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The xt 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(ηt+1, ut+1)=σηu<0, which translates, as per below, into a statement about risky return autocorrelations:

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

${\mathrm{var}}_{t}\left({\stackrel{˜}{r}}_{t+1}+{\stackrel{˜}{r}}_{t+2}\right)=2{\mathrm{var}}_{t}\left({\stackrel{˜}{r}}_{t+1}\right)+2{\mathrm{cov}}_{t}\left({\stackrel{˜}{r}}_{t+1},{\stackrel{˜}{r}}_{t+2}\right)<2{\mathrm{var}}_{t}\left({r}_{t+1}\right)$
$\frac{{\mathrm{var}}_{t}\left({\stackrel{˜}{r}}_{t+1}+{\stackrel{˜}{r}}_{t+2}+\cdots +{\stackrel{˜}{r}}_{t}}{}$