A 30-year maturity bond making annual coupon payments with a coupon rate of 7.5% has duration of 12.27 years and convexity of 216.28. The bond currently sells at a yield to maturity of 8%.
e-1. Find the price of the bond if its yield to maturity increases to 9%. (Do not round intermediate calculations. Round your answers to 2 decimal places.)
e-2. What price would be predicted by the duration rule? (Do not round intermediate calculations. Round your answers to 2 decimal places.)
e-3. What price would be predicted by the duration-with-convexity rule? (Do not round intermediate calculations. Round your answers to 2 decimal places.)
e-4. What is the percent error for each rule? (Negative answers should be indicated by a minus sign. Do not round intermediate calculations. Round your answers to 2 decimal places.
e-1
New bond price @ YTM =9 | |||||||||
K = N | |||||||||
Bond Price =∑ [(Annual Coupon)/(1 + YTM)^k] + Par value/(1 + YTM)^N | |||||||||
k=1 | |||||||||
K =30 | |||||||||
Bond Price =∑ [(7.5*1000/100)/(1 + 9/100)^k] + 1000/(1 + 9/100)^30 | |||||||||
k=1 | |||||||||
Bond Price = 845.9 | |||||||||
e-2
New bond price @ YTM =9 using duration | |||||||||
Mod.duration prediction = -Mod. Duration*Yield_Change*Bond_Price | |||||||||
=-12.27*0.01*943.71 | |||||||||
=-115.793217 | |||||||||
New bond price = bond price+Modified duration prediction | |||||||||
=943.71+-115.793217 | |||||||||
=827.92 |
e-3
New bond price @ YTM =9 using duration and convexity | |||||||||
Convexity adjustment = 0.5*convexity*Yield_Change^2*Bond_Price | |||||||||
=0.5*216.28*0.01^2*943.71 | |||||||||
=10.20527994 | |||||||||
New bond price = bond price+Mod.duration pred.+convex. Adj. | |||||||||
=943.71+-115.79+10.21 | |||||||||
=838.13 |
e-4
Percentage error for YTM =9 and duration rule | |||||||||
Difference in price predicted and actual | |||||||||
=predicted price-actual price | |||||||||
=827.916783-845.9 | |||||||||
=-17.983 | |||||||||
%age difference = difference/actual | |||||||||
=-17.983/845.9 | |||||||||
=-2.13% | |||||||||
Percentage error for YTM =9 and duration & convexity rule | |||||||||
Difference in price predicted and actual | |||||||||
=predicted price-actual price | |||||||||
=838.13-845.9 | |||||||||
=-7.77 | |||||||||
%age difference = difference/actual | |||||||||
=-7.77/845.9 | |||||||||
=-0.92% | |||||||||
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