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Expected excess returns and risk premia

The instantaneous conditional expected excess return is
\begin{displaymath} r^{xs}\mathrm{d}t = \mathsf{E}\left(\left.\frac{\mathrm{d}P}{P}-r\,\mathrm{d}t\right\vert\mathcal{F}_t\right)\quad, \end{displaymath} (24)

with

\begin{eqnarray*} \frac{\mathrm{d}P}{P} &=& \frac{\mathrm{d}u^\prime}{\mathrm{d... ...\left[\lambda(\xi)\,\mathrm{d}t-\Sigma\mathrm{d}W\right]\quad . \end{eqnarray*}

Since $a^\star-a = \lambda^1$, $A^\star-A = \Lambda$ the conditional expected excess return is
\begin{displaymath} r^{xs} = u^\prime\lambda^1 + u^\prime\,\Lambda\,\xi\quad , \end{displaymath} (25)

and the conditional instantaneous excess return variance is
\begin{displaymath} \mathsf{var}\left(r^{xs}\right) = u^\prime\,V\,u\quad . \end{displaymath} (26)

For a choice of $n$ key rates ${\mathsf y_{\mathrm{kr}}}$ the conditional expected excess return is

\begin{eqnarray*} \mathsf E\left[r^{xs}\vert{\mathsf y_{\mathrm{kr}}}\right] ... ...sf U\Lambda\mathsf U_{\mathrm{kr}}^- {\mathsf v}\right] \quad . \end{eqnarray*}


Markus Mayer 2009-06-22