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?
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
Get Answers For Free
Most questions answered within 1 hours.