Question

A 13.05-year maturity zero-coupon bond selling at a yield to maturity of 8% (effective annual yield) has convexity of 157.2 and modified duration of 12.08 years. A 40-year maturity 6% coupon bond making annual coupon payments also selling at a yield to maturity of 8% has nearly identical modified duration—-12.30 years—-but considerably higher convexity of 272.9a. Suppose the yield to maturity on both bonds increases to 9%. What will be the actual percentage capital loss on each bond? What percentage capital loss would be predicted by the duration-with-convexity rule?b. Suppose the yield to maturity on both bonds decreases to 7%. What will be the actual percentage capital gain on each bond? What percentage capital gain would be predicted by the duration-with-convexity rule?

Answer 1

(A) Actual loss =7.92%

Predicted loss = 10.94%

(B) Actual gain = 8.87%

Predicted gain = 13.66%

Step-by-step explanation:

Part A

Price of zero coupon bond at 8%=(1000/(1.08^13.05))

=366.29

Price of zero coupon bond at 9%=(1000/(1.09^13.05))

=324.78

Price of 6% coupon bond at 8%= (60*((1-(1.08^-13.05))/0.08)+(1000/(1.08^13.05))

=841.57

Price of 6% coupon bond at 9%=(60*((1-(1.09^-13.05))/0.09)+(1000/(1.09^13.05))

=774.93

Zero coupon bond

Actual loss =(324.78 - 366.29)/366.29

=11.33%

Predicted loss =((-12.08*0.01))+(0.5*157.20*(0.01^2))

=11.29%

Price of 6% coupon bond

Actual loss =(774.93-841.57)/841.57

=7.92%

Predicted loss = (-12.30*0.01)+(0.5*272.9*(0.01^2))

=10.94%

Part B

Price of zero coupon bond at 8%=(1000/(1.08^13.05))

=366.29

Price of zero coupon bond at 7%=(1000/(1.07^13.05))

=413.56

Price of 6% coupon bond at 8%= (60*((1-(1.08^-13.05))/0.08)+(1000/(1.08^13.05))

=841.57

Price of 6% coupon bond at 7%=(60*((1-(1.07^-13.05))/0.07)+(1000/(1.07^13.05))

=916.22

Zero coupon bond

Actual gain =(413.56 - 366.29)/366.29

=12.91%

Predicted gain =((12.08*0.01))+(0.5*157.20*(0.01^2))

=12.87%

Price of 6% coupon bond

Actual gain =(916.22-841.57)/841.57

=8.87%

Predicted gain = (12.30*0.01)+(0.5*272.9*(0.01^2))

=13.66%

Answer 2

Part A

Price of zero coupon bond at 8%=(1000/(1.08^13.05))=366.29

Price of zero coupon bond at 9%=(1000/(1.09^13.05))=324.78

Price of 6% coupon bond 8%= (60*((1-(1.08^-13.05))/0.08)+(1000/(1.08^13.05))=841.57

Price of 6% coupon bond 9%=(60*((1-(1.09^-13.05))/0.09)+(1000/(1.09^13.05))=774.93

Zero coupon bond

Actual loss =(324.78 - 366.29)/366.29=11.33%

Predicted loss =((-12.08*0.01))+(0.5*157.20*(0.01^2))=11.29%

Price of 6% coupon bond

Actual loss =(774.93-841.57)/841.57=7.92%

Predicted loss = (-12.30*0.01)+(0.5*272.9*(0.01^2))=10.94%

Part B

Price of zero coupon bond at 8%=(1000/(1.08^13.05))=366.29

Price of zero coupon bond at 7%=(1000/(1.07^13.05))=413.56

Price of 6% coupon bond at 8%= (60*((1-(1.08^-13.05))/0.08)+(1000/(1.08^13.05))=841.57

Price of 6% coupon bond at 7%=(60*((1-(1.07^-13.05))/0.07)+(1000/(1.07^13.05))=916.22

Zero coupon bond

Actual gain =(413.56 - 366.29)/366.29=12.91%

Predicted gain =((12.08*0.01))+(0.5*157.20*(0.01^2))=12.87%

Price of 6% coupon bond

Actual gain =(916.22-841.57)/841.57=8.87%

Predicted gain = (12.30*0.01)+(0.5*272.9*(0.01^2))=13.66%