Question

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 $600.000 per year. The interest rate is 8%. You plan to fully fund the obligation using 5-year and 20-year maturity zero-coupon bonds. How much market value of each of the zeros will be necessary to fund the plan if you desire an immunized position?

Answer 1

The present value of the Perpetuity = Perpetuity / Rate of return

The present value of the Perpetuity = 600,000 / 8%

The present value of the Perpetuity = 7,500,000

The duration of perpetuity = (1 + 8%) / 8%

The duration of perpetuity = 13.50

Let

x = Weight of 5 years Zero coupon bonds

1-x = Weight of 20 years Zero coupon bonds

13.50 = 5x + 20 (1-x)

13.50 = 5x + 20 - 20x

13.50 = -15x + 20

-15x = 13.50 - 20

-15x = -6.50

x = -6.50 / -15

x = 0.4333

Market values

5 years Zero coupon bonds = 0.43333 * 7,500,000

5 years Zero coupon bonds = 3,249,975

20 years Zero coupon bonds = 7,500,000 - 3,249,975

20 years Zero coupon bonds = 4,250,025