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 $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. (lo 11-2) a. how much market value of each of the zeros will be necessary to fund the plan if you desire an immunized position? b. what must be the face value of each of the two zeros to fund the plan?

Answer 1

To fund the plan with an immunized position, you will need approximately $620,921.61 of the 5-year zero and $148,627.76 of the 20-year zero. The face value of each zero will be approximately $1 million.

a. To fund the plan with an immunized position, you will need to use a combination of 5-year and 20-year maturity zero-coupon bonds.

Let's calculate the market value of each zero.

For the 5-year zero-coupon bond:

Future Value (FV) = $1 million

Interest Rate (r) = 10%

Number of periods (n) = 5 years

Present Value (PV) = FV / ((1 + r)^n)

PV = $1 million / ((1 + 0.10)^5) = $1 million / 1.61051

PV ≈ $620,921.61

For the 20-year zero-coupon bond:

Number of periods (n) = 20 years

PV = $1 million / ((1 + 0.10)^20) = $1 million / 6.7275

PV ≈ $148,627.76

Therefore, you will need approximately $620,921.61 of the 5-year zero and $148,627.76 of the 20-year zero to fund the plan with an immunized position.

b. To calculate the face value of each zero to fund the plan, you need to consider the present value of each zero at a 10% interest rate.

For the 5-year zero-coupon bond:

PV = $620,921.61

Face Value (FV) = PV * ((1 + r)^n)

FV = $620,921.61 * ((1 + 0.10)^5)

FV ≈ $1 million

For the 20-year zero-coupon bond:

PV = $148,627.76

FV = PV * ((1 + r)^n)

FV = $148,627.76 * ((1 + 0.10)^20)

FV ≈ $1 million