Question

Given Par rates p(1)=2.00%, p(2) =3.25%, p(3)=4.25%. p(4)=5.10%.

1. Use the bootstrapping methodology to derive the theoretical value of a Spot rate curve, and calculate Spot rates r(1)= ?, r(2) = ?, r(3)=? r(4)=?  (4 Points)
2. Based on the Spot rates you calculated above, what is the price of a 4 year annual coupon paying bond, with a coupon rate of 6% and a par value of a $1,000 (3 Points)
3. Based on the price you obtained in b, what is the Yield to Maturity of this bond. (3 Points)

Answer 1

1) Since the cash-flows for period 1 are directly discounted at rate p(1)

r(1) = p(1) = 2.00%

r(2) = [(1+p(1))*(1+p(2))^(1/2) ] -1

r(2) = [(1.02*1.0325)^(1/2)]-1 = 0.0262

r(2) = 2.62%

r(3) = [(1+p(1))*(1+p(2)*(1+p(3))^(1/3) ] -1

r(3) = [(1.02*1.0325*1.0425)^(1/3)]-1 = 0.0316

r(3) = 3.16%

r(4) = [(1+p(1))*(1+p(2)*(1+p(3)*(1+p(4))^(1/4) ] -1

r(4) = [(1.02*1.0325*1.0425*1.0510)^(1/4)]-1 = 0.0364

r(4) = 3.64%

2)

The bonds pays a coupon of $60 annually for 4 years and par value $1000 at the end of 4 years

The bond price can be calculated by discounting the annual cassh-flows by the spot rates

Year	Cash-flow	Spot rate for the year	PV of the cash-flow	PV of the cash-flow formula
1	60	2%	58.82352941	Y6/((1+Z6)^X6)
2	60	2.62%	56.97537991	Y7/((1+Z7)^X7)
3	60	3.16%	54.65340822	Y8/((1+Z8)^X8)
4	1060	3.64%	918.7476643	Y9/((1+Z9)^X9)
			1089.199982

The total of Present value (PV) is the bond-price = $1089.20

3)

The YTM is the rate at which all the cash-flows of the bond are discounted to arrive at the current price

Solve this using a financial calculator

PV= -1089.2

PMT= 60

FV=1000

N=4

CPT I/Y, we get

I/Y =3.568

Calculating for YTM, we get YTM= 3.568%