Question

A 27-year maturity bond making annual coupon payments with a coupon rate of 9% has duration of 11.5 years and convexity of 191.2. 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 rule	Duration-with- convexity 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 rule	Duration-with- convexity rule
Percent error for 7% YTM	%	%
Percent error for 9% YTM	%	%

  

(d)	What do you conclude about the accuracy of the two rules?

Answer 1

K = N

Bond Price =∑ [(Annual Coupon)/(1 + YTM)^k]     +   Par value/(1 + YTM)^N

k=1

K =27

Bond Price =∑ [(9*1000/100)/(1 + 8/100)^k]     +   1000/(1 + 8/100)^27

k=1

Bond Price = 1109.35

a)

New bond price @ YTM =7

K = N

Bond Price =∑ [(Annual Coupon)/(1 + YTM)^k]     +   Par value/(1 + YTM)^N

k=1

K =27

Bond Price =∑ [(9*1000/100)/(1 + 7/100)^k]     +   1000/(1 + 7/100)^27

k=1

Bond Price = 1239.73

New bond price @ YTM =9

K = N

Bond Price =∑ [(Annual Coupon)/(1 + YTM)^k]     +   Par value/(1 + YTM)^N

k=1

K =27

Bond Price =∑ [(9*1000/100)/(1 + 9/100)^k]     +   1000/(1 + 9/100)^27

k=1

Bond Price = 1000

b)

New bond price @ YTM =7 using duration

Mod.duration prediction = -Mod. Duration*Yield_Change*Bond_Price

=-11.5*-0.01*1109.35

=127.57525

New bond price = bond price+Modified duration prediction

=1109.35+127.57525

=1236.93

New bond price @ YTM =7 using duration and convexity

Convexity adjustment = 0.5*convexity*Yield_Change^2*Bond_Price

=0.5*191.2*-0.01^2*1109.35

=10.605386

New bond price = bond price+Mod.duration pred.+convex. Adj.

=1109.35+127.58+-10.61

=1247.54

New bond price @ YTM =9 using duration

Mod.duration prediction = -Mod. Duration*Yield_Change*Bond_Price

=-11.5*0.01*1109.35

=-127.57525

New bond price = bond price+Modified duration prediction

=1109.35+-127.57525

=981.77

New bond price @ YTM =9 using duration and convexity

Convexity adjustment = 0.5*convexity*Yield_Change^2*Bond_Price

=0.5*191.2*0.01^2*1109.35

=10.605386

New bond price = bond price+Mod.duration pred.+convex. Adj.

=1109.35+-127.58+10.61

=992.38

c)

Percentage error for YTM =7 and duration rule

Difference in price predicted and actual

=predicted price-actual price

=1236.92525-1239.73

=-2.805

%age difference = difference/actual

=-2.805/1239.73

=-0.23%

Percentage error for YTM =7 and duration & convexity rule

Difference in price predicted and actual

=predicted price-actual price

=1247.54-1239.73

=7.81

%age difference = difference/actual

=7.81/1239.73

=0.63%

Percentage error for YTM =7 and duration rule

Difference in price predicted and actual

=predicted price-actual price

=981.77475-1000

=-18.225

%age difference = difference/actual

=-18.225/1000

=-0.23%

Percentage error for YTM =7 and duration & convexity rule

Difference in price predicted and actual

=predicted price-actual price

=992.38-1000

=7.81

%age difference = difference/actual

=-7.62/1000

=-0.76%

d. Duration rule is more accurate than convexity +duration rule