Consider a bond with $2$ year to maturity, $6\%$ coupon rate, face value of $$1000,$ and $8\%$ YTM$.$

Assume the bond pays coupon twice a year, please calculate the $(1)$ Macaulay Duration, $(2)$

Modified Duration, and $(3)$ convexity. $(15$pt$)$

Suppose the YTM of a $6\%$ coupon, $2-$year bond increases from $5\%$ to $8\%.$ The bond has par

value of $$1000.$ What is $(1)$ the actual percentage price change, $(2)$ the estimated percentage price

change using Modified Duration, $(3)$ the estimated percentage price change using Modified

Duration and convexity? $(15$pt$)$