Classical financial theory states that the average return of a stock is the payoff to investors for taking on risk. One way of expressing this risk-reward relationship is through a factor model. A factor model can be used to decompose the returns of a security into factor-specific and asset-specific returns,

where β_{i,1}, β_{i,2}, …, β_{i,K} are the factor exposures of stock *i, f*_{1,t}, *f*_{2,t}, …, *f*_{K,t} are the factor returns, α* _{i}* is the average abnormal return of stock

This factor model specification is *contemporaneous*, that is, both left- and right-hand side variables (returns and factors) have the same time subscript, *t*. For trading strategies one generally applies a *forecasting* specification where the time subscript of the return and the factors are *t + h* (*h* ≥ 1) and *t*, respectively. In this case, the econometric specification becomes

How do we interpret a trading strategy based on a factor model? The explanatory variables represent different factors that forecast security returns, each factor has an associated factor premium. Therefore, future security returns are proportional to the stock's exposure to the factor premium,

and the variance of future ...

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