Pension funds pay lifetime annuities to recipients. If a firm remains in busines

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 20%.

If the duration of 5-year maturity bonds with coupon rates of 16% (paid annually) is 4 years and the duration of 25-year maturity bonds with coupon rates of 9% (paid annually) is 16 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?

What will be the par value of your holdings in the 25-year coupon bond?

If the duration of 5-year maturity bonds with coupon rates of 16% (paid annually) is 4 years and the duration of 25-year maturity bonds with coupon rates of 9% (paid annually) is 16 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?

What will be the par value of your holdings in the 25-year coupon bond?

Explanation / Answer

Present value of perpetuity = 2.6/0.2 = 13 million

duration of obligation = 1.2/0.2 = 6 years

let w be the weight of 5-year bond

=>

w * 4 + (1-w) * 16 = 6
=>
w = 5/6

amount invested in 5-year bond = 5/6 * 13 = $10,833,333 .33

amount invested in 25 year bond = 13000000 - 10833333.33 = $2,166,666.67

b)

price of 25 year bond
=>
90 * [1-(1.2)^-25]/0.2 + 1000/(1.2)^25 = 455.77

par value = 2166666.67/455.77 * 1000 = 4,753,859.78

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