Suppose that the price of interest rate risk is constant and that the spot rate $r_t$ follows the first$-$order autoregressive process under the risk$-$neutral measure:

$r_{t+1}=(1-{}^N){}^N+{}^N{r}_t+u_{t+1}^N,$

where $u_{t+1}^N$ i$.$i$.$d$.$N $(0,{}^2),$ so that the logarithm of a zero$-$coupon bond price with $n$ periods to maturity $)=(log{D}_{n,t}$ if an affine function of the short rate:

$-d_{n,t}=a_n+b_n{r}_t.$

$($a$)$ Derive a formula for the coefficients $a_n$ and $b_n$ in terms of the parameters of the short rate dynamics.

$($b$)$ Evaluate:

$E_t^N[d_{n-1,t+1}]-d_{n,t}.$

$($c$)$ Assuming that derive the process for a one$-$period forward rate $f_{n,t}$ in terms of the risk$-$neutral parameters for a very long maturity, $n.$ What are the implications of the model for the long forward rate? $[$Hint: Note that $f_{n,t}=[$:$d_{n,t}-d_{n+1,t}*\}$

$($d$)$ Now assume that the short rate follows a random walk:

$r_{t+1}=r_t+u_{t+1}^N.$

Derive the process for a one$-$period forward rate $f_{n,t}$ for a very long maturity, $n.$ What are the implications of the model for the long forward rate?