n\n4 Relative Pricing\n\nRoll and Ross (1980) view the APT as having implications for approximate asset pricing\n\nrelative to any set of factors. They were very far-sighted, but unfortunately they did not persist\n\nin moving further in the direction of relative pricing.\n\n4.1 Approximate Factor Pricing\n\nUnder the setting of an approximate factor structure and a continuous pricing function, the\n\nupper bound for the overall pricing errors is determined by the pricing kernel.\n\nTheorem 3: In the factor model (2.2) defined in L2 with an approximate factor structure (2.4).\n\nIf the pricing function is continuous, such that the pricing kernel Ψ ∈ L2 and E(ΨRn) = 1,\n\nn = 0, 1, 2, · · · , then the pricing errors in (2.3) satisfy\n\n∥d∥2\n2 =\n\n∞∑\nn=1\n\nd2\nn ⩽ R2\n\n0A2 E(Ψ2) < +∞ (4.1)\n\nProof: This proof is inspired by Back (2017, §6.5). The pricing function ℘(X) = E(ΨX) is\n\ncontinuous on L2, so Ψ ∈ L2, and E(Ψ2) < ∞. Let vn = E(Ψen) = −dn/R0, then by extremal\n\nequality\n\n∥v∥2 = max\ny∈ℓ2,∥y∥2=1\n\n∣∣∣∣∣\n∞∑\n\nn=1\nynvn\n\n∣∣∣∣∣ = sup\ny∈ℓ2,∥y∥2=1\n\n∣∣∣∣∣\n∞∑\n\nn=1\nynvn\n\n∣∣∣∣∣ supremum attained\n\n= sup\ny∈ℓ2,∥y∥2=1\n\n∣∣∣∣∣\n∞∑\n\nn=1\nyn E(Ψen)\n\n∣∣∣∣∣ = sup\ny∈ℓ2,∥y∥2=1\n\n∣∣∣∣∣E\n(\n\nΨ\n∞∑\n\nn=1\nenyn\n\n)∣∣∣∣∣ by (2.5)\n\n⩽ sup\ny∈ℓ2,∥y∥2=1\n\nE\n(∣∣∣∣∣Ψ\n\n∞∑\nn=1\n\nenyn\n\n∣∣∣∣∣\n)\n\n⩽ sup\ny∈ℓ2,∥y∥2=1\n\n√√√√√E\n\n( ∞∑\nn=1\n\nenyn\n\n)2\nE(Ψ2)\n\n= A\n√\n\nE(Ψ2)\n\nthus ∥d∥2\n2 = R2\n\n0 ∥v∥2\n2 ⩽ R2\n\n0A2 E(Ψ2) < +∞.\n\nIn the proof, the extremal equality converts the overall pricing error into the pricing error\n\nof the portfolio residual. Then, using the Cauchy-Schwarz inequality, the pricing kernel and\n\nthe variance of the portfolio residual are separated. This corresponds to the two assumptions\n\n16\n\n\n\nof APT: the absence of asymptotic arbitrage ensures the continuity of the pricing function, thus\n\nthe pricing kernel is bounded; and the approximate factor structure requires the variance of the\n\nportfolio residual to be bounded. Thus, an upper bound on the total pricing error is obtained.\n\nThe finiteness of the overall pricing error implies that most of the approximationmistakes are\n\nsmall, although nothing is implied about any specific approximation mistake. Since the pricing\n\nfunction is unknown, the value of E(Ψ2) is unknown, which permits arbitrarily large expected\n\nreturn deviations for any given finite set of assets, and this finite set can contain an arbitrarily\n\nlarge number of primitive assets.\n\n4.2 Beta Representation\n\nIn an exact factor structure, by permuting the rows of B, we may assume that BK is full\n\nrank. Now, from\n\nrK = µK + BK(f − E(f)) + eK\n\nwe see that factors f are perfectly replicated by\n\nf − E(f) = B−1\nK (rK − µK − eK)\n\nUsing rK as factors, we obtain a new exact factor structure:\n\nrN = µN + LN(rK − µK) + uN\n\nwhere LN = BNB−1\nK and uN = eN − BNB−1\n\nK eK .\n\nRelationship (1.2) shows that there always exists an exact one-factor pricing: In (1.2), since\n\nE(Run) = E(Rn − Rfn) = 0 and ℘(Run) = ℘(Rn − Rfn) = ℘(Rn) − ℘(Rfn) = 1 − 1 = 0 for\n\nall n, (1.2) defines an exact factor structure. For any asset m with βm ̸= 0, define\n\nβnm = βn\n\nβm\n\nSince Rπ − E(Rπ) = (Rm − E(Rm) − Run)/βm, we can replace the factor Rπ by Rm in (1.2),\n\nand obtain a new exact factor structure6\n\nRn = E(Rn) + βnm(Rm − E(Rm)) + en n = 1, 2, · · · (4.2)\n\nwhere en = Run − βnmRum . Due to the arbitrariness of factor space, Gilles and LeRoy (1991)\n\nargue the theorems relating expected returns to factor risk require substantial reinterpretation.