Income Security prep Question 1 A bond with a face value of $1,000 is currently selling for $950. The bond has a coupon rate of 6%, with annual payments, and will mature in 4 years. Calculate the approximate yield to maturity (YTM). Solution: Step 1: Calculate the annual coupon payment: C = 1, 000 · 0.06 = 60 Step 2: Use the approximate YTM formula: YTM ≈ Face Value−Price Years Face Value+Price 2 Coupon Payment + Substitute the values: YTM ≈ YTM ≈ 1,000−950 4 1,000+950 2 60 + 60 + 12.5 72.5 ≈ ≈ 0.0744 (7.44%) 975 975 Answer: The approximate YTM is 7.44%. — Question 2 A bond with a face value of $1,000 has a coupon rate of 5%, with annual payments, and matures in 3 years. The market yield is 6%. Calculate the price of the bond. Solution: 1 Step 1: Calculate the annual coupon payment: C = 1, 000 · 0.05 = 50 Step 2: Use the bond pricing formula: P = n X t=1 C F + (1 + r)t (1 + r)n Substitute the values: P = 50 50 50 1, 000 + + + (1.06)1 (1.06)2 (1.06)3 (1.06)3 Step 3: Calculate each term: 50 ≈ 47.17, (1.06)1 50 ≈ 44.50, (1.06)2 50 ≈ 41.98 (1.06)3 1, 000 ≈ 839.62 (1.06)3 Step 4: Add the components: P = 47.17 + 44.50 + 41.98 + 839.62 = 973.27 Answer: The price of the bond is approximately $973.27. — Question 3 A bond with a face value of $1,000 pays an annual coupon of 4% and matures in 2 years. The market yield is 5%. Calculate the duration of the bond. Solution: Step 1: Calculate the annual cash flows: • t = 1: Cash flow = $40. • t = 2: Cash flow = $1,040 (coupon + face value). 2 Step 2: Calculate the present value of each cash flow: 40 ≈ 38.10, (1.05)1 P V1 = P V2 = 1, 040 ≈ 943.40 (1.05)2 Step 3: Calculate the weighted time for each cash flow: t · P Vt : 1 · 38.10 = 38.10, 2 · 943.40 = 1, 886.80 Step 4: Sum the weighted times and divide by total present value: D= 38.10 + 1, 886.80 1, 924.90 ≈ ≈ 1.96 years 38.10 + 943.40 981.50 Answer: The duration of the bond is approximately 1.96 years. 3