Under the assumptions of Sect. 6.5, suppose that the representative agent is risk neutral.

(i) Consider a security paying the following dividend stream:

\[d_{t}=\bar{d}+\varrho\left(d_{t-1}-\bar{d}\right)+\varepsilon_{t}, \quad \text { for all } t \in \mathbb{N}\]

with \(\mathbb{E}\left[\varepsilon_{t} \mid \mathscr{F}_{t-1}\right]=0\), for all \(t \in \mathbb{N}\). Show that the fundamental value of this security is given by

\[s_{t}^{*}=\left(\frac{\delta}{1-\delta}-\frac{\delta \varrho}{1-\delta \varrho}\right) \bar{d}+\frac{\delta \varrho}{1-\delta \varrho} d_{t}, \quad \text { for all } t \in \mathbb{N}\]

(ii) Consider a security paying the following dividend stream:

\[d_{t}=\beta d_{t-1}, \quad \text { for all } t \in \mathbb{N}\]

for some \(\beta>1\) such that \(\beta \delta<1\). Show that the fundamental value of this security is given by

\[s_{t}^{*}=\frac{\delta \beta}{1-\delta \beta} d_{t}\]

(iii) Consider a security paying the following dividend stream:

\[\log d_{t}=\mu+\log d_{t-1}+\varepsilon_{t}, \quad \text { for all } t \in \mathbb{N}\]

for some \(\mu \in \mathbb{R}\) and where \(\left(\varepsilon_{t}\right)_{t \in \mathbb{N}}\) is a sequence of independent and identically distributed normal random variables with zero mean and variance \(\sigma^{2}\). Show that, if \(\log \delta+\mu+\frac{\sigma^{2}}{2}<0\), the fundamental value of this security is given by

\[s_{t}^{*}=\frac{\delta \mathrm{e}^{\mu+\frac{\sigma^{2}}{2}}}{1-\delta \mathrm{e}^{\mu+\frac{\sigma^{2}}{2}}}, \quad \text { for all } t \in \mathbb{N}\]