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 $3.0 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 7% (paid annually) is 6.5 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.) Holdings million 5-year bond 20-year bond million 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.) Par value

Answer 1

Given,

Perpetual payments = $ 3.0 million per year

Yield to maturity (y) = 20% or 0.20

Solution :-

(a pv of Perpetual obligation Perpetual obligation – $ 15 million $3.0 million O.RO Duration of obligation - lty I t 0.20 0. 20 = 1.20 = 6 years Duration of S-year maturity bords = 3.7 years (Os) Duration of 20 year maturity bords = 6.5 Years

het 'W' be the weight of 5 year maturity bonds. weight of 20-year matunitez bonds - - o Now, (W x Ds) + [11_w) x D20] - Duration of Obligation (w x 3.7) + [ (1-2) x 6.5) = 6 (3.70 +6.5-16.500 = 6 (3.770 - (6.5)w = 6-6.5 (-2.8) w = -0.5 W = – O. S – O. 17857

Therefore, market value of 5-year bond - PU of Perpetual obligation & = $ 15 million & O. 17857 - - 3.า ๓) 2507 w market value of 20-year bond - PV of Perpetual obligation & (1-w) & is million X (1- 0.17857) $ 15 million x 0.82143 = $ 12.3 million (b) Suppose the par value of 20-year bond is $ 1000 Coupon rate = 7% p.a. Coupon payment (C) = $1000 x 7% i $70

Vield to maturity ly) = 20% or 0.20 Time to maturity (n)= 20 year Now, Price of 20- year bond T I 7 par value = 350[1-19120330] + lo - 350[1-30.3325099) + 35999 38.335999 looo 38.3375999 350 x 37.3375999) 38.3375999 looo 38.3375999 340.87058 t 26.084053 $366.95 So, 20 year bond is selling 0.36695 times ($66.95) its par Value. $1000

Thus, Price of 20 yead - Par value x 0.36695 bond $ 15 million x 0.82143 = Par Value x 0.36695 $ 12.32145 = par Value x 0.36695 $12.32145 - par Value 0.366 95 $33.578008 million - Par value par Value = $33.5 78008 million or $33578008