Question

You manage a pension fund that will provide retired workers with lifetime annuities. You determine that the payouts of the fund are going to closely resemble level perpetuities of $2.1 million per year. The interest rate is 10%. You plan to fully fund the obligation using 5-year and 20-year maturity zero-coupon bonds.

a. How much market value of each of the zeros will be necessary to fund the plan if you desire an immunized position? (Do not round intermediate calculations. Enter your answers in millions. Round your answers to 1 decimal place.)

	Market Value Five-year million Twenty-year million

b. What must be the face value of each of the two zeros to fund the plan? (Do not round intermediate calculations. Enter your answers in millions rounded to 2 decimal places.)

Market Value Five-year million Twenty-year million

Answer 1

a. Market value of each of the zeros necessary to fund the plan

Level perpetuities = $2.1 million per year

Interest rate is 10%

5-year and 20-year maturity zero-coupon bonds.

Step 1 : Find Present Value of Perpetual Annuities

Present Value of Perpetual Annuities = Level perpetuities / Interest Rate = $2.1 Million / 10% = $21 Million

Step 2: Find the Duration of Perpetuity

Duration of Perpetuity = (1+Interest rate )/Interest Rate = (1+10%)/10% = 11 years

Step 3: Find the weights of investment in 5 year zero coupon bond and 20 year zero coupon bond to equal Perpetuity duration of 11 years

Let the Investment in 5 year Zero Coupon Bond be X.

Thus, Investment in 20 year Zero Coupon Bond will be (1-X)

Thus, Duration of Perpetuity = (Duration of 5 year Zero Coupon Bond * Investment in 5 year Zero Coupon Bond )+(Duration of 20 year Zero Coupon Bond * Investment in 20 year Zero Coupon Bond )

11 =(5*X)+(20*(1-X))

=11=5X+20-20X

=11-20 = 5X-20X

=-9 = -15X

X = -9/-15 = 60%

Investment in 5 year Zero Coupon Bond = 60%

Investment in 20 year Zero Coupon Bond = 1-60% = 40%

Step 4:  Market value of each of the zeros necessary to fund the plan

Present Value of Perpetual Annuities = $21 Million

Market value needed of 5 year Zero Coupon Bond =  $21 Million*60% = $12,600,000 or $12.60 Million

Market value needed of 20 year Zero Coupon Bond =  $21 Million*40% = $8,400,000 or $8.40 Million

b. Face Value of each of the two zeros to fund the plan

Since these are Zero Coupon Bonds, no periodic coupon will be paid and instead only the face value be repaid at the end of the maturity period. Thus, the face value will equal the future value at the end of maturity period of the market value now.

Face value needed of 5 year Zero Coupon Bond = Market Value of 5 year Zero Coupon Bond * (1+Interest Rate)^Duration = $12,600,000*(1+10%)^5 = $12,600,000*1.61051 = $20,296,426 or $20.29 Million

Face value needed of 20 year Zero Coupon Bond = Market Value of 20 year Zero Coupon Bond * (1+Interest Rate)^Duration = $8,400,000*(1+10%)^20 = $8,400,000*6.72750= $56,511,000 or $56.51 Million