Question

Q. You manage a pension fund that will provide retired workers with lifetime annuities. You determine that the payouts of the fund are essentially going to resemble level perpetuities of $1 million per year. The interest rate is 10%. You plan to fully fund the obligation using 5-year and 20-year maturity zero-coupon bonds.

a. How much market value of each of the zeros will be necessary to fund the plan if you desire an immunized position?

b. what must be the face value of the two zeros to fund the plan?

Answer 1

a. How much market value of each of the zeros will be necessary to fund the plan if you desire an immunized position?

Present Value of Annuities = $1 million / 0.10 = $10 million

Duration = 1.10 / 0.10 = 11 years

Let x be weight of 5 years Zero coupon bonds and

Let (1 - x) be weight of 20 years Zero coupon bonds

So equation will be:-

5x + 20 (1 - x) = 11

5x + 20 - 20x = 11

15x = 9

x = 0.60

1 - x = 1 - 0.60 = 0.40

Hence 5 years Zero coupon bonds = $10 million * 0.60 = $6 million and

20 years Zero coupon bonds = $10 million * 0.40 = $4 million

  

b. what must be the face value of the two zeros to fund the plan?

Face value of 5 years Zero coupon bonds = $6 million * [(1.10) ^ 5 ]

= $6 million * 1.61

= $9.66 million

Face value of 20 years Zero coupon bonds = $4 million * [(1.10) ^ 20 ]

= $4 million * 6.73

= $26.92 million