# 1.3.2 Risk-factor pricing

Beyond single- or multi-factor CAPMs, the second building block of modern asset pricing is Ross' (1976) arbitrage pricing theory (APT). Under arbitrage pricing, pure arbitrage profits are not possible, and differences between expected returns (or prices) and actual returns are the result of individual assets' relative exposure (betas) to a combination of zero-mean risk factors (or 'surprises').

Hence,

where .

All available information at the time of pricing is already incorporated in expected returns, and the impact of factors on returns *ex-post* is the combination of individual assets' exposure to these factors (the ) and any unexpected change or 'surprise' in the realisation of the, which have an expected value of zero.

Here again, expected excess returns are written as the linear combination of multiple risk-factor exposures and their risk premia or prices on the measurement date, that is,

where is the price of risk or premium of the risk factor with , and is the re-defined measure of an asset's exposure to the risk factor, which is equal to if factors are standardised to have a standard deviation of .

That is, expected (excess) returns are the sum of the risk-factor exposures of asset at time times the respective price of each risk factor to which asset is exposed.

Realised excess returns are thus written:

Thus, realised excess returns for asset are generated by its exposure to risk factors and their unexpected realisation, as well as firm-specific risk. Because we will work only with standardised factors, there is no distinction between and , and for notational convenience, we will only use to write expected returns, so that:

where .

The price of asset today should thus equal the sum of all future cash flows discounted at the APT rate defined by equation (3), in which expected returns for asset are a linear function of various factors and the sensitivity to changes in each factor, as represented by each asset-specific beta coefficient.

Multi-factor models focus on finding robust and persistent statistical effects that can explain and predict the price preferences of buyers and sellers over time. Modelling discount factors reflecting the fair value of financial assets is thus a matter of identifying relevant priced risk factors and determining both individual asset's exposure to each factor as well as the market price or required premia of pure exposure to each such factor taken independently.

In effect, CAPM and APT are the only two theoretical frameworks that provide a solid foundation for computing the trade-off between risk and returns. APT, in particular, leaves the identification of the relevant factors open and justifies the need to explicitly identify and test the factors impacting returns for different types of financial assets.

_{Ross, S. (1976). The arbitrage theory of capital asset pricing. }_{The Journal of Economic Theory. 13(3), }_{341-360.}