# Expectation of Value Function

I am not sure whether this is the right place to ask this.

I have solved a standard Bellman equation problem. The Value Function $V$ depends on 3 state variables: $K_t$, $X_t$, $Z_t$.

The variables $X$ and $Z$ follow an exogenous process and both have transition probability matrices. The variable $K_{t+1}$ is a choice variable. The problem can be setup in the following fashion:

\begin{equation} V(K_{t}, X_t, Z_{t}) = \max_{k_{t+1}} [\pi_{j,t} + E_t[M_{t+1} V(K_{t+1}, X_{t+1}, Z_{t+1})]] \end{equation}

where $\pi_{j,t}$ is a function of state variables and $M_{t+1}$ is a standard discount factor.

I have numerically solved this problem and I got a policy function for $V$ and $K_{t+1}$, i.e. $V_t = f(K_t, X_t, Z_t)$ and $K_{t+1} = g(K_t, X_t, Z_t)$ where $g$ and $f$ are functions.

What I am looking for is how to get: $E_t[V(K_{t+12}, X_{t+12}, Z_{t+12})]$.

I know how to get: $E_t[V(K_{t+1}, X_{t+1}, Z_{t+1})]$, I can just use the transition probability matrices for $X$ and $Z$ and the policy function for $K_{t+1}$. But how should I go about getting: $E_t[V(K_{t+12}, X_{t+12}, Z_{t+12})]$?