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 $$1$ million per year. The interest

rate is $10\%.$ You plan to fully fund the obligation using $5-$year and $20-$year

maturity zero$-$coupon bonds.

a$.$ How much market value of each of the zeros will be necessary to fund

the plan if you desire an immunized position?

b$.$ What must be the face value of the two zeros to fund the plan?

Perpetuity CF

mil

Bond $1$ matruity

Bond $2$ maturity

Perpetuity YTM

Solution

a$.$

Duration of perpetuity" #DIV$/$OI

Weight $of5yr$ HDI$\frac{V}{01}$

Weightof$20yr\frac{,}{\text{\#}}DI\frac{V}{0}!$

Value of $5yr,"$ HDIV$/0!$ mil

Value of $20tr,$ "DIV$/$o$!$ mil

b$.$

Face value of $5$ yr zero

#DIV$/0$I mil

Face value of $20yr$ zero

#DIV$/0!$ mil