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 $3.6 million per year to beneficiaries. The yield to maturity on all bonds is 20%.

a. If the duration of 5-year maturity bonds with coupon rates of 16% (paid annually) is 3.7 years and the duration of 20-year maturity bonds with coupon rates of 10% (paid annually) is 6.2 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

Duration of the perpetuity = (1+r)/r where r = YTM on all bonds

= (1+20%)/20% = 6 years

a). For immunization, duration of the liability has to match the sum of the weighted durations of the bonds.

Let the weight of the 5-year bond be w. Then the weight of the 20-year bond will be (1-w).

The duration-matching equation will be

6 = 3.7w + 6.2*(1-w)

6 = 3.7w + 6.2 - 6.2w

-0.2 = -2.50w

w = 0.08 or 8%

So, weight of 5-year bond = 8%; weight of 20-year bond = 100%-8% = 92%

Present Value of perpetual liability = Annual payment/r = 3,600,000/20% = 18,000,000

So, market value of the 5-year bond which should be held for immunization = weight of the bond*PV of perpetual liability

= 8%*18,000,000 = 1,440,000.0

Market value of the 20-year bond which should be held for immunization = weight of the bond*PV of perpetual liability

= 92%*18,000,000 = 16,560,000.0

b). Market price of the 20-year bond: FV (or par value) = 1,000; coupon rate = 10%; PMT (or annual coupon) = 10%*1,000 = 100; N = 20; rate (or YTM) = 20%, CPT PV. PV = 513.04

Market price/par value = 513.04/1,000 = 0.51304

Market price = par value*0.51304

par value of the holdings in the 20-year bond = market price/0.51304 = 16,560,000/0.51304 = 32,278,057