Term Structure theory: $(20$ points$)$ Suppose you have identified the following spot

interest rates:

$($a$)$ Assume that one$-$ and two$-$year pure discount $($zero coupons$)$ bonds each with

face value $$1,000$ are available. If you want to guarantee receipt of $$3,000$ each

at the end of Year $1$ and Year $2,$ how much money would you have to invest

today? $(5$ points$)$

$($b$)$ A coupon bond with a face value of $$1,000,$ three years to maturity, and annual

coupon payments can be purchased at $$1,300.$ What is the annual coupon rate

of the bond? $(5$ points$)$

$($c$)$ Suppose today a bank offers you a four$-$year forward contract covering years $2,$

$3,4,$ and $5$ at a rate $(f_{25})$ of $5\%$ per annum, which allows you to either borrow

or lend $$150,000$ at the beginning of year $2.$ Is there an arbitrage opportunity?

If yes, how can you make an arbitrage profit that is generated only at the last

period $,$ year $5)?$ Please describe the exact transactions that you need

to make in order to obtain an arbitrage profit. Assume that one$-,$ two$-,$ three$-,$

four$-,$ and five$-$year pure discount $($zero$-$coupon$)$ bonds each with $$1,000$ face

value $($labeled as Bond $1,$ Bond $2,$ Bond $3,$ Bond $4,$ and Bond $5,$ respectively$)$ are

available for trading at their fair values today. Note that the strategy can only

use the four$-$year forward contract at a rate $(f_{25})$ and the pure discount bonds.

$(5$ points$)$

$($d$)$ Without doing any calculations, explain whether a $3-$year $4\%$ coupon bond or a

$5-$year $4\%$ coupon bond should have a higher yield$-$to$-$maturity. $(5$ points$)$