Question

My pension plan will pay me $15,000 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 9.5% per year. (Do not round intermediate calculations. Round your answer to 4 decimal places.)

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? (Do not round intermediate calculations. Round your answers to the nearest whole dollar amount.)

c. What will be the face value of the holdings in each zero? (Do not round intermediate calculations. Round your answers to the nearest whole dollar amount.)

Answer 1

As a first step we need to determine the PV of the payments under the pension plan and then the duration. Please see the table below. Please be guided by the second row to understand the mathematics. Figures in parenthesis, if any, mean negative values. All financials are in $.

Year	Cash flows	Interest rate	Discount factor	PV of cash flows	PV x n
n	C	r	DF = (1+r)-n	PV = C x DF
0	-	9.50%	1.0000	-	-
1	-	9.50%	0.9132	-	-
2	-	9.50%	0.8340	-	-
3	-	9.50%	0.7617	-	-
4	-	9.50%	0.6956	-	-
5	15,000	9.50%	0.6352	9,528.41	47,642.07
6	15,000	9.50%	0.5801	8,701.75	52,210.49
7	15,000	9.50%	0.5298	7,946.80	55,627.62
8	15,000	9.50%	0.4838	7,257.35	58,058.83
9	15,000	9.50%	0.4418	6,627.72	59,649.48
10	15,000	9.50%	0.4035	6,052.71	60,527.13
11	15,000	9.50%	0.3685	5,527.59	60,803.51
12	15,000	9.50%	0.3365	5,048.03	60,576.35
13	15,000	9.50%	0.3073	4,610.07	59,930.94
14	15,000	9.50%	0.2807	4,210.11	58,941.56
Total				65,510.56	573,967.98

PV of the pension payments = P = 65,510.56

Part (a)

And duration of the payments = D = Total of (PV x n) / P =  573,967.98 /  65,510.56 = 8.7615 years

Part (b)

Let W be the weight of the portfolio placed in the 5-year zero-coupon bonds (ZCBs) and (1 - W) be the weight placed in the 20-year zero-coupon bonds to construct the immunized position.

The duration of the immunized position = Weighted average duration of the ZCBs in the portfolio = Duration of the payments under pension plan = D

Hence, W x 5 + (1 - W) x 20 = D = 8.7615

Hence, W = (20 - 8.7615) / (20 - 5) =  0.7492

Hence, the money placed in 5 years ZCBs = W x P = 0.7492 x 65,510.56 = $ 49,083

and that placed in 20 years ZCBs = (1 - W) x P = (1 - 0.7492) x 65,510.56 = $ 16,428

Part (c)

The face value will be = FV = Current value x (1 + r)t where Current value = money placed in the respective bond and t = time to maurity in years, r = intererst rate per annum = 9.5%

Hence, face value of the holdings in 5 years ZCB = 49,083 x (1 + 9.5%)5 = $ 77,268

and the face value of the holdings in 20 years ZCB = 16,428 x (1 + 9.5%)20 = $ 100,892