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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

Respuesta :

Solution :

The PV  "perpetual" obligation of the firm  = [tex]$\frac{\$ 2 \text{ million}}{0.16}$[/tex]

                                                                     = $ 12.5 million

Also based on duration of the perpetuity, duration of this obligation = [tex]$\frac{1.16}{0.16}$[/tex]

                                                                                                                  = 7.25 years

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

[tex]$w \times 4 +(1-w) \times 11 = 7.25$[/tex]

[tex]$w=0.5357$[/tex]

Therefore,

[tex]$0.5357 \times \$ 12.5 = \$ 6.7$[/tex] million in the [tex]$5$[/tex] year bond

[tex]$0.4643 \times \$12.5=\$5.8$[/tex] million in the [tex]$2$[/tex] year bond.

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

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