Portfolio A consists of a one-year zero-coupon bond with a face value of $2,000 and a 10-year zero-coupon bond with a face value of $6,000. Portfolio B consists of a 7-year zero-coupon bond with a face value of $5,000. The current yield on all bonds is 4.51% per annum (continuously compounded).

(a) Show that both portfolios have (almost) the same duration.

(b) Show that the percentage changes in the values of the two portfolios for a 0.15% per annum increase in yields are the same.

(c) What are the percentage changes in the values of the two portfolios for a 6% per annum increase in yields?

(d) What are the convexities of the two portfolios?

(e) To what extent does (a) duration and (b) convexity explain the difference between the percentage changes calculated in part (c)? Show with calculations