Question

A 13.35-year maturity zero-coupon bond selling at a yield to maturity of 8% (effective annual yield) has convexity of 164.2 and modified duration of 12.36 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 272.9.

a. 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? (Do not round intermediate calculations. Round your answers to 2 decimal places.)

	Zero-Coupon Bond		Coupon Bond
Actual loss		%		%
Predicted loss		%		%

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? (Do not round intermediate calculations. Round your answers to 2 decimal places.)

	Zero-Coupon Bond		Coupon Bond
Actual gain		%		%
Predicted gain		%		%

Answer 1

MD = Modified duration; C = Convexity

Change in price of the bond predicted by the duration and convexity rule = - MD x %age change in yield + ½ x C x (%age change in yield)²

Please see the table below. Cells highlighted in yellow contain your answers. Par value = $ 1,000 is an assumption, that will not impact the final answer.

Part (a) is shown in Scenario 1

Part (b) is shown in Scenario 2

	Linkage	Zero-Coupon Bond	Coupon Bond
Rate = YTM		8%	8%
Nper		13.35	40
Coupon		0%	6%
PMT		0	60
Par value, FV		1,000	1,000
Current Price, PV	P0 = - PV (Rate, Nper, PMT, FV)	$357.93	$761.51
Modified Duration	MD	12.36	12.30
Convexity	C	164.20	272.90
Scenario 1	Change in yield 9% - 8% = 1%
Rate = YTM changes to		9%	9%
Price	P1 = - PV (Rate = 9%, Nper, PMT, FV)	$316.49	$677.28
Actual loss	P1/P0 - 1	-11.58%	-11.06%
Predicted loss	= - MD x %age change in yield + ½ x C x (%age change in yield)^2	-11.54%	-10.94%
Scenario 2	Change in yield 7% - 8% = -1%
Rate = YTM changes to		7%	7%
Price	P2 = - PV (Rate = 9%, Nper, PMT, FV)	$405.25	$0.00
Actual loss	P2 / P0 - 1	13.22%	-100.00%
Predicted loss	'= - MD x %age change in yield + ½ x C x (%age change in yield)^2	13.18%	13.66%