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 $2.6 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 6% (paid annually) is 7.9 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

Present Value of Perpetual Obligation = Obligation/YTM = $2.6 million / 0.16   = $16.25 M

Duration of Perpectual Obligation = (1+YTM)/YTM = (1+0.16)/0.16 = 1.16/0.16 = 7.25 years

a)

Now, Duration of 5 year bond is 4 year & duration of 20 years bond is 7.9 years.

For immunization purpose, duration of the liability should be match the sum of the weighted durations of both bonds.

Let's Consider the weightage of 5 year bond = W

it means, weightage of 20 year bond should be = 1 - W

So That,

[W * duration of 5 year bond] + [(1-W) * duration of 20 year bond] = Duration of Obligation

W * 4 + (1-w) * 7.9 = 7.25

4W + 7.9 - 7.9W = 7.25

4W -7.9W = 7.25 - 7.9

- 3.9W = -0.65

W = (-0.65)/(-3.9)

W = 0.16666666

So, weightage of 5 year bond = 0.16666667

Then Weightage of 20 year bond = 1- 0.16666667 = 0.83333333

So, Market value of 5 year bond for immunization = Weightage of bond * Present Value of Perpetual Obligation

                                                                               = 16,250,000 * 0.16666667

                                                                               = 2,708,333.4 (Rounded off)

So, Market value of 20 year bond for immunization = Weightage of bond * Present Value of Perpetual Obligation

                                                                               = 16,250,000 * 0.83333333

                                                                               = 13,541,666.6 (Rounded off)

b)

Now,

Lets consider the Face value or Par value of 20 year coupon bond is = P

then its coupon amount = P*6% = 0.06P

YTM = 16% or 0.16

n = 20 years

Price of 20 year bond (calculated above) = 13,541,666.6

Now,

So, Par Value of 20 years bond = $33,262,436 (Rounded off to dollar)