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.