\n\n6Shanken (1982) and Gilles and LeRoy (1991) demonstrate that virtually any variables could serve as the factors\n\nin a well-defined exact factor model.\n\n17\n\n\n\n4.3 Reinterpretation\n\nIn (1.2), the factor Rπ is a frontier portfolio, which can effectively represent the movement\n\nof the entire market, while in (4.2), Rm is an individual asset, which may not even represent the\n\nindustry it operates in, let alone reflect the movement of the entire market. The systematic risk\n\ninterpretation for APT doesn’t work.\n\nIn APT, the expected return of an asset is a linear function of its factor sensitivities and\n\nthe risk premiums associated with those factors. The factor structure only provides betas; to\n\ncalculate an asset’s expected return, we also need to know the factor risk premium. Factor risk\n\npremiums λ = E(f) − R0 E(Ψf) cannot be determined by the APT itself, even if the factor is a\n\nportfolio return: By (2.2) the factor loading of Fk is ik\n\nFk = E(Fk) + (Fk − E(Fk)) = E(Fk) + i′\nk(f − E(f)) + 0\n\nby (2.3), with factor loading ik and E(ΨFk) = 1, we obtain\n\nE(Fk) = R0 + i′\nk(E(f) − R0 E(Ψf)) = R0 + E(Fk) − R0 E(ΨFk) = E(Fk)\n\nE(Fk) a free variable within APT, its risk premium λk = E(Fk) − R0 is indeterminate.7 Factor\n\nrisk premiums are exogenous data. Given the factor risk premium, we can calculate the expected\n\nreturn of any asset n. In other words, assets in (4.2) are priced relative to the given price system\n\nof the factor Rπ or Rm. No matter what variables are selected as factors, those that can represent\n\nthe market and those that cannot, as long as their risk premium is known, they are serving as a\n\nreference price system to calculate prices for other assets.\n\nFactor loadings and factor risk premiums are a pair of partners, they work together to de-\n\ntermine an asset’s excess return. From an equilibrium pricing perspective, the entire pricing\n\nprocess is divided into two steps:\n\n1. The first stage is to calculate the factor risk premium\n\nλ = −R0℘(f − E(f)) = E(f) − R0 E(Ψf)\n\n2. APT undertakes the second stage of pricing, using the factor risk premium λ as input to\n\ncalculate the asset’s excess return β′\nnλ, and then yielding\n\nE(Rn) = R0 + β′\nnλ\n\nor Pn = (E(Xn) − b′\nnλ)/R0.\n\n7The market risk premium cannot be directly determined within the Capital Asset Pricing Model (CAPM) itself\n\nbecause it is an input to the CAPM, not an output derived from the model.\n\n18\n\n\n\nTherefore, systematic risk functions as a pricing proxy, the APT performs relative pricing.\n\n5 Numerical Examples\n\nThe APT applies to a market with infinitely many assets. For a finite market, the main result\n\nof APT is self-evident and unconditional. Besides, finite markets can be solved analytically,\n\nfacilitating numerical demonstrations. Our first example uses a finite market to illustrate that\n\nChamberlain-Rothschild bound is not an upper bound on overall pricing errors and that the pric-\n\ning errors of individual assets can be arbitrarily large. Subsequently, we use an infinite market\n\nexample to demonstrate that although APT rules out asymptotic arbitrage, it does not eliminate\n\narbitrage opportunities.