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?

(A) Actual loss = 7.92%

Predicted loss = 10.94%

(B) Actual gain = 8.87%

Predicted gain = 13.66%

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%

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%