A 30-year maturity bond making annual coupon payments with a coupon rate of 14.5

Question

A 30-year maturity bond making annual coupon payments with a coupon rate of 14.5% has duration of 10.64 years and convexity of 164.5. The bond currently sells at a yield to maturity of 9%.

a. Find the price of the bond if its yield to maturity falls to 8%. (Do not round intermediate calculations. Round your answer to 2 decimal places.)

b. What price would be predicted by the duration rule? (Do not round intermediate calculations. Round your answer to 2 decimal places.)

c. What price would be predicted by the duration-with-convexity rule? (Do not round intermediate calculations. Round your answer to 2 decimal places.)

d-1. 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.)

d-2. What do you conclude about the accuracy of the two rules?

e-1. Find the price of the bond if it's yield to maturity rises to 10%. (Do not round intermediate calculations. Round your answer to 2 decimal places.)

e-2. What price would be predicted by the duration rule? (Do not round intermediate calculations. Round your answer to 2 decimal places.)

e-3. What price would be predicted by the duration-with-convexity rule? (Do not round intermediate calculations. Round your answer to 2 decimal places.)

e-4. 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.)

e-5. Are your conclusions about the accuracy of the two rules consistent with parts (a) – (d)?

Percent Error YTM Duration Rule Duration-with-
Convexity Rule 8% % %

Explanation / Answer

Assuming Bond Par Value = $ 100

a.Using BAII PLUS CALCULATOR
100 FV, 30 N, 14.5 PMT, 8 I/Y, CPT PV

PV = $ 173.18

If Yeild is 8%, Price of the bond is $173.18

b. Bond Duration(D) is 10.64 years and Price volatility is given by Modified duration (MD)= D/1+r

Here r = 9%, MD = 10.64/1.09= 9.76%( It means if yeild changes by 1%, Bond Price will change by 9.76% in the opposite direction ignoring convexity)

Current price of bond: 30N, 100FV, 9I/Y, 14.5 pmt CPT PV= 156.51

Suppose Yeilds now becomes 8% / 10%, then bond price will rise /Fall by 9.76% of 156.51 i.e., by $15.28

In case of yeild 8% , Price will be 156.51+9.76% of 156.51= $171.79

In case of yeild 10%, Price will be 156.51-9.76% of 156.51= $141.23

c.Duration shows the linear relationship between Bond price and its yeild, it does not captures convexity. hence it does not show the actual relation between bond price and its yeild. Correct price of bond is given by Duration + Convexity effect.

Calculation of Bond Price using both Duration and convexity effect

Duration Effect ( as shown in ans b.) D = 9.76%

Convexity adjustment is given by : 1/2* Convexity*(Dy)2*100, where Dy = change in yeild ( in decimal form)

= 1/2* 164.5*(0.01)2*100

= 0.8225%( extra effect due to convexity)

Total effect= Duration Effect + Convexity effect

= 9.76%+ 0.8225%= 10.5825%

10.5825% means if yeild change by 1%, Bond Price will change by 10.5825%

In case of yeild 8% , Price will be 156.51+10.5825% of 156.51= $173.07

In case of yeild 10%, Price will be 156.51-10.5825% of 156.51= $139.95

d.Duration does not show the actual relation between bond Price and its yeild.Therefore Price of bond are different in question a.(173.18) & question b. (171.19) at the same yeild of 8%

Also Duration + convexity Effects have some error because bond are are still diferent in answer a(173.18) and in c.(173.07)

Percentage Error:

(173.18-171.09) /171.09*100

YTM Duartion Rule Duartion with convexity Rule 8%

(173.18-171.09) /171.09*100

(173.18-173.07) /173.07*100 1.22% 0.0635%

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