\n\n5.1 Arbitrarily Large Error and Bound\n\nMarkets with a finite number of assets can be considered a special case of infinite markets.\n\nThis is because we can set Rn = 0 and dn = 0 for all assets n > N .\n\nExample 5.1: Assume that the market has four states with uniform probability distribution. Let\n\nthe risk-free return be R0 = 21\n20 , and the factor model\n\nR1 = 29\n25\n\n+ 5\n7\n\n(F1 − µF 1) + 9\n8\n\n(F2 − µF 2) + e1\n\nR2 = 107\n100\n\n+ 6\n5\n\n(F1 − µF 1) + (F2 − µF 2)\n\nwhere (random variables can be expressed as a vector in R4 by isomorphism, linear algebra\n\nand probability theory can be used interchangeably at one’s convenience)\n\nF1 =\n\n\n\n0\n\n0\n\n2\n\n5\n\n F2 =\n\n\n\n0\n\n0\n\n−1\n\n3\n\n R1 =\n\n\n\n−1949\n3600\n\n−2749\n3600\n\n− 977\n2800\n\n17 623\n2800\n\n R2 =\n\n\n\n−153\n100\n\n−153\n100\n\n− 13\n100\n\n747\n100\n\n e1 =\n\n\n\n1\n9\n\n−1\n9\n\n0\n\n0\n\n\n\nWe have µF 1 = 7\n4 , µF 2 = 1\n\n2 , E(e1) = 0, var(e1) = 1\n162 , and\n\nµ =\n\n 29\n25\n107\n100\n\n V =\n\n8948 753\n1016 064\n\n6173\n560\n\n6173\n560\n\n699\n50\n\n B =\n\n5\n7\n\n9\n8\n\n6\n5 1\n\n S = var(f) =\n\n67\n16\n\n19\n8\n\n19\n8\n\n9\n4\n\n\n\n19\n\n\n\nUsing Huberman’s method,\n\nγH0 = (B′B)−1B′(µ − R01) =\n\n− 49\n356\n\n412\n2225\n\n =\n\n−0.137 64\n0.185 17\n\n\nthen d = µ − R01 − BγH0 = 0, the minimum sum of squared deviations is indeed obtained.\n\nUsing Chamberlain-Rothschild’s method, we need to first transform the factors appropriately\n\nso that the variance matrix of the transformed factors is a unit matrix. By Cholesky decomposi-\n\ntion S = LL′, we have\n\nL =\n√\n\n67\n\n 1\n4 0\n19\n134\n\n11\n√\n\n2\n134\n\n and BC = BL =\n√\n\n67\n\n2537\n7504\n\n99\n√\n\n2\n1072\n\n148\n335\n\n11\n√\n\n2\n134\n\n\nThus\n\nγC0 = (B′\nCBC)−1B′\n\nC(µ − R01) =\n√\n\n67\n\n − 49\n5963\n\n78 483\n√\n\n2\n3279 650\n\n =\n\n−0.06726 2\n0.277 01\n\n\nand d = µ − R01 − BCγC0 = 0. The choice of transformation matrix isnot unique. If we use\n\nS1/2 = 1\n\n4\n√\n\n103 + 44\n√\n\n2\n\n67 + 22\n√\n\n2 38\n38 36 + 22\n\n√\n2\n\n\nthen we obtain\n\nBC = BS1/2 = 1√\n44\n\n√\n2 + 103\n\n55\n14\n\n√\n2 + 2537\n\n112\n99\n16\n\n√\n2 + 947\n\n56\n33\n5\n\n√\n2 + 148\n\n5\n11\n2\n\n√\n2 + 102\n\n5\n\n\nγC0 = 1\n\n24 475\n√\n\n44\n√\n\n2 + 103\n\n−26 681\n√\n\n2 − 13 475\n78 483\n\n2\n\n√\n2 + 18 128\n\n =\n\n−0.162 77\n0.234 02\n\n\nand again d = µ − R01 − BCγC0 = 0.\n\nThe coefficients γH0 and γC0 calculated above are regression coefficients in the sense of\n\nminimizing the sum of squared deviations, not factor risk premiums. This is because factor risk\n\npremiums are determined by the market, not entirely by factor loadings.\n\nLet the pricing kernel be\n\nΨ = 4\n\n\nx\n\n117 318\n271 145 − 9875\n\n7747x\n2520\n7747x + 15 842\n\n54 229\n185 116\n813 435 − 392\n\n7747x\n\n\nwhere the real number x is chosen by the market. The mimicking payoff is\n\nXπ =\n\n\n5898 567 244\n8611 874 145\n7385 786 572\n8611 874 145\n266 265 064\n191 374 981\n7540 857 904\n8611 874 145\n\n\nIt is easy to verify that ℘(Rn) = E(ΨRn) = E(XπR) = 1 for n = 0, 1, 2. Since e1 /∈\n\nsp(R0, R1, R2), Xπ does not price e1 correctly\n\nE(Xπe1) = − 5901 664\n1230 267 735\n\n̸= E(Ψe1) = 1958\n7747\n\nx − 39 106\n813 435\n\n20\n\n\n\nNote that d2\n1 = (−R0 E(Ψe1))2 = R0\n\n(\n1958\n7747x − 39 106\n\n813 435\n\n)2\ncan be arbitrarily large.