Question

Consider the following. a. What is the duration of a four-year Treasury bond with a 4 percent semiannual coupon selling at par? b. What is the duration of a three-year Treasury bond with a 4 percent semiannual coupon selling at par? c. What is the duration of a two-year Treasury bond with a 4 percent semiannual coupon selling at par? (For all requirements, do not round intermediate calculations. Round your answers to 2 decimal places. (e.g., 32.16))

Answer 1

Assume that the par value of Treasury bond is $1000

As the Treasury bond is selling at par therefore yield to maturity of the bond is equal to its coupon rate

Coupon rate is 4% per year therefore yield to maturity is also 4% per year

a. What is the duration of a four-year Treasury bond with a 4 percent semiannual coupon selling at par?

Duration Calculation:

Year (t)	Payments (n)	Cash Flow from coupon payments (4%/2 of $1000)	Cash Flow from maturity amount	Total Cash Flow from coupon payments and maturity amount (CF)	Present value (PV) discounted at 4%/2 = 2% semiannual yield to maturity	PV *t
0.5	1.0	$20.00		$20.00	$19.61	$9.80
1.0	2.0	$20.00		$20.00	$19.22	$19.22
1.5	3.0	$20.00		$20.00	$18.85	$28.27
2.0	4.0	$20.00		$20.00	$18.48	$36.95
2.5	5.0	$20.00		$20.00	$18.11	$45.29
3.0	6.0	$20.00		$20.00	$17.76	$53.28
3.5	7.0	$20.00		$20.00	$17.41	$60.94
4.0	8.0	$20.00	$1,000.0	$1,020.00	$870.56	$3,482.24
Sum					$1,000.00	$3,736.00
					Bond's Price↑
				Duration = sum of (PV*t)/sum of PVs =	$3,736.00/$1000	3.74

Duration of the bond is 3.74 years

b. What is the duration of a three-year Treasury bond with a 4 percent semiannual coupon selling at par?

Year (t)	Payments (n)	Cash Flow from coupon payments (4%/2 of $1000)	Cash Flow from maturity amount	Total Cash Flow from coupon payments and maturity amount (CF)	Present value (PV) discounted at 4%/2 = 2% semiannual yield to maturity	PV *t
0.5	1.0	$20.00		$20.00	$19.61	$9.80
1.0	2.0	$20.00		$20.00	$19.22	$19.22
1.5	3.0	$20.00		$20.00	$18.85	$28.27
2.0	4.0	$20.00		$20.00	$18.48	$36.95
2.5	5.0	$20.00		$20.00	$18.11	$45.29
3.0	6.0	$20.00	$1,000.0	$1,020.00	$905.73	$2,717.19
Sum					$1,000.00	$2,856.73
					Bond's Price↑
				Duration = sum of (PV*t)/sum of PVs =	$2856.73/$1000	2.86

Duration of the Bond is 2.86 years

c. What is the duration of a two-year Treasury bond with a 4 percent semiannual coupon selling at par?

Year (t)	Payments (n)	Cash Flow from coupon payments (4%/2 of $1000)	Cash Flow from maturity amount	Total Cash Flow from coupon payments and maturity amount (CF)	Present value (PV) discounted at 4%/2 = 2% semiannual yield to maturity	PV *t
0.5	1.0	$20.00		$20.00	$19.61	$9.80
1.0	2.0	$20.00		$20.00	$19.22	$19.22
1.5	3.0	$20.00		$20.00	$18.85	$28.27
2.0	4.0	$20.00	$1,000.0	$1,020.00	$942.32	$1,884.64
Sum					$1,000.00	$1,941.94
					Bond's Price↑
				Duration = sum of (PV*t)/sum of PVs =	$1941.94/$1000	1.94

Duration of the Bond is 1.94 years

Formula used in excel calculation: