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**

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