A bond with a par value of $$1,000$ has a coupon rate of $9\%.$ The bond pays interest annually and it has $4$ years until maturity. The investors require a return $($Yield$)$ of $10\%$ at the present time.

a$.$ Calculate the duration and convexity of the bond.

b$.$ Assume that yield goes up from $10\%$ to $10\mathrm{.}5\%.$ Calculate the price change due to duration and convexity.