You observe the following prices of Treasury securities per $$100$ of par value:

$$ a $6-$month T$-$bill sells for $96\mathrm{.}1538$

$$ a $1-$year T$-$bill sells for $94\mathrm{.}2596$

$$ a $1\mathrm{.}5-$year $10\%$ coupon T$-$bond sells for $104\mathrm{.}9142$

Assume that the pure expectations theory of the term structure holds.

a$.)$ Calculate the $6-$month, $1-$year, and $1\mathrm{.}5-$year spot rates on a bond$-$equivalent yield basis.