Example: A 30-year maturity bond making annual coupon payments with a coupon rat
ID: 2757978 • Letter: E
Question
Example:
A 30-year maturity bond making annual coupon payments with a coupon rate of 15.5% has duration of 10.58 years and convexity of 162.6. The bond currently sells at a yield to maturity of 9% a. Find the price of the bond if its yield to maturity falls to 8% or rises to 10%. (Do not round intermediate calculations. Round your answers to 2 decimal places.) YTM 8% 10% Price b. What prices for the bond at these new yields would be predicted by the duration rule and the duration- with-convexity rule? (Do not round intermediate calculations. Round your answers to 2 decimal places.) Duration-with- Convexity Rule Duration Rule 8% 10% c. What is the percent error for each rule? (Do not round intermediate calculations. Round "Duration Rule" to 2 decimal places and "Duration-with-Convexity Rule" to 3 decimal places.) Percent Error Duration-with- Convexity Rule Duration Rule 8% 10% d. What do you conclude about the accuracy of the two rules? The duration-with-convexity rule provides more accurate approximations to the actual change in price C The duration rule provides more accurate approximations to the actual change in priceExplanation / Answer
Par Value 1,000 Annual Interest 155.00 YTM 9% Maturity 30 Year Bond Price calculation Year Interest+Maturity Pv Factor @9% PV of Cash flows @YTM 9% Pv Factor @8% PV of Cash flows @YTM 8% Pv Factor @10% PV of Cash flows @YTM 10% 1 155 0.9174 142.20 0.926 143.52 0.909 140.91 2 155 0.8417 130.46 0.857 132.89 0.826 128.10 3 155 0.7722 119.69 0.794 123.04 0.751 116.45 4 155 0.7084 109.81 0.735 113.93 0.683 105.87 5 155 0.6499 100.74 0.681 105.49 0.621 96.24 6 155 0.5963 92.42 0.630 97.68 0.564 87.49 7 155 0.5470 84.79 0.583 90.44 0.513 79.54 8 155 0.5019 77.79 0.540 83.74 0.467 72.31 9 155 0.4604 71.37 0.500 77.54 0.424 65.74 10 155 0.4224 65.47 0.463 71.79 0.386 59.76 11 155 0.3875 60.07 0.429 66.48 0.350 54.33 12 155 0.3555 55.11 0.397 61.55 0.319 49.39 13 155 0.3262 50.56 0.368 56.99 0.290 44.90 14 155 0.2992 46.38 0.340 52.77 0.263 40.82 15 155 0.2745 42.55 0.315 48.86 0.239 37.11 16 155 0.2519 39.04 0.292 45.24 0.218 33.73 17 155 0.2311 35.82 0.270 41.89 0.198 30.67 18 155 0.2120 32.86 0.250 38.79 0.180 27.88 19 155 0.1945 30.15 0.232 35.92 0.164 25.34 20 155 0.1784 27.66 0.215 33.25 0.149 23.04 21 155 0.1637 25.37 0.199 30.79 0.135 20.95 22 155 0.1502 23.28 0.184 28.51 0.123 19.04 23 155 0.1378 21.36 0.170 26.40 0.112 17.31 24 155 0.1264 19.59 0.158 24.44 0.102 15.74 25 155 0.1160 17.98 0.146 22.63 0.092 14.31 26 155 0.1064 16.49 0.135 20.96 0.084 13.01 27 155 0.0976 15.13 0.125 19.40 0.076 11.82 28 155 0.0895 13.88 0.116 17.97 0.069 10.75 29 155 0.0822 12.73 0.107 16.64 0.063 9.77 30 1,155 0.0754 87.05 0.099 114.78 0.057 66.19 Total $ 1,667.79 $ 1,844.33 $ 1,518.48 a YTM Bond Price 8% $ 1,844.33 10% $ 1,518.48 b Duration Rule Price change =-Modified duration*% change in interets Duration with convesxity Rule= -Normal duration*% change in interets +1/2*Convexity*Interest cahneg ^2 Given Duration = 10.58 years Modified Duration = 10.58/1.09= 9.71 Years Convexity 162.60 Current Price 1,667.79 Duration rule Duration with Convexity YTM % Change in Price Changed Price % Change in Price Changed Price 8% 9.71% $ 1,829.7 10.52% $ 1,843.2 10% -9.71% $ 1,505.8 -8.90% $ 1,519.4 c Percent Error YTM Duration rule Duration with Convexity 8% 0.79% 0.06% 10% 0.83% -0.06% d The duration with convexity rule provides more accurate approximation to the Actual change in price.
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