Risk Premium Contribution

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.

$$//<![CDATA[ \begin{array}{l}\displaystyle \hat{\gamma_i}=\sum_k \hat{\lambda_{k}} \beta_{i,k}\end{array} //]]>$$

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

$$//<![CDATA[ \begin{array}{l}\displaystyle \Delta\hat{\gamma_{i,f}}=\sum_k \hat{\lambda_{k}} \beta_{i,k} - \sum_k \hat{\lambda_{k}} \beta_{i,k} (with \beta_{i,f} = 0)\end{array} //]]>$$

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

$$//<![CDATA[ \begin{array}{l}\displaystyle MC_{f,t} = \sum_{i=1}^{n} (w_{i,t-1} \times \Delta\hat{\gamma_{i,f}})\end{array} //]]>$$

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.

$$//<![CDATA[ \begin{array}{l}\displaystyle PMC_{f,t} = \frac{MC_{f,t} \times 100}{\sum_f MC_{f,t}}\end{array} //]]>$$