Question

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 4.0 years and the duration of 20-year maturity bonds with coupon rates of 4% (paid annually) is 8.6 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? (Do not round intermediate calculations. Enter your answers in millions rounded to 1 decimal place.)

b. What will be the par value of your holdings in the 20-year coupon bond? (Enter your answer in dollars not in millions. Do not round intermediate calculations. Round your answer to the nearest dollar amount.)

Answer 1

a.

PV of the firm’s “perpetual” obligation = ($2.0 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 8.6 = 7.25,

4w + 8.6 - 8.6w = 7.25

1.35 = 4.6w

which implies that w = 0.29348. Therefore,

0.29348 x $12.50 = $3.7 million in the 5-year bond and
0.70652 x $12.50 = $8.8 million in the 20-year bond.

b.

The price of the 20-year bond is 40 x PA(16%, 20) + 1000 x PF(16%, 20) = $288.54

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 0.28854 times its par value, and
Market value = Par value x 0.28854
$8.8 million = Par value x 0.28854
Par value = $30.4983 or 30.50 million.