Pension funds pay lifetime annuities to recipients. If a firm will remain in bus

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 $2.5 million per year to beneficiaries. The yield to maturity on all bonds is 10.5%.

If the duration of 5-year maturity bonds with coupon rates of 12.2% (paid annually) is 4 years and the duration of 20-year maturity bonds with coupon rates of 5% (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. Omit the "$" sign in your response.)

What will be the par value of your holdings in the 20-year coupon bond? (Do not round intermediate calculations. Enter your answer in millions rounded to 2 decimal places. Omit the "$" sign in your response.)

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 $2.5 million per year to beneficiaries. The yield to maturity on all bonds is 10.5%.

Explanation / Answer

Amount of perpetual obligation =D/r =$ 2,500,000/0.105 =$23,809,524

Duration of perpetuity = (1+y)/y =(1+0.105)/0.105 = 1.105/0.105 = 10.5238 years

Let w be the weight of 5-year bond and (1-w) is the weight of 20-year bond in the bond portfolio.

Portfolio duration = weighted average duration of holdings

10.5238 = w*4 + (1-w)*11

10.5238 = 4w + 11 -11w

7w = 0.4762

w=0.0680

a.) Amount of 5-Year bond =0.068x23,809,524 = 1.6 million

Amount of 20-Year bond =0.932x23,809,524 = 22.2 million

b.) Price of 20-Year 5%coupon bond with 10.5% yield,

P = 50x{(1-(1+0.105)-20)/0.105} + 1000/(1+0.105)20

= 50x{(1-(1.105)-20)/0.105} + 1000/(1.105)20

= 411.54 + 135.75

= 547.30

Number of bonds held =22,190,476/547.30 =40,545.36

Hence, the face value =$1,000 x 40545.36=$40.5 million

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