Question

Assume continuous compounding.

Year (t)	Bond A's Cash Flows	Bond B's Cash Flows
1	0	100
2	0	100
3	0	100
4	0	100
5	1000	1100

The yield to maturity on both bonds is 5.5%.

a. What are the current prices for the bonds?

b. What is the duration for each of the bonds?

c. If the yield to maturity of both bonds were to increase 85 basis points, what would be the approximate percentage change in the prices of both the bonds? (Do not recalculate the prices of the two bonds.)

I have to show work. Thank you!

Answer 1

Bond A

K = N

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

k=1

K =5

Bond Price =∑ [(0*1000/100)/(1 + 5.5/100)^k]     +   1000/(1 + 5.5/100)^5

k=1

Bond Price = 765.13

Duration

Period	Cash Flow	Discounting factor	PV Cash Flow	Duration Calc
0	($765.13)	=(1+YTM/number of coupon payments in the year)^period	=cashflow/discounting factor	=PV cashflow*period
1	-	1.06	-	-
2	-	1.11	-	-
3	-	1.17	-	-
4	-	1.24	-	-
5	1,000.00	1.31	765.13	3,825.67
			Total	3,825.67

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

=3825.67/(765.13*1)

=5

Modified duration = Macaulay duration/(1+YTM)

=5/(1+0.055)

=4.74

Using only modified duration

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

=-4.74*0.0085*765.13

=-30.82

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

=-30.82/765.13

=-4.03%

New bond price = bond price+Modified duration prediction

=765.13-30.82

=734.31

Bond B

K = N

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

k=1

K =5

Bond Price =∑ [(10*1000/100)/(1 + 5.5/100)^k]     +   1000/(1 + 5.5/100)^5

k=1

Bond Price = 1192.16

Period	Cash Flow	Discounting factor	PV Cash Flow	Duration Calc
0	($1,192.16)	=(1+YTM/number of coupon payments in the year)^period	=cashflow/discounting factor	=PV cashflow*period
1	100.00	1.06	94.79	94.79
2	100.00	1.11	89.85	179.69
3	100.00	1.17	85.16	255.48
4	100.00	1.24	80.72	322.89
5	1,100.00	1.31	841.65	4,208.24
			Total	5,061.09

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

=5061.09/(1192.16*1)

=4.25

Modified duration = Macaulay duration/(1+YTM)

=4.25/(1+0.055)

=4.02

Using only modified duration

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

=-4.02*0.0085*1192.16

=-40.78

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

=-40.78/1192.16

=-3.42%

New bond price = bond price+Modified duration prediction

=1192.16-40.78

=1151.38