Question

1. You have a zero coupon bond with ten years until maturity with $1000 face value. Yield to maturity is 5%. What is the duration of the bond? Show all calculations.

2. You have a 2% bond paying annual coupons, 3 years until maturity with $1000 face value. Yield to maturity is 2.5%. What is the duration of the bond? Show all calculations.

Answer 1

1

K = N

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

k=1

K =10

Bond Price =∑ [(0*1000/100)/(1 + 5/100)^k]     +   1000/(1 + 5/100)^10

k=1

Bond Price = 613.91

Period	Cash Flow	Discounting factor	PV Cash Flow	Duration Calc
0	($613.91)	=(1+YTM/number of coupon payments in the year)^period	=cashflow/discounting factor	=PV cashflow*period
1	-	1.05	-	-
2	-	1.10	-	-
3	-	1.16	-	-
4	-	1.22	-	-
5	-	1.28	-	-
6	-	1.34	-	-
7	-	1.41	-	-
8	-	1.48	-	-
9	-	1.55	-	-
10	1,000.00	1.63	613.91	6,139.13
			Total	6,139.13

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

=6139.13/(613.91*1)

=10.000053

Modified duration = Macaulay duration/(1+YTM)

=10/(1+0.05)

=9.52386

2

K = N

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

k=1

K =3

Bond Price =∑ [(2*1000/100)/(1 + 2.5/100)^k]     +   1000/(1 + 2.5/100)^3

k=1

Bond Price = 985.72

Period	Cash Flow	Discounting factor	PV Cash Flow	Duration Calc
0	($985.72)	=(1+YTM/number of coupon payments in the year)^period	=cashflow/discounting factor	=PV cashflow*period
1	20.00	1.03	19.51	19.51
2	20.00	1.05	19.04	38.07
3	1,020.00	1.08	947.17	2,841.51
			Total	2,899.10

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

=2899.1/(985.72*1)

=2.941098

Modified duration = Macaulay duration/(1+YTM)

=2.94/(1+0.025)

=2.869364