Question

Both Bond A and Bond B have 7.1 percent coupons and are priced at par value. Bond A has 8 years to maturity, while Bond B has 19 years to maturity. If interest rates suddenly rise by 2 percentage points, what is the difference in percentage changes in prices of Bond A and Bond B? (i.e., Bond A - Bond B). The bonds pay coupons twice a year. (A negative value should be indicated by a minus sign. Do not round intermediate calculations. Enter your answers as a percent rounded to 2 decimal places.)

Answer 1

Here both the bonds are priced at par , hence coupon rate = YTM. Thus YTM = 7.1%

Now if YTM rises by 2%, then new YTM = 9.1%

Price of bond A

Here face value = $1000 ,
Interest = face value x coupon rate
= 1000 x 7.1% x 1/2
= 35.50 $
n = no of coupon payments= 8 x 2 = 16
YTM = 9.1%/2 =4.55%
Value of bond = Interest x PVIFA(YTM%,n) + redemption value x PVIF(YTM%,n)
PVIFA(YTM%,n) = [1-(1/(1+r)^n / r ]
PVIFA(4.55%,16) = [1-(1/(1+4.55%)^16 / 4.55%]
=[1-(1/(1+0.0455)^16 / 0.0455]
=[1-(1/(1.0455)^16 / 0.0455]
=[1-0.490699 / 0.0455]
=0.50930/0.0455
=11.19342
PVIF(4.55%,16) = 1/(1+4.55%)^16
=1/(1.0455)^16
= 0.490699
Value of bond = 35.5 x 11.19342 + 1000 x 0.490699
=397.37 + 490.70
= 888.07 $

Thus % Change in price of bond A = 888.07-1000/1000
= -111.93/1000
= -11.19 %

Price of bond B

Here face value = $1000 ,
Interest = face value x coupon rate
= 1000 x 7.1% x 1/2
= 35.50 $
n = no of coupon payments= 19 x 2 = 38
YTM = 9.1%/2 =4.55%
Value of bond = Interest x PVIFA(YTM%,n) + redemption value x PVIF(YTM%,n)
PVIFA(YTM%,n) = [1-(1/(1+r)^n / r ]
PVIFA(4.55%,38) = [1-(1/(1+4.55%)^38 / 4.55%]
=[1-(1/(1+0.0455)^38 / 0.0455]
=[1-(1/(1.0455)^38 / 0.0455]
=[1-0.184368 / 0.0455]
=0.81563/0.0455
=17.92597
PVIF(4.55%,38) = 1/(1+4.55%)^38
=1/(1.0455)^38
= 0.184368
Value of bond = 35.5 x 17.92597 + 1000 x 0.18437
=636.37 + 184.37
= 820.74 $

Thus % Change in price of bond B = 820.74-1000/1000
= -179.26/1000
= -17.93 %

Difference in percentage changes in prices of Bond A and Bond B = -11.19 % - (-17.93%)

= -11.19 % + 17.93%

= 6.74%