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 $1.1 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? (Do not round intermediate calculations. Enter your answers in millions. Round your answers to 1 decimal place.)

Five-year

$  million

Twenty-year

$  million

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

Face Value
Five-year	$  million
Twenty-year	$  million

Answer 1

a). PVA = $1.1/0.1 = $11

Duration = 1.10/0.1 = 11 years

Let x = Weight of 5-Year zeros, and

1 - x = Weight of 20-Year zeros. Then

11 = 5x + 20(1 - x)

So, x = 0.60 (in 5 year zeros), and

1 - x = 0.4 (in 20 year zeros).

Market Value of 5 year zeros = $11 x 0.6 = $6.6 million

Market Value of 20 year zeros = $11 x 0.4 = $4.4 million

b). Face Value of 5 year zeros = $6.6 million x (1.10)⁵ = $6.6 million x 1.61051 = $10.63million

Face Value of 20 year zeros = $6.6 million x (1.10)²⁰ = $6.6 million x 6.7275 = $44.40 million