A 30-year maturity bond making annual coupon payments with a coupon rate of 14.3
ID: 2756044 • Letter: A
Question
A 30-year maturity bond making annual coupon payments with a coupon rate of 14.3% has duration of 11.34 years and convexity of 185.7. The bond currently sells at a yield to maturity of 8%.
Find the price of the bond if its yield to maturity falls to 7% or rises to 9%. (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.)
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
A 30-year maturity bond making annual coupon payments with a coupon rate of 14.3% has duration of 11.34 years and convexity of 185.7. The bond currently sells at a yield to maturity of 8%.
a.Find the price of the bond if its yield to maturity falls to 7% or rises to 9%. (Do not round intermediate calculations. Round your answers to 2 decimal places.)
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.)
YTM Duration Rule Duration-with Convexity Rule 7% 9%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
YTM Duration Rule Duration-with-Convexity Rule 7% n/r % n/r % 9% n/r % n/r %
Explanation / Answer
Answer (a)
YTM
Bond Price
7%
1905.86
9%
1544.50
Answer (b)
YTM
Duration Rule
Duration with convexity rule
7%
1888.71
1904.58
9%
1529.77
1545.64
Answer (c)
Percent Error
YTM
Duration Rule
Duration with convexity rule
7%
8.72%
0.651%
9%
8.94%
0.692%
Coupon Rate = 14.3%
Coupon payment = Annual
Annual coupon amount = 1000 * 14.3% = 143
Time to maturity = 30 years
Ytm = 8%
Duration = 11.34 years
Convexity = 185.7
Modified Duration = Mecaulay’s Duration /(1+r)
= 11.34/1.08 = 10.50
Bond current price = 143* [(1-(1/(1+0.08)^30)/0.08] + 1000/(1+0.08)^30
= 143*[(1-(1/10.06266))/0.08]+1000/10.06266
= 143*(1-0.099377)/0.08] + 1000 * 0.099377
= 143 * (0.900623/0.08) + 1000 * 0.099377
= 143 * 11.25778 + 1000 * 0.099377
= 1609.8630 + 99.37733
= 1709.2404 or 1709.24
If the Bond yield falls to 7%
Bond Price = 143 * [(1-(1/(1+0.07)^30)/0.07] + 1000/(1+0.07)^30
= 143*[(1-(1/7.612255))/0.07] + 1000 / 7.612255
= 143* [(1-0.131367)/0.07] + 1000 * 0.131367
= 143*(0.868633/0.07) + 1000 * 0.131367
= 143 * 12.40904 + 1000 * 0.131367
= 1774.4929 + 131.3671
= 1905.86
Change in Price = 1905.86 – 1709.24 = 196.62
Calculation of bond price with duration rule
Change in Price = - Modified Duration * Change in yield * Bond Price
= - 10.5 * (0.07-0.08) * 1709.24
= -10.5 * -0.01 * 1709.24
= 179.4702 or 179.47
Price for the bond = 1709.24 + 179.47 = 1888.71
% error in estimation of bond price change = (difference in price change / actual price change)*100
= ((196.62 - 179.47)/196.62) * 100
= (17.15/196.62)*100
= 8.7224 or 8.72%
Change in price using duration and convexity rule
Change in price =[-Modified Duration * change in yield)+ (1/2)*Convexity*(change in yield)^2]*Bond Price
Change in Price = [-10.5 * - 0.01 + 0.5 * 185.7 * (-0.01)^2] * 1709.24
= [0.105 + 0.5*185.7* 0.0001] * 1709.24
= [0.105 + 0.009285]*1709.24
= 0.114285 * 1709.24
= 195.3404934 or 195.340 (rounded off)
New Bond price = 1709.24 + 195.34 = 1904.58
% error in estimation of price change = ((196.62-195.340)/196.62)*100
= 0.65100193 or 0.651%
If the bond yield rises to 9%
Bond price = 143 * [(1-(1/(1+0.09)^30)/0.09] + 1000/(1+0.09)^30
= 143*[(1-(1/13.26768))/0.09] + 1000 / 13.26768
= 143* [(1-0.075371)/0.09] + 1000 * 0.0.075371
= 143*(0.924629/0.09) + 1000 * 0.0.075371
= 143 * 10.27365 + 1000 * 0.0.075371
= 1469.1325 + 75.37114
= 1544.5037 or 1544.50 (rounded off)
Change in Price = 1544.50 – 1709.24 = -164.74
Change in price using duration rule
Change in price = -10.5 * (0.09-0.08) * 1709.24
= -10.5 * 0.01 * 1709.24
= -179.4702 or -179.47
New Bond Price = 1709.24 – 179.47 = 1529.77
% error in estimation of change in price = ((-164.74+179.47)/-164.74) * 100
= (-14.7302/-164.74)*100
= 8.94148 or 8.94%
Change in price using duration and convexity
Change in price = [-10.5*0.01 + (1/2)* 185.7* (0.01)^2] * 1709.24
= [-0.105 + 0.009285] * 1709.24
= -0.095715 * 1709.24
= -163.5999066 or -163.600 (rounded off)
New Bond Price = 1709.24 – 163.60 = 1545.64
% error in estimation of price = ((-164.74 + 163.60)/-164.74)*100
= 0.69205% or 0.692%
YTM
Bond Price
7%
1905.86
9%
1544.50
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