Suppose that you purchase a bond that matures in five years and pays a $13\mathrm{.}76$ percent annual coupon rate. The bond is priced to yield $10$ percent.

a$.$ Show that the duration is equal to $4$ years.

b$.$ You plan to sell the bond after holding $4$ years. Due to the Fed$$s monetary policy, the interest rates rise to $11\%$ and you reinvest the coupon income at $11\%$ four times. Show that you will still earn a $10$ percent yield on your investment $($the realized rate of return$).$