Question

My pension plan will pay me $18,500 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 13.5% per year.

b. If the plan uses 5-year and 20-year zero-coupon bonds to construct the immunized position, how much money ought to be placed in each bond?

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

Answer 1

Answer-A

Duration of Obligations =

PV= Present value of cash flows

Year	Cash flows	Interest rate	Discount factor	PV of cash flows	PV x n
n	C(In $)	r	[email protected]%	PV = C x DF
0	0	13.50%	1	0	0
1	0	13.50%	0.88106	0	0.00
2	0	13.50%	0.77626	0	0.00
3	0	13.50%	0.68393	0	0.00
4	0	13.50%	0.60258	0	0.00
5	18,500	13.50%	0.53091	9821.83	49109.15
6	18,500	13.50%	0.46776	8653.59	51921.57
7	18,500	13.50%	0.41213	7624.31	53370.19
8	18,500	13.50%	0.36311	6717.46	53739.65
9	18,500	13.50%	0.31992	5918.46	53266.17
10	18,500	13.50%	0.28187	5214.51	52145.05
11	18,500	13.50%	0.24834	4594.28	50537.06
12	18,500	13.50%	0.21880	4047.82	48573.86
13	18,500	13.50%	0.19278	3566.36	46362.72
14	18,500	13.50%	0.16985	3142.17	43990.38
Total				59,300.80	503,015.80

Hence Duration of the Obligation =D = 503015.80/59300.80 = 8.48 Years

---------------------------------------------------------------

Answer-b

Duration of the Zero coupon bond is its maturity period.

Hence Duration of the 5 year Zero Coupon Bond = 5 Years

Hence Duration of the 20 year Zero Coupon Bond = 20 Years

Let the weight of the 5year Zero Coupin bond = W

Hence Weight of the 20 Year Zero Coupon Bond = 1-W

Immunised Duration = Weighted duration of the Zero Coupon bonds in the portfolio

=>8.48 = 5*W + [20*(1-W)]

=>8.48 = 5W +20-20W

=>11.52 = 15W

=>W = 0.768

Hence weight of 5 year Zero coupon bond = 0.768

Weight of 20 year Zero coupon Bond = 1-0.768 = 0.232

Money to be placed in 5 year Zero Coupon bond = $59300.80*0.768 = $45543.01

Money to be placed in 20 year Zero Coupon bond = $59300.80*0.232 = $13757.79

----------------------------------------------------------------------------------------

AnswerC

Face value = Current present value * (1+r)^n

Hence face value of 5 year ZCB = $45543.01* (1.135)^5 = $85782.96

face value of 20 year ZCB = $13757.79* (1.135)^20 = $173167.31

-----------------------------------------------------------------------------