Pension funds pay lifetime annuities to recipients. If a firm remains in business indefinitely, the pension obligation will resemble a perpetuity. Suppose, therefore, that you are managing a pension fund with obligations to make perpetual payments of $2.0 million per year to beneficiaries. The yield to maturity on all bonds is 16%.
| a. | If the duration of 5-year maturity bonds with coupon rates of 12% (paid annually) is 4 years and the duration of 25-year maturity bonds with coupon rates of 4% (paid annually) is 16 years, how much of each of these coupon bonds (in market value) will you want to hold to both fully fund and immunize your obligation?
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Answer:
(a) If the duration of 5-year maturity bonds with coupon rates of 12% (paid annually) is 4 years and the duration of 25-year maturity bonds with coupon rates of 4% (paid annually) is 16 years, how much of each of these coupon bonds (in market value) will you want to hold to both fully fund and immunize your obligation?
PV of the firm’s “perpetual” obligation
= ($2 million/0.16)
= $12.5 million. ·
Based on the duration of a perpetuity, the duration of this obligation
= (1.16/0.16)
= 7.25 years.
Denote by w the weight on the 5-year maturity bond, which has duration of 4 years.
Then, w x 4 + (1 – w) x 11
= 7.25, which implies that w = 0.5357
. Therefore, 5 0.5357 x $12.5 = $6.7 million in the 5-year bond and 0.4643 x $12.5 = $5.8 million in the 25-year bond. The total invested amounts to $(6.7+5.8) million = $12.5 million, fully matching the funding needs
(b)
The price of the 20-year bond is 60 x PA(16%, 20) + 1000 x PF(16%, 20) = $407.11. where PA(x%, n) is the present value of an annuity that has $1 par value, yields x% yearly and has n years to maturity, and PF(x%, n) is the corresponding number for a zero coupon bond. Therefore, the bond sells for 0.4071 times its par value, and Market value = Par value x 0.4071 => $5.8 million = Par value x 0.4071 => Par value = $14.25 million. Another way to see this is to note that each bond with a par value of $1,000 sells for $407.11. If the total market value is $5.8 million, then you need to buy 14,250 bonds, which results in total par value of $14,250,000.
