Arbitrage pricing theory ($\mathbb{E}{r}_i-R={\sum }_j{b}_{ij}{\lambda }_j$)

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.

For centered residual random variables ${ϵ}_i$ with covariance matrix $\Gamma$, the residual of a portfolio has variance $w^T\Gamma w$. Pairwise uncorrelated residuals with uniformly bounded variances and weights $|w_i|\le W/n$ give $\mathrm{Var}(w^Tϵ)\le s^2{W}^2/n$, hence convergence to zero in mean-square convergence. For nonnegative weights summing to one, the sharper bound is $s^{2}W/n$. With correlated residuals, $|w_i|\le W/n$ and ${\sum }_{i,j}|{\Gamma }_{ij}|=o(n^2)$ also suffice. Individual bounded variances alone do not: if every residual equals the same nondegenerate centered $Z$, every fully invested portfolio retains residual $Z$. A one-sided upper bound on signed weights is insufficient even for uncorrelated residuals because it allows a fixed large short selling.

Suppose asset returns have the exact representation $r=a+Bf$, allowing unrestricted long and short portfolios. A position $v$ with $1^{T}v=0$ and $B^{T}v=0$ costs zero and has constant financial payoff $a^{T}v$. Absence of arbitrage therefore forces $a^{T}v=0$, since either sign of $v$ is allowed. The orthogonal complement identity gives $a\in \mathrm{span}(1,\mathrm{col}B)$. Taking expectations proves the displayed pricing relation, with $\lambda$ equal to the coefficients of $B$ in $a$ plus $\mathbb{E}f$. When $[1B]$ has full column rank, ${\lambda }_0$ is the return of a unit-cost zero-exposure portfolio and ${\lambda }_j$ is the expected return of a zero-cost portfolio with exposure one to factor $j$ and zero to the others. Without that rank condition the coefficients need not be unique, and a traded zero-exposure portfolio need not exist.

For returns $r_i=a_i+{\sum }_j{b}_{ji}{f}_j$, fund a risky position $v$ by a risk-free asset. Its zero-cost payoff is $(a-R1{)}^{T}v+{\sum }_j(Bv{)}_j{f}_j$. If $Bv=0$, a nonzero constant term would create an arbitrage after choosing the sign of $v$. Hence $a-R1$ annihilates $\ker B$ and lies in the row span of $B$. Taking expectations gives exact factor-premium pricing. Full row rank makes the coefficients unique; if exposures have smaller rank, only the effective spanned factors are identified.