$10\mathrm{.}3$ Extracting spot rates from coupon$-$paying bonds.

The price of a one$-$year, $3$ percent government bond is $$1,004\mathrm{.}88,$ the price of a two$-$year, $3\mathrm{.}5$ percent government bond is $$1,009\mathrm{.}73,$ and the price of a three$-$year, $4$ percent government bond is $$1,014\mathrm{.}75.$

a$.$ What are the $1-$year, $2-$year, and $3-$year spot rates?

b$.$ What is the yield to maturity of each of the three bonds? Compare each yield to maturity to its corresponding spot rate of the same maturity.