A 30-year maturity bond making annual coupon payments with a coupon rate of 16.0
ID: 2709429 • Letter: A
Question
A 30-year maturity bond making annual coupon payments with a coupon rate of 16.0% has duration of 10.55 years and convexity of 161.7. The bond currently sells at a yield to maturity of 9%.
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.)
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.)
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
A 30-year maturity bond making annual coupon payments with a coupon rate of 16.0% has duration of 10.55 years and convexity of 161.7. The bond currently sells at a yield to maturity of 9%.
Explanation / Answer
Answer (a)
YTM
Price
8%
190.06
10%
156.56
Answer (b)
YTM
Price using Duration rule
Price using Duration with Convexity Rule
8%
188.56
189.95
10%
155.28
156.67
Answer (c)
YTM
Percent Error
Duration Rule
Duration with Convexity Rule
8%
0.789%
0.058%
10%
0.818%
0.070%
Answer d
The percentage error in estimation of bond price is less for Duration with convexity rule
The duration with convexity rule provides results more accurate approximations to the actual change in price.
Coupon Rate = 16%
Time to maturity = 30 years
Ytm = 9%
Duration = 10.55 years
Convexity = 161.7
Current Price = 16 * [(1-(1/(1.09)^30))/0.09] + 100/1.09^30
= 16 * [(1-(1/(13.26768))/0.09] + 100/13.26768
= 16 * [(1-0.075371)/0.09] + 100 * 0.075371
= 16 * 0.924629/0.09 + 7.537
= 16 * 10.27365 + 7.537
= 164.3785 + 7.537
= 171.9156 or 171.92 (rounded off)
Modified Duration = change in bond price/change in yield
= -Duration /(1+ytm/no of payments per year)
Modified Duration = 10.55 /(1+9%/1) =10.55/1.09 = 9.678899
If Bond yield falls to 8%
Price = 16 * [(1-(1/(1.08)^3))/0.08] + 100/1.08^30
= 16 * [(1-(1/(10.06266))/0.08] + 100/10.06266
= 16 * (1-0.099377)/0.08 + 100 * 0.099377
= 16 * 0.900623/0.08 + 9.938
= 16 * 11.25778 + 9.938
= 180.1245 + 9.938
= 190.0623 or 190.06
Price using Duration Rule
Bond Price change = yield change * Modified Duration * Bond Price
= (0.08 – 0.09) * -9.678899 * 171.92
= -0.01 * -9.678899 * 171.92
= 16.6399 or 16.64
Bond price = 171.92 + 16.64 = 188.56
Percentage error in Price estimation = (190.06 – 188.56)/190.06 * 100 = 0.78922 or 0.789%
Change in Bond Price/Bond Price = - Modified Duration * change in yield + ((change in yield)^2/2) * convexity
Change in Bond Price / Price = - 9.678899 * -0.01 + ((-0.01)^2/2)*161.7
Change in bond price/ price = 0.09678899 + 0.008085
Change in Bond Price = 0.10487399 * 171.92 = 18.0299 or 18.03
Bond price at 8% ytm = 171.92 + 18.03 = 189.95
Percentage error in price estimation = (190.06 – 189.95)/190.06 * 100 = 0.05787 or 0.0579%
YTM = 10%
Bond Price = 16 * [(1-(1/(1.10)^3))/0.10] + 100/1.10^30
= 16 * [(1-(1/(17.4494))/0.10] + 100/17.4494
= 16 * (1-0.057309)/0.10 + 100 * 0.057309
= 16 * 0.942691/0.10 + 5.731
= 16 * 9.426914 + 5.731
= 150.8306 + 5.731
= 156.5615 or 156.56
Price using Duration Rule
Price Change = Modified Duration * yield change * Bond Price
= -9.678899 * 0.01 * 171.92
= - 16.63996 or -16.64
Bond Price at 10% Yield = 171.92 – 16.64 = 155.28
Percentage error in price estimation = (156.56- 155.28)/156.56 * 100
= 0.8175 or 0.818%
Estimation of Bond Price with Convexity
Change in Bond Price/Bond Price = - Modified Duration * change in yield + ((change in yield)^2/2) * convexity
Change in Bond Price / 171.92 = -9.678899 * (0.10-0.09) + (0.01)^2/2) * 161.7
Change in Bond Price / 171.92 = -9.678899*.01 +0.00005 * 161.7
Change in Bond Price = 171.92 * (-0.09678899+0.008085)
Change in Bond Price = 171.92 * -0.08870399
= -15.2499 or -15.25
Bond Price with convexity adjustment = 171.92 – 15.25 = 156.67
Percentage error in estimation = (156.67 – 156.56)/156.56 = 0.0702%
YTM
Price
8%
190.06
10%
156.56
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