Question

A company faces a stream of obligations over the next 4 years as shown below, where the numbers denote thousands of dollars.

Year	1	2	3	4
CF ('000)	200	0	500	300

The spot rate curve is given below:

Year	1	2	3	4	5	6	7	8	9	10
Spot Rate (%)	6.68	7.27	7.81	8.31	8.75	9.16	9.52	9.85	10.15	10.42

The company decides to invest in two bonds (each with face value $1,000) described as follows:

Bond 1 is a 4-year 8% coupon bond, and

Bond 2 is a 3-year zero-coupon bond.

(a) Calculate the prices of the two bonds and the obligation. (Keep 2 decimal places, e.g xx.12)

Bond 1:  _____________ Bond 2: ______________ Obligation: __________

(b) Calculate the quasi-modified durations for the two bonds and the obligation. (Keep 2 decimal places, e.g xx.12)

Bond 1:  ____________ Bond 2: ______________ Obligation: _____________

(c) Find a portfolio P₁x₁ + P₂x₂, where x₁ and x₂ denote the number of units of bonds 1 and 2, that has the same present value as the obligation stream and is immunized against a parallel shift in the spot rate curve. (Keep 2 decimal places, e.g xx.12)

x₁: _____________ x₂: __________________  

Answer 1

(a) Bond 1 coupon, C = 8% = 8% x 1,000 = $ 80

Price of bond 1 = C / (1 + s1) + C / (1 + s2)² + C / (1 + s3)³ + (C + Face value) / (1 + s4)⁴ = 80 / (1 + 6.68%) + 80 / (1 + 7.27%)² + 80 / (1 + 7.81)³ + (80 + 1,000) / (1 + 8.31%)⁴ = $ 1,220.91

Price of bond 2 = Face value / (1 + s3)³ = 1,000 / (1 + 7.81%)³ = $  798.04

Obligation = PV of the obligations = 200,000 / (1 + s1) + 500,000 / (1 + s3)³ + 300,000 / (1 + s4)⁴ = 200,000 / (1 + 6.68%) + 500,000 / (1 + 7.81%)³ + 300,000 / (1 + 8.31%)⁴ = $ 804,490

(b) Yield of bond 1 = RATE (Period, PMT, PV, FV) = RATE(4,80,-1220.91,1000) = 2.17%

Yield of the obligation = calculated using IRR function of excel

I have resorted to excel to calculate the quasi modified duration.

Bond	Settlement	Maturity	Coupon	Yield	Quasi Modified Duration
	A	B	C	D	F = MDURATION (A, B, C, D, 1)
Bond 1	1/1/2019	1/1/2023	8.00%	2.17%	3.54
Bond 2	1/1/2019	1/1/2022	0.00%	7.81%	2.78

Modified duration of the obligation:

Please see the table below. The figure of -804,490 is the PV of obligation calculated in part (a). Adjacent cells in blue contain the formula in excel I have used to get the final output.

Part (c)

PV of obligation = 804,490 = PV of bonds = P1x1 + P2x2 = 1,220.91x1 + 798.04x2 --------------- equation (1)

PV of obligation x Modified duration of obligation = 804,490 x 2.60 = P1x1D1 + P2x2D2 = 1,220.91x1 x 3.54 + 798.04x2 x 2.78

Hence, 2,091,101.45 = 4,326.44x1 + 2,220.68x2 --------- equation (2)

Perform: 2.78 x equation (1) - equation (2) gives:

2.78 x 804,490 - 2,091,101.45 = (2.78 x 1,220.91 - 4,326.44)x1 + (2.78 x 798.04 - 2,220.68)x2

Hence, 147,531.79 = -929.038x1

hence, x1 = - 158.80

and x2 = [804,490 - 1,220.91 x (-18.80)] / 798.04 = 1,251.03