Question

You manage a pension fund that will provide retired workers with lifetime annuities. You determine that the payouts of the fund are going to closely resemble level perpetuities of $1.6 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? (Do not round intermediate calculations. Enter your answers in millions. Round your answers to 1 decimal place.)

b. What must be the face value of each of the two zeros to fund the plan? (Do not round intermediate calculations. Enter your answers in millions rounded to 2 decimal places.)

Answer 1

We will first calculate present value fo annuities:

Payout = 1.6 million

Rate of interest = 10%

Present value = 1.6 million/ 10%

Present value = 16 million

Duration of perpetuity = (1 + y)/ y

y = 10%

Duration of perpetuity = (1 + 10%)/ 10%

Duration of perpetuity = 11 years

Now assume x = weight of 5 year zeroes, and

1-x = weight of 20 year zeroes.

11 = 5 * x + 20 * (1 - x)

15 * x = 9

x = 3/5

1 - x = 2/5

5 year zeroes: $16M * 3/5 = $9.6 million market value

20 year zeroes: $16M * 2/5 = $6.4 million market value

Part B:

Face value of 5 year zeroes = 9.6 * (1.1)5

Face value of 5 year zeroes = $15.46 million

Face value of 20 year zeroes = 6.4 * (1.1)20

Face value of 20 year zeroes = $43.06 million