1v and vi more detailed solutions

Question 1 Consider the four bonds in the table below. All of them pay coupons annually and all have the same yield to maturity of 15%; Bond Face value Coupon rate Years to maturity W $1000 0.12 3 X $1000 0.03 3 Y $1000 0 3 Z $1000 0 1 i) Price each of these bonds to the nearest cent. [4 marks] ii) iii) Consider the Macaulay durations of these bonds. Without relying on a set of computations, discuss the ranking of Macaulay durations that these bonds must exhibit. Explain why the ranking must be as you say. [3 marks] Now compute (or state) the precise duration of each bond and describe what these values imply for the sensitivity of bond prices to the yield to maturity. [3 marks] If bond W's yield to maturity was to rise to 20%, estimate the change in its price using the duration. Give a written and graphical description of how the price is estimated using the duration, why estimation error arises and then explain how you might modify the estimation procedure to make it more accurate. [6 marks] iv) v) You have a portfolio containing 2 different bonds, each with 3 years to maturity, $1000 face value and the same yield to maturity of y. You have bought N, units of the first bond and N units of the second bond. Thus the total value of your bond portfolio is; V = N P+ N P2 where P, and P are the prices of the two bonds. Show that the Macaulay duration of the portfolio is equal to; DM = (N P D + N P2 D2)N where D, D and DM are the durations of the two bonds and the portfolio respectively. [6 marks] vi) Use your result from (v) to create a bond portfolio with Macaulay duration exactly equal to 2.5 from bonds X and Z above. [3 marks] 33 SOLUTION a) i) to iii) iv) v) vi) yield Bond W X Y Z CF DCF Duration 0.15 Face value Coupon rate 1000 0.12 1000 0.03 1000 0 1000 0 1 W X 120 30 120 30 1120 1030 1000 104.35 26.09 0.00 869.57 90.74 22.68 0.00 0.00 736.42 677.24 657.52 0.00 931.50 726.01 657.52 869.57 104.35 26.09 0.00 869.57 181.47 45.37 0.00 0.00 2209.25 2031.73 1972.55 0.00 2.68 2.90 3.00 1.00 0.05 Delta y Delta P/P -0.11645864 Delta P -108.481602 V=N1P1+ N2P2 Take derivative w.r.t (1+y) and multiply by (1+y)/V to get duration of portfolio This is easily rearranged to give the result 2.5 Target Weight X D_P 0.79076749 2.5 1 2 3 1 2 3 Price 1 2 3 D Maturity 333 Y 0 0 Price 931.50 726.01 657.52 869.57 Z 1000 0 0 We W ai As W Thus Question 1 Consider the four bonds in the table below. All of them pay coupons annually and all have the same yield to maturity of 15%; Bond Face value Coupon rate Years to maturity W $1000 0.12 3 X $1000 0.03 3 Y $1000 0 3 Z $1000 0 1 i) Price each of these bonds to the nearest cent. [4 marks] ii) iii) Consider the Macaulay durations of these bonds. Without relying on a set of computations, discuss the ranking of Macaulay durations that these bonds must exhibit. Explain why the ranking must be as you say. [3 marks] Now compute (or state) the precise duration of each bond and describe what these values imply for the sensitivity of bond prices to the yield to maturity. [3 marks] If bond W's yield to maturity was to rise to 20%, estimate the change in its price using the duration. Give a written and graphical description of how the price is estimated using the duration, why estimation error arises and then explain how you might modify the estimation procedure to make it more accurate. [6 marks] iv) v) You have a portfolio containing 2 different bonds, each with 3 years to maturity, $1000 face value and the same yield to maturity of y. You have bought N, units of the first bond and N units of the second bond. Thus the total value of your bond portfolio is; V = N P+ N P2 where P, and P are the prices of the two bonds. Show that the Macaulay duration of the portfolio is equal to; DM = (N P D + N P2 D2)N where D, D and DM are the durations of the two bonds and the portfolio respectively. [6 marks] vi) Use your result from (v) to create a bond portfolio with Macaulay duration exactly equal to 2.5 from bonds X and Z above. [3 marks] 33 SOLUTION a) i) to iii) iv) v) vi) yield Bond W X Y Z CF DCF Duration 0.15 Face value Coupon rate 1000 0.12 1000 0.03 1000 0 1000 0 1 W X 120 30 120 30 1120 1030 1000 104.35 26.09 0.00 869.57 90.74 22.68 0.00 0.00 736.42 677.24 657.52 0.00 931.50 726.01 657.52 869.57 104.35 26.09 0.00 869.57 181.47 45.37 0.00 0.00 2209.25 2031.73 1972.55 0.00 2.68 2.90 3.00 1.00 0.05 Delta y Delta P/P -0.11645864 Delta P -108.481602 V=N1P1+ N2P2 Take derivative w.r.t (1+y) and multiply by (1+y)/V to get duration of portfolio This is easily rearranged to give the result 2.5 Target Weight X D_P 0.79076749 2.5 1 2 3 1 2 3 Price 1 2 3 D Maturity 333 Y 0 0 Price 931.50 726.01 657.52 869.57 Z 1000 0 0 We W ai As W Thus