Question

My pension plan will pay me$5,000 once a year for a 10 -year period. The first payment will come in exactly five years. The pension fund wants to immunize its position.

a. What is the duration of its obligation to me? The current interest rate is 8% per year.

b.If the plan uses 3-year and 30-year zero-coupon bonds to construct the immunized position, how money ought to be placed in each bond?

c.What will be the face value of the holdings in each zero?

Answer 1

Part (a)

Year	Cash flows	Disc rate	Disc. Factor	PV of CF	t x PV
t	CF	r	DF = (1 + r)^(-t)	CF x DF
5	5000	8%	0.6806	3,403	17,015
6	5000	8%	0.6302	3,151	18,905
7	5000	8%	0.5835	2,917	20,422
8	5000	8%	0.5403	2,701	21,611
9	5000	8%	0.5002	2,501	22,511
10	5000	8%	0.4632	2,316	23,160
11	5000	8%	0.4289	2,144	23,589
12	5000	8%	0.3971	1,986	23,827
13	5000	8%	0.3677	1,838	23,900
14	5000	8%	0.3405	1,702	23,832
Total				24,661	218,771

Duration of obligation = 218,771 / 24,661 = 8.871

Part (b)

PV of obligation, P = 24,661; Duration of obligation, D = 8.871

PV of 3 year zero coupon bond = P1 = 1,000 / (1 + 8%)³ = 793.83; Duration D1 = 3; assume N1 number in the immunized portfolio

PV of 30 year zero coupon bond = P2 = 1,000 / (1 + 8%)³⁰ = 99.38; Duration, D2 = 30; assume N2 number in the immunized portfolio

N1 x P1 + N2 x P2 = P; Hence, 793.83N1 + 99.38N2 = 24,661 ----- eqn (1)

N1 x P1 x D1 + N2 x P2 x D2 = P x D; Hence, 793.83 x 3 x N1 + 99.38 x 30 x N2 = 24,661 x 8.871 -------eqn (2)

30 x eqn (1) - eqn (2) results in:

793.83 x (30 - 3) x N1 = 24,661 x (30 - 8.871)

Hence, N1 = 24.31

and N2 = (24,661 - 793.83 x 24.31) / 99.38 = 53.96

Part (c) Face value in 3 year zero = N1 x 1000 = 24,310

Face value in 30 year zero = N2 x 1000 = 53,962