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 $0.8 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.) Market Value 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

Duration of Perpetuity = [1 + r] / r = 1.10 / 0.1 = 11 years

PV of Payment = Perpetual CF / r = 0.8 / 0.10 = $8 million

Let w be the weight of the five­-year zero­-coupon bond and therefore (1 – w) is the weight of the twenty-­year zero-coupon bond. Then we find w by solving:

(w × 5) + [(1 – w) × 20] = 11

5w + 20 - 20w = 11

15w = 9

w = 9/15 = 0.60

So, 60% of the portfolio will be invested in the five­-year zero-­coupon bond and 40% in the twenty-­year zero­-coupon bond.Therefore, the market value of the five­ year zero is:

$8 million x 0.6 = $4.8 million

Similarly, the market value of the twenty year zero is:

$8 million x 0.4 = $3.2 million

b). FV of 5-year Zero-coupon Bond = $4.8 million x 1.10⁵ = $7.73 million

FV of 20-year Zero-coupon Bond = $3.2 million x 1.10²⁰ = $21.53 million