$($a$)$ Explain why a zero$-$coupon bond has duration equal to its maturity date.

$($b$)$ Find the duration of a $5$ year bond with a face value of $1,000,$ which is sold at par with $12\%$ coupon per year paid semi$-$annually.

$($c$)$ Derive the formulae for estimating the volatility of bond price and thereby calculate the percentage change in bond price and the new bond price if the yield to maturity falls to $11\%.,2+2+(2+4)$