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 $1.8 million per year to beneficiaries. The yield to maturity on all bonds is 11.0%.

a. If the duration of 5-year maturity bonds with coupon rates of 8.5% (paid annually) is four years and the duration of 20-year maturity bonds with coupon rates of 4% (paid annually) is 11 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.)

5 year=

20 year=

b. What will be the par value of your holdings in the 20-year coupon bond?

Par Value=

Answer 1

a. Duration of Obligation = (1+Yield) / (1 + Yield) = (1.11)/0.11 = 10.09 Years

Present Value of Obligations = Annual Payment / YTM = 1800000 / 11% = $16363636.36

Duration of Obligation = Weight of 5 Years Maturity * Duration of 5 year Maturity + Weight of 20 Years Maturity * Duration of 20 year Maturity

10.09 = Weight of 5 Years Maturity * 4 + (1 - Weight of 5 Years Maturity) * 11

10.09 = Weight of 5 Years Maturity * 4 + 11 - 11 * weight of 5 year maturity

- 7 * weight of 5 year maturity = - 0.91

Weight of 5 Year Maturity Bond = 0.13

Weight of 20 Year Maturity Bond = 1 - 0.13 = 0.87

5 year= Weight of 5 year Bond * PV of Obligation = 0.13 * 16.36 M = $2.13 Million

20 year = Weight of 20 year Bond * PV of Obligation = 0.87 * 16.36 M = $14.24 Million

b. What will be the par value of your holdings in the 20-year coupon bond?

Market Value = Par Value * Market rate %

$14.24 Million = Par Value * 0.44257 (Calculated using PV function)

Par Value = $32.17 Million

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