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Arbitrage pricing theory relates expected returns to systematic factor loadings and their risk premiums. With idiosyncratic noise, a diversification or asymptotic argument needs additional assumptions. With no idiosyncratic noise and spanned factors, the formula is an exact consequence of law of one price: portfolios matching factor exposures can differ only by a constant payoff, which absence of arbitrage forces to be zero.

Use unrestricted long and short portfolios, as required by the usual arbitrage pricing theory argument. A vector $v\in {\mathbb{R}}^n$ of initial monetary positions with $1^{T}v=0$ costs zero. Its terminal financial payoff, in the exact one-factor model, isIf also $v^{T}b=0$, the financial payoff is the constant $v^{T}a$. A nonzero value would be an arbitrage after choosing the sign of $v$. Absence of arbitrage thus says that $a$ annihilates $\ker [1b{]}^T$. By finite-dimensional linear algebra,Write $a={\lambda }_{0}1+\kappa b$. Taking expectations, assuming the factor has finite mean, provesThis is exact factor pricing without a traded risk-free asset. The PDF prints an asset-indexed ${\lambda }_i$; the stronger valid result has the same ${\lambda }_1$ for every asset, so it also satisfies the printed relation by taking all its ${\lambda }_i$ equal. If every loading is equal, the spanning vectors are dependent and the coefficients need not be unique. For a nondegenerate factor and a zero-exposure unit-cost portfolio, ${\lambda }_0$ is that portfolio's certain return; if a risk-free asset is explicitly traded, law of one price makes it the risk-free asset's return. No equilibrium preferences are needed for this argument.

The capital asset pricing model givesThus it has the same affine relation between expected returns and a single systematic exposure. When the factor is the market portfolio's excess return, the loadings are the beta of an asset, ${\lambda }_0=r_f$, and ${\lambda }_1=\mathbb{E}{r}_M-r_f$. More generally, for a nondegenerate exact one-factor model and a market portfolio with loading $b_M\ne 0$, ${\beta }_i=b_i/b_M$, and the pricing relation can be rewritten using that exposure.

The interpretations and assumptions differ. The CAPM is an equilibrium relation obtained with a mean-variance optimization framework, common beliefs, and suitable borrowing/lending and market-clearing assumptions. Arbitrage pricing theory uses absence of arbitrage, factor structure, and in realistic models diversification; it does not identify a generic factor with the market portfolio. Moreover the CAPM permits asset-specific risk, whereas the exact model in part (a) has no residual risk. The two formulas coincide for a suitable market factor, but no-arbitrage factor pricing by itself does not establish the CAPM's equilibrium assumptions.

For weights summing to one, collect the exact portfolio return asThe desired approximation means that $e_n\to 0$, for example in mean-square convergence. Centering the residuals gives $\mathbb{E}{e}_n=0$, but its variance isThis identifies an omission in the printed hypotheses: bounded individual variances do not control the cross terms. Indeed, take ${ϵ}_i=Z$ for every $i$, where $Z$ is a nondegenerate centered random variable with $\mathrm{Var}Z<s^2$. Equal weights $1/n<W/n$ for any $W>1$ retain $e_n=Z$ at every $n$. The claimed diversification conclusion does not follow from the printed assumptions alone.

Under the usual extra assumption that the residuals are pairwise uncorrelated and the weights are nonnegative, the advertised calculation isThe Chebyshev inequality then gives $\mathrm{Pr}(|e_n|>\delta )\le s^{2}W/(n{\delta }^2)\to 0$. Alternatively, signed weights satisfying $|w_i|\le W/n$ give the bound $s^2{W}^2/n$. These are instances of the covariance criterion for diversification of factor residuals. More generally, with that absolute weight bound, ${\sum }_{i,k}|\mathrm{Cov}({ϵ}_i,{ϵ}_k)|=o(n^2)$ suffices.

If short selling are allowed, the printed one-sided weight bound is another insufficiency. Choose $0<c<W-1$, put $w_1=-c$ and $w_i=(1+c)/(n-1)$ for $i\ge 2$, and let only ${ϵ}_1$ be a nondegenerate centered residual. For all sufficiently large $n$ the one-sided bounds hold, but $e_n=-c{ϵ}_1$ never vanishes. Thus nonnegative weights or an absolute bound is needed.

With the repaired assumptions, well-diversified portfolios become approximately exposed only to the common factors, and their residual standard deviation is $O(n^{-1/2})$ in the uncorrelated case. The averaged coefficients may depend on $n$; convergence to fixed coefficients needs additional assumptions. In the associated asymptotic arbitrage pricing theory, diversifiable risk cannot command a persistent premium in well-diversified portfolios under absence of asymptotic arbitrage, while common-factor exposure can. This does not prove an exact pricing equation for every individual noisy asset from the finite-market hypotheses alone.

When the two factor-exposure rows are independent, every three-column excess-coefficient matrix being singular means that the entire coefficient matrix has rank at most two. The constant excess-return row must therefore be in the span of the two exposure rows. There are common constants ${\theta }_1,{\theta }_2$ such thatTo see that the constants are common, fix two assets with independent exposure columns, solve their two equations for ${\theta }_1,{\theta }_2$, and apply the zero determinant with any third asset; its constant entry is forced to satisfy the same equation. Taking expectations gives the exact two-factor arbitrage pricing theory formulaAn asset with unit loading on one factor and zero loading on the other has excess expected return equal to that factor's premium. This recovers part (i).

The rank qualification matters: singularity by itself would not identify two premiums if both exposure rows were dependent. More generally, write the exposure matrix as $B$. A position $v\in \ker B$ has a constant excess payoff $(a-R1{)}^{t}v$. Absence of arbitrage forces this constant to vanish, so $a-R1$ annihilates $\ker B$ and belongs to the row space of $B$. This proves exact factor pricing without idiosyncratic risk also in a reduced-rank model, though factor premiums need not then be unique. Independence of the random factors is not a substitute for their being spanned by traded exposures.

Form a portfolio with weights $\beta$ in the first factor-exposure asset, $\gamma$ in the second, and $1-\beta -\gamma$ in the risk-free asset. Its total initial investment is one. Its two factor exposures match those of asset $k$, so subtracting its return from $r_k$ leaves a constant. Any nonzero constant difference would allow a zero-cost long-short position with a strictly positive certain payoff after choosing its direction, an arbitrage. Under absence of arbitrage, the returns must therefore have the same expectation. ThusThe coefficients are the two factor risk premiums relative to the risk-free return, not the expected factor levels themselves. This reasoning permits negative weights, as assumed in the unconstrained arbitrage pricing theory model.