The *x*_{t} random variable thus moves slowly (depending on *ϕ*), with a tendency to return to its mean value. Lastly, mean reversion is captured by assuming cov(*η*_{t+1}, *u*_{t+1})=*σ*_{ηu}<0, which translates, as per below, into a statement about risky return autocorrelations:

$\begin{array}{l}0>{\sigma}_{\eta u}=\mathrm{cov}({\tilde{u}}_{t+1},{\tilde{\eta}}_{t+1})={\mathrm{cov}}_{t}[({\tilde{r}}_{t+1}-{E}_{t}{\tilde{r}}_{t+1}),({\tilde{x}}_{t+1}-\overline{x}-\varphi ({x}_{t}-\overline{x}))]\hfill \\ ={\mathrm{cov}}_{t}({\tilde{r}}_{t+1},{\tilde{x}}_{t+1})\hfill \\ ={\mathrm{cov}}_{t}\left({\tilde{r}}_{t+1},{E}_{t}{\tilde{r}}_{t+2}-{r}_{f}+\frac{{\sigma}_{u}^{2}}{2}\right)\hfill \\ ={\mathrm{cov}}_{t}\left({\tilde{r}}_{t+1},{\tilde{r}}_{t+2}-{\tilde{u}}_{t+2}-{r}_{f}+\frac{{\sigma}_{u}^{2}}{2}\right)\hfill \\ ={\mathrm{cov}}_{t}({\tilde{r}}_{t+1},{\tilde{r}}_{t+2})\hfill \end{array}$

a high return today reduces expected returns next period. Thus,

${\mathrm{var}}_{t}({\tilde{r}}_{t+1}+{\tilde{r}}_{t+2})=2{\mathrm{var}}_{t}({\tilde{r}}_{t+1})+2{\mathrm{cov}}_{t}({\tilde{r}}_{t+1},{\tilde{r}}_{t+2})<2{\mathrm{var}}_{t}({r}_{t+1})$

in contrast to the independence case. More generally, for all horizons *k*,

$\frac{{\mathrm{var}}_{t}({\tilde{r}}_{t+1}+{\tilde{r}}_{t+2}+\cdots +{\tilde{r}}_{t}}{}$

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