Question

A 33-year maturity bond making annual coupon payments with a coupon rate of 15% has duration of 10.8 years and convexity of 1916 . The bond currently sells at a yield to maturity of 8% Required (a) 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 "$" sign in your response.) Yield to maturity of 7% Yield to maturity of 9% (b) What prices for the bond at these new yields would be predicted by the duration rule and the duration with-convexity rule? (Round your answers to 2 decimal places. Omit the "$" sign in your response.) Duration-with- convexity rule Duration rule YTM falls to 7% YTM increases to 9% (c) What is the percent error for each rule? (Round your answers to 3 decimal places. Omit the "%" sign in your response.) Duration-with- convexity rule Duration rule Percent error for 7% YTM Percent error for 9% YTM (d)What do you conclude about the accuracy of the two rules? (Click to select) The duration-with-convexity rule provides more accurate approximations to the actual change in price The duration rule provides more accurate approximations to the actual change in price

Answer 1

a). Par value = 1,000; coupon rate = 15%; coupon = 15%*1,000 = 150; N = 33

Using PV function, price of the bond is found as:

FV = 1,000; PMT = 150; N = 33; I (given YTM) = 7%, PV = 2,020.30

FV = 1,000; PMT = 150; N = 33; I (given YTM) = 9%, PV = 1,627.87

Current bond price: FV = 1,000; PMT = 150; N = 33; I (current YTM) = 8%, PV = 1,805.97

Using duration,

Duration x AyxPo Predicted price change-(- la n)-ay

	(a)	(b)	('c)	(d = -a*b*c)	(e = d +b)	(f)	(e-f)/f
YTM	Duration	P0	Change in YTM/(1+8%)	Predicted price change	Predicted price	Actual price	%age error
7%	10.8	1,805.97	(0.01)	180.60	1,986.57	2,020.30	-1.670%
9%	10.8	1,805.97	0.01	(180.60)	1,625.37	1,627.87	-0.153%

Using duration-with-convexity,

Predicted price change-DurationAy+05xConvexity x(ay) P

	(a)	(b)	('c)	(d)	e = {(-a*d/1.08) + (0.5*c*d^2)}*c	f = c + e	g	(f-g)/g
YTM	Duration	Convexity	P0	Change in yield	Change in price	Predicted price	Actual price	%age error
7%	10.8	191.6	1,805.97	(0.01)	197.90	2,003.87	2,020.30	-0.813%
9%	10.8	191.6	1,805.97	0.01	(163.30)	1,642.68	1,627.87	0.910%

b). Prices of the bond:

	Price
YTM	Duration rule	Duration with convexity rule
7%	1,986.57	2,003.87
9%	1,625.37	1,642.68

c). %age errors:

	%age error
YTM	Duration rule	Duration with convexity rule
7%	-1.670%	-0.813%
9%	-0.153%	0.910%

d). The duration with convexity rule provides more accurate approximations to the actual change in price.