Question

 

You are responsible for managing the following liability: 29-year bond, 6.5% annual coupon, when the market interest rate is 5%.

You want to consider immunizing the liability using 14-year and 16-year zero-coupon ?bonds.

What are the investment weights needed for the two bonds? ?

What are the present values of the two bonds needed to immunize the liability? ?

What are the face values of the two bonds needed to immunize the liability? ?

Build a sensitivity table showing the results of changes in interest rates, with the following format:

	weight	3%	4%	5%	6%	7%
Liability
Bond (14 yrs)
Bond (16 yrs)
Portfolio sum

Answer 1

Assuming coupon bond and each of the zero-coupon bond have a face value of $100:

1. Immunization is the process of duration matching of the liabilities with assets, such that any change in the value of liabilities over time is covered with the equivalent change in the value of the assets. Thus, we need to find out the duration of libilities and assets:

Step 1: Duration calculation of liability (coupon bond):

Duration of a bond = , where:

• PV(CF_t*t) = present value of payments on bond at period t
• t = time for each cash flow

Year	Cash Flow, CFt	PVF @ 5%	PV	CFt*t	PV(CFt*t)
(a)	(b)	( C)	(d) = (b*C)	( e)= (a*b)	(f) = (e*c)
1	6.50	0.95	6.19	6.50	6.19
2	6.50	0.91	5.90	13.00	11.79
3	6.50	0.86	5.61	19.50	16.84
4	6.50	0.82	5.35	26.00	21.39
5	6.50	0.78	5.09	32.50	25.46
6	6.50	0.75	4.85	39.00	29.10
7	6.50	0.71	4.62	45.50	32.34
8	6.50	0.68	4.40	52.00	35.20
9	6.50	0.64	4.19	58.50	37.71
10	6.50	0.61	3.99	65.00	39.90
11	6.50	0.58	3.80	71.50	41.80
12	6.50	0.56	3.62	78.00	43.43
13	6.50	0.53	3.45	84.50	44.81
14	6.50	0.51	3.28	91.00	45.96
15	6.50	0.48	3.13	97.50	46.90
16	6.50	0.46	2.98	104.00	47.64
17	6.50	0.44	2.84	110.50	48.21
18	6.50	0.42	2.70	117.00	48.62
19	6.50	0.40	2.57	123.50	48.87
20	6.50	0.38	2.45	130.00	49.00
21	6.50	0.36	2.33	136.50	49.00
22	6.50	0.34	2.22	143.00	48.88
23	6.50	0.33	2.12	149.50	48.67
24	6.50	0.31	2.02	156.00	48.37
25	6.50	0.30	1.92	162.50	47.99
26	6.50	0.28	1.83	169.00	47.53
27	6.50	0.27	1.74	175.50	47.01
28	6.50	0.26	1.66	182.00	46.43
29	106.50	0.24	25.87	3,088.50	750.34
	Market price of bond		122.71	? PV(CFt*t)	1,855.39
	Duration of bond = 1855.39/122.71 = 15.12

step 2: Duration of zeros: Since zero-coupon bonds do not have any coupon payments during its life time, its duration is equal to life of zero-coupon bond:

• Duration of 14-year zero-coupon: 14
• Duration of 16-year zero-coupon: 16

Step 3: Calculate weights needed for duration matching:

Portfolio duration = ?(weighted average duration of each of the assets)

15.12 = (WA*14)+(WB*16), where: WA = weight of 14-year zero-coupon and WB = weight of 16-year zero-coupon

Substituting WA = 1-WB, since total of weights should be 1:

15.12 = ((1-WB)*14)+(16*WB), WB = 0.56, thus WA = 1-0.56=0.44

2. Present value of zeros:

• 14-year = Face value * PVF(5%,14 years) = 100*0.5051 = $50.51
• 16-year = Face value * PVF(5%,16 years) = 100*0.4581 = $45.81

3. Face values of zeros:

• 14-year: $100
• 16-year: $100

4. Sensitivity table:

Step1: Calculation of PV of bonds in each of the scenarios:

Calculated using financial calculator:

• coupon bond: N=29, I/Y=rate per period(3%,4%....,7%),PMT=-6.50,FV=-100, CPT--->PV
• 14-year zero: N=14, I/Y=rate per period(3%,4%....,7%),PMT=0,FV=-100, CPT--->PV
• 16-year zero: N=16, I/Y=rate per period(3%,4%....,7%),PMT=0,FV=-100, CPT--->PV

	3%	4%	5%	6%	7%
Liability	-167.16	-142.46	-122.71	-106.80	-93.86
Bond (14 year)	66.11	57.75	50.51	44.23	38.78
Bond (16 year)	62.32	53.39	45.81	39.36	33.87

Step 2: Calculate effect on portfolio:

	Weight	3%	Weight *PV	4%	Weight *PV	5%	Weight *PV	6%	Weight *PV	7%	Weight *PV
Liability	-1.00	167.16	-167.16	142.46	-142.46	122.71	-122.71	106.80	-106.80	93.86	-93.86
Bond (14 year)	0.44	66.11	29.09	57.75	25.41	50.51	22.22	44.23	19.46	38.78	17.06
Bond (16 year)	0.56	62.32	34.90	53.39	29.90	45.81	25.65	39.36	22.04	33.87	18.97
Portfolio sum	0.00		-103.17		-87.15		-74.83		-65.29		-57.83