Question $2.$

$2-$a$)$ Consider a bond with a face value of $$100$ that matures in $10$ years and pays quarterly coupons of

$$1$ each. Suppose the bond is trading at par, and its price remains constant. If you invest $$1000$ today and

reinvest the coupon payments to buy more bonds $($including fractional amounts$),$ how many bonds will you

hold at $t=0,1,2,$dots,$10$ years?

$[$Hint: Find a recursive relation for the number of bonds you have at each quarter.$]$

$2-$b$)$ Suppose there are two zero$-$coupon bonds with face values of $$100$ and maturities of $1$ and $2$ years.

As of today $(t=0),$ the $1-$year bond is priced at $$95\mathrm{.}38,$ and the $2-$year bond is priced at $$91\mathrm{.}78.$ These are

real market prices as of January $4,2024.$

If you plan to buy a $1-$year zero$-$coupon bond one year from today $(t=1)$ and the price is agreed upon

today, what should that price be$?$

$[$Hint: To solve this question, construct two portfolios: one that buys $1-$year maturity bonds twice, and

another that buys a certain amount of $2-$year maturity bonds. Use the replicating lemma to conclude.$]$