A 30-year maturity bond making annual coupon payments with a coupon rate of 7.5%

Question

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%. Find the price of the bond if its yield to maturity falls to 7% or rises to 9%. (Round your answers to 2 decimal places. Omit the "$" sign in your response.) What prices for the bond at these new yields would be predicted by the duration rule and the duration-with-convexity rule? (Round your answers to 2 decimal places. Omit the "$" sign in your response.) What is the percentage error for each rule? (Negative answers should be indicated by a minus sign. Round your answers to 2 decimal places. Omit the "%" sign in your response.) What do you conclude about the accuracy of the two rules?

Explanation / Answer

Answer:a Calculation of the Actual Price: IF YTM is 8%.

=75*PVIFA(8%,30)+1000PVIF(8%,30)

=$943.44

IF YTM falls to 7%. then bond price is:

=75*PVIFA(7%,30)+1000PVIF(7%,30)

=$1062.36

If YTM rises to 9% then bond price is:

=75*PVIFA(9%,30)+1000PVIF(9%,30)

=$845.21

Answer:b

Using the Duration Rule, assuming yield to maturity falls to 7%

Predicted price change = – Duration*Change in YTM/(1+y)*P0

=-12.27*(-0.01/1.08)*$943.44

=$107.19

Therefore: Predicted price = $107.19 + $943.44 = $1,050.63

The actual price at a 7% yield to maturity is $1062.36 Therefore

% Error=($1062.36-$1,050.63)/$1062.36=1.104%

Using the Duration Rule, assuming yield to maturity increases to 9%:

Predicted price change = – Duration*Change in YTM/(1+y)*P0

=-12.27*(0.01/1.08)*$943.44

=-$107.19

Therefore: Predicted price = -$107.19 + $943.44 = $836.25

The actual price at a 9% yield to maturity is $845.21 . Therefore

% Error=($845.21-$836.25)/$845.21=1.06%

Using Duration-with-Convexity Rule, assuming yield to maturity falls to 7%

Predicted price change = [(– Duration*Change in YTM/(1+y))+(0.5*convexity*(Change in YTM)2)]*P0

=[(-12.27*(-0.01/1.08))+(0.5*216.28*(-0.01)2)]*$943.44=[0.113611+0.010814]*$943.44

=117.387522

Therefore: Predicted price = 117.387522 + $943.44 = $1060.83

The actual price at a 7% yield to maturity is $1062.36 Therefore

% Error=($1062.36-$1,060.83)/$1062.36=0.1440% (too low)

Using Duration-with-Convexity Rule, assuming yield to maturity rises to 9%:

Predicted price change = [(– Duration*Change in YTM/(1+y))+(0.5*convexity*(Change in YTM)2)]*P0

=[(-12.27*(0.01/1.08))+(0.5*216.28*(0.01)2)]*$943.44=[-0.113611+0.010814]*$943.44

=-96.9828

Therefore: Predicted price = -96.9828 + $943.44 = $846.46

The actual price at a 9% yield to maturity is $845.21 . Therefore

% Error=($845.21-$846.46)/$845.21=-0.147% (too low)

Answer:c

Answer:d The duration-with-convexity rule provides more accurate approximations to the actual change in price. In this example, the percentage error using convexity with duration is less than one-tenth the error using duration only to estimate the price change

Answer:b Particulars Duration rule Duration with Convexity rule YTM falls to 7% $1,050.63 $1,060.83 YTM rises to 9% $836.25 $846.46

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