Example:
Let's say we have an AR(1) model for which we are trying to find conditional and unconditional risk.
$$ x_{t+1} = \phi \times x_t + \epsilon_{t+1}, x_0 = 0, \epsilon_{t+1} \sim i.i.d(0, \sigma_\epsilon^2) $$
The conditional 1-step ahead risk measure is:
$$ \begin{align*}Var(x_{t+1}|\mathcal{F}{t}) &\overbrace{=}^{\text{AR(1)}} Var(\phi \times x{t} + \epsilon_{t+1} |\mathcal{F}t) \\\\& = \phi^2 Var(x_t|\mathcal{F}t) + Var(\epsilon{t+1}|\mathcal{F}t) \\\\& = \phi^2 \times 0 + \sigma^2{\epsilon} \\\\& = \sigma^2{\epsilon}.\end{align*} $$
The conditional 2-step ahead risk measure is
$$ \begin{align*} Var(x_{t+2}|\mathcal{F}{t}) & \overbrace{=}^{\text{AR(1)}} Var(\phi \times x{t+1} + \epsilon_{t+2} | \mathcal{F}{t}) \\\\ & = Var(\phi^2 \times x_t + \phi \times \epsilon{t+1} + \epsilon_{t+2} |\mathcal{F}{t} ) \\\\ & = \phi^4 Var(x_t|\mathcal{F}t) + \phi^2 Var(\epsilon{t+1}|\mathcal{F}t) + Var(\epsilon{t+2}|\mathcal{F}t) \\\\ & = 0 + \phi^2 \sigma^2{\epsilon} +\sigma^2{\epsilon} \\\\ & = (1 + \phi^2) \times \sigma^2_{\epsilon}. \end{align*} $$
The conditional k-step ahead risk measure is
$$ \begin{align*} Var(x_{t+k} | \mathcal{F}t) &= (\phi^0+\phi^2 + ... + \phi^{2(k-1)} ) \times \sigma^2_{\epsilon}. \end{align*} $$
The unconditional risk measure can be found using the stationarity condition (time-invariance variances):
$$ \begin{align*}Var(x_{t+1}) &= \phi^2 Var(x_t) + \sigma^2_{\epsilon} \\\\Var(x_t) &= \frac{\sigma^2_{\epsilon}}{1-\phi^2}.\end{align*} $$
AR with
$|\phi| = 1$
TODO How to deal with unit roots