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 $2.5 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 6.3 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

first we have to calculate amount of perpetual obligation:

perpetual obligation = perpetual payments / YTM

= 2,500,000 / 0.2

= 12,500,000

duration of perpetuity = (1+Y) / Y

where Y = YTM

duration = 1.2 / 0.2 = 6 years

Portfolio duration = weighted average duartion

let W1 = weight of 5 year bond

W2 = weight of 20 year bond

W1+W2 = 1

so W2 = 1 - W1

6 = W1*3.7 + (1 - W1)*6.3

6 = 3.7W1 + 6.3 - 6.3W1

W1 = 0.115385

so W2 = 1 - 0.115385

= 0.884615

a)

5 - year bond = 0.115385 x 12,500,000

= 1,442,308

20 - year bond = 0.884615 x 12500000

= 11,057,692

b)

first we have to calculate price of 20 year 10% coupon bond.yield is 20%

coupon = 1000 x 10% = $100

price of the bond can be calculated using PV excel function.it will be as follows:

YTM = 20%

nper = number of periods = 20

coupon = PMT = 100

future value = $1000

so bond value =

number of these bonds should be held = 11,057,692 / 513.04

= 21,553.19

so par value of holdings = 1000 x 21,553.19 = 21,553,190