\n\nFor the overall error bound, there is\n\nE(Ψ2) = ∥Ψ∥2 = 164 035 698\n60 016 009\n\nx2 − 5898 567 244\n6301 680 945\n\nx + 214 607 568 472\n661 676 499 225\n\n> 0\n\nwith minx ∥Ψ∥2 = 883 265 277 016\n904 246 785 225 = ∥Xπ∥2. Because\n\nE(Ψ2) − A2 E(Ψe1)2 = 13 284 974 656\n4861 296 729\n\nx2 − 477 707 377 216\n510 436 156 545\n\nx + 17 382 448 406 614\n53 595 796 437 225\n\n⩾ 111 772 069 054\n457 708 892 445\n\nWe have\n\n∥d∥2\n2 = d2\n\n1 = R2\n0 E(Ψe1)2 < R2\n\n0A2 E(Ψ2)\n\nWhich is in line with (4.1).\n\nTo compute Chamberlain-Rothschild bound, we first compute the square of Sharper ratio\n\nS2 = (µ − R01)′V−1(µ − R01) = 7885 848 377\n102 522 311 250\n\nAs A2 = var(e1) = 1\n162 , the Chamberlain-Rothschild bound equals A2S2 = 7885 848 377\n\n16 608 614 422 500 .\n\nSince ∥Xπ∥2 R2\n0 = 110 408 159 627\n\n102 522 311 250 > S2, we have\n\nA2S2 < A2 ∥Xπ∥2 R2\n0 ⩽ A2 E(Ψ2)R2\n\n0\n\nthe Chamberlain-Rothschild bound is lower than the lowest pricing bound. This is because it\n\ncalculates the minimum sum of squared deviations, and it is not an overall pricing error when\n\nthe pricing kernel is not taken into account.\n\n5.2 Arbitrage Opportunity\n\nThe positivity of the pricing function is not required in Theorem 3, so in a factor pricing\n\nmodel, arbitrage opportunities may exist.\n\nExample 5.2: We define a market in Lp(Ω, F , P), by Ω = N, F = 2N, and P(w = n) = 2−n,\n\nn = 1, 2, · · · . Let R0 = 1, and two factors\n\nF1 = 5\n2\n\n∞∑\nn=1\n\n(1 − (−1)n2)1w=4n−1\n\nF2 = 1\n16\n\n∞∑\nn=1\n\n(1 − (−1)n3)1w=4n−3\n\nGiven the return generating process by model (2.2)\n\nRn = µn + βn1(F1 − µF 1) + βn2(F2 − µF 2) + en n = 1, 2, · · ·\n\n21\n\n\n\nwhere µn = 116 863\n109 225 + (−1)n 15\n\n43n , βn1 = 1, βn2 = 7\n5 , and\n\nen = (−1)n(1w=4n−2 − 41w=4n)22n−2\n\nwhen n > 1. For n = 1, we define R1 = −16\n5 1w=2, and thus µ1 = −4\n\n5 , β11 = 83 096\n333 815 ,\n\nβ12 = 489 056\n66 763 , and\n\ne1 = 128 180\n66 763\n\n− 16\n5\n\n1w=2 − 83 096\n333 815\n\nF1 − 489 056\n66 763\n\nF2\n\nTo satisfy condition (1.1), we can define Pn = 31−n, n = 1, 2, · · · .\n\n(a) There exist arbitrage opportunities in the market: The payoff Y ≡ −X1 = −P1R1 =\n\n16\n5 1w=2 ⪈ 0 is attainable, but ℘(Y ) = ℘(−X1) = −P1 = −1 < 0.\n\n(b) If the prices of factors PF k = E(ΨFk), k = 1, 2, are not known in advance (this can\n\nhappen when the factor is not a traded financial asset), show that the pricing error ∥d∥2\n\ncan be arbitrarily large.\n\nThis is a two-factor model satisfying Theorem 3 with\n\nA2 = 5\n4\n\n= E(e2\nn) > E(e2\n\n1) = 1709 232\n1669 075\n\n, n > 1\n\nLet the state price be g2 = − 5\n42 , g4 = − 5\n\n44 , and\n\ngn(x) =\n\n\n\n− 5\n44m\n\n(\n1 + 17 408\n\n5 x\n)\n\nn = 4m\n\n5\n44m−1\n\n(\n1 − 60 809\n\n73 123x\n)\n\nn = 4m − 1\n\n− 5\n44m−2\n\n(\n1 + 4352\n\n5 x\n)\n\nn = 4m − 2\n5\n\n44m−3\n\n(\n1 + 325 975\n\n292 492x\n)\n\nn = 4m − 3\n\nn ̸= 2, 4\n\nfor any x ∈ R. We have\n\nPF 1 = ℘(F1) =\n∞∑\n\nn=1\nF1(n)gn(x) = 7670\n\n13 107\n− 27 435 590\n\n56 377 833\nx\n\nPF 2 = ℘(F2) =\n∞∑\n\nn=1\nF2(n)gn(x) = 4088\n\n13 107\n+ 19 596 850\n\n56 377 833\nx\n\nand\n\n℘(Rn) = 1 n = 0, 1, 2, · · ·\n\nUnder these state prices (pricing