Question

Immunization with Coupon Bonds Suppose that you are managing a pension fund with obligations to make perpetual payments of $2.6 million per year to beneficiaries. The required rate of return 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 6% (paid annually) is 7.9 years, how much of each of these coupon bonds (in $ market value) will you want to hold to fully fund and immunize your obligation? The yield to maturity on both bonds is 16%. [Hint: compute the present value and duration of liabilities first. This provides the dollar amount and target duration you will need to match with an investment in the two coupon bonds above.] (Do not round intermediate calculations. Enter your answers in millions rounded to 1 decimal place.) Holdings 5-year bond $ million 20- year bond $ million b. What will be the par value of your holdings in the 20-year coupon bond? (Do not round intermediate calculations. Round your answer to 2 decimal places.) Par value $ million

Answer 1

ANSWER 1

THE PRESENT VALUE OF THE ANNUITIES IS = 2.6/0.16 = $ 16.25 MILLION

DURATION OF PERPTUITY = (1 + RATE)/RATE = (1+0.16)/0.16 = 7.25 YEARS

SAY W = WEIGHT OF 5 YEAR MATURITY BOND (WITH DURATION OF 4 YEARS)

SO (W X 4) + ((1-W) X 7.9) = 7.25

SO W = 0.1667 ( 5 YEAR MATURITY BOND), 1-W = 0.8334 (20 YEAR MATURITY BOND) ANSWER 1

THEREFORE,

0.1667 X $ 16.25 = $ 2.7089 MILLION IN 5 YEAR MATURITY BOND  ANSWER 1

0.8333 X $ 16.25 = $ 13.5411 MILLION IN 20 YEAR MATURITY BOND  ANSWER 1

ANSWER 2 : PAR VALUE OF HOLDING IN BOTH THE BONDS =

THE PRICE OF 5-YEAR BOND =

120 X PVIFA @16%, 5 YEARS + 1000 X PVIF @16%, 5 YEARS

120 X 3.2743 + 1000 X 0.476 = 392.916 + 476 = 868.916

THEREFORE

PAR VALUE = $ 2.7089 MILLION  X (868.916/1000) = $ 2.3538 MILLION = $ 2.35 MILLION (ANSWER 2)

THE PRICE OF 20-YEAR BOND =

60 X PVIFA @16%, 20 YEARS + 1000 X PVIF @16%, 20 YEARS

60 X 5.9288 + 1000 X 0.051 = 355.728 + 51 = 406.728

THEREFORE

PAR VALUE = $ 13.5411  X (406.728/1000) = $ 5.5075 MILLION = $ 5.51 MILLION (ANSWER 2)

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