Factor investing is a strategy that chooses assets on attributes that are associated with higher returns.
Examples of factor models include CAPM, Fama/French, APT, and smart beta strategies. The idea is to expose the investor's portfolio to more factor risk to achieve higher expected returns.
Smart Beta strategies often are based on having long and short positions on overperforming/underperforming subindices.
Examples:
Trustworthy 'smart beta' risk premium factors should:
There is also a significant share of academic studies that argue that behavioral biases are the cause behind several factor premiums.
The following models see expected return and risk as a linear relationship. Risk is modeled by the respective factor-beta.
$$ E[r_i] - r_f = \underbrace{ \lambda }{\text{MPR}} \; \times \; \underbrace{ \frac{cov(f,r_i)}{var(f)} }{\text{factor risk exposure}} \\\\ \equiv \lambda \times \beta $$
$MPR$ is the market price of risk and $\beta$ is the amount of risk factor present in asset $i$, $f$ is some abstract factor
If we choose the factor risk to be market risk $f = r_M - r_f$ and the $MPR$ to be the market return in excess of the risk-free rate $E[r_M] - r_f$, we get the Security Market Line of the CAPM.
$$ E[r_i] - r_f = \left(E[r_M]-r_f\right) \, \times \, \frac{cov(r_M-r_f, r_i)}{var(r_M-r_f)} $$
The idea is basically the same as the Single Factor Beta Model, but we now have multiple factors and their respective MPR to take into account. That's why we change the formula to matrix form. Given $F = (f_1, ..., f_k)$ risk factors: