Imagine a market where two bonds are traded. Both have a face value of $$1,000$ and a single

annual payment. The first bond is a $5\%$ bullet bond maturing in one year and is traded

at $$1,034\mathrm{.}48.$ The second bond is a $4\%$ bullet bond maturing in two years and is traded at

$$1,019\mathrm{.}71.$ Assume that fractions of bonds can be traded.

a$.$ Using these bonds, show how you can construct a zero$-$coupon bond with face value

$$1,000$ maturing in one year and a zero$-$coupon bond with face value $$1,000$ maturing

in two years.

Answer:

$1-$Year zero$-$coupon bond: buy $0\mathrm{.}95$ of $1-$Year $5\%$ bullet bond and zero of $2-$Year $4\%$

bullet bond.

$2-$Year zero$-$coupon bond: buy $0\mathrm{.}96$ of $2-$Year $4\%$ bullet bond and sell $0\mathrm{.}037$ units of

$1-$Year $5\%$ bullet bond.

b$.$ What are the one$-$ and two$-$year zero$-$coupon yields?

Answer:

P$0,1=982\mathrm{.}76$ r$0,1=1\mathrm{.}75\%$

P$0,2=941\mathrm{.}67$ r$0,2=5\mathrm{.}08\%$

Suppose now that a third bond is introduced in the market. It is a $6\%$ annuity bond maturing

in two years, also having a single annual payment and a face value of $$1,000.$

c$.$ What is the unique no$-$arbitrage price?

Answer:

Buy $0\mathrm{.}52$ of $2-$Year $4\%$ bullet bond and $0\mathrm{.}5$ of $1-$Year $5\%$ bullet bond $$ PA $=1047\mathrm{.}48.$ please show how they got to these answers