Question

Pension funds pay lifetime annuities to recipients. If a firm will remain 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 $2.3 million per year to beneficiaries. The yield to maturity on all bonds is 14.2%. a. If the duration of 5-year maturity bonds with coupon rates of 11.5% (paid annually) is four years and the duration of 20-year maturity bonds with coupon rates of 6% (paid annually) is 11 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?

Answer 1

PV of the firm’s “perpetual” obligation = ($2.3 million/0.14) = $16.4 million.

Based on the duration of a perpetuity, the duration of this obligation = (1.14/0.14) = 8.14 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 = 8.14, which implies that w = 0.4085.

w x 4 + (11-11w) =8.14

- 7W + 11 = 8.14

7W = 2.86

w = 0.4085

Therefore, 0.4085 x $16.4 = $6.7 million in the 5-year bond and

0.4643 x $12.5 = $5.8 million in the 20-year bond.

The total invested amounts to $(6.7+5.8) million = $12.5 million, fully matching the funding needs.