Question

Consider two bonds: bond XY and bond ZW . Bond XY has a face value of $1,000 and 10 years to maturity and has just been issued at par. It bears the current market interest rate of 7% (i.e. this is the yield to maturity for this bond). Bond ZW was issued 5 years ago when interest rates were much higher. Bond ZW has face value of $1,000 and pays a 13% coupon rate. When issued, this bond had a 15-year, so today its remaining maturity is 10 years. Both bonds make annual coupon payments.

a) (5 points) What is the price of Bond ZW , given that market interest rates are 7%? b) (15 points) Compute the duration for bond bonds (use Excel).

Answer 1

Price of the bond can be calculated by discounting the future cash flows at the current rate of interest:

Coupon Par value oupon oupon Price CouonCoup 10 10

0.13 × 1000 0.13 1000 0.13 1000 1000 PriceBondzw_ (1.07)1 (1.07)10(1)10 (1.07)2 1.07)1

Price Bondzw $1,421.41

Note: when the coupon rate > than the discount rate, the price > par and vice versa

Duration of the two bonds are calculated below:

Bond ZW
Year	CF	PV		PV x t
1	130	130*1/1.09^(1)=	119.266055	119.266055045872*1=	119.266055
2	130	130*1/1.09^(2)=	109.4183991	109.418399124653*2=	218.8367982
3	130	130*1/1.09^(3)=	100.3838524	100.383852407938*3=	301.1515572
4	130	130*1/1.09^(4)=	92.09527744	92.0952774384755*4=	368.3811098
5	130	130*1/1.09^(5)=	84.49108022	84.4910802187849*5=	422.4554011
6	130	130*1/1.09^(6)=	77.51475249	77.5147524942981*6=	465.088515
7	130	130*1/1.09^(7)=	71.11445183	71.1144518296312*7=	497.8011628
8	130	130*1/1.09^(8)=	65.24261636	65.2426163574599*8=	521.9409309
9	130	130*1/1.09^(9)=	59.85561134	59.8556113371191*9=	538.700502
10	130	130*1/1.09^(10)=	54.9134049	54.9134048964396*10=	549.134049
10	1000	1000*1/1.09^(10)=	422.4108069	422.410806895689*10=	4224.108069
		SUM	1256.706308	SUM	8226.86415
				Macaulay's duration	8226.86415/1256.706308 = 6.546369742
				Modified duration	Macaulay's duration/(1.07)= 6.118102563

Bond XY
Year	CF	PV		PV x t
1	70	70*1/1.09^(1)=	64.22018349	64.2201834862385*1=	64.22018349
2	70	70*1/1.09^(2)=	58.91759953	58.9175995286592*2=	117.8351991
3	70	70*1/1.09^(3)=	54.0528436	54.0528436042745*3=	162.1585308
4	70	70*1/1.09^(4)=	49.58976477	49.5897647745638*4=	198.3590591
5	70	70*1/1.09^(5)=	45.49519704	45.4951970408842*5=	227.4759852
6	70	70*1/1.09^(6)=	41.73871288	41.7387128815451*6=	250.4322773
7	70	70*1/1.09^(7)=	38.29239714	38.2923971390322*7=	268.04678
8	70	70*1/1.09^(8)=	35.13063958	35.1306395770938*8=	281.0451166
9	70	70*1/1.09^(9)=	32.22994457	32.2299445661411*9=	290.0695011
10	70	70*1/1.09^(10)=	29.56875648	29.5687564826982*10=	295.6875648
10	1000	1000*1/1.09^(10)=	422.4108069	422.410806895689*10=	4224.108069
		SUM	871.646846	SUM	6379.438266
				Macaulay's duration	6379.438266/871.646846=7.318833649
				Modified duration	Macaulay's duration/(1.07)= 6.840031448