A 30-year maturity bond making annual coupon payments with a coupon rate of 15.3
ID: 455616 • Letter: A
Question
A 30-year maturity bond making annual coupon payments with a coupon rate of 15.3% has duration of 9.97 years and convexity of 144.9. The bond currently sells at a yield to maturity of 10%.
the price of the bond if its yield to maturity falls to 9% is $1647.24.
a) What price would be predicted by the duration rule? (Do not round intermediate calculations. Round your answer to 2 decimal places.)
b) What price would be predicted by the duration-with-convexity rule? (Do not round intermediate calculations. Round your answer to 2 decimal places.)
c) What is the percent error for each rule? (Enter your answer as a positive value. Do not round intermediate calculations. Round "Duration Rule" to 2 decimal places and "Duration-with-Convexity Rule" to 3 decimal places.)
Percent Error
d) If the price of the bond if it's yield to maturity rises to 11%, it would be $1373.83. What price would be predicted by the duration rule? (Do not round intermediate calculations. Round your answer to 2 decimal places.)
e) What price would be predicted by the duration-with-convexity rule? (Do not round intermediate calculations. Round your answer to 2 decimal places.)
f) 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
Percent Error
YTM Duration Rule Duration-with-Convexity Rule 9% % %
d) If the price of the bond if it's yield to maturity rises to 11%, it would be $1373.83. What price would be predicted by the duration rule? (Do not round intermediate calculations. Round your answer to 2 decimal places.)
e) What price would be predicted by the duration-with-convexity rule? (Do not round intermediate calculations. Round your answer to 2 decimal places.)
f) 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 11% % %
Explanation / Answer
Lets calculate the bond price
Bond price P0 = C* [ 1- 1/(1+i)^n] /i + M / (1+i)^n
Where
C = coupon payment = 15.3% of $1000 = $153
n = number of payments = 30
i = interest rate, or required yield = 10% or 0.1
M = value at maturity, or par value = $ 1000
Bond Price = 153 * [1 – 1 /(1+0.1)^30] /0.1 + 1000 / (1+0.1)^30
= $ 1442.32 + $ 57.31 = $ 1499.63
the price of the bond if its yield to maturity falls to 9% is $1647.24.
(a)the price predicted by the duration rule
we have duration = 9.97 years and y is yield to maturity = 10%
Predicted price change = – Duration * (change in y)/(1+y)* P0
= - 9.97 *( -0.01)/(1.10) * $ 1499.63
= $ 135.92
Therefore, predicted new price = $135.92 + $1499.63 = $1635.55
(b)
Using Duration-with-Convexity Rule, assuming yield to maturity falls to 9%
Predicted price change
= [( – Duration * change in y/(1+y) ) + (0.5 * convexity *(change in y)^2)] * P0
=[ ( - 9.97 * (-0.01) /(1 +0.1) ) + (0.5 * 144.9 * (-0.01)^2)] * 1499.63
= 146.79
Therefore, predicted price = 146.79 + 1499.63 = $1646.42
(c) What is the percent error for each rule
Percent Error
YTM
Duration Rule
Duration-with-
Convexity Rule
9%
[(1647.24 – 1635.55)/1647.24] * 100 = 0.71%
[(1647.24 – 1646.42)/1647.24] * 100 = 0.05%
the price of the bond if its yield to maturity falls to 9% is $1647.24.
(d) If the price of the bond if it's yield to maturity rises to 11%, it would be $1373.83. The price predicted by the duration rule
we have duration = 9.97 years and y is yield to maturity = 10%
Predicted price change = – Duration * (change in y)/(1+y)* P0
= - 9.97 *( 0.01)/(1.10) * $ 1499.63
= - $ 135.92
Therefore, predicted new price = - $135.92 + $1499.63 = $1363.71
(e)
Using Duration-with-Convexity Rule, assuming yield to maturity rises to 11%
Predicted price change
= [( – Duration * change in y/(1+y) ) + (0.5 * convexity *(change in y)^2)] * P0
=[ ( - 9.97 * (0.01) /(1 +0.1) ) + (0.5 * 144.9 * (0.01)^2)] * 1499.63
= - 125.06
Therefore, predicted price = - 125.06 + 1499.63 = $1374.57
(f) What is the percent error for each rule
Percent Error
YTM
Duration Rule
Duration-with-
Convexity Rule
9%
[(1373.83 – 1363.71)/1373.83] * 100 = 0.74%
[(1373.83 – 1374.57)/1373.83] * 100 = - 0.054%
Percent Error
YTM
Duration Rule
Duration-with-
Convexity Rule
9%
[(1647.24 – 1635.55)/1647.24] * 100 = 0.71%
[(1647.24 – 1646.42)/1647.24] * 100 = 0.05%
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