A 20-year U.S. Treasury bond with a face value of $1,000 pays a coupon of 5.50%

Question

A 20-year U.S. Treasury bond with a face value of $1,000 pays a coupon of 5.50% (2.750% of face value every six months). The reported yield to maturity is 5.2% (a six-month discount rate of 5.2/2 2.6%). (Do not round intermediate calculations. Round your answers to 2 decimal places.) a. What is the present value of the bond? Present value b. If the yield to maturity changes to 1%, what will be the present value? Present value C. If the yield to maturity changes to 8%, what will be the present value? Present value d. If the yield to maturity changes to 15%, what will be the present value? Present value

Explanation / Answer

Bond Price= C x [1-{1/(1+r)n}]/r + M/(1+r)n

M= Face Value = $1,000

C = Coupon amount= $1,000 x 2.75%=$27.50

n= no of periods = 20 yrs x 2=40

Bond Price= C x [1-{1/ (1+r)n}]/r + M/(1+r)n

                      =$27.50 x [1 - {1/ (1+0.026)40}]/0.026 + $1,000/ (1+0.026)40

                      =$27.50 x [1 - {1/ (1.026)40}]/0.026 + $1,000/ (1.026)40

                                     =$27.50 x [1 - {1/ 2.791865108}]/0.026 + $1,000/2.791865108

                                    =$27.50 x [1 - {0.358183494}]/0.026 + $ 358.18

                                    =$27.50 x (0.641816506)/0.026 + $ 358.18

                                   =$27.50 x 24.68525022 + $ 358.18

                                     = $ 678.84 + $ 358.18

                                Bond Price   = $ 1,037.03

Bond Price= C x [1-{1/ (1+r)n}]/r + M/(1+r)n

                      =$27.50 x [1 - {1/ (1+0.005)40}]/0.005 + $1,000/ (1+0.005)40

                      =$27.50 x [1 - {1/ (1.005)40}]/0.005 + $1,000/ (1.005)40

                                     =$27.50 x [1 - {1/ 1.220794236}]/0.005 + $1,000/ 1.220794236

                                    =$27.50 x [1 – (0.819138861)]/0.005 + $ 819.1388607

                                    =$27.50 x (0.180861139)/0.005 + $ 819.1388607

                                   =$27.50 x 36.17222786 + $ 819.1388607

                                     = $ 994.7362663 + $ 819.1388607

                Bond Price   = $ 1,813.88

c)At interest rate 8 % or 0.08 i.e. 0.04 six months

Bond Price= C x [1-{1/ (1+r)n}]/r + M/(1+r)n

                      =$27.50 x [1 - {1/ (1+0.04)40}]/0.04+ $1,000/ (1+0.04)40

                      =$27.50 x [1 - {1/ (1.04)40}]/0.04 + $1,000/ (1.04)40

                                     =$27.50 x [1 - {1/ 4.801020628}]/0.04 + $1,000/ 4.801020628

                                    =$27.50 x [1 – (0.208289045)]/0.04 + $ 208.2890447

                                    =$27.50 x [(0.791710955)/0.04]+ $ 208.2890447

                                   =$27.50 x 19.79277388 + $ 208.2890447

                                     = $ 544.3012818 + $ 208.2890447

                                Bond Price   = $ 752.59

d) At interest rate 15 % or 0.15 i.e. 0.075 six months

Bond Price= C x [1-{1/ (1+r)n}]/r + M/(1+r)n

                      =$27.50 x [1 - {1/ (1+0.075)40}]/0.075 + $1,000/ (1+0.075)40

                      =$27.50 x [1 - {1/ (1.075)40}]/0.075 + $1,000/ (1.075)40

                                     =$27.50 x [1 – (1/ 18.04424)]/0.075 + $1,000/18.04424

                                    =$27.50 x (1 - 0.055419)/0.075 + $ 55.41935

                                    =$27.50 x (0.944581)/0.075 + $ 55.41935

                                   =$27.50 x 12.59441 + $ 55.41935

                                     = $ 346.3462 + $ 55.41935

                                Bond Price   = $ 401.77

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