Question

Bond A	Bond B
Time to maturity (years)	5	6
Annual yield to maturity	4.00%	4.00%
Annual coupon payment	40.00	65.94
Current price	-1000	-1000
Face value	1000	826.02

So we start with two bonds of equal price ($1000) and annual yield to maturity (4.00%)

a)What is the Macaulay duration (in years) for Bond A and B?

b) Assume annual yield to maturity drops from 4.00% to 3.00%. What is the total ending wealth of Bond A and B?

Answer 1

(a)

Bond A
Year (t)	Payments (P)	PV of Payments = P/(1+4%)t	PV x t
1	40	38.46	38.46
2	40	36.98	73.96
3	40	35.56	106.68
4	40	34.19	136.77
5	1040*	854.80	4,274.02
Total		1,000.00	4,629.90
Macaulay Duration		4.63

*Annual coupon + Face Value repaid = 40 + 1,000 = $ 1,040

Macaulay Duration = 4,629.90 / 1,000 = 4.63 years

Bond B
Year (t)	Payments (P)	PV of Payments	PV x t
1	65.94	63.40	63.40
2	65.94	60.97	121.93
3	65.94	58.62	175.86
4	65.94	56.37	225.46
5	65.94	54.20	270.99
6	891.96*	704.93	4,229.57
Total		998.48	5,087.22
Macaulay Duration		5.09

*Annual coupon + Face Value repaid = 65.94 + 826.02 = $ 891.96

Macaulay Duration = 5,087.22 / 998.48 = 5.09 years

Part (b)

Price when yield is 3% can be calculated using PV function

Bond A: New Price = - PV(Rate, Period, PMT, FV) = - PV(3%, 5, 40, 1000) = $ 1,045.80 = Ending wealth of Bond A

Bond B: New Price = - PV(Rate, Period, PMT, FV) = - PV(3%, 6, 65.94, 826.02) = $1,048.99 = Ending wealth of Bond B