#$2$a A $6\%$ coupon bond paying interest annually has a modified duration of $10$ years, sells for $$800,$ and is priced at a yield to maturity of $8\%.$ If the YTM increases to $9\%,$ what is the predicted change in price based on the bond's duration?

$2$b A $6\%$ coupon bond with semiannual coupons has a convexity $($in years$)$ of $120,$ sells for $80\%$ of par, and is priced at a yield to maturity of $8\%.$ If the YTM increases to $9\mathrm{.}5\%,$ what is the predicted contribution of convexity to the percentage change in price due to convexity?

$2$c A bond with annual coupon payments has a coupon rate of $8\%,$ yield to maturity of $10\%,$ and Macaulay's duration of $9$ years. What is the bond's modified duration?