Question

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 $3.5 million per year to beneficiaries. The yield to maturity on all bonds is 17.5%.

Required:
a. 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?
b. What will be the par value of your holdings in the 25-year coupon bond?

Answer 1

Duration of liability (perpetual) = (1 + y) / y

= (1 + 17.5%) / 17.5%

= 6.71

Value of liability = Cash Flow / yield

= $3.5 million / 17.5%

= $20 million

a. Assume you invest w in 5-year bond and 1-w in 25-year bond such that the duration of the portfolio is 6.71

6.71 = w x 4 + (1 - w) x 16

w = (16 - 6.71) / (16 - 4)

w = 77% in 5-year bond

1 - w = 28% in 25 year bond

Market Value of 5 year bond = 77% * $20 million = $15.4 million

Market Value of 20 year bond = 23% * $20 million = $4.6 million

b. Market Price of 20 year bond can be calculated using PV function on a calculator

N = 25, I/Y = 17.5%, PMT = 9, FV = 100

Price = Present Value (25,17.5%, 9 ,100)

Price = 52.29042644

Price = $52.30

Par Value of 25 year bond = Market Value /% Price

Par Value of 25 year bond = $4.6 million / 50.83%

Par Value of 25 year bond = $9,049,774