Visualization of Macaulay duration as a point of total immunization
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• A Numerical Example
In this section we consider a basic numerical immunization example. Suppose you are trying to immunize a year-10 obligation whose present value is $1,000; that is, at the current interest rate of 6 percent, its future value is: $1,000[pic][pic]= $1,790.85 You intend to immunize the obligation by purchasing $1,000 worth of a bond or a combination of bonds. You consider three bonds:
i. Bond 1 has 10 years until maturity, a coupon rate of 6.7 percent, and a face value of $1,000. ii. Bond 2 has 15 years until maturity, a coupon rate of 6.988 percent, and a face value of $1,000. iii. Bond 3 has 30 years until maturity, a coupon rate of 5.9 percent and a face value of $1,000.
If the yield to maturity doesn’t change, then you will be able to reinvest each coupon at 6 percent. [pic]
The upshot of this table is that purchasing $1,000 of any of the three bonds will provide—10 years from now—funding for your future obligation of $1,790.85, provided the market interest rate of 6 percent doesn’t change.
Now suppose that, immediately after you purchase the bonds, the yield to maturity changes to some new value and stays there. This change will obviously affect the calculation we just did. For example, if the yield falls to 5 percent, the table will now look as follows:
Thus, if the yield falls, bond 1 will no longer fund our obligation, whereas bond 3 will overfund it. Bond 2’s ability to fund the obligation—not surprisingly, in view of the fact that its duration is exactly 10 years—hardly changes.