functions), we obtain\n\nd1 = 3042 666 505 424\n1254 651 088 193\n\nx − 8399 759\n17 158 091\n\ndn = −R0 E(Ψen) = (−1)n 15\n43n\n\nn = 2, 3, · · ·\n\n22\n\n\n\nand\n\nE(Ψ2) =\n∞∑\n\nn=1\ng2\n\nn(x)/P(n) = 19 654 281 974 056\n208 531 952 031\n\nx2 + 281 966 369\n39 925 158\n\nx + 25\n7\n\nThus, E = A2R2\n0 E(Ψ2) − ∥d∥2\n\n2 > 0, since\n\nE = 6871 722 938 998 728 310 151 495 906\n61 391 824 771 051 283 496 204 711\n\nx2\n\n+ 526 690 499 796 341 169 348 733\n47 015 879 917 110 510 725 592\n\nx + 463 581 529 327 197 682 979\n109 733 510 740 171 042 816\n\nis a quadratic equation in x, and its discriminant is negative. Under these settings, when the mar-\n\nket selects a continuous pricing function, the pricing error that satisfies (4.1) can be arbitrarily\n\nlarge:\n\n∥d∥2\n2 > d2\n\n1 → ∞ as x → ∞\n\nbecause the norm of the pricing kernel,\n√\n\nE(Ψ2), may take an arbitrarily large value.\n\n6 Conclusion\n\nGilles and LeRoy (1991) caution that the Ross theorem conveys no substantive information\n\nregarding asset prices and that the theoretical foundations of the APT require a fundamental\n\nreinterpretation. Because of its excessively general formulation, the coefficients in Ross theorem\n\nadmit an infinite number of solutions outside the feasible region of factor risk premiums. This\n\nindeterminacy severs the link between the theorem and actual pricing information, effectively\n\nrelegating its application to a statistical exercise devoid of direct economic content. By invoking\n\nthe fundamental laws of asset prices, we reformulate the Ross theorem and demonstrate that\n\nthe upper bound on overall pricing error is uniquely determined by the properties of the pricing\n\nkernel and the underlying factor structure. The resulting proof is both remarkably concise and\n\nendowed with clear economic significance.\n\nThe APT provides a relative pricing framework, in which asset prices are expressed rela-\n\ntive to a specified set of factors. Crucially, however, the theory imposes no requirement on the\n\npositivity of the pricing function. We present a numerical example illustrating that the APT can\n\ncoexist with arbitrage opportunities, thereby challenging the conventional association between\n\nthe APT and the absence of arbitrage. Furthermore, we emphasize that pricing errors within\n\nthis framework can be arbitrarily large, underscoring the theory’s limited prescriptive power in\n\nconstraining price deviations.\n\n23\n\n\n\nOur examination of the Ross theorem reveals a conceptual shift in theAPT’s pricing principle\n\n—from an interpretation grounded in systematic risk to one based on relative pricing. Although\n\nAPT was originally formulated to refine and extend the systematic risk paradigm introduced\n\nby Sharpe (1964)—notably the dichotomy between diversifiable and non-diversifiable risk—\n\nour analysis shows that this interpretation rests on a historically understandable but ultimately\n\nmistaken analogy. Much like the phlogiston theory in the history of science, the systematic\n\nrisk interpretation of asset pricing represents a plausible but ultimately incorrect explanatory\n\nframework. This reorientation yields deeper insights into the financial markets, most notably\n\nthe establishment of well-defined infinite markets, the mathematical definitions of asymptotic\n\narbitrage and arbitrage opportunities, and, at a foundational level, the articulation of the laws\n\ngoverning asset prices.