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 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 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

The PV "perpetual" obligation of the firm =
`$\frac{\$ 2 \text{ million}}{0.16}$`

= $ 12.5 million

Also based on duration of the perpetuity, duration of this obligation =
`$(1.16)/(0.16)$`

= 7.25 years

Let
`$w$` be the
`$\text{weight}$` on the
`$5$` year maturity bond, which has a duration of
`$4$`years. Then :

`$w * 4 +(1-w) * 11 = 7.25$`

`$w=0.5357$`

Therefore,

`$0.5357 * \$ 12.5 = \$ 6.7$` million in the
`$5$` year bond

`$0.4643 * \$12.5=\$5.8$` million in the
`$2$` year bond.

Therefore, the total invested amounts to $
`$(6.7+5.8)$` million =
`$\$12.5$` million, which fully matches the funding needs.