Question

A newly issued bond has a maturity of 10 years and pays a 7% coupon rate (with coupon payments coming once annually). The bond sells at par value.

a. What are the convexity and the duration of the bond? Use the formula for convexity in footnote 7. (Round your answers to 3 decimal places.)

Convexity
Duration

b. Find the actual price of the bond assuming that its yield to maturity immediately increases from 7% to 8% (with maturity still 10 years). Assume a par value of 100. (Round your answer to 2 decimal places.)

Actual Price of the Bond (%)

c. What price would be predicted by the modified duration rule ΔPP=−D*Δy?ΔPP=−D*Δy? What is the percentage error of that rule? (Negative answers should be indicated by a minus sign. Round your answers to 2 decimal places.)  

Percentage Price change (%)
Percentage Error (%)

d. What price would be predicted by the modified duration-with-convexity rule ΔPP=−D*Δy+12×Convexity×(Δy)2?ΔPP=−D*Δy⁢+12×Convexity×(Δy)2? What is the percentage error of that rule? (Negative answers should be indicated by a minus sign. Round your answers to 2 decimal places.)

Percentage Price Change (%)
Percentage Error (%)

Answer 1

K = N

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

k=1

K =10

Bond Price =∑ [(7*1000/100)/(1 + 10/100)^k]     +   1000/(1 + 10/100)^10

k=1

Bond Price = 815.66

a

Period	Cash Flow	Discounting factor	PV Cash Flow	Duration Calc	Convexity Calc
0	($815.66)	=(1+YTM/number of coupon payments in the year)^period	=cashflow/discounting factor	=PV cashflow*period	=duration calc*(1+period)/(1+YTM/N)^2
1	70.00	1.10	63.64	63.64	105.18
2	70.00	1.21	57.85	115.70	286.87
3	70.00	1.33	52.59	157.78	521.57
4	70.00	1.46	47.81	191.24	790.26
5	70.00	1.61	43.46	217.32	1,077.63
6	70.00	1.77	39.51	237.08	1,371.53
7	70.00	1.95	35.92	251.45	1,662.46
8	70.00	2.14	32.66	261.24	1,943.14
9	70.00	2.36	29.69	267.18	2,208.11
10	1,070.00	2.59	412.53	4,125.31	37,502.85
			Total	5,887.95	47,469.61

Macaulay duration =(∑ Duration calc)/(bond price*number of coupon per year)

=5887.95/(815.66*1)

=7.219

Modified duration = Macaulay duration/(1+YTM)

=7.22/(1+0.1)

=6.562

Convexity =(∑ convexity calc)/(bond price*number of coupon per year^2)

=47469.61/(815.66*1^2)

=58.2

b

Actual bond price change

K = N

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

k=1

K =10

Bond Price =∑ [(7*100/100)/(1 + 11/100)^k]     +   100/(1 + 11/100)^10

k=1

Bond Price = 76.44

c

Using only modified duration

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

=-6.56*0.01*81.57

=-5.35

%age change in bond price=Mod.duration prediction/bond price

=-5.35/81.57

=-6.56%

New bond price = bond price+Modified duration prediction

=81.57-5.35

=76.22

Difference in price predicted and actual

=predicted price-actual price

=76.22-76.44

=-0.22

%age difference = difference/actual-1

=-0.22/76.44

=-0.3%

d

Using convexity adjustment to modified duration

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

0.5*58.19*0.01^2*81.57

=0.24

%age change in bond price=(Mod.duration pred.+convex. Adj.)/bond price

=(-5.35+0.24)/81.57

=-6.27%

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

=81.57-5.35+0.24

=76.45

Difference in price predicted and actual

=predicted price-actual price

=76.45-76.44

=0.015

%age difference = difference/actual-1

=0.01/76.44

=0.02%