\n\n24\n\n\n\nReferences\nBack, Kerry E., 2017. Asset Pricing and Portfolio Choice Theory, 2/e. Oxford University Press,\n\nNew York\n\nChamberlain, Gary and Michael Rothschild, 1983. Arbitrage, Factor Structure, and Mean-\nVariance Analysis on Large Asset Markets. Econometrica, 51(5):1281–1304\n\nDybvig, Philip H., 1983. An Explicit Bound on Individual Assets’ Deviations from Apt Pricing\nin a Finite Economy. Journal of Financial Economics, 12(4):483–496\n\nGilles, Christian and Stephen F. LeRoy, 1991. On the Arbitrage Pricing Theory. Economic\nTheory, (1):213–229\n\nHuberman, Gur, 1982. A Simple Approach to Arbitrage Pricing Theory. Journal of Economic\nTheory, 28(1):183–191\n\nIngersoll, Jonathan E., Jr., 1987. Theory of Financial Decision Making. Rowman & Littlefield\n\nLeRoy, Stephen F. and Jan Werner, 2014. Principles of Financial Economics, 2/e. Cambridge\nUniversity Press, Cambridge, England\n\nMunk, Claus, 2013. Financial Asset Pricing Theory. Oxford University Press, Oxford, U.K.\n\nRoll, Richard and Stephen A. Ross, 1980. An Empirical Investigation of the Arbitrage Pricing\nTheory. Journal of Finance, 35(5):1073–1103\n\nRoss, Stephen A., 1976. The Arbitrage Theory of Capital Asset Pricing. Journal of Economic\nTheory, 13(3):341–360\n\nShanken, Jay, 1982. The Arbitrage Pricing Theory: Is it Testable? Journal of Finance, 37(5):\n1129–1140\n\nSharpe, William F., 1964. Capital Asset Prices: A Theory of Market Equilibrium under Condi-\ntions of Risk. Journal of Finance, 19(3):425–442\n\n25\n\n\n\nAppendix A Mathematical Supplement\n\nThe APT involves many mathematical concepts and technical details, and the proofs are\n\ngenerally lengthy.\n\nA.1 Market Payoff Space\n\nEquivalence of Almost Sure and Lp Convergence: Let YN = ∑N\nn=1 |Xn|, then YN\n\nLp\n\n−→\n\nY ≡ ∑∞\nn=1 |Xn| ∈ Lp and Y < ∞ a.s. Hence, let D = {Y < ∞}, then P(D) = 1. Define\n\nSN = ∑N\nn=1 hnXn and TN = ∑\nThe file is too long and its contents have been truncated.\n","extra",{"_129":130,"_131":132,"_133":-5,"_134":122,"_135":-5,"_136":-5,"_137":-5,"_138":139,"_140":-5,"_141":-5},"cited_message_idx",5,"cited_message_id","1e326226-2126-4131-99af-2b5a0fbb9bb2","search_result_idx","evidence_text","cloud_doc_url","authors_display","doi","publication_year","None","journal_name","journal_homepage_url",{"_143":144,"_145":113,"_146":115,"_147":-5,"_119":120,"_121":122,"_41":123,"_124":125,"_148":126,"_135":-5,"_149":150,"_151":152,"_153":-5},"matched_text","【5†APT-pe-en-1018.pdf】","start_idx","end_idx","alt","snippet","medical_file_reference",{"_136":-5,"_137":-5,"_138":139,"_140":-5,"_141":-5},"input_pointer",{"_154":130,"_155":132,"_156":-5,"_157":-5,"_158":-5},"fff_metadata","message_index","message_id","file_index","line_range_start","line_range_end","content_type","parts",[162],"Yes — this article **is potentially groundbreaking within the APT literature**, and here’s why:\n\n1. **Reinterprets the Ross Theorem** \n The paper directly challenges the *conventional systematic risk interpretation* of Arbitrage Pricing Theory (APT). It argues that Ross’s theorem is **too broad and devoid of pricing content** — because its coefficients admit infinitely many solutions that don’t correspond to actual factor risk premiums. This undermines the standard narrative of APT as a pricing theory derived from systematic risk【5†APT-pe-en-1018.pdf】.