 Solution

Let $B$ be the $n\times m$ matrix of factor loadings, so $r=a+Bf$. A zero-cost portfolio $v$ with $1^{T}v=0$ and $B^{T}v=0$ again has constant financial payoff $v^{T}a$. Absence of arbitrage impliesTaking expectations givesIf $[1B]$ has full column rank, a unit-cost portfolio with zero exposure to all factors exists and earns the certain rate ${\lambda }_0$. For each $j$, choose a zero-cost portfolio with loading one on factor $j$ and zero on the others; its expected return is ${\lambda }_j$. Hence these coefficients are the rewards per unit systematic exposure, or factor risk premiums, with the financial payoff measured per unit of the chosen factor normalization. A traded risk-free asset fixes ${\lambda }_0$ to its return. Changing the scale or basis of the factors changes the coordinates of the risk premiums but leaves asset prices unchanged. With deficient rank, only the spanned factor directions are identified and the coefficients need not be unique.