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

**Solution**

The present value of the annuities is $0.6 million / 0.08 = $7.5M

The duration is 1.08 / 0.08 =13.5 years

Let x = Weight of 5 years Zeros and

1-x = Weight of 20 years Zeros

13.5= 5x+20(1-x)

13.5 = -15x +20

x = 6.5/15

and so x = 0.4333(in 5 year zeros) and 1-x = 0.56667 (in 20 year zeros)

**5 year zeros : $7.5 M *0.4333 = $3.25M market
value**

**20 year zeros : $7.5 M *0.56667 = $4.25 M market
value**

**Face value of 5 year zeros : $3.25 M *(1.08) ^{5}
= 4.78 M**

**Face value of 20 year Zeros : $4.25 M *
(1.08) ^{20} = 19.81 M**

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