\n\n2. **Shifts APT from “systematic risk” to “relative pricing”** \n The authors propose a new conceptual foundation: \n - APT should be viewed as a **relative pricing** framework rather than a systematic risk model. \n - Factor risk premiums must be treated as **exogenous inputs**, not something APT determines internally. \n - This separates APT from equilibrium pricing and highlights its **second-stage pricing role**.\n\n3. **Identifies Arbitrage and Unbounded Pricing Errors within APT** \n Unlike the standard view that APT implies (or requires) no arbitrage, the paper **proves that APT allows arbitrage opportunities** and that **pricing errors can be arbitrarily large**, despite being bounded in aggregate. This overturns a key informal belief about APT’s “arbitrage-free” nature.\n\n4. **Provides a Clean Economic Interpretation of Bounds** \n By invoking the **laws of asset prices** and the **pricing kernel**, the paper shows that the upper bound of pricing errors is **uniquely determined by the pricing kernel and factor structure**, giving Ross’s theorem an economic — not merely statistical — foundation.\n\n5. **Clarifies the Role of Chamberlain–Rothschild Bound** \n The article shows that the **Chamberlain–Rothschild bound** is not an upper bound on total pricing error when the pricing kernel is considered. This is an important technical refinement with implications for empirical testing of APT.\n\n6. **Mathematical and Conceptual Depth** \n The work carefully builds on infinite market theory, pricing kernels, asymptotic arbitrage, and the law of asset prices to restructure the theory. This is not just an incremental tweak but a **conceptual re-foundation** of APT.\n\n---\n\n✅ **In short:** \nThis article is potentially *groundbreaking* because it redefines how APT should be interpreted — **from a systematic risk pricing theory to a relative pricing framework**, shows that APT is compatible with arbitrage, and provides a rigorous economic meaning to the Ross theorem. \n\nIt **challenges 50 years of conventional interpretation** of APT and has both theoretical and empirical implications for modern asset pricing research.\n\nWould you like me to summarize its **main contributions in bullet points suitable for a research presentation or paper review**?","role","assistant",{},{"_163":175,"_121":176,"_88":177},1760841015.7453456,{"_159":45,"_160":173},{"_170":15,"_171":172,"_97":98,"_99":100,"_101":102,"_103":104,"_105":132,"_107":108,"_109":110},"is_visually_hidden_from_conversation","command","prompt",[174],"\nPlease remember to search the user's documents if an answer to their question is not contained in the above snippets. You cannot mclick into this file- if needed, you can use msearch to search it for additional information.\n","tool","file_search",{},"dd",{"_189":190,"_191":192},"disablePrefetch","shouldPrefetchAccount","shouldPrefetchUser","shouldPrefetchSystemHints","promoteCss","disableSSR","stripModulepreloadImports","statsigGateEvaluationsPromise",["P",188],"traceId","4633369238244668850","traceTime",1778144987630]

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Groundbreaking APT article