My pension plan will pay me $11,500 once a year for a 10-year period. The first

Question

My pension plan will pay me $11,500 once a year for a 10-year period. The first payment will come in exactly 5 years. The pension fund wants to immunize its position.

  

What is the duration of its obligation to me? The current interest rate is 6.0% per year. (Do not round intermediate calculations. Round your answer to 4 decimal places.)

  

  

If the plan uses 5-year and 20-year zero-coupon bonds to construct the immunized position, how much money ought to be placed in each bond? What will be the face value of the holdings in each zero? (Do not round intermediate calculations. Round your answers to the nearest whole dollar amount. Omit the "$" sign in your response.)

  

My pension plan will pay me $11,500 once a year for a 10-year period. The first payment will come in exactly 5 years. The pension fund wants to immunize its position.

Explanation / Answer

Part a)

The table for calculation of modified duration is shown below:

Duration = 9.0220 Years.

Important Information:

1) Present value is calculated by dividing the cash flow with (1+Discount Rate)^n where discount rate is 6% and n is years. For example, for year 5, present value = 11,500/(1+6%)^5 and for year 6 = 11,500/(1+6%)^6 and so on.

2) Weight for each year is calculated by dividing the present value of that with year with the total of present value ($67,043.60). For example, for year 5, weight = 8,593.469/67,043.60 and for year 6 = 8,107.046/67,043.60 and so on.

3) Duration is calculated by summing up total of weight*time.

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Part b)

Step 1: Calculate Weight of Each Type of Bond in the Portfolio:

Let us assume that the weight of 5 year zero coupon bond in the portfolio is X

The weight of 20 year zero coupon bond will therefore be = 1-X

We will use the target duration equation to calculate the weight of each type of bond. The equation for target duration:

Target Duration = Weight of Bond 1*Maturity of Bond 1 + Weight of Bond 2*Maturity of Bond 2

Using weights and maturity as provided to us in the question we get,

9.0220 = 5*X + 20*(1-X)

9.0220 = 5X + 20 - 20X

X (Weight of 5 Year Bond) = (20 -9.0220)/15 = .7319

Weight of 20 Year Bond = 1-.7319 = .2681

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Step 2: Calculate the Investment in Each Type of Bond:

Investment in 5 Year Bond = Weight of 5 Year Bond*Total Present Value = .7319*67,043.60 = $49,069

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Investment in 20 Year Bond = Weight of20 Year Bond*Total Present Value = .2681*67,043.60 = $17,974

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Step 3: Calculate the Face Value of the Holdings:

We have calculated the present value of each type of bond in step 2 already. Face value is the future value of each type of bond. The formula for calculating future value is:

Face/Future Value = Present Value*(1+r)^n where r is rate of interest and n is period.

__________

Face Value of 5 Year Bond = 49,069*(1+6%)^5 = $65,665

Face Value of 20 Year Bond = 17,974*(1+6%)^20 = $57,645

Time Cash Flow Present Value Weight of Present Value Weight*Time 0 0 0.000 0 0 1 0 0.000 0 0 2 0 0.000 0 0 3 0 0.000 0 0 4 0 0.000 0 0 5 11,500 8,593.469 0.12818 0.6408865954 6 11,500 8,107.046 0.12092 0.7255319948 7 11,500 7,648.157 0.11408 0.7985415037 8 11,500 7,215.242 0.10762 0.8609611899 9 11,500 6,806.832 0.10153 0.9137559799 10 11,500 6,421.540 0.09578 0.9578154926 11 11,500 6,058.057 0.09036 0.9939594734 12 11,500 5,715.148 0.08525 1.0229428543 13 11,500 5,391.649 0.08042 1.0454604643 14 11,500 5,086.461 0.07587 1.0621514151 Total 67,043.601 Duration 9.0220

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