$[20$ points$]$ It is December $12,2016.$ You can trade five government securities: a

$6-$month T$-$bill, $1-$year note, a $2-$year note that expires in $2$ years, a $2-$year note that

expires on June $12,2018,$ and $30-$year coupon bond that matures on December $12,$

$2018$ with a $4\%$ semi$-$annual coupon. The bond has just made a coupon payment,

and it pays on Dec $12$ and June $12.$ Assume that all prices are "clean." The bond

has a face value of $$1,000,$ the T$-$bills and the notes have face value $$10,000.$ There

are no other securities available to you. There are no commissions. Use $P_{6m},P_{1y},$

$P_{2ya},P_{2yb}$ and $P_{30yc}$ for the $6-$month T$-$bill, the $1-$year note, the two$2-$year notes,

and the $30-$year bond respectively.

$($a$)$ Provide one no$-$arbitrage condition that must hold regarding the prices of these

five securities$.$

$($b$)$ Describe an algorithm that uses market orders to take advantage of an arbitrage

opportunity among the five securities$,$ should it arise.