Question

Pension funds pay lifetime annuities to recipients. If a firm remains 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 $2.0 million per year to beneficiaries. The yield to maturity on all bonds is 16%. a. If the duration of 5-year maturity bonds with coupon rates of 12% (paid annually) is 4.0 years and the duration of 20-year maturity bonds with coupon rates of 5% (paid annually) is 8.2 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?

Answer 1

To fully fund and immunize the $2.0 million annual pension obligation, you would want to hold approximately $4.4 million of the 5-year coupon bond and $8.1 million of the 20-year coupon bond in market value.

Step-by-step explanation:

To fully fund and immunize the $2.0 million annual pension obligation, we need to find the market value of each coupon bond by solving a system of equations. Let's assume the market value of the 5-year bond is X and the market value of the 20-year bond is Y.

The present value of the perpetual payments is:

Present Value = $2.0 million / 0.16 = $12.5 million

From the duration, we know that the 5-year bond accounts for 4.0/12.5 = 0.32 and the 20-year bond accounts for 8.2/12.5 = 0.66 of the present value. Therefore, we can write the following equations:

X + Y = $12.5 million (equation 1)

0.32X + 0.66Y = $2.0 million (equation 2)

Solving these equations, we find that the market value of the 5-year bond should be approximately $4.4 million and the market value of the 20-year bond should be approximately $8.1 million.

Answer 2

The total investment of $12.5 million is adjusted between 5-year and 25-year bonds are $9.115 and $3.385 respectively

Step-by-step explanation:

Present value of the perpetual obligation of the firm is calculated as follows.

Perpetual obligation = $2 million / 0.16

= $12.5 million

Now, the duration of the perpetual obligation computed as follows.

Duration of the obligation = 1.16 / 0.16

= 7.25 years

Let us calculate the weight (W) of the 5 year maturity bond as below by substituting the values.

7.25 years = (W × 4) + (1-W) × 16

7.25 = W4 + 16 – W16

W of 5 year bond is = 0.7292

And

W of 25 year bond is = 0.2708.

Now, with the help of computed weights, let us find the fully fund obligation as follows.

5-year bond = 0.7292 × $12.5

= $9.115

25-year bond = 0.2708 × $12.5

= $3.385

Therefore, the total investment of $12.5 million is adjusted between 5-year and 25-year bonds are $9.115 and $3.385 respectively