Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 32 / 1 / c

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.

Let $M$ denote the displayed matrix of excess-return coefficients. If it were invertible, choose $v$ with $Mv=(1,0,0{)}^t$. Hold $v_i$ units of the three risky investments and short ${\sum }_i{v}_i$ units of the risk-free asset. Initial net cost is zero. The terminal payoff isThis is a guaranteed arbitrage, contradicting the assumption. HenceThe contradiction does not require guessing probabilities for the factors: invertibility would allow simultaneous cancellation of both random exposures while keeping a nonzero constant gain.

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.