Question

2. Pension funds pay lifetime annuities to recipients. If a firm will remain 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 million per year to beneficiaries. The yield to maturity on all bonds is 16%. Note: duration of perpetuity is (1+i)/i.

a (0.25’). What is the present value and duration of the pension obligations?

b (0.25’). To fund the liabilities, you decide to choose two types of bonds to invest: the 5-year maturity bonds with coupon rates of 12% (paid annually) and the 20-year maturity bonds with coupon rates of 6% (paid annually). What is the duration of the 5-year coupon bond and 20-year coupon bond?

c (0.25’). To immunize your obligation, what is the asset weight (w) of the 5-year coupon bond and what is the asset weight of the 20-year coupon bond?

d (0.25’). To fully fund and immunize your obligation, how much of each of these coupon bonds you want to hold?

e (0.25’). What will be the par value of your holdings in the 20-year coupon bond? Assume the par value for each bond contract is $1000. Hint: first use financial calculator to calculate the PV of individual 20-year coupon bond. Then calculate # of bond contracts to hold.

Answer 1

a What is the present value and duration of the pension obligations?

Present Value = Perpetual payment / i = $ 2 mn / 16% = $ 12.50 mn

Duration = (1+i) / i = (1 + 16%) / 16% = 7.25 years

b To fund the liabilities, you decide to choose two types of bonds to invest: the 5-year maturity bonds with coupon rates of 12% (paid annually) and the 20-year maturity bonds with coupon rates of 6% (paid annually). What is the duration of the 5-year coupon bond and 20-year coupon bond?

The duration can be calculated using the DURATION function of excel. Please see the table below. The settlement date has been arbitrarily assumed to be 1/1/2019. The maturity date will be settlement date + years of maturity. The duration values are shown in yellow colored cells. The formula used to produce the duration is shown in the adjacent cell. The formula has been highlighted in blue color.

Hence, the duration of the 5-year coupon bond = 3.96 years and that of 20-year coupon bond = 7.91 years

c To immunize your obligation, what is the asset weight (w) of the 5-year coupon bond and what is the asset weight of the 20-year coupon bond?

In order to immunize, the duration of liabilities = duration of assets

Hence, 7.25 = w x 3.96 + (1 - w) x 7.91

Hence, w = (7.91 - 7.25) / (7.91 - 3.96) = 16.71%

d To fully fund and immunize your obligation, how much of each of these coupon bonds you want to hold?

5 years coupon bonds held for an amount = PV of obligation x w = $ 12.5 mn x 16.71% = $  2.09 mn

20 years coupon bonds held for an amount = PV of obligation x (1 - w) = $ 12.5 mn x (1 - 16.71%) = $ 10.41 mn

e What will be the par value of your holdings in the 20-year coupon bond? Assume the par value for each bond contract is $1000. Hint: first use financial calculator to calculate the PV of individual 20-year coupon bond. Then calculate # of bond contracts to hold.

Use financial calculator with following inputs to get PV of the 20 years coupon bond as:

i = 16%, n = 20, PMT = 6% x 1,000 = 60, FV = Future value = par value = 1000

Hence, PV = PV(i, n, PMT, FV) = PV(16%, 20, 60, 1000) = - $ 407.12

Hence, current price of the bond = -PV = $ 407.12

Nos. of bonds required = Amount of 20 years coupon bonds held / Price of 1 bond = $ 10.41 mn / 407.12 =  0.025574 mn

Hence, the par value of your holdings in the 20-year coupon bond = the par value of 1 bond x nos. of bonds held = $ 1,000 x 0.025574 mn = $ 25.57 mn