The risk premium for an individual asset is computed from the factor prices and factor loadings of a company. See "The cross section of factor prices" for more details.

\hat{\gamma_i}=\sum_k \hat{\lambda_{k}} \beta_{i,k}

The marginal contribution of a factor f to the risk premium of an asset j, \Delta\hat{\gamma_{i,f}}, is the difference between \hat{\gamma_i} and the risk premium computed with \beta_{i,f} set to 0 (or equivalent). We are essentially computing the premium without the effect of factor f to see the impact that it has.

\Delta\hat{\gamma_{i,f}}=\sum_k \hat{\lambda_{k}} \beta_{i,k} - \sum_k \hat{\lambda_{k}} \beta_{i,k} (with \beta_{i,f} = 0)

The marginal contribution of a factor f for an index is the weighted average of the asset marginal contributions.

MC_{f,t} = \sum_{i=1}^{n} (w_{i,t-1} \times \Delta\hat{\gamma_{i,f}})

This can be expressed as a percentage of the total factor marginal contributions as per the below. This is a measure of how much a given factor drives the risk premium of an index relative to other factors.

PMC_{f,t} = \frac{MC_{f,t} \times 100}{\sum_f MC_{f,t}}


Note on contribution signs

Risk factors can have a positive or negative contribution to the risk premia: e.g. the profit factor has a negative effect because more profitable infrastructure companies tend to have lower risk premia. Conversely, the leverage factor has a positive effect: higher leverage leads to higher risk premia ceteris paribus.