Consider a bond with annual payments of $$100,$ a principal payment of $$1,000$ in $10$ years, and a cost of $$1000.$ Assume a flat yield curve with a $10\%$ yield$-$to$-$maturity and annual compounding.

$($a$)$ What is the Duration of the bond?

$($b$)$ What is the Convexity of the bond?

Suppose the yield of the bond now drops to $9\mathrm{.}5\%.$

$($c$)$ What is the estimated percentage price change using both the Duration and Convexity measures?

$($d$)$ What is the actual percentage price change?