Expected excess returns and risk premia

The instantaneous conditional expected excess return is

$$r^{xs}\mathrm{d}t = \mathsf{E}\left(\left.\frac{\mathrm{d}P}{P}-r\,\mathrm{d}t\right\vert\mathcal{F}_t\right)\quad,$$ (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

$$r^{xs} = u^\prime\lambda^1 + u^\prime\,\Lambda\,\xi\quad ,$$ (25)

and the conditional instantaneous excess return variance is

$$\mathsf{var}\left(r^{xs}\right) = u^\prime\,V\,u\quad .$$ (26)

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

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