Suppose that the log stochastic discount factor $($SDF$)$ in the economy is an AR$(1)$ process:

$log{m}_{t+1}+=(log{m}_t+)+{}_{t+1},{}_{t+1}N(0,{}_{}^2).$ Recall that with this SDF$,$ the log $1-$year

yield, $log{R}_{t,1},$ is also an AR$(1)$ process, and all other yields are expressed in terms of $log{R}_{t,1}.$

Suppose that the investor's utility discount rate is $=0\mathrm{.}05,$ the persistence parameter is $=0\mathrm{.}8,$ and the

volatility of shocks to the log stochastic discount factor is ${}_{}=0\mathrm{.}05.$ Finally, let $t=0$ be today, $t=1$

be one year from now, etc.

Questions:

a$.$ Using the unconditional distribution of the log SDF in the slides, compute the unconditional

distribution of the $log1-$year yield $($this involves computing $E[log{R}_1]$ and Var$[log{R}_1],$ and

identifying a family of distributions to which the unconditional distribution of $log{R}_1$ belongs$).$

b$.$ Suppose that the current $($spot$)1-$year yield is $r_{0,1}=2\%.$ Using Excel $($preferred$)$ or Matlab,

construct the current zero$-$coupon yield curve for maturities of $1,2,3,4,$ and $5$ years. Present

the result in the form of a graph $($preferred$)$ or table.

c$.$ You decide to purchase today a $5-$year government bond with $2\%$ annual coupon and $$100$ face

value. What is the bond's current price, $p_0?$

d$.$ Suppose that one year from now, the new spot $1-$year yield is $r_{1,1}=4\%.$ What is the new price

of your government bond, $p_1?$ What is the realized return that you have earned over the year?

e$.$ For this part of the question, disregard part d$.$ Suppose that today there is a sudden increase in

the current $($spot$)1-$year yield from $r_{0,1}=2\%$ to $4\%.$ Is the modified duration informative in

figuring out how your government bond's price will change in response to this sudden change?

Why or why not? You can explain your answer intuitively and$/$or using calculations.

f$.($difficult$,$ but try to get as far as you can: partial credit will be given$)$ What is the current

expected next year's price of your government bond, $E_0[p_1]?$ And what is the expected return

you can earn over the year? Comment on similarities$/$differences of your answers to parts $d.$